Semi-Blind Strategies for

Int erference Suppression

in DS-CDMA Systems

Ryan A. Pacheco

.A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Graduate Department of Electrical and Computer Engineering University of Toronto

@ Copyright by Ryan A. Pacheco, 2000

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Semi-Blind Strategies for Interference Suppression

in DS-CDMA Systems

Ryan -4. Pacheco

Master of Applied Science, 2000

Graduate Department of Electrical and Computer Engineering

University of Toronto

Abstract

The scrni-blind algorithms presented in this work attempt to irnprove interference sup-

pression for a given number of training and information symbols by combining training-

based and blind estimation techniques. CVe first derive the minimum number of training

symbols the training-based least squares (LS) estimator requires for complete suppression

of interference (in the noiseless case). This provides a. benchmark which the semi-blind

algorithms are compared against. Next. a framework for semi-blind iterative interference

suppression is developed which is then used to enhance the LS estimator by adding the

constant modulus and cyclostationar-y properties used in blind algorithms. Finally. a

decision-aided (multistage) semi-blind algorithm that jointly processes the signals from

d l intra-ce11 users is presented. Simulation results indicate a significant improvernent

over training-baçed and blind estimation techniques. Application of this work is in the

reverse-link of third generation short-burst aspchronous DS-CDhIA systems subject to

both ISI and MAI.

Acknowledgement s

First. 1 would like to thank my advisor. Professor Hatzinakos. for his patience and helpful

feedbnck over the past few years, and for the financial support he provided through a

research assistantship. I would also like to thank Dr. Kuzminskiy and Professor Sousa

for several useful discussions on signal estimation and CD'ILA systems respectively. 1 am

gratefiil for the financial support provided by the University of Toronto. NSERC. and

CITO. Finaliy. 1 want to thank rny farnily for their love and support. and a special thanks

to Justin for letting me use his PC rlrhenever 1 needed to.

Dedicat ion

To my mother, and to the memory of my father.

Contents

Abstract

Acknowledgement s

Dedicat ion

List of Figures

List of Tables

Principle Symbols and Abbreviations

1 Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . 1.1 LIultiaccess Communications

. . . . . . . . . . . . . . . . . . . . . . . . 1.2 bfodeling of Wireless Channels

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Rayleigh Fading

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Delay Spread

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Angle Spread

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Coherence Time

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Previous Work

. . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Time-Only Processing

iii

. . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Space-Time Processing 10

1.4 MotivationandObjective . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Signal Model. Eqiialization and Blind Identification 1.5

2.1 Signal Mode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 FIR Zero-Forcing Eqiialization . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Blind Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 CM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Blind Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.41 Subspace Methods . . . . . . . . . . . . . . . . . . . . . . . . . . II

2.4.3 Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Summnry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Serni-B lind Algorit hms and Simulation Examples 29

3.1 Linear NMSE Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Beharior at High SNR . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Iterative Semi-Blind Framework . . . . . . . . . . . . . . . . . . . . . . . i3-l

3.2.1 hIOE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.2 LSRChI.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.3 Ch1 p:! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Subspace Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.1 Training-based and Semi-Blind Channel Estimation . . . . . . . . 42

3.3.2 SBCXIACI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -44

3.4 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4.1 Cornparison for Different Channel Lengths . . . . . . . . . . . . . 49

3.42 Dependence on Xumber of Data Symbols (A$) . . . . . . . . . . . 49

3.4.3 Dependence on Amount of Spatial Diversity (A) . . . . . . . . . . 52

3.4.4 Dependence on Number of L.sers (JI) . . . . . . . . . . . . . . . . 52

3 . 4 Probability of Error Cornparison . . . . . . . . . . . . . . . . . . . 53 - C 3.5 Multistage Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . aa

3.6 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 Conclusion 62

4.1 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2

4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Bibliography

vii

List of Figures

. . . . . . . . . . . . . . . . . . . . Multiaccess Communications Scenario I

Simple DS-CDMA Systern . . . . . . . . . . . . . . . . . . . . . . . . 3 .

Multipath Signal with PSD . . . . . . . . . . . . . . . . . . . . . . . . . 3

. . . . . . . . . . . . . . . . . . Histogram of hlultipath Signal Envelope 3

. Rake Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

Bank of Matched Filters as a Front-end . . . . . . . . . . . . . . . . . . . S

. . . . . . . . . . . . . . . . . . . . . . . Space-Time Cascade Processing I l

. . . . . . . . . . . . . . . . . . . . . . . . Space-Tirne Joint Processing I l

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Burst structure 12

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Baseband system 16

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Penalizing Criteria 22

3.1 >ISE Performance for Diagonal Loading: WmSLS . . . . . . . . . . . . . . 33

3.2 LISE Performance of LSRCMA for Different SNR . . . . . . . . . . . . . 39

3.3 &ISE Performance for Different versions of LSRCMA . . . . . . . . . . . 40

3.4 Channel Estimation Error for Increasing o! and Different . . . . . . . 43

3.5 Channel Est.imation Performance for Different .V (a = 3) . . . . . . . . . 44

3.6 Channel Estimation Performance for Different û . . . . . . . . . . . . . . 45

3.7 Cornparison of LSRCàM with SBChIACI using same initialization . . . . - L i

. . . . . . . . . . . . . . . . . 3.8 Performance of SBCMACI for Different û!

. . . . . . . . . . . . . . 3.9 LISE Comparison for Increasing L: Synchronous

3.10 LISE Cornparison for Increasing L: .-\ synchronous . . . . . . . . . . . . .

3.1 1 SISE Comparison for Increasing Number of Data Symbols: Synchronous

3.12 SISE Comparison for Increasing Nurnber of Data Symbols: .4 synchronous

. . . . . . . . . . . . . 3.13 4ISE Cornparison for Increasing Spatial Diversity

. . . . . . . . . . . . . 3.14 USE Comparison for Increasing Number of Users

3.13 Probability of Error Comparison: Synchronous . . . . . . . . . . . . . . .

3.16 Probability of Error Comparison: .-\ synchronous . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . 3.17 III LSRChIA Performance Improvement

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18 Llultistage Example

List of Tables

. . . . . . . . . . . 1.1 BER vs . Data Length for Blind Algorithm: Temporal 9

1.3 BER vs . Data Length for Blind Algorithm: Spatial-Temporal . . . . . . 11

2.1 Rank estimate using MDL . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 LSRCIIA - - + . * . * * . . - . - . - . - . . . . - - . . . . . . . . . . . . 35

3.2 SBCI.I.4CI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Simulation Parameters for SBChIACI . . . . . . . . . . . . . . . . . . . . 49

3.4 Siniulation Parameters for LSRCSJA and LS . . . . . . . . . . . . . . . . 49

- - 3 5 Probability of Error for Blind Methoci . . . . . . . . . . . . . . . . . . . JJ

3.6 Probability of Error for SBCbIACI . . . . . . . . . . . . . . . . . . . . . 56

3.7 Probebility of global convergence for multistage algorithm . . . . . . . . 58

Principle Symbols and Abbreviations

DS-CD L1.A

h1.U

ISI

hlhISE

LS

CXl.4

LSRCkI.1

SBChIACI

MSBChl.JLCI

MLSRC;\IA

Space-time weight vector for mth user

Received signal vector for kth symbol (dim(r,(k))=.-Np)

