1

Chapter 6

Code Division Multiple Access

2

bull We can achieve code division multiple access (CDMA) based on spread spectrum techniques

bull In particular we can assign different spreading codes to different users in a DSSS system so that the users can share the communication channel

bull For a DS-SS based CDMA (DSCDMA) system multiple access interference (MAI) is the major factor limiting the performance and hence the capacity of the system

bull Therefore analyses of the effect of MAI on the system performance as well as ways to suppress MAI have been the major focus of CDMA research

3

bull Roughly speaking there are two different approaches to the problem

1 The first approach is based on the concept of single-user detection

bull In this approach we identify one of the users in the system as the desired user and treat all signals from the other users as interference

bull The receiver (for the desired user) detects only the desired user signal

2 The second approach is called multiuser detection in which all signals from all users are detected jointly and simultaneously by the receiver

4

bull A common receiver based on the single-user detection approach is the matched filter receiver matched to the desired userrsquos spreading signal

bull We note that the matched filter receiver is not optimal (in the sense of maximizing the likelihood function) in the presence of MAI

ndash Optimal receivers of such a kind are discussed in [1]

bull However due to their complexity we will omit the optimal receivers here and focus on the simple matched filter receiver

bull As a starting point we will refine the crude analysis of the symbol error performance

5

bull In general it is difficult to conduct an exact symbol error performance analysis of a DS-CDMA system based on the matched filter receiver even in an AWGN channel

bull Usually we have to resort to bounds and approximations

bull We will discuss several common techniques to calculate the approximate symbol error probability of a DS-CDMA system

bull These approximate analyses are important because they provide us simple ways to obtain the symbol error probability which is crucial in determining the capacity of the system

bull Towards the end of the chapter we will also present some techniques based on the matched filter receiver to suppress MAI so that performance of the system can be improved

6

61 Asynchronous DS-CDMA model

bull In many cases the transmissions from different users in a DS-CDMA system are not synchronized

bull One of such examples is given by the uplink (reverse link) of IS-95 [2]

bull In order to include these asynchronous cases we consider a general asynchronous model of the DS-CDMA system here

bull We assume that there are K actively transmitting users in the system

bull We associate the kth user with a data signal bk(t) and a spreading signal ak(t) where

7

ndash is the (transmitted) power of the kth user signal

ndash is the sequence of data symbols for the kth user

ndash is the spreading sequence assigned to the kth user

bull For simplicity we assume that BPSK modulation is employed ie bi

(k) are iid binary random variables taking values from the set +1 -1 with equal probabilities

bull We also assume that the spreading sequence is periodic with period N where T = NTc and the chip waveform is time-limited to [0 Tc) and is normalized such that

bull Extensions to other types of modulations and long sequences are straightforward

8

bull The received signal at the receiver matched to the spreading signal of the mth user for 1≦m ≦ K is

ndash n(t) AWGN with power spectral density N0

ndash the amplitude response

ndash the phase response

ndash the delay (with respect to some time reference) of the channel from the transmitter of the kth user (we will call it the kth transmitter) to the receiver of the mth user (we will call it the mth receiver)

ndash Also where the term is the phase difference due to the delay

9

bull We employ to model the asynchronous nature of the system

bull We assume that synchronization with the mth userrsquos signal has been achieved at the mth receiver

bull Hence we can assume without loss of generality

bull For k ne m we model as independent uniform random variables on the intervals [0 T) and [0 2π) respectively

bull Moreover we also make the assumption that all the random variables associated with different users are independent

10

bull Now let us look at the mth matched filter receiver which is designed to detect the mth userrsquos signal

bull Without loss of generality we consider the detection of the symbol b0

(m)

bull Let be the received power of the kth userrsquos signal at the mth receiver

11

bull The decision statistic for the 0th symbol is given by

ndash ikm is the interference component due to the kth signal

ndash ηis the component due to the AWGN that is a zero-mean Gaussian random variable with variance N0T

12

bull On the other hand is given by

where is decomposed into

13

bull are the aperiodic even and odd cross correlation functions between the sequences respectively

14

62 Error analysis of matched filter receiver

bull As mentioned before it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI

bull Obviously it would be most desirable if we could obtain the exact average symbol error probability

bull Unfortunately this task is exceedingly complex for most practical scenarios in which many users are actively transmitting

bull Therefore bounds and approximations are typically employed

bull It is the goal of this section to give an introduction to some common bounding and approximation techniques

bull We will focus on one of the receiver namely the mth receiver

bull To simplify notation we write

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

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2

bull We can achieve code division multiple access (CDMA) based on spread spectrum techniques

bull In particular we can assign different spreading codes to different users in a DSSS system so that the users can share the communication channel

bull For a DS-SS based CDMA (DSCDMA) system multiple access interference (MAI) is the major factor limiting the performance and hence the capacity of the system

bull Therefore analyses of the effect of MAI on the system performance as well as ways to suppress MAI have been the major focus of CDMA research

3

bull Roughly speaking there are two different approaches to the problem

1 The first approach is based on the concept of single-user detection

bull In this approach we identify one of the users in the system as the desired user and treat all signals from the other users as interference

bull The receiver (for the desired user) detects only the desired user signal

2 The second approach is called multiuser detection in which all signals from all users are detected jointly and simultaneously by the receiver

4

bull A common receiver based on the single-user detection approach is the matched filter receiver matched to the desired userrsquos spreading signal

bull We note that the matched filter receiver is not optimal (in the sense of maximizing the likelihood function) in the presence of MAI

ndash Optimal receivers of such a kind are discussed in [1]

bull However due to their complexity we will omit the optimal receivers here and focus on the simple matched filter receiver

bull As a starting point we will refine the crude analysis of the symbol error performance

5

bull In general it is difficult to conduct an exact symbol error performance analysis of a DS-CDMA system based on the matched filter receiver even in an AWGN channel

bull Usually we have to resort to bounds and approximations

bull We will discuss several common techniques to calculate the approximate symbol error probability of a DS-CDMA system

bull These approximate analyses are important because they provide us simple ways to obtain the symbol error probability which is crucial in determining the capacity of the system

bull Towards the end of the chapter we will also present some techniques based on the matched filter receiver to suppress MAI so that performance of the system can be improved

6

61 Asynchronous DS-CDMA model

bull In many cases the transmissions from different users in a DS-CDMA system are not synchronized

bull One of such examples is given by the uplink (reverse link) of IS-95 [2]

bull In order to include these asynchronous cases we consider a general asynchronous model of the DS-CDMA system here

bull We assume that there are K actively transmitting users in the system

bull We associate the kth user with a data signal bk(t) and a spreading signal ak(t) where

7

ndash is the (transmitted) power of the kth user signal

ndash is the sequence of data symbols for the kth user

ndash is the spreading sequence assigned to the kth user

bull For simplicity we assume that BPSK modulation is employed ie bi

(k) are iid binary random variables taking values from the set +1 -1 with equal probabilities

bull We also assume that the spreading sequence is periodic with period N where T = NTc and the chip waveform is time-limited to [0 Tc) and is normalized such that

bull Extensions to other types of modulations and long sequences are straightforward