Autocorrelation matrix calculated using .Vb data symbols

Cross-correlation vector for mth user calculated using .\i, training

symbols

Number of data symbols per burst

Xumber of training syrnbols per burst

Number of antenna elements (diversity channels)

Processing gain

Smoothing factor ( p 2 1)

Channel length

Xumber of active users

Direct Sequence Code Division hIultiple Access

Multiple Access Interference

Intersyrnbol Interference

Minimum 'vlean Squared Error

Least Squares

Constant Modulus Algorithm

Least Squares Regularized by the CM;\

Semi-Blind CM.4 wit h Channel Identification

Multistage SBCIIACI

1Iultistage LSRCMA

Chapter 1

Introduction

1.1 Mult iaccess Communications

In rnultiaccess communications (Fig. 1.1). channel resources are allocatecl in \v-s t hat

require either strict cooperation arnong users. such a s in time or frequency division miil-

tiple access (TDkI.4 or FDhlA). or not. as in code or space division multiple access [301

User 1

User 2

. User

M

M usen coordinated or One or more independent umrdinated mrurnujion observetlons in noise

Figure 1.1: hlultiaccess Communications Scenario

DS-CD hIA and SDMA systems generally require more complex receivers than TDhl.4

or FDMA since the received data may

int erference. also called mu1 t iple access

contain interference from other users (multiuser

interference (MAI)). The amount of interference

1.1. Multiaccess Communications

that one user contributes to another is dependent on the orthogonality between their

received signals. In DS-CDMA systems, orthogonality may exist on the transmitting-

end by possibly assigning orthogonal spreading-codes to users but will not exist on the

receiving-end due to propagation effects ( Fig. 1 2)).

User 1 Spreadng Code User 1

Noise

2 Receiver .

M User M

M Orthogonal Codes Users

One or more independem observations in noise

Figure L 2: Simple DS-CDbI.4 System

1 The figure above depicts a simple DS-CD hl;\ system. Here. rl. . . T . J ~ and p i . . . p.,,

indicate potentially different transmission delays and propagation channels for each user

(as seen by the first sensor). If the transmit ted signals are orthogonal. it is not guaranteed

that the received signals will be orthogonal alter convolution. For mobile systems. the

problem is cornpounded by the fact that the propagation environment is always changing.

making it v e - difficult to design spreading codes that Nil1 remain orthogonal.

Figure 1.2 illustrates the ongin of two forms of interference in DS-CDM.4 systerns. We

have already discussed multiple access interference (XIAI), which is caused by nonorthog-

onality. There is also intersymbol interference (ISI). or "self-interference" . which is caused

by multipath propagation, a characteristic of wireless channels. ISI is negligible in low

rate DS-CDhl.4 applications but is becoming more of an issue in third generation systems

(where the symbol duration is on the order of the multipath delay spread [2]:[4]). Con-

1.2. içlodeling of Wireless Channels

sequently. this work will be concerned with the mitigation of two forms of interference in

DS-CDSIA systems: MAI and ISI.

The remainder of this chapter will review sorne important facts about mireless sys-

tems. discuss previous work in this area. explain the motivation for this thesis and its

contribution. and finally. descri be its organizat ion.

Modeling of Wireless

There are certain characteristics of wireless

Channels

channels w hich

signal processing met hodology for wireless systerns. The four

have significant impact on

most significant are fading.

t irne-sprcading ( leading to ISI) . angle-spreading , and t ime-variat ions. as descri bed below .

The time-spreading and time-variations of the channel are characterized by the delay

spread and coherence t ime respect ively.

Figure 1.1: Histogram of Multipath Signal Figure 1.3: Multipath Signal with PSD

Envelo pe

1.2. ilbdeling of Wzreless Channels

1.2.1 Rayleigh Fading

In Fig. 1.2 propagation channels p i ? . . . p h cause distortions of the transmitted sig-

nals. The nature of these distortions is described by their respective impulse responses.

Traditionally. impulse responses for wireless channels are statistically rnodeled using the

conplcs Gaussian random proccss. Thc cnrclopc of thc chnnncl impulse rcsponsc ~ i l l

then have a Rayleigh distribution. This is applicable when there are a large number of

scatterers in the transmission path. causing multiple copies of the original signal to be

received over a certain time interval.

An example of a rnultipath signal is given in Fig. 1.3. and the histogram of its envelope

in Fig. 1.4. The signal is the sum of 100 sinusoids with independent phase

LOO

where 8, and O, are uniformly distributed over [O. 2 ~ 1 . and fd is used to mode1 the Doppler

spread (section 1 .U). The histogram mas computed using 10000 independent trials of

the envelope of y(t)

where y ( t ) is the Hilbert transform of y ( t ) . It has the shape of the Rayleigh pdf as

expected.

One can easily see that the envelope of the signal changes with time. At tirnes when

the amplitude is very low (a deep fade) detection becomes very difficult. In this case

some form of diversity is usually required (section 1.2.3). In this work we assume that

each user transmits through a Rayleigh fading channel, and that fading statistics are

independent between users.

1.2.2 Delay Spread

.i series of narrow pulses sent by the transmitter will be received with different delays

at the receiver. If we plot the average received power versus delay, the profile will likely

resemble an esponential probability distribution. .inalyticallp modeling the power-del-

pronie using an exponenriai random variable is cornmon practice. 111 this case. the dei-

spread corresponds to the expected value of the random variable. The inverse of the

del- spread is called the coherence bandwidth of the channel.

The effects of delay spread on the transmitted signal depends on the bandwidth

of the transmitted signal in comparison to the coherence bandwidth. If the coherence

bandwidth is larger than the transmitted signal's bandmidth. then Aat fading will occur

(al! lrequency components will fade by the same amount). Frequency selective fading

occiirs if the coherence bandwidth is srnaller t han the transmitted signal's bandwidt h.

and ISI will occur if the coherence bandwidth is snialler than the information signal's

bandwid t h.

We make the distinction between these ttvo cases because in DS-CDMA systems

it is possible to have frequency selective fading, but negligible ISI. This is because in

DS-CDhI.1 systems the information signal is spread to a much Iarger bandwidth before

transmission by multiplying it with a high bandwidth spreading waveform. The spreading

waveform consists of a sequence of narrow pulses called chips. If the symbol rate is low

(say 9600 bps) then the information signal has a small bandwidth and delay introduced

by the channel will mainly be Felt at the chip level. This is true for first generation

DS-CDhIA systems, such as 1s-95. It is expected that next generation systems will

offer higher data rates [2]. meaning the information signal Nil1 have a higher bandwidth.

In this work it is assumed that the delay spread is greater than the symbol duration.

meaning that lrequency selective fading and ISI are both present.

1.2. Modelzng of Wzreless Channels

1.2.3 Angle Spread

To mitigate the effects of fading, it is often necessary to design wireless systems a i th

some form of redundancy. Diversity, either in tirne? frequency. or space is meant to pro-

vide separate paths to transmit redundant information (as figures 1.1 and 1.2 indicate).

Space diversity, implemented with an antenna array, is cornmonly used at basestations.

An important factor for any diversity scheme is the correlation between the redundant

paths. For an antenna array this is described by the angle spread. A high degree of

scattering close to the antenna array will cause a large angle spread. helping to decrease

the correlation between antenna elements. and increase the diversity gain. In this tvork

we assume that fading is independent between sensors.

1.2.4 Coherence Time

Tinie-variations of the communication channel causes a broadening of the transmitted

signal's spectrum. In other words. motion of the trançmitter in relation to the receiver

causes the transmitted signal to spread in frequency. as shomn in the PSD of y(t) (Fig.

1.3). The Doppler spread is used to characterize this effect in the frequency domain. In

eq. (1.1), a Doppler spread of 0.2 Hz is present ( fd = 0.2). The maximum Doppler shift

can be estimated based on vehicle speed, v

where fc is the carrier frequency. and c is the speed of light. A vehicle traveling a t 100

km/h tvill have a maximum Doppler shift of about 100 Hzo with a 900 MHz carrier.

The inverse of the Doppler spread is knomn as the coherence time. If the symbol

duration is longer than the coherence time, then the channel is said to be fast time-

varying. In this work we will assume that the coherence time is longer than the symbol

duration. meaning that more than one s p b o l s "observes" the same channel.

1.3 Previous Work

In a single user scenario, assurning negligible ISI, the best detection strategy is to use

a receive filter that is "matched" to the transmitted signal [31]. In DS-CDBIA systems

the matched filter corresponds to the spreading code for the desired user. If there is

niultipath propagation. but thc del- sprcad is much lcss than thc s:;rnbol duration. thc::

a R W E receiver [31] is used (Fig. 1.3). -1 R W E is matched to the overall impulse

response of channel plus spreading code.

Spreading A,) Waveforrn 1 4 I. pl ] 1 % 1 [ TL 1

Figure 1.5: Rake Receiver Wide band

Signai Requires t hat muitipat h channel is known. and that ISI is negligi- ble. 1s not near/far resistant.

lntegrate and sum

In either case. it has been established [43] that these strategies are not viable for rnul-

tiuser situations. Correlation-type receivers such as these have an irnpractical probability

of error in cases where interfering usen are received with much higher power than the

desired user! called the near/far problem. They also require that the number of users be

rnuch Iess than the processing gain. constraining the capacity of the network. A large

body of work. classified as multiuser detection. is devoted to studying how detection can

be efficiently performed in multiuser enviro~iments. This section briefly covers the major

work in this area. with an ernphasis towards blind solutions.

1.3.1 Time-Only Processing

Early work in rnultiuser detection [G]: 1231 [24]? [13], concentrated on time-only pro-

cessing and focussed on removing XIAI at the receiver. ISI was largely ignored, the

1.3. Previous Work

predominant channel mode1 being the AWGN channel.

Verdu's paper [43] was the first in this area. He showed that by incorporating infor-

mation about the interference from other users. optimal multiuser detection is possible

through a maximum likelihood detection strategy. The prohibitively high computational

and informational complexi ty of t his strategy has motivated researchers to look for lower

complexity solutions.

'VI'VISE [24] and decorrelating (zero-forcing) [23] detectors were later developed that

have linear cornputational complexity. These solutions, however. still have high informa-

tional compiexity. The hlhLSE detector. for example. needs to know the spreading codes

and received SNR for al1 active users. The front-end to such detection strategies gener-

ally consists of a bank of matched filters. which can then be followed bu man- different

algorithms (Fig. 1.6)

Matched Filter + user 1 k \ j I I Process I

Matched Filter 1 User2 tJ ( Viterbi Alg.

Zero-Forcing MMSE ... ) l

Matched Filter U w r M F'j

Figure 1.6: Bank of Matched Filters as a Front-end

A detection strategy was later developed that could approach the performance of the

MkISE detector but only required knowledge of the desired user's spreading waveform

[13]. It was titled blznd multiuser detection because of the fact that interference could

1.3. Previous Work 9

be suppressed without needing explicit knowledge about other users (spreading codes.

SNR. etc.). similar to traditional blind equalization algorithms that remove ISI without

knowing the input or the channel [Il].

Honig et al's paper [13] is the motivation for much of the recent work on blind

multiuser detection. It attracted the attention of researchers in the signal processing area

mho were working on blind equalization. Of particular interest is the work of Tsatsanis

and Giannakis [39], [10]. who extended the algorithms in [13]. [23], and (241 to the case of

frequency-selective channels. After this. there has been a steady shift torvards considering

how to perform multiuser detection when there is .LIAI and [SI. Wang and Poor have

published recent work that apply SVD-based blind algorithrns to the problem of joint

equalizat ion and MAI suppression [U].

The driving force behind the majority of blind algorithms for DS-CDAIA systenis is

the work of Tong et al [37]. They showed that blind equalization of non-minimuni phase

channels is possible with second-order statistics. Later work. based on that of 1371. hy

Sloulines et al [27] is what Cl'ang and Poor and others generally apply

Blind multiuser detection algorithms are popular because they do not use any more

information then the correlation-type detectors but are not near/far lirnited. It is ques-

tionable. however, whether they can be used in mobile systems since the amount of data

t hey require may be more than what can be transmitted over quasi-stationary conditions.

For example, Table 1.1 shows the probability of error for different lengths of data using

the offline blind algorithm proposed by Wang and Poor [U]. In this case the DS-CDbI.4

system ivas a t 50 % capacity. there was a delay spread of one symbol and SXR of 10 dB.

1 Num. of Bits 1 50 1 100 1 150 1 100 1 300 1 400 1 450 1

Table 1.1: BER vs. Data Length for Blind Algorithm: Temporal

1.3. Preuiozls Work 10

The probability of error was estimated using 500 independent trials for each length of

data. In this case the algorithm needs approximately -400 symbols to achieve a practical

probability of error. If this was a GSM system, we would have to do this with only 150

symbols.

1.3.2 S pace-Time Processing

Space division multiple access (SDiLI-4) is a general term that can be applied to TDMX

or DS-CDhl.1 systems. The early work on SDMA for DS-CDXI.4 was on space-time

RAKE reception. first by Naguib and Paulraj [28] and later by Zoltowski and Ramos

[A;]. The space-time RAKE (or 2-D R X E ) consists of an antenna array followed by a

RAKE receiver. This is an example of spacctime cascade processing jspace-only process-

ing followed by time-only processing). Optimal cascading is possible by using opt inial

space processing followed by optimal temporal processing. Figure 1.7 depicts a general

space-tirne cascading structure. The beamformer weights UT;. --. . ut can be adjusted to

mavimize SINR. and temporal processing can use maximum likelihood sequence detec-

tion. In [45] Wang and Poor extend the optimal detection strategy of Verdu to include

space-t ime cascading. They also present hI LISE and blind techniques.

Alternatively. one can consider joint domain processing (Fig. 1.8). Joint domain

processing can provide more degrees of freedom than cascade processing, meaning that

it should be able to provide better performance. In [5] the capacity of a DS-CDiLI.4

system using cascade processing \vas cornpared with joint dornain processing. Simulations

showed that joint domain processing was able to accommodate more users than cascade

processing for the same probability of error. In this work we d l use joint domain

processing.

There is a large amount of work on joint domain processing that uses a general SDM\

Beamformer

Temporal

Processing

(ML, MMSE)

Figure 1.8: Space-Ti~ne Joint Process- Figure 1 .T: S pace-Time Cascade Processing

ing

frarnework that can be applied to TDLIA or DS-CDhLA systems. In particular. Slock

[36] and van der Veen et ut [Q] have published work on zero-forcing equalizntion and

blind subspace-based identification for multiuser SDSIA systems respectively. The blind

method of Wang and Poor [44] that was discussed in the previous section is essentially

an application of (421 to the case of DS-CDSIA. Their original work does not consider

the use of an antenna array. but it is a simple extension to incorporate space-time joint

processing. The probability of error for different data lengths is given in Table 1.2 using

the same simulation parameters as before but now cvith a two element antenna a r t .