8

bull The received signal at the receiver matched to the spreading signal of the mth user for 1≦m ≦ K is

ndash n(t) AWGN with power spectral density N0

ndash the amplitude response

ndash the phase response

ndash the delay (with respect to some time reference) of the channel from the transmitter of the kth user (we will call it the kth transmitter) to the receiver of the mth user (we will call it the mth receiver)

ndash Also where the term is the phase difference due to the delay

9

bull We employ to model the asynchronous nature of the system

bull We assume that synchronization with the mth userrsquos signal has been achieved at the mth receiver

bull Hence we can assume without loss of generality

bull For k ne m we model as independent uniform random variables on the intervals [0 T) and [0 2π) respectively

bull Moreover we also make the assumption that all the random variables associated with different users are independent

10

bull Now let us look at the mth matched filter receiver which is designed to detect the mth userrsquos signal

bull Without loss of generality we consider the detection of the symbol b0

(m)

bull Let be the received power of the kth userrsquos signal at the mth receiver

11

bull The decision statistic for the 0th symbol is given by

ndash ikm is the interference component due to the kth signal

ndash ηis the component due to the AWGN that is a zero-mean Gaussian random variable with variance N0T

12

bull On the other hand is given by

where is decomposed into

13

bull are the aperiodic even and odd cross correlation functions between the sequences respectively

14

62 Error analysis of matched filter receiver

bull As mentioned before it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI

bull Obviously it would be most desirable if we could obtain the exact average symbol error probability

bull Unfortunately this task is exceedingly complex for most practical scenarios in which many users are actively transmitting

bull Therefore bounds and approximations are typically employed

bull It is the goal of this section to give an introduction to some common bounding and approximation techniques

bull We will focus on one of the receiver namely the mth receiver

bull To simplify notation we write

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

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3

bull Roughly speaking there are two different approaches to the problem

1 The first approach is based on the concept of single-user detection

bull In this approach we identify one of the users in the system as the desired user and treat all signals from the other users as interference

bull The receiver (for the desired user) detects only the desired user signal

2 The second approach is called multiuser detection in which all signals from all users are detected jointly and simultaneously by the receiver

4

bull A common receiver based on the single-user detection approach is the matched filter receiver matched to the desired userrsquos spreading signal

bull We note that the matched filter receiver is not optimal (in the sense of maximizing the likelihood function) in the presence of MAI

ndash Optimal receivers of such a kind are discussed in [1]

bull However due to their complexity we will omit the optimal receivers here and focus on the simple matched filter receiver

bull As a starting point we will refine the crude analysis of the symbol error performance

5

bull In general it is difficult to conduct an exact symbol error performance analysis of a DS-CDMA system based on the matched filter receiver even in an AWGN channel

bull Usually we have to resort to bounds and approximations

bull We will discuss several common techniques to calculate the approximate symbol error probability of a DS-CDMA system

bull These approximate analyses are important because they provide us simple ways to obtain the symbol error probability which is crucial in determining the capacity of the system

bull Towards the end of the chapter we will also present some techniques based on the matched filter receiver to suppress MAI so that performance of the system can be improved

6

61 Asynchronous DS-CDMA model

bull In many cases the transmissions from different users in a DS-CDMA system are not synchronized

bull One of such examples is given by the uplink (reverse link) of IS-95 [2]

bull In order to include these asynchronous cases we consider a general asynchronous model of the DS-CDMA system here

bull We assume that there are K actively transmitting users in the system

bull We associate the kth user with a data signal bk(t) and a spreading signal ak(t) where

7

ndash is the (transmitted) power of the kth user signal

ndash is the sequence of data symbols for the kth user

ndash is the spreading sequence assigned to the kth user

bull For simplicity we assume that BPSK modulation is employed ie bi

(k) are iid binary random variables taking values from the set +1 -1 with equal probabilities

bull We also assume that the spreading sequence is periodic with period N where T = NTc and the chip waveform is time-limited to [0 Tc) and is normalized such that

bull Extensions to other types of modulations and long sequences are straightforward

8

bull The received signal at the receiver matched to the spreading signal of the mth user for 1≦m ≦ K is

ndash n(t) AWGN with power spectral density N0

ndash the amplitude response

ndash the phase response

ndash the delay (with respect to some time reference) of the channel from the transmitter of the kth user (we will call it the kth transmitter) to the receiver of the mth user (we will call it the mth receiver)

ndash Also where the term is the phase difference due to the delay

9

bull We employ to model the asynchronous nature of the system

bull We assume that synchronization with the mth userrsquos signal has been achieved at the mth receiver

bull Hence we can assume without loss of generality

bull For k ne m we model as independent uniform random variables on the intervals [0 T) and [0 2π) respectively

bull Moreover we also make the assumption that all the random variables associated with different users are independent

10

bull Now let us look at the mth matched filter receiver which is designed to detect the mth userrsquos signal

bull Without loss of generality we consider the detection of the symbol b0

(m)

bull Let be the received power of the kth userrsquos signal at the mth receiver

11

bull The decision statistic for the 0th symbol is given by

ndash ikm is the interference component due to the kth signal

ndash ηis the component due to the AWGN that is a zero-mean Gaussian random variable with variance N0T

12

bull On the other hand is given by

where is decomposed into

13

bull are the aperiodic even and odd cross correlation functions between the sequences respectively

14

62 Error analysis of matched filter receiver

bull As mentioned before it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI

bull Obviously it would be most desirable if we could obtain the exact average symbol error probability

bull Unfortunately this task is exceedingly complex for most practical scenarios in which many users are actively transmitting

bull Therefore bounds and approximations are typically employed

bull It is the goal of this section to give an introduction to some common bounding and approximation techniques

bull We will focus on one of the receiver namely the mth receiver

bull To simplify notation we write

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

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4

bull A common receiver based on the single-user detection approach is the matched filter receiver matched to the desired userrsquos spreading signal

bull We note that the matched filter receiver is not optimal (in the sense of maximizing the likelihood function) in the presence of MAI

ndash Optimal receivers of such a kind are discussed in [1]

bull However due to their complexity we will omit the optimal receivers here and focus on the simple matched filter receiver

bull As a starting point we will refine the crude analysis of the symbol error performance

5

bull In general it is difficult to conduct an exact symbol error performance analysis of a DS-CDMA system based on the matched filter receiver even in an AWGN channel

bull Usually we have to resort to bounds and approximations

bull We will discuss several common techniques to calculate the approximate symbol error probability of a DS-CDMA system

bull These approximate analyses are important because they provide us simple ways to obtain the symbol error probability which is crucial in determining the capacity of the system

bull Towards the end of the chapter we will also present some techniques based on the matched filter receiver to suppress MAI so that performance of the system can be improved

6

61 Asynchronous DS-CDMA model

bull In many cases the transmissions from different users in a DS-CDMA system are not synchronized

bull One of such examples is given by the uplink (reverse link) of IS-95 [2]

bull In order to include these asynchronous cases we consider a general asynchronous model of the DS-CDMA system here

bull We assume that there are K actively transmitting users in the system

bull We associate the kth user with a data signal bk(t) and a spreading signal ak(t) where