1 Xum. of Bits 1 50 Prob. of Err. 0.2583

Table 1.2: BER vs. Data Length for Blind Algorithm: Spatial-Temporal

Spatial-temporal processing helped to reduce the required data length but not to the

point at which GSM sets its benchmark. We will find that by incorporating a small

nurnber of training symbols we are able to meet this mark.

1.4. lk t ivat ion and Objective

1.4 Motivation and Objective

This work is motivated by the need for receiver algorithms that can be applied in high-

rate DS-CDMA systerns subject to ISI and iLIAI. Also. it is desired that these algorithms

should provide interference suppression far bet ter t han correlation-type receivers. but

üse the samc amount of information. Blind algorithms hr:e sho-n to Se srccrssfii! in

this regard [39]. [U] but do not provide adequate performance for small data lengths

(see Tables 1.1 and 1.2 br an esample). This precludes their application in niobile

communication systems. Semi-blind techniques (171. (1 81 ? [2 11 have gained recent interest

because of their ability to improve the convergence behavior of blind algorithms by aclding

information commonly found in communication systems (such as training symbols [l;].

spreading codes [14]. and transmitter filter knowledge [Tl).

In DS-CDhI.4 syterns the spreading code of one or more users is typically known.

and for third generation systems it has been proposed that a short sequence of training

symbols be acidecl. periodically. to user information before spreading. as s h o w in Fig.

1.9. The time duration of the burst, 0.625 ms. is chosen to be much less than the

coherence time of the channel. as in GSM systems 121.

I I Oata l Symbols

S. Spreading (h ChipYsymbal)

I \

Figure 1.9: Burst structure

Training symbols can be used for channel estimation, interference cancellation. and

1 . 5 Contributions 13

adaptation of antenna arrays. Least squares algorithms are popular training-based ap-

proaches [16]. However. in practice, depending on the severity of the interference en-

vironment (number of users, fading, etc.) the available amount of training used for

conventional LS estimation may not be sufficient [18], [29]. The objective of this work

is to develop semi-blind algorithms for mobile DS-CDLIA systems which proride ro-

bust interference suppression with a relatively small amount of information symbols and

training-syrnbols. The intended application is in the reverse-link (mobile to base) of third

generation asynchronous short-burst DS-CDM-4 systems. It will be shown. via simula-

tions. how properties of comrnunicatioii signals. such as cyclostationarity and constant

modulus. can be used to significantly reduce t.he number of training symbols required for

accurate signal estimation in cornparison to the classical LS estimator. By reducing the

required number of training syrnbols we are able to improve the throughput and rapacity

of the wireless network.

1.5 Contributions

Our contributions are as follows

0 In section 3.1 we discuss the minimum number of training symbols required for

conventional LS estimation. It is shown that as the SNR +P CQ this is the nurnber

of training symbols needed for complete suppression of ISI and MAI. This result

will provide a benchmark which the semi-blind algorithms are compared against.

In section 3.2 the LS estimator is generalized to include constant modulus infor-

mat ion. The resulting semi-blind algorithm, called LSRCM.4 (Least Squares Reg-

ularized by the Constant Modulus ?ilgorithm), is s h o m to belong to much larger

family of optimization functions. This novel Framework helps CO understand how

the LSRCMA works and how it can be improved.

0 In section 3.3.1 a new second-order serni-blind multiuser channel estimation tech-

nique is developed that acts as a front-end for the interference suppression algorithm

presented in section 3.3.2.

0 -4 novel semi- blind interference suppression algorithm t hat uses the cyclost at ion-

arity and constant-modulus properties of the information symbols is presented in

section 3.3.2. The resulting algorit hm. called SBCMACI (Semi-Blind Constant

Xlodulus Algorit hm wit h Channel Identification). greatly outperforms the LSR-

0 In section 3.3 multistage versions of the LSRCh1.A and SBCIIIACI is presented

(called 3ILSRCSI.A and hISBCXIAC1). where joint processing of the signals from

al1 intra-ce11 users is performed. It is shown that joint processing helps to reduce

the required number of training symbols even further.

1.6 Organization of Thesis

This chapter tvas an introduction to the problem of interference suppression for DS-

CDlLIA systerns. In Chapter 2, the signal model. FIR zero-forcing equalization. and sonie

pertinent background material on blind equalization and identification will be discussed.

Semi-blind algorit hms and simulation examples are presented in Chapter 3. Chapter 4

draws conclusions and presents directions for future work.

Chapter 2

Signal Model, Equalizat ion and

Blind Identification

This chapter is a collection of varioiis topics which provide a background for material

presented in the nest chapter.

2.1 Signal Model

In DS-CDhIA systems each user is assigned a unique code that is used to spread the

'Yb - 1 information bandwidth. Let {b,[n]},,, and {cm[n]), denote. respectivelu. the in-

formation symbols and spreading code for the mth user (rn = 1. - . * . AI). where .Yb is

the number of data symbols transmitted per burst (it is assurned that the channel CO-

efficients do not change while & data symbols are being transmitted), and .V is the

spreading factor (N chips/symbol. also called the processing gain). Hence,

k=O

represents the baseband discrete-time transmitted signal. .Assurning there are -4 diversitp

channels (implemented Nith -4 antenna elements). the received baseband signal at the

2.1. Signal Model

ath antenna elernent (ath sensor in Fig. 1.2) is modeled as:

where p & ( t ) is the combined channel impulse response for the rnth user (including trans-

mit filter, multipath channel, and receive filter, as shown in Fig. 1.1)

l /Tc is the transmission rate (chip rate)? T, is the propagation delay. ;Il is the number

of active users. and ua( t ) is additive noise:

Noise n (t) a

Figure 2.1: Baseband system

It is useful to write r,(t) in terrns of the information s~mbols. using (2.1):

where g:( t ) is the overall impulse response of channel plus spreading code.

n=O

For spread-spectrum signals. multipath components with delays greater than the chip

duration (Tc) can be resolved. So the multipath channel hm(t) is modeled as a tapped

delay line wit h tap spacing Tc [31]:

2.1. Signal filode1 17

L, is the number of resolvable rnultipath cornponents for the mth user (Lm = r?]' where Tm is the delay spread experienced by the mth user), and {3:[1]}f2-~ are? in

general. complex random variables. Finally, the received signal is sampled a t the chip

rate. to yield the following discrete- time signal:

Throughout this work the follorving standard asçumptions will be made about the

previously defined quant ities:

1. Spreading codes cm = [c,[O]~ . - - . çn[X - l]lT. m = 1. - - - . .II are linearly indepen-

dent.

2. ( b , [ n ~ } , ~ is a complex i.i.d. sequence with uniform pmf. taking values from a

-4-QAhl alphabet: (kl* j ) /&. The sequences for different users are independent.

3. pO,[k] is FIR with order q, such that Lm - 1 5 qm 5 ( L - 1 ) N for some integer L

and p",O] # O if rn is the desired user (implies synchronization for the desired user.

Tm = O)-

Lm-L 4. (,&$]},=, are complex zero-mean Gaussian random variables (Rayleigh fading).

that remain fixed during the transmission of .Lb data symbols.

5 . Fading is independent from user to user. and between antenna elernents.

6. v,[n] is circular complex white Gaussian noise. uncorrelated with b,[n] Q m, with

zero rnean and variance a2.

2.2. FIR Zero-Forcing Equalzzation 18

Assumption 4 places a restriction on the size of the data burst (&). For GSM

systems. .Vb is 1-44 (including tail bits). In our simulations we will use either Zib =

100 or 150. .ksumption 5 is an ideal assurnption about the antenna a r r q In practice

there will be some correlation between the fading channels observed at each antenna

element depending on the spacing between the elernents and the degree of scattering

near the array (as discussed in section 1.2.3). We also assume that there is no coupling

between the signals at each sensor.

2.2 FIR Zero-Forcing Equalization

In traditional multirate signal processing, a filter bank is designed to analyze a signal

(analysis bank) and then to reconstmct it (synthesis bank). .-\ DS-CDhl.4 system can

be riewed as the opposite of this. We are given the reconstructed signal and look for

an analysis bank to separate the individual signals. In the case of DS-CDMA systerns.

we want to know under what conditions FIR filters can be founci that will remove both

ISI and MAI (zero-forcing) from the received data [do]. Obviousl- in the noiseless case.

these 6lters are optimal. In this section the filter length that is needed for zero-forcing

d l be derived. The result can be found in many papers dealing with equalization in

MIMO systems [40], (361: [42], [44], but is repeated to establish the notation and receiver

structure used in Chapter 3.

Using (2.5) the received discrete-time signal, r. [n], can be written as

where R [ k ] = g k ( t = kTc)- Assume that me are interested in recovering the kth symbol.

To do this Ive first collect the chips from al1 antenna elements to form the vector r(k)

which is espressed in terms of the data syrnbols, b(k) and channel impulse responses.

2.2. FIR Zero- Forcing Equalzzation

G(1)- for al1 users plus additive noise:

r (k ) = [G(L - 1). , G(O)] . [b(k - L + 1)*, - ! b ( l ~ ) ~ ] ~ + ~ ( k )

where.

[(k + 1 ) J - 11

in some cases. depending on the channel length (L), number of users (dl). processing

gain (X). ancl amount of spatial diversity (A). it rnight be necessary to process more than

one receivecl vector a t a time in order to estirnate the k th symbol. In general. stacking ,u

consecutive symbols. the vector that will be processed, r,(k). is written as

where.

2.3. Blind Eqaalitation

It is desired that the received signal be filtered such that. in the noiseless case:

wliere the weight vector W, is the space-time equalizer for the mth user' and 6 is an

arbitrary delay. To satisfy the zero-forcing result of (2.13) it is necessary that Gp have

fidl column rank [36]. [42], hence the required number of symbols we need to stack is

given by the following inequality

The parameter p is sornetimes called the snioothing factor. When performing direct

channel inversion. i.e. equalization wit hout channel estimation first . t hen i t is irnport ant

that (2.14) is satisfied. If channel identification needs to be performed than further

constraintç on p must be made as will be seen in section 2.4.

2.3 Blind Equalizat ion

Blind equalization dates back to the iate 70's and early 80's with the work of Sato [33].

Godard [9] and Treichler [38]. Up until 1994 blind algorit hms performed equalization

by computing higher-order statistics (HOS) of the baud-rate sampled channel output.

This was done either expl ic i t l~ through the use of cumulants [Hl. or irnplicitly. b -

using nonlinear cost functions [33], (91. The popular constant modulus algorit hm (CIVIA)

2.3. Blind Equalizatzon 21

irnplicitly uses HOS. It has significantly lower cornputational complexity than the explicit

methods, and has a simple adaptive implementation.

In 1994, Tong et al showed that blind identification and equalization of non-minimum

phase channels is possible by using second order statistics (SOS) if the channel input is

cyclostationary (as is the case for communication signals) [37]. This was a significant

discovery which created a lot of interest in the use of blind algorithms for fractionally-

samplecl receivers, including DS-CD hI.4 spstems. By using second-order stat ist ics. such

algorithms are able to perform better with less data than HOS-based techniques.

In this work the CbI.4 and SOS-based identification will be used. so these techniques

are introduced in the following sections.

2.3.1 CM Algorithm

The constant modulus algorithm works by penalizing the receked signal for deviations

froni a constant modulus (the unit circle in oiir case). Figure 2.2. for esample. displays

how the filtered symbol WEr,(k) niight be compared with the unit circle using the CM\.

or with the source symbol in a training-based algorithm.

In its generôl form. the equalizer coefficients are computed according to the folloaing

cost function

w&! = arg min - C(l~*rJk)l~-q* w 2pi\,

k=O

whcre.

The most popular versions are for p = lo the CMA 1-2. and p = 2. the CMA 2-2 or

just CM.-\. There is no closed form solution for the weight vector mhich rninimizes the

CS1 cost function, so typically some method of gradient descent is atternpted. For batch

8.3. Blind Equalzzation

Source Symbol

Training-based criteria

Constant modulus criteria

Filtered symbol

Figure 2.2: Penalizing Criteria

applications' Yemton's method [16]. [34] caii be used. To do this we need to compute the

Hessian and gradient of the cost function. It can be shown [SI t h the gradient G(W).

ancl Hessian H(W) for (2.15) are

w here

The case of p = 1 is attractive for offline implementations since the Hessian is inde-

pendent of W. rneaning it only needs to be computed once per burst.

2.4- Blind Identzfication 23

The major drawback to using the CMA is the presence of undesirable local minima. It

can be shown (251, [22] that when there is an infinite ainount of data and no noise. for any

initialization we will get the ideal equalizer (zero- forcing) with an FIR filter (asstiming

that the length condition (2.14) is met). Howvever for finite amounts of data there is no

giiarantee. Improvement of the convergence characteristics can be accomplished through

accurate initialization (from a short training sequence for example).

2.4 Blind Identification

The CM.-\ performs direct channel equalization (channel equalization cvithout identifica-

tion). It is also possible to perform channel equalization by first identifying the channel.

Blind channel identification can be perforrn using either higher-order statistics or second-

order statistics. Blind channel identification through second-order statistics is made pos-

sible by oversampling the received signal and/or by using antenna arrays. Indeed. this is

what makes G p have full column rank, a necessary condition for channel identification

The subspace based identification technique in [Z] is generally favored over that of

Tong's [37] because it is computationally more efficient. and it has been shown to have

Iower estimation variance [27]. In [42] the rnethod of [27] was extended to a general

multiuser case, we will now outline how it can be applied to DS-CDMA systems.

2.4. Blind Identification

2.4.1 Subspace Methods

The autoconelation matriu, R = E{r,(k)rF(k)), can be expressed in terms of its signal

and noise su bspaces via eigendecomposit ion:

where U = [ U , U, 1. and .\ = diag(.\,, -1,). U, = lul. . . . u,] contains the or-

thonormal eigenvectors ahich span the signal space. i.e. range(U,)=range(G,). U, =

[u,,~. - . . . u . - ~ ~ ~ ] contains the orthonormal eigenvectors which span the noise space and

is orthogonal to the signal space: UFG, = 0. .\, = diag(XI. . - . A,) contains the K

eigenvalues of the signal space and .\, = CT~I~~,~,-, the eigenvalues of the noise space. h:

is the rank of G,' cvhich from section 2.2 is known to be .\I(p + L - 1).