7

ndash is the (transmitted) power of the kth user signal

ndash is the sequence of data symbols for the kth user

ndash is the spreading sequence assigned to the kth user

bull For simplicity we assume that BPSK modulation is employed ie bi

(k) are iid binary random variables taking values from the set +1 -1 with equal probabilities

bull We also assume that the spreading sequence is periodic with period N where T = NTc and the chip waveform is time-limited to [0 Tc) and is normalized such that

bull Extensions to other types of modulations and long sequences are straightforward

8

bull The received signal at the receiver matched to the spreading signal of the mth user for 1≦m ≦ K is

ndash n(t) AWGN with power spectral density N0

ndash the amplitude response

ndash the phase response

ndash the delay (with respect to some time reference) of the channel from the transmitter of the kth user (we will call it the kth transmitter) to the receiver of the mth user (we will call it the mth receiver)

ndash Also where the term is the phase difference due to the delay

9

bull We employ to model the asynchronous nature of the system

bull We assume that synchronization with the mth userrsquos signal has been achieved at the mth receiver

bull Hence we can assume without loss of generality

bull For k ne m we model as independent uniform random variables on the intervals [0 T) and [0 2π) respectively

bull Moreover we also make the assumption that all the random variables associated with different users are independent

10

bull Now let us look at the mth matched filter receiver which is designed to detect the mth userrsquos signal

bull Without loss of generality we consider the detection of the symbol b0

(m)

bull Let be the received power of the kth userrsquos signal at the mth receiver

11

bull The decision statistic for the 0th symbol is given by

ndash ikm is the interference component due to the kth signal

ndash ηis the component due to the AWGN that is a zero-mean Gaussian random variable with variance N0T

12

bull On the other hand is given by

where is decomposed into

13

bull are the aperiodic even and odd cross correlation functions between the sequences respectively

14

62 Error analysis of matched filter receiver

bull As mentioned before it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI

bull Obviously it would be most desirable if we could obtain the exact average symbol error probability

bull Unfortunately this task is exceedingly complex for most practical scenarios in which many users are actively transmitting

bull Therefore bounds and approximations are typically employed

bull It is the goal of this section to give an introduction to some common bounding and approximation techniques

bull We will focus on one of the receiver namely the mth receiver

bull To simplify notation we write

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

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5

bull In general it is difficult to conduct an exact symbol error performance analysis of a DS-CDMA system based on the matched filter receiver even in an AWGN channel

bull Usually we have to resort to bounds and approximations

bull We will discuss several common techniques to calculate the approximate symbol error probability of a DS-CDMA system

bull These approximate analyses are important because they provide us simple ways to obtain the symbol error probability which is crucial in determining the capacity of the system

bull Towards the end of the chapter we will also present some techniques based on the matched filter receiver to suppress MAI so that performance of the system can be improved

6

61 Asynchronous DS-CDMA model

bull In many cases the transmissions from different users in a DS-CDMA system are not synchronized

bull One of such examples is given by the uplink (reverse link) of IS-95 [2]

bull In order to include these asynchronous cases we consider a general asynchronous model of the DS-CDMA system here

bull We assume that there are K actively transmitting users in the system

bull We associate the kth user with a data signal bk(t) and a spreading signal ak(t) where

7

ndash is the (transmitted) power of the kth user signal

ndash is the sequence of data symbols for the kth user

ndash is the spreading sequence assigned to the kth user

bull For simplicity we assume that BPSK modulation is employed ie bi

(k) are iid binary random variables taking values from the set +1 -1 with equal probabilities

bull We also assume that the spreading sequence is periodic with period N where T = NTc and the chip waveform is time-limited to [0 Tc) and is normalized such that

bull Extensions to other types of modulations and long sequences are straightforward

8

bull The received signal at the receiver matched to the spreading signal of the mth user for 1≦m ≦ K is

ndash n(t) AWGN with power spectral density N0

ndash the amplitude response

ndash the phase response

ndash the delay (with respect to some time reference) of the channel from the transmitter of the kth user (we will call it the kth transmitter) to the receiver of the mth user (we will call it the mth receiver)

ndash Also where the term is the phase difference due to the delay

9

bull We employ to model the asynchronous nature of the system

bull We assume that synchronization with the mth userrsquos signal has been achieved at the mth receiver

bull Hence we can assume without loss of generality

bull For k ne m we model as independent uniform random variables on the intervals [0 T) and [0 2π) respectively

bull Moreover we also make the assumption that all the random variables associated with different users are independent

10

bull Now let us look at the mth matched filter receiver which is designed to detect the mth userrsquos signal

bull Without loss of generality we consider the detection of the symbol b0

(m)

bull Let be the received power of the kth userrsquos signal at the mth receiver

11

bull The decision statistic for the 0th symbol is given by

ndash ikm is the interference component due to the kth signal

ndash ηis the component due to the AWGN that is a zero-mean Gaussian random variable with variance N0T

12

bull On the other hand is given by

where is decomposed into

13

bull are the aperiodic even and odd cross correlation functions between the sequences respectively

14

62 Error analysis of matched filter receiver

bull As mentioned before it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI

bull Obviously it would be most desirable if we could obtain the exact average symbol error probability

bull Unfortunately this task is exceedingly complex for most practical scenarios in which many users are actively transmitting

bull Therefore bounds and approximations are typically employed

bull It is the goal of this section to give an introduction to some common bounding and approximation techniques

bull We will focus on one of the receiver namely the mth receiver

bull To simplify notation we write

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

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6

61 Asynchronous DS-CDMA model

bull In many cases the transmissions from different users in a DS-CDMA system are not synchronized

bull One of such examples is given by the uplink (reverse link) of IS-95 [2]

bull In order to include these asynchronous cases we consider a general asynchronous model of the DS-CDMA system here

bull We assume that there are K actively transmitting users in the system

bull We associate the kth user with a data signal bk(t) and a spreading signal ak(t) where

7

ndash is the (transmitted) power of the kth user signal

ndash is the sequence of data symbols for the kth user

ndash is the spreading sequence assigned to the kth user

bull For simplicity we assume that BPSK modulation is employed ie bi

(k) are iid binary random variables taking values from the set +1 -1 with equal probabilities

bull We also assume that the spreading sequence is periodic with period N where T = NTc and the chip waveform is time-limited to [0 Tc) and is normalized such that

bull Extensions to other types of modulations and long sequences are straightforward

8

bull The received signal at the receiver matched to the spreading signal of the mth user for 1≦m ≦ K is

ndash n(t) AWGN with power spectral density N0

ndash the amplitude response

ndash the phase response

ndash the delay (with respect to some time reference) of the channel from the transmitter of the kth user (we will call it the kth transmitter) to the receiver of the mth user (we will call it the mth receiver)

ndash Also where the term is the phase difference due to the delay

9

bull We employ to model the asynchronous nature of the system

bull We assume that synchronization with the mth userrsquos signal has been achieved at the mth receiver

bull Hence we can assume without loss of generality

bull For k ne m we model as independent uniform random variables on the intervals [0 T) and [0 2π) respectively

bull Moreover we also make the assumption that all the random variables associated with different users are independent