The orthogonalitp between the noise and signal spaces is exploited to estimate the

channel. In the general multiuser scenario [42] blind channel identification up to a non-

singular matrix factor is possible (given certain conditions are met). but channel clas-

sification (which channel corresponds to which user) is not. In DS-CDMA systems the

spreading code for the desired user is known (and is assumed to be unique) and this can

be used for channel classification. making the impulse response known to a mu1tiplicati1-e

constant.

Define the following:

2.4. Blind Identification

where F i ((.4iVp - r i ) x ;LN matrix) is a partition of F. I t is easily seen that

So. G can be identifieci. from the right nul1 space of F. up to a nonsingular JI x .\.!

matriv if F is tail [-El. This requires that Ir be chosen such that:

This result applies to the general rnultiuser case. The .I.I x ,LI arnbiguity matrix poses

a Limitation on the use of this channel identification method in practice. In the single

user case? the ambiguity matrix reduces to a complex scaler. Using the spreading code

for the desired user' the multiuser situation can be reduced to that of dealing with a

single user. For example. for the mth user

'-& where Cm = diag(C,. - - - . c , , . ) .qLnx .4(q,+1) and

2.4. Blind Identification

Hence. p, can be identified. up to a multiplicative factor, from the right nul1 space

of FC, if p is chosen such that

this is the second condition that p must satisfy (see section 2.2). Once p, has been

identified any number of equalization strategies can be used.

2 A.2 Implementation Issues

The ort hogonality betaeen the signal and noise su bspace is approximately sat isfied in

practice because only a time-average of the autocorrelation m a t r k is available

meaning that (2.25) is typically solved in the Ieast squares sense [24]. tvhich Leads to the

following minimization problem

0, = arg min I IFC,~ I I~ l l~I I=l

- H - = arg min p H ( ~ m ~ F C , ) ~

IIpII=l - - arg min pH@,p

IIPII=~

2.4. Blind Identification 27

It is known that the vector which minirnizes (2.28) is given by the eigenvector corre-

sponding to the smallest eigenvalue of 4,. In section 3.3. (2.28) will be used. together

with training-based optimization to improve the channel estimate, and remove the mul-

tiplicat ive ambigui t y

Also, in practice the dimensionality of the signal space is not known a priori. -4s

Yb + X. with white noise present. this can be easily determined because the eigenralues

due to noise will d l be the same. When Nb is finite the noise eigenvalues will not be

the same. There are many techniques which can be used in this case, based on the

assumption of white Gaussian noise [3], [32].

-4s an example. consider iising Rissanen's Minimum Description Length (NDL) prin-

ciple [32]

where

The dimensionality of the signal space (rank of G,) is estimated as the value of k which

rninirnizes (2.29). Table 2.1 gives the mean and standard deviation (averaged over 200

iterations) for the estimated rank using the following simulation parameters: -& = 100.

-4 = 2. A1 = 8. N = 16. L = 2- p = 2. In this case the rank is M(p + L - 1) = 24. As

expected the estimate improves with increasing SNR.

2.5. Summary

1 STD 1 0.8068 1 0.5766 1 0.3685 1 0.1969 1 0.1407 1 O 1 Table 2.1: Rank estimate using MDL

SNR ( d ~ ) h1EAN

2.5 Summary

4 23.98

In this chapter we have presented the signal model. discussed FIR zero-forcing equdiza-

tion. and introduced two blind techniques: the constant modulus algorit hm and subspace-

based identification. Along the way we defined the smoothing factor. p. through equations

(2.14). and (2.26). which is an important factor for both equalization and identification.

We know p effects the dimensionality of the signal space. in Chapter 3 it d l be seen

how this is related to the number of training syrnbols required for interference suppres-

sion. The Hessian and gradient of the C'XI.\ cost function. equations (2.18) and (2.17)

respectively. will be needed to derive the first senii-blind algori thm preçented in Chapter

3. the LSRCM.4 in section 3.2. Finally. equation (2.28) tvhich describes the estimation of

the channe1 for the mth user' will be needed in section 3.3.1 when we present semi-blind

channei estimation-

a 24

O 22.66

3 23.96

1 23.53

2 23.84

Chapter 3

Semi-Blind Algorit hms and

Simulation Examples

In this chapter rnany different approaches towards determining the space-tinie weight

vector . W, (for the mth user)? will be presented. As we move from one section to

the next the algorithms progressively use more information about the communication

system and its signals. First training symbols (3.1) then constant modulus (3.2)). then

cylostationarity (3.3). and finally the spreading codes and training symbols for other

users (3.5).

3.1 Linear MMSE Estimation

in the presence of noise. zero-forcing equalizers are not optimal. and can cause severe

noise amplification if the channel has deep spectral niills. I t is generally preferred to

design W, to minimize the following cost function:

3 . Linear R/IVSE Estimation

for which the well known solution is

R is called the autocorrelation rnatrix and Pm the cross-correlation vector. Given the

independence assumptions from section 2.1 it is clear that

where

In this work it is assumed that the channel impulse response is unknown. hence R

and Pm must be estimated h m the received data and/or a small number of training

symbols. in this section. and the next. the use of regularized LS estimators to do this

will be considered.

Using a time-average to approsimate the ensemble average in (3.1). the classical LS

estiniator is obtained

1 Nt - 1

Wm,LS = arg min - C lbm[k] - W,r,(k) 1' w m Nt

k=O

where 3, < Yb is the number of transmitted training symbols. If 'i, 2 -4.L'p then the

solution is

3.1. Linear MMSE Estimation 31

..ssurning the channel does not change, as Nt + m, RN, ( ~ 7 ~ ) + R(P,) respectively.

In ma- cases of practical interest it would not be reasonable to assume that ;Vt 2 - 4 3 ~ .

For example. a systern with a processing gain of 64 (iV = 64), using a three element

antenna array (A = 3): and a smoothing factor of one ( p = 1). would require at least 192

training syrnbols pet burst! I t might not even be likely that this many data symbols (&)

will be available. It seems clear that to use LS-type algorithms in DS-CDMA systems

regularization of the correlation matrix is needed to avoid the singularit ies present when

iV, < .-L.Vp. Henceforth it is assumed that :V, < dNp.

The simplest kind of regularization is by diagonal loading, basically an approximation

of the pseudo-inverse

where & is a srna11 positive constant (SV 0'). The number of eigenvalues of RHi that

are zero is A.Vp - .K. These will be set to 6. and the significant eigenvalues of R.~.~.

those that correspond to the signal space. will be little effected (assuming sufficiently

high SNR).

3.1.1 Behavior at High SNR

The nurnber of significant eigenvalues is given by the dimensionality of the signal space.

From (3.4) it is seen that this corresponds to the rank of G,Gr, which. from section 2.2.

is simply JI(p + L - 1). This provides us with a benchmark lrom which regularized LS

estimators can be cornpared. In the noiseless case, only M ( p + L - 1) training symbols

e l 1 be needed to completely suppress ISI and bIAI. If the minimum value of p is chosen.

then the minimum number of training symbols is

3.1. Linear MMSE Estimation

The proof is as follows.

The solution to (3.7) is' in general

where Rlrft indicates the pseudo-inverse of RN, [Z!O]. Defiae the following quantities

inverse [20].