10

bull Now let us look at the mth matched filter receiver which is designed to detect the mth userrsquos signal

bull Without loss of generality we consider the detection of the symbol b0

(m)

bull Let be the received power of the kth userrsquos signal at the mth receiver

11

bull The decision statistic for the 0th symbol is given by

ndash ikm is the interference component due to the kth signal

ndash ηis the component due to the AWGN that is a zero-mean Gaussian random variable with variance N0T

12

bull On the other hand is given by

where is decomposed into

13

bull are the aperiodic even and odd cross correlation functions between the sequences respectively

14

62 Error analysis of matched filter receiver

bull As mentioned before it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI

bull Obviously it would be most desirable if we could obtain the exact average symbol error probability

bull Unfortunately this task is exceedingly complex for most practical scenarios in which many users are actively transmitting

bull Therefore bounds and approximations are typically employed

bull It is the goal of this section to give an introduction to some common bounding and approximation techniques

bull We will focus on one of the receiver namely the mth receiver

bull To simplify notation we write

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

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• Slide 78

7

ndash is the (transmitted) power of the kth user signal

ndash is the sequence of data symbols for the kth user

ndash is the spreading sequence assigned to the kth user

bull For simplicity we assume that BPSK modulation is employed ie bi

(k) are iid binary random variables taking values from the set +1 -1 with equal probabilities

bull We also assume that the spreading sequence is periodic with period N where T = NTc and the chip waveform is time-limited to [0 Tc) and is normalized such that

bull Extensions to other types of modulations and long sequences are straightforward

8

bull The received signal at the receiver matched to the spreading signal of the mth user for 1≦m ≦ K is

ndash n(t) AWGN with power spectral density N0

ndash the amplitude response

ndash the phase response

ndash the delay (with respect to some time reference) of the channel from the transmitter of the kth user (we will call it the kth transmitter) to the receiver of the mth user (we will call it the mth receiver)

ndash Also where the term is the phase difference due to the delay

9

bull We employ to model the asynchronous nature of the system

bull We assume that synchronization with the mth userrsquos signal has been achieved at the mth receiver

bull Hence we can assume without loss of generality

bull For k ne m we model as independent uniform random variables on the intervals [0 T) and [0 2π) respectively

bull Moreover we also make the assumption that all the random variables associated with different users are independent

10

bull Now let us look at the mth matched filter receiver which is designed to detect the mth userrsquos signal

bull Without loss of generality we consider the detection of the symbol b0

(m)

bull Let be the received power of the kth userrsquos signal at the mth receiver

11

bull The decision statistic for the 0th symbol is given by

ndash ikm is the interference component due to the kth signal

ndash ηis the component due to the AWGN that is a zero-mean Gaussian random variable with variance N0T

12

bull On the other hand is given by

where is decomposed into

13

bull are the aperiodic even and odd cross correlation functions between the sequences respectively

14

62 Error analysis of matched filter receiver

bull As mentioned before it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI

bull Obviously it would be most desirable if we could obtain the exact average symbol error probability

bull Unfortunately this task is exceedingly complex for most practical scenarios in which many users are actively transmitting

bull Therefore bounds and approximations are typically employed

bull It is the goal of this section to give an introduction to some common bounding and approximation techniques

bull We will focus on one of the receiver namely the mth receiver

bull To simplify notation we write

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

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• Slide 77
• Slide 78

8

bull The received signal at the receiver matched to the spreading signal of the mth user for 1≦m ≦ K is

ndash n(t) AWGN with power spectral density N0

ndash the amplitude response

ndash the phase response

ndash the delay (with respect to some time reference) of the channel from the transmitter of the kth user (we will call it the kth transmitter) to the receiver of the mth user (we will call it the mth receiver)

ndash Also where the term is the phase difference due to the delay

9

bull We employ to model the asynchronous nature of the system

bull We assume that synchronization with the mth userrsquos signal has been achieved at the mth receiver

bull Hence we can assume without loss of generality

bull For k ne m we model as independent uniform random variables on the intervals [0 T) and [0 2π) respectively

bull Moreover we also make the assumption that all the random variables associated with different users are independent

10

bull Now let us look at the mth matched filter receiver which is designed to detect the mth userrsquos signal

bull Without loss of generality we consider the detection of the symbol b0

(m)

bull Let be the received power of the kth userrsquos signal at the mth receiver

11

bull The decision statistic for the 0th symbol is given by

ndash ikm is the interference component due to the kth signal

ndash ηis the component due to the AWGN that is a zero-mean Gaussian random variable with variance N0T

12

bull On the other hand is given by

where is decomposed into

13

bull are the aperiodic even and odd cross correlation functions between the sequences respectively

14

62 Error analysis of matched filter receiver

bull As mentioned before it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI

bull Obviously it would be most desirable if we could obtain the exact average symbol error probability

bull Unfortunately this task is exceedingly complex for most practical scenarios in which many users are actively transmitting

bull Therefore bounds and approximations are typically employed

bull It is the goal of this section to give an introduction to some common bounding and approximation techniques

bull We will focus on one of the receiver namely the mth receiver

bull To simplify notation we write

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
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• Slide 77
• Slide 78

9

bull We employ to model the asynchronous nature of the system

bull We assume that synchronization with the mth userrsquos signal has been achieved at the mth receiver

bull Hence we can assume without loss of generality

bull For k ne m we model as independent uniform random variables on the intervals [0 T) and [0 2π) respectively

bull Moreover we also make the assumption that all the random variables associated with different users are independent

10

bull Now let us look at the mth matched filter receiver which is designed to detect the mth userrsquos signal

bull Without loss of generality we consider the detection of the symbol b0

(m)

bull Let be the received power of the kth userrsquos signal at the mth receiver

11

bull The decision statistic for the 0th symbol is given by

ndash ikm is the interference component due to the kth signal

ndash ηis the component due to the AWGN that is a zero-mean Gaussian random variable with variance N0T

12

bull On the other hand is given by

where is decomposed into

13

bull are the aperiodic even and odd cross correlation functions between the sequences respectively

14

62 Error analysis of matched filter receiver

bull As mentioned before it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI

bull Obviously it would be most desirable if we could obtain the exact average symbol error probability

bull Unfortunately this task is exceedingly complex for most practical scenarios in which many users are actively transmitting

bull Therefore bounds and approximations are typically employed

bull It is the goal of this section to give an introduction to some common bounding and approximation techniques

bull We will focus on one of the receiver namely the mth receiver

bull To simplify notation we write

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
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• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

10

bull Now let us look at the mth matched filter receiver which is designed to detect the mth userrsquos signal

bull Without loss of generality we consider the detection of the symbol b0

(m)

bull Let be the received power of the kth userrsquos signal at the mth receiver

11

bull The decision statistic for the 0th symbol is given by

ndash ikm is the interference component due to the kth signal

ndash ηis the component due to the AWGN that is a zero-mean Gaussian random variable with variance N0T

12

bull On the other hand is given by

where is decomposed into

13

bull are the aperiodic even and odd cross correlation functions between the sequences respectively

14

62 Error analysis of matched filter receiver

bull As mentioned before it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI

bull Obviously it would be most desirable if we could obtain the exact average symbol error probability

bull Unfortunately this task is exceedingly complex for most practical scenarios in which many users are actively transmitting

bull Therefore bounds and approximations are typically employed

bull It is the goal of this section to give an introduction to some common bounding and approximation techniques

bull We will focus on one of the receiver namely the mth receiver

bull To simplify notation we write

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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• Slide 77
• Slide 78

11

bull The decision statistic for the 0th symbol is given by

ndash ikm is the interference component due to the kth signal

ndash ηis the component due to the AWGN that is a zero-mean Gaussian random variable with variance N0T

12

bull On the other hand is given by

where is decomposed into

13

bull are the aperiodic even and odd cross correlation functions between the sequences respectively

14

62 Error analysis of matched filter receiver

bull As mentioned before it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI

bull Obviously it would be most desirable if we could obtain the exact average symbol error probability

bull Unfortunately this task is exceedingly complex for most practical scenarios in which many users are actively transmitting

bull Therefore bounds and approximations are typically employed

bull It is the goal of this section to give an introduction to some common bounding and approximation techniques

bull We will focus on one of the receiver namely the mth receiver

bull To simplify notation we write

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

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• Slide 2
• Slide 3
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• Slide 61
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• Slide 64
• Slide 65
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• Slide 68
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• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

12

bull On the other hand is given by

where is decomposed into

13

bull are the aperiodic even and odd cross correlation functions between the sequences respectively

14

62 Error analysis of matched filter receiver

bull As mentioned before it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI

bull Obviously it would be most desirable if we could obtain the exact average symbol error probability

bull Unfortunately this task is exceedingly complex for most practical scenarios in which many users are actively transmitting

bull Therefore bounds and approximations are typically employed

bull It is the goal of this section to give an introduction to some common bounding and approximation techniques

bull We will focus on one of the receiver namely the mth receiver

bull To simplify notation we write

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

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13

bull are the aperiodic even and odd cross correlation functions between the sequences respectively

14

62 Error analysis of matched filter receiver

bull As mentioned before it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI

bull Obviously it would be most desirable if we could obtain the exact average symbol error probability

bull Unfortunately this task is exceedingly complex for most practical scenarios in which many users are actively transmitting

bull Therefore bounds and approximations are typically employed

bull It is the goal of this section to give an introduction to some common bounding and approximation techniques

bull We will focus on one of the receiver namely the mth receiver

bull To simplify notation we write

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

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14

62 Error analysis of matched filter receiver

bull As mentioned before it is important to determine the error performance of a DS-CDMA system using the matched filter receiver with the presence of MAI

bull Obviously it would be most desirable if we could obtain the exact average symbol error probability

bull Unfortunately this task is exceedingly complex for most practical scenarios in which many users are actively transmitting

bull Therefore bounds and approximations are typically employed

bull It is the goal of this section to give an introduction to some common bounding and approximation techniques

bull We will focus on one of the receiver namely the mth receiver

bull To simplify notation we write

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

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• Slide 75
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• Slide 77
• Slide 78

15

621 Error bounds

bull Let us assume that the set of spreading sequences for the K users are given

bull For convenience we define a set of system parameters

bull Then the conditional symbol error probability given Sm and b0

(m) = 1 and that given Sm and b0(m) = -1 are respectively

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
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• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

16

Where

is the MAI component bull Hence the average symbol error probability is

where the expectation is taken over all the random variables in the set Sm

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

17

bull The second equality in (610) is due to the fact that the data symbols b0

(k) and b-1(k) for k ne m are symmetrically distributed about 0 i

e the distribution of Im is symmetric about zero

bull In general the complexity of calculating of the expectation in (610) is exceedingly high even when there are only a moderate number of users

bull In many cases we have to resort to bounds which can be calculated with a practical level of computational complexity

18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

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18

bull A simple bound based on (610) is that

where the maximization is over all possible choices of values of the system parameters in the set Sm

bull For example with the rectangular chip waveform ie

and BPSK spreading for k ne m

Where

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
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• Slide 77
• Slide 78

19

bull Hence

bull Therefore

where is the symbol energy of the mth user

bull Although the bound in (611) or (615) can be calculated easily given the set of sequences used this bound is often not tight and hence its usefulness is limited

m mP T

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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• Slide 55
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• Slide 75
• Slide 76
• Slide 77
• Slide 78

20

bull Another way to bound the average symbol error probability is to first determine and bound the distribution function of the MAI contribution Im and then obtain bounds on the average symbol error probability by taking the expectation in (610) using the bounds on the distribution function of Im

bull Using this approach we can obtain very tight upper and lower bounds on the average symbol error probability with a complexity which increases linearly with K [3]

bull In general it is difficult to determine the distribution function of Im

bull Only the distribution functions for some simple cases such as BPSK spreading and QPSK spreading have been worked out

bull Yet another way to bound the average symbol error probability is to make use of the idea of moment-space bounds [4]

21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
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• Slide 4
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• Slide 6
• Slide 7
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21

622 Gaussian approximations

bull Instead of obtaining bounds on the average symbol error probability we can assume the MAI contribution Im as a Gaussian random variable and obtain an approximation to the average symbol error probability based on this assumption

bull There are mainly two variations to this Gaussian-approximation approach

ndash Standard Gaussian approximation

ndash Improved Gaussian approximation

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 61
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• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

22

Standard Gaussian approximation

bull The first method is to assume Im as a zero-mean Gaussian random variable

bull This method is usually known as the standard Gaussian approximation (SGA) [5] and is applicable to situations in which there are a large number of users with similar received powers in the system

bull Its validity is justified by the central limit theorem based on the fact that Im is a summation of independent random variables Re[ik

m]

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
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• Slide 7
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• Slide 77
• Slide 78

23

bull When the number of users K is large the distribution of Im approaches Gaussian

bull With SGA the approximate average symbol error probability is given by

bull Therefore all we need is to calculate the mean and variance of Im

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
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• Slide 6
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• Slide 75
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• Slide 77
• Slide 78

24

bull Hence the problem reduces to the evaluation of the variance of Re[ikm]

bull To do this we use the following simple identity

bull Then for k ne m

bull Let us write

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
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• Slide 74
• Slide 75
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• Slide 77
• Slide 78

25

bull From (65) we have

bull The second and third equalities in (623) are due to the independence of the random variables involved

bull Substituting (621) and (622) into (623) and making use of the fact that are independent

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

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• Slide 77
• Slide 78

26

bull Let us define

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
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• Slide 24
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• Slide 26
• Slide 27
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• Slide 48
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• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
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• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
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• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

27

bull For rectangular chip waveform ie as in BPSK or QPSK spreading

bull Hence (626) reduces to

bull Combining (617) (618) and (629)

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
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• Slide 74
• Slide 75
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• Slide 77
• Slide 78

28

bull Given the set of sequences the standard Gaussian approximation PSGA can be calculated as easily as the simple bound in (615)

bull Sometimes it is more convenient to have an approximation to the average symbol error probability which does not depend on the set of sequences employed

bull One reasonable way to obtain such an approximation is to assume that all the sequence elements are zero-mean iid random variables with and replace the terms in (630) by their expectations