Hence. assuming we are interested in recovering the kth symbol for the mth user

It can be easily seen that b;,,, corresponds to row vector at row nurnber M ( L - 1) +m

of BPTN. or column number M ( L - 1) + m of B K ' ~ ~ . If Nt >_ M ( p + L - 1): then br.,,

cannot be written as a linear combination of any other rom (coliimns) of B,,.vt (B:'~). Hence. if *V, 2 M ( p + L - 1)

3.1. Linear MMSE Estimation

where e r r c ~ - i , + ~ is a vector of length M(p+L-1) with a one in row number d l (L-l)+rn.

and O everywhere else. Hence, if :\i, 2 M ( p + L - l), then

so the k th symbol for the mth user has been recovered exactly.

As an ~xnrnplc~ Fig. 3.1 is a plot of the LISE performance (a.reraged omr 3.00 indepen- A

dent trials) of WmVLS for increasing SNR = 10 l ~ g , ~ ( l / o ' ) . The simulation parameters

are: .V = 16, M = 8. L = 2. p = 1, -4 = 2. = 150. 6 = 02. and r, = O V m (syn-

chronous). The channel coefficients. ( L - 1) -i of them for each user. \vas generated from

a complex Gaussian distribution with unit variance. In this case the required number of

training syinbols in the noiseless case (O' = O ) is W ( p + L - 1) = 16. The figure confirms

t his resiil t.

MSE vs. Nt for diagonal loadlng regulafization

1 t 1 1 1 1 t 1 4 6 8 10 12 14 16 18

Number of Training Symbols (Nt)

Figure 3.1: M E Performance for Diagonal Loading: w ~ , ~ ~

In the presence of noise. slightly more training sqmbols nrill be needed, depending on

3.2. Iterative Semi- Bla'nd Frarnework 34

how accuratel- the signal space c m be est imated through regularizat ion. Obviously (3.9)

does not make any attempt to improve signal space estimation. As such. it provides an

upper bound that other algorithms can be compared to. In the nest section it will be

seen that a more effective regularization can be obtained by using the constant modulus

cost function.

3.2 Iterative Semi-Blind Frarnework

Newton's algorithm is commonly used to solve nonlinear LS problems [16]: [34]. It will be

used to provide a general framework for semi-blind regularized estimation. The algorithm

iteratively cornputes the space-tirne weight vector according to the following rule:

where H(w") and G(w$)) are. respectively. the Hessian and gradient of the corre-

sponding cost function using the weight vector coefficients computed at the ith iteration

Iterat ions stop once the weight vector satisfies

for some C > 0.

If the cost function is linear. as in ( 3 3 , then the algorithm converges to the optimal

solution in a single step. For example. consider the following modified linear LS cost

func t ion

3.2. It erata've Serni- Blind Framework

then

This is simple diagonal loading. It is desired to find an improved regularization. It

is reasonable to assume that this would be achieved by replacing the diagonal matris by

the autocorrelation rnatrix estimated from the entire data sequence. RLVb. thus giving a

far bet ter estimnte of the signal space.

Csing this as the starting point. the Hessian and gradient are

where p is a positive constant. and @(W,) is some vector such that V ~ ; @ ( W , ) ~ = 0.

@(W,) is the unknown that we are free to choose (given that the gradient is zero to

satisfy (3.32)). To incorporate blind techniques into this framework. @(W,) is formed

using the Bussgang method [Il]

where gburs(.) is some memoryless nonlinearity that returns a scaler given the received

vector. The usefulness for t his framework cornes from the fact that if gk,, (r,(k)) = b& [k]

then @(W,) 2: PT^, and

3.2. [terat ive Semi- Blind Frarnework

So the gradient is similar to that which would be obtained assuming al1 information

symbolç were known a priori: -PT* + Rivbwm The drawback is that there iç no known

rnemoryless nonlinear funct ion gbuss (a) t hat will give global convergence for finite data

sets. The two most popular Bussgang functions are Sato's [33] and Godard's [9]. Sato's

is not differentiable. so it will not be considered, but Godard's is and will form the basis

for the LSRC'vi-1. as section 3.2.2 esplains.

If @(W,) = O? then the corresponding space-time equalizer and cost function are

It is seen that this is a combination of the ordinary LS cost function with the poptilar

Minimum Output Energy (b1OE) cost function of Honig et al [13]. The SIOE cost

function gained popularity for the following reasons:

1. Blind cost function without local minima.

2. In the absence of multipath. if properly constrained. can achieve close to XIhISE

performance by only knowing the spreading code for the desired user.

It has been shown, however. that in the presence of multipath, such constraints on

the equalizer can cause a degradation in performance. and complete cancellation of the

desired signal may even occur [13]. kloreover, it is known that the performance of the

LS estimator cannot be improved, but actually degrades, if the estirnate of the autocor-

relation ma t r~u is more accurate than the cross-correlation vector as in (3.36) [l]. So the

use of the MOE will not be pursued further.

3.2- Iteratiue Semi-Blind Framework

3.2.2 LSRCMA

From section 2.1 it is known tha t the source s p b o l s belong to the following set

Since it is known that the source symbols have unit magnitude al1 that remains is an

estimate of the phase. to do this we propose the use of the following nonlinearity

üsing these as source symbols. @(Wm) is computed in accordance with the cross-

correlation vector as in (3.34)

, Arb - 1

It can be shown that v ~ ~ @ ( w ~ ) ~ = O. The corresponding gradient and cost

which is a combination of the LS cost function nrith Godard's Constant Modulus (CM)

1-2 cost function [9], [ls]. This is Kuzminskiy's so called LSRCàIA (Least Squares

Regularized by the Constant Modulus Algorithm) which was first presented in [li]. Later.

in [29], the algorithm was applied in DS-CDhLA systems.

3.2. Iterative Semi- Blind Framework 38

The constant modulus cost function is the most popular blind cost function. Its major

drawback are local minima that cannot be eliminated with finite data sets. Accurate

initialization using training data can help to reduce the probability of local convergence

[ l i ] . Multistage detection c m also help, as will be seen in section 3.5. The LSRCS1.A is

summarized in Table 3.1.

n H = (R,vt + P~,vb ) - l . X [~Jo) . . . . PJX* - I)]

Choose w:' = VirmaLs = (Rlvt + J I ) -~PF~

for i = 0.1, . . -

a. kW = ( X " W : ) / ~ W ~ ~ ~ X I ~ : k' = XEp/;Vb

b. Z = H . (R,V, w$' - + p ( ~ L v b ~ $ - Y)) c, w p ) = w;) -

untii ( Z H Z)/(W$)~W(')) na < c Table 3.1: LSRCMA

-4s initialization we use the LS estirnate (3.9)' w:' = wmvLS In this case the LSR-

ChlA behaves similar to w ~ , ~ ~ . Figure 3.2 shows the improvement in &ISE performance

for increasing SNR, using the same simulation parameters as previously.

If Vw- na @(w,)* is not required to be zero. then the LSRCMA can be generalized to any

version of the CM

3.2. Iterative Semi-Blind Framework

MSE Performance of LSRCMA for Ditferent SNR

2 4 6 8 1 O 12 14 16 t8 20 Number of Training Symbols

Figure 3.2: hISE Performance of LSRCXIX for Different SNR

but it can be shown that only for p = 1 does the Hessian have the form of (3.32) (see

section 2.3.1,[29]). In Fig. 3.3 the XISE performance for different d u e s of p is shown

(using the same parameters as in section 3.1, and SNR=10 dB. p = 1. C = IO-"). The

results were obtained after averaging over 500 independent trials.

I t is seen that the CS1 1-2 has a better transient response over the CM 2-2 and CM

3-2 and lower computational complexity (the Hessian and its inverse only needs to be

computed once per burst).

3.2. Itera tive Semi- Blind Frarnework

MSE perfomance for different versions of SBCMA 10' 1 I I L I r 1 1 I I

CM 1-2

CM 3-2

I O - ~ L 1 1 1 1 I 1 I 1 4 I

O 2 4 6 8 1 O 12 14 16 18 20 Number of Training Syrnbols (Nt)

Figure 3.3: SISE Performance for Different versions of LSRC'IIA

3.2.4 Comments

The initial motiwttion for the LSRChIA came from the need for improved regularization

of the autocorrelation matrix. However, the quantity that requires greater attention is

the cross-correlation vector. Computation of the cross-correlation vector is dependent on

training symbols. the autocorrelation matrix is not. It is the training symbols which n e

seek to minimize.

The LSRCb1.4 uses constant modulus knowledge to create pseudo training symbols

using a memoryless nonlinearity (3.39). Performance. however. is very dependent on

init ialization. In the general SD blA system considered by Kuzminskiy [17]. the best

initialization available is with wmVLs, but Nith DS-CDR.I.1 systerns a more accurate

initialization can be used if channel estimation is first perfomed. Indeed: the cross-

correlation vector, Pm, is simply the channel for the desired user (3.5). Our strategy in

3.3. Subspace Approach -41

the next section rvill be to estimate the cross-correlation vector direct15 at the chip level.

by using semi-blind channel estimation. Using this as initialization, simulation examples

will show a significant reduction in the required number of training symbols.

3.3 Subspace Approach

When dealing with DS-CDMA systems oversampling is a natural function. mhere at least

.V samples per symbol are taken. In this section a semi-biind algorithm that uses second-

order subspace-based channel identification will be presented. The channel identification

technique is essentially an extension of [IO] to the case of DS-CDbIA systerns. By first

performing semi-biind second-order channel identification this algorithm is able to ex-

ploit cyclostationarity and constant modulus prop~rties. so it is reasonable to espect an

improvement. In b c t . it leads to a dramatic rediiction in training symbols cornparecl

with the regularized estimators in sections 3.1 and 3.2.

The optimal ('vi31SE) weight vector can be expressed in terms of the signal space

components

where the last equality follows from the ort hogonality between the signal space and the

noise space (gm E range(G,)). Once the signal and noise spaces have been found. g,

(the channel) needs to be estimated before Wm,YSE can be determined. In section 2.4 it

was shown that U;G, = O can be used to perform blind channel identification. In the

next section it is shown how training symbols can be incorporated into this procedure.

3.3. Subspace Approach 43

The resulting semi-blind channel identification technique does not have the corn plex

multiplicative ambiguity factor that the blind version has.

3.3.1 Training-based and Serni-Blind Channel Estimation

The multipath channel for the mth user is estimated through the following regularized

LS op t imizat ion

Training- based Blind 1 -

p, = arg min - P .4iVNt Ilr.v, - X?lt~1I2 +a P **P

- - where X;t = diagO(yt. - . . XT:) ..i,v,v, .qq, + 1). O is sorne positive constant and

r.4 [O]

the solution can be easily shown to be

This is a modification of Gorokov and Loubaton7s semi-blind channel estimation tech-

nique [IO]. Their work is only applicable to single-user systems. tvhereas this algorithm

can be used in a multiuser system. This is possible by exploiting knowledge of the desired

user's spreading code, as explained in section 2.4.

The choice for a depends on the number of information symbols in relation to the

number of training symbols (see [6] for a discussion of the asyrnptotic behavior). In Fig.

3.3. Subspace A pproach 43

3.4 the channel estimation is compared for different a and three different lengths of data

iv, = 100. -Vb = 125' and Tb = 150. The other simulation parameters are .V = 16, -4 = '2.

SNR = 10 dB. hrt = 5, L = 2, hl = 8, and T, = O V m. The MSE is defined as

Channel Estimation Error M. alpha for Different Nb

Figure 3.4: Channel Estimation Error for Increasing a and Different .Vb

Based on these results we Ml1 use a = 3 in al1 simulations to follow. In Fig. 3.5

the channel estimation is compared for increasing and three different lengths of data

Nb = 100. Jb = 125 and Nb = 130 al1 other simulation parameters the sarne as the

previous example.

When a = O then (3.46) is just the training-based estimator. In this case. if ;V, 2

[(q, + 1)/N] ! then X8 wiI1 have full column rank and p, = (X~~X'i;,)-L~~~~!vt- In

3.3. Subspace Appmach

I l 1 1 l 1 t I 2 4 6 8 10 12 14

Number of Training Symbols

Figure 3.5: Channel Estimation Performance for Different ';b (o = 3)

Fig. 3.6 channel estimation is compared for cr = O. a = 0.5 and a = 3.0 using the same

simulation parameters as in Fig. 3.4. We can see the large improvernent of semi-blind

channel estimation over the training-only case.

3.3.2 SBCMACI

Once p, has been estirnated. the overall impulse response of channel plus spreading code

can be calculated using Cm

g, = c m p ,

which is then used to calculate the MhISE weight vector

- H - w,,,, = u,*~;~u, gm

3.3. Subspace Approach

IO-'; 3 1 4 I 5 I 6 I 7 I 8 I 9 I 1 1 O

Number of Training Symbols

Channel Estimation Performance for different alpha 1 0" T I 1 - atphaEO (training-onîy) - - alpha=0.5 (semi-blind) : - alphaS.0 (semi-blind) .

-

Figure 3.6: Channel Estimation Performance for Different a

where Û,. and i, are eçtimated using an eigendecomposition of the tirne-averaged aiito-

correlation rnatris RN*.

f

i

W

2

It is possible to use (3.50) as initiakation for the LSRCh1.4, i.e. set w:) = VLmVrub and cornpute H(w$) and G ( w ~ ) ) as outlined in Table 3.1. However. it would be more

convenient if the LSRCMA could be rnodified so that Û,. and .i, can be used. because

they have already been calculated For channel estimation. Our method for making these

modifications is related to how the LSRCIv1.A was derived in section 3.2.

1 0 - ~

- A Let bm = (b,[O], . - - * y b,[.Vt - 11' pghSs(r,(:Vt)) - - -* , pghsr(rIi(-Vb - 1))) be a sequence

that contains the k n o m X, source symbols. and the iVb - Nt estimated. via (3.39) source

çyrnbolç (we cal1 these pseudo training symbols). Shen W, is computed assuming 6m

7 -

* m.

. - - ... - ... 7-

A. -

3.3. Subspace A pproach

contains Nb source syrnbols

the complete algorithm is given in Table 3.2. It is an LS algorithm with a portion of

pseudo training symbols provided by constant modulus knowledge and channel 1dent.i-

fication. The algorithm is called SBCSIACI (Semi-Blind Constant SIodulus Algorithm

wit h Channel Identification)

Out: W, n a H = U & ' Ù ~ ~ ~ x = [r&Vt). --- . rp(.vb - l)]

Choose w;) =

for i = 0.1.. -.

a. 1- = (w~)*x) /~w~"x(: b, = [bm[O]. . - - . b,[.Vt - 11.1'1

b. = (X * b:)/ivb