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

29

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
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• Slide 64
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• Slide 68
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• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

30

bull Replacing the term in (630) by the expectation in (632) we have the approximation

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
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• Slide 51
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• Slide 55
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• Slide 57
• Slide 58
• Slide 59
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• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
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• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

31

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
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• Slide 16
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• Slide 19
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• Slide 29
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• Slide 33
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• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
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• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

32

bull Figure 62 shows the plots of the approximate symbol error probability given by the SGA in (633) for the case in which all the users have equal received powers and N = 31

bull Note that for K = 1 the symbol error probability is exact

bull When the received powers of all users are the same and the signal-to-noise ratio as indicated by the plots in Figure 62 we can further approximate (633) as

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
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• Slide 77
• Slide 78

33

bull Comparing to the approximation of the average symbol error probability given by (249) The standard Gaussian approximation in (634) gives a more optimistic estimate on the system capacity

bull For example in order to achieve an average symbol error probability of 10-3 K≦N3 approximately based on (634)

bull Hence a DS-CDMA system with a processing gain N can accommodate N3 users (compared to N5 users predicted by (250))

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
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• Slide 53
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• Slide 55
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• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

34

Improved Gaussian approximation

bull In general the SGA is reasonably accurate if the number of users is large and the received powers of the users are similar

bull However when either the number of active users is not large or there are a few users with received powers much higher than those of the others the SGA does not give an accurate approximation to the symbol error probability and another form of approximation is needed

bull We would still like to approximate the MAI component Im in the decision statistic as a Gaussian random variable

bull However we can no longer to do so by the reasoning employed before since Im contains of only a few (significant) terms Re[ikm] now

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 13
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• Slide 38
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• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
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• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

35

bull This difficulty can be circumvented by noting that each ikm is a summation of a large number of terms involving the sequence elements (refer to (65) and (631)) when the processing gain N is large

bull We can make use of this property to obtain the desired Gaussian approximation for the MAI by modeling the sequence elements as iid zero-mean random variables with

bull To aid our discussion and to conform to the notation in the literature let us define

to be the set of all the phases and delays of the other users and normalize the decision statistic zm in (64) by the factor

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
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• Slide 21
• Slide 22
• Slide 23
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• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
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• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

36

bull Then the real part of the normalized decision statistic becomes

ndash

ndash is a zero-mean (real) Gaussian random variable with variance

bull It can be shown [6 7 8] that the normalized MAI terms are conditional independent given and the desired userrsquos spreading sequence

bull Based on this one can further show [6 7 8] that conditioning on the set the normalized MAI component in (636) approaches a zero-mean Gaussian random variable with variance Vm as N approaches infinity

37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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37

bull The conditional variance of the limiting Gaussian random variable is given by

ndash where corresponds to the term due to the kth user and is a simple function (depending on the spreading technique) of and the chip waveform [7 8]

bull For example with BPSK spreading

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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• Slide 68
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• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

38

bull While with QPSK spreading

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
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• Slide 33
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• Slide 37
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• Slide 39
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• Slide 41
• Slide 42
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• Slide 44
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• Slide 46
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• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
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• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
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• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

39

bull We note that are iid random variables given our model of the delays and phases

bull The discussion above implies that we can accurately approximate the MAI component in the normalized decision statistic as a zero-mean Gaussian random variable with variance Vm when the processing gain N is large

bull Hence the approximate conditional symbol error probability given is

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

40

bull Averaging this over the delays and phases we obtain an accurate approximation to the (unconditional) symbol error probability

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
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• Slide 38
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• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
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• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

41

bull The approximation in (641) is known as the improved Gaussian approximation (IGA)

bull We note that the IGA is accurate regardless of the number of active users in the system as long as the processing gain is large

bull On the other hand the SGA is accurate regardless of the processing gain when there are a large number of active users with equal received powers

bull We notice that the conditional error probability depends on the delays and phases only through Vm

bull Hence an efficient way to calculate PIGA in (641) is to first obtain the probability density function pVm(v) of the random variable V

m and then evaluate the expectation by the integral

42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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42

bull The density function pVm(v) of Vm in turns can be easily obtained as the (K-1)-fold convolution of the density functions of the independent random variables

bull Compared to the SGA in (631) the computational complexity of (642) is still significant higher

bull We can reduce the computational complexity of the IGA by further approximating the expectation in (641) based on a Taylor series approximation [7 9] of the conditional symbol error probability as below

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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• Slide 44
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• Slide 48
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• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
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• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

43

where are the mean and variance of the random variable Vm respectively

bull From (637)

since are iid

bull For example with BPSK spreading

while with QPSK spreading

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
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• Slide 22
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• Slide 24
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• Slide 26
• Slide 27
• Slide 28
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• Slide 30
• Slide 31
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• Slide 33
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• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

44

bull Putting these into (643) we obtain an approximation of the symbol error probability which is as simple computationally as the SGA in (630)

bull It is shown in [7 9] that the approximated IGA in (643) is almost as accurate as the original IGA in (641) in many situations

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
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• Slide 31
• Slide 32
• Slide 33
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• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

45

bull In summary we point out that the IGA generally gives a more accurate approximation to the symbol error probability than the SGA does when the spreading gain N is reasonably large as in most practical DS-CDMA systems

bull To illustrate this point let us consider the symbol error probabilities of two DS-CDMA systems with BPSK spreading and QPSK spreading respectively

bull From (633) the SGA predicts that the symbol error probabilities of the two systems are the same

bull On the other hand the IGA (643) states that the system with QPSK spreading has a smaller symbol error probability than the system with BPSK spreading

bull The latter is in fact true for randomly selected sequences

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
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• Slide 19
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• Slide 21
• Slide 22
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• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
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• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

46

63 Near-far problem

bull Based on the SGA in (630) we see that the signal-to-noise ratio (SNR) of a user employing the matched filter receiver in a DS-CDMA system with K active users is degraded by the factor

as compared to the case in which only the user is active

bull When the received powers of all users are the same and the set of spreading sequences are properly chosen the degradation in SNR is relatively small if there are a moderate number of users

bull However when the received powers of some of the interferers are much larger than that of the desired user the performance degradation is large

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
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• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

47

bull In the context of wireless communications this situation occurs when some of the interferers are located close to the base station while the desired user is far away

bull This problem is known as the near-far problem in CDMA systems

bull A common measure of the robustness of a receiver against the near-far problem is the near-far resistance defined in [10] of the receiver

bull For now we argue the intuitive idea of the near-far resistance measure

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

48

bull To understand the concept of near-far resistance let us imagine that only one user say the mth user were active in the DS-CDMA system considered previously

bull In this case the optimal symbol error probability (using the matched filter receiver) would be which decreases exponentially with rate approximately equal to the SNR when the SNR is large

bull We employ this as a benchmark to compare performance of different receivers in the multiuser scenario

bull Going back to the realistic situation of multiple active users the performance of any receiver will be poorer than that of the optimal receiver of the single-user scenario just described because of the existence of MAI

bull In fact the larger the received powers of the interferers the poorer is the performance

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
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• Slide 19
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• Slide 22
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• Slide 24
• Slide 25
• Slide 26
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• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

49

bull We look at the exponential rate of decrease of the symbol error probability given by a certain receiver as the SNR increases to some very large values

bull The ratio of exponential decrease rate to tells us how efficient the receiver is compared to the optimal receiver of the single-user scenario

bull If the received MAI power increases this ratio will get smaller bull The ratio of the exponential decrease rate to in the limiting

case of extremely large MAI power is the near-far resistance of the receiver

bull A receiver with near-far resistance close to 1 is almost as efficient in any near-far situation as the optimal receiver of the single-user scenario (the best that could be done)

bull A near-far resistance of 0 indicates that the receiver will break down in a near-far situation

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
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• Slide 33
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• Slide 35
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• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
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• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

50

bull For the matched filter receiver we can employ (631) or (643) to conclude that the symbol error probability levels off when the SNR increase (for example see Figure 62)

bull Hence the exponential decrease rate is 0 and the near-far resistance is 0

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
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• Slide 21
• Slide 22
• Slide 23
• Slide 24
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• Slide 26
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• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

51

bull When the matched filter receiver does not give an adequate level of performance there are basically two ways to tackle the near-far problem

ndash One of the ways is to control the transmitted powers of all the users so that the received powers are the same This method is known as power control and is employed in all current practical CDMA systems such as IS-95

ndash Another way to tackle the near-far problem is to notice that the matched filter receiver is not optimal in the presence of MAI and try to develop better receivers that are near-far resistant

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

52

64 Multiple access interference suppression

bull In this section we discuss receivers based on the single-user detection approach

bull Receivers based on the multiuser detection approach will be introduced in Chapter 7

bull The optimal signal-user receiver described in [1] is near-far resistant but it is too complex for practical implementation

bull Its development is rather involved and interested readers are referred to [1]

bull Here we focus on suboptimal receivers that are less complex than the optimal signal-user receiver

bull The basic working principle of these receivers is to exploit some structures of the MAI that are different from the desired signal and to utilize the difference to remove or suppress the MAI component from the received signal

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
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• Slide 51
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• Slide 55
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• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
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• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

53

bull The structures of MAI we can utilize depend the design of the spreading sequences and the availability of resources such as multiple receive antennas

bull A main dichotomy on MAI suppression receivers can be obtained by distinguishing between short-sequence-based and long-sequence-based DS-CDMA systems

bull In a short-sequence-based system a simple structural differentiation between the desired signal and the MAI is the difference in the sequences employed ndash Since the sequences repeat every symbol period this

structural differentiation is invariant from symbol to symbol ndash As a result we can easily extract and utilize this structural

difference by using for example some standard adaptive signal processing techniques making the receiver implementation simple and desirable

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

54

bull It turns out that most of these short-sequence-based single-user MAI suppression techniques can be interpreted as special cases of their multiuser counterparts

bull To avoid repetition we leave their development to Chapter 7 where we will develop the general multiuser techniques and specialize them to single-user receivers

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 13
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• Slide 39
• Slide 40
• Slide 41
• Slide 42
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• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

55

bull For DS-CDMA systems employing long sequences although the sequences are still different the structural differentiation mentioned above is not very useful since the structural difference varies from symbol to symbol making it hard to extract and utilize

bull Therefore we have to employ some other forms of structural differentiation between the MAI and the desired signal that are invariant from symbol to symbol

bull To illustrate this idea we will present a simplified development of the MAI suppression receiver suggested in [11]

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

56

bull The MAI suppression receiver in [11] is designed for asynchronous long-sequence-based DS-CDMA systems

bull It is based on the matched filter receiver bull Its working principle is to exploit the structural difference betwee

n the desired signal and the MAI caused by the fact that the delays of the signals are different

bull The problem of symbol-by-symbol varying nature of the spreading sequences is solved by obtaining the averaged (over all possible choices of sequences) structural difference instead of the instantaneous one

bull This is possible since segments (over a symbol duration) of the long sequences look like random

bull Hence the statistical averaged structural difference can be approximated by the ldquoeasy-to-obtainrdquo time-averaged structural difference

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
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• Slide 16
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• Slide 33
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• Slide 35
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• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

57

bull The abstract statement above can made precise by looking at the output signal from the matched filter

bull First the signal model we consider here is exactly the same as the one described in Section 61except that the period of the spreading sequences is now much large than the spreading gain N

bull To simplify our discussion we model the sequence elements al(k)

as zero-mean iid random variables withbull In the context of long-sequence-based systems this model basica

lly means that the sequences are aperiodic and are picked randomly from the set of all possible sequences

bull Of course this is an (good) approximation to the actual set of sequences used in practice

bull Let us focus on the detection of b0(m) the 0th symbol of the mth u

ser

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
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• Slide 21
• Slide 22
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• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
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• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

58

bull In this case the impulse response of the matched filter is given by

bull Note that we have chosen to employ a non-causal filter here to simplify our notation later

bull Of course we have to use a causal filter in practice and the amount of delay in the causal filter for example T has to be added to all the results we are going to present

bull Let us denote the output signal of the matched filter as

ndash is the component due to the desired user ndash is the MAI plus thermal-noise component

with the subscripts I and W standing for MAI and thermal noise respectively

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

59

bull Conditioning on the set of delays and phases defined in (635) it is easy to show that the autocorrelation function of the matched filter output is given by

where

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
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• Slide 19
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• Slide 26
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• Slide 29
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• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

60

bull The function is the autocorrelation of the chip waveform defined by

bull Also is just a scaled version of

bull We note that since the chip waveform is time-limited to [0 Tc) both are time-limited to (- Tc Tc) and hence the summation in (650) contains only a finite numbers of terms for any fixed values of t and s

bull For example when the chip waveform is rectangular

then is the triangular waveform stretching from -Tc to Tc

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 51
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• Slide 55
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• Slide 57
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• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

61

bull Now the structural difference between the desired signal and the MAI can be readily observed by considering the difference between the autocorrelation functions of the desired signal component and the MAI component nI(t) respectively

bull The autocorrelation functions are made up of products of different delayed versions of the functions

bull This difference can be easily visualized by considering the intuition provided in Figure 63 which shows the delayed versions of the function that make up the desired signal and MAI autocorrelation functions for the two-user case (K = 2) with rectangular chip waveform

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
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• Slide 35
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• Slide 37
• Slide 38
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• Slide 41
• Slide 42
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• Slide 44
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• Slide 47
• Slide 48
• Slide 49
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• Slide 51
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• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

62

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
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• Slide 31
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• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
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• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

63

bull The blue triangle corresponds to the due to the desired signal bull The four red triangles correspond to the delayed versions of

due to the interferer bull The reason that there are four triangles corresponding to the

interferer is that for as shown in the figure each interferer contributes exactly four non-zero terms in the second summation in (650) for the observation interval of 1048576

in which the desired signal autocorrelation function is non-zero

bull For there is only one triangle corresponds to the interferer This triangle coincides with the blue triangle corresponding to the desired user

bull In this case there is no structure difference between the desired signal and the MAI and hence the MAI cannot be suppressed

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

64

bull The discussion above reveals the structural difference between the desired signal and the MAI induced by the different delays of the signals

bull We can utilize this structure to suppress the MAI component in the received signal by observing the matched filter output

bull Since the structure does not depend on the spreading sequences it is invariant from symbol to symbol and hence simple adaptive algorithms can be employed

bull A simple way [11] to suppress MAI based on this structure is to sample the matched filter output about the peak at each symbol (see Figure 63) and then weigh the samples to form a decision statistic for that symbol as shown in Figure 64

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

65

bull We note that the matched filter receiver is a special case of this method with only one sample taken

bull The weights are chosen so that the mean-squared error (MSE) between the decision statistic and the actual symbol is minimized

bull The suppression of the MAI is performed implicitly in the process of minimizing the MSE

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
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• Slide 31
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• Slide 33
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• Slide 35
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• Slide 37
• Slide 38
• Slide 39
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• Slide 41
• Slide 42
• Slide 43
• Slide 44
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• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

66

bull More precisely suppose that we sample the matched filter output 2M + 1 times at

where Ts lt Tc for the detection of the 0th symbol of the mth user

ndash For example Figure 63 shows that case in which Ts = Tc2 and M = 2

bull For convenience we arrange the samples into the vector and work using vector and matrix notation

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
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• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

67

bull An estimate of the transmitted symbol is obtained as the weighted sum of the samples taken at the output of the matched filter ie

where is the sample at time is the weight for that sample

bull The weight vector is chosen to minimize the MSE defined by

where the expectation is conditioned over the set

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

68

bull It can be easily shown [11] that the optimal choice of weight vector is the solution of the following set of equations

where are samples of the autocorrelation functions of the matched filter output and the signal respectively

bull As the existence of the thermal noise component in the signal guarantees the invertibility of the correlation matrix the optimal weight vector is given by

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

69

bull Writing the samples of the MAI-plus-thermal-noise autocorrelation function as the matrix

we can employ the matrix inversion lemma to show that

Where

bull It can also be shown [11] that this optimal choice of the weight vector maximizes the SNR at the decision statistic

bull The maximum SNR value achieved is exactly given by (658)

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
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• Slide 41
• Slide 42
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• Slide 44
• Slide 45
• Slide 46
• Slide 47
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• Slide 51
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• Slide 55
• Slide 56
• Slide 57
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• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

70

bull Suppose that we can employ the improved Gaussian approximation here to approximate the MAI component in the decision statistic as a Gaussian random variable

bull Then the conditional (conditioned on ) symbol error probability is given by

bull For example Figure 65 gives the plot of the symbol error probability (based on the Gaussian approximation) obtained by the receiver with a sampling scheme as shown in Figure 63 in the case where there are two active users in the system the fractional delay of the interferer is Tc2 and the received power of the interferer is 20dB above that of the desired user

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

71

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

72

bull Also plotted in the figure are the symbol error probability (IGA) obtained the matched filter receiver and the symbol error probability of the single-user scenario

bull We see that the presence of the strong MAI causes the error rate of the matched filter levels off as the SNR increases

bull With the MAI suppression receiver the symbol error probability decreases exponentially as the SNR increases showing that the MAI is suppressed by the receiver

bull In fact with the particular set of system parameters the performance of the MAI suppression receiver is only 3dB worse than the single-user scenario

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

73

bull We evaluate the robustness of the receiver against the near-far problem based on a measure similar to the near-far resistance suggested in Section 63

bull First we note that the SNR obtained by the receiver in any situation cannot be larger than the SNR obtained when the mth user is the only active user ie

bull Following the idea of the near-far resistance in Section 63 we define the ldquonear-far efficiencyrdquo (NFE) as

bull We note that 0 lt NFE lt 1 and it is a function of the delays and phases in the set

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Slide 60
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Slide 66
• Slide 67
• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

74

bull For example in the two-user case (K = 2) it can be shown [11] that the NFE of the receiver discussed here is upper bounded by (k ne m)

where this bound can be achieved in the limit by sampling finer and finer

bull For example the NFEinfin for the rectangular chip waveform is shown in Figure 66 as a function of (k ne m)

bull We see that since the NFE is positive when the system is not chip-synchronous (ie ) the receiver is robust against the near-far problem when the two users are asynchronous

bull In general the robustness of this receiver degrades when more and more strong interferers are in the system

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
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• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

75

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
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• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

76

bull As a closing note we emphasize that as the structural difference between the MAI and the desired signal is invariant from symbol to symbol we can employ standard adaptive algorithms to obtain the optimal weight vector described in (656)

bull For example when a training data sequence is available we can employ the LMS algorithm to obtain the weight vector

bull In fact it is shown in [11] that a blind adaptive algorithm which does not require the availability of a training sequence can be developed to obtain the weight vector

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
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• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

77

65 References

[1] H V Poor and S Verdu ldquoSingle-user detectors for multiuser channelsrdquo IEEE Trans Commun vol 36 no 1 pp 50ndash60 Jan 1988

[2] TIAEIAIS-95 Interim Standard Mobile Station-Base Station Compatibility Standard for Dual Mode Wideband Spread Spectrum Cellular System Telecommunications Industry Association Washington DC Jul 1993

[3] J S Lehnert ldquoAn efficient technique for evaluating direct-sequence spread-spectrum multipleaccess communicationsrdquo IEEE Trans Commun vol 37 no 8 pp 851ndash858 Aug 1989

[4] K Yao ldquoError probability of asynchronous spread spectrum multiple access communication systemsrdquo IEEE Trans Commun vol 25 no 8 pp 803ndash809 Aug 1977

[5] M B Pursley ldquoPerformance evaluation for phase-coded spread-spectrum multiple-access communicationmdash Part I System analysisrdquo IEEE Trans Commun vol 25 no 8 pp 795ndash799 Aug 1977

[6] R K Morrow and J S Lehnert ldquoBit-to-bit error dependence in slotted DSSSMA packet systems with random signature sequencesrdquo IEEE Trans Commun vol 37 no 10 pp 1052ndash1061 Oct 1989

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
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• Slide 68
• Slide 69
• Slide 70
• Slide 71
• Slide 72
• Slide 73
• Slide 74
• Slide 75
• Slide 76
• Slide 77
• Slide 78

78

[7] T M Lok and J S Lehnert ldquoError probabilities for generalized quadriphase DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Commun vol 44 no 7 pp 876ndash885 Jul 1996

[8] TM Lok and J S Lehnert ldquoAn asymptotic analysis of DSSSMA communication systems with random signature sequencesrdquo IEEE Trans Inform Theory vol 42 no 1 pp 129ndash136 Jan 1996

[9] J M Holtzman ldquoA simple accuratemethod to calculate spread-spectrum multiple-access error probabilitiesrdquo IEEE Trans Commun vol 40 no 3 pp 461ndash464 Mar 1992

[10] S Verdu Multiuser Detection Cambridge University Press 1998

[11] T F Wong T M Lok and J S Lehnert ldquoAsynchronous Multiple Access Interference Suppression and Chip Waveform Selection with Aperiodic Random Sequencesrdquo IEEE Trans Commun vol 47 no 1 pp 103ndash114 Jan 1999

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 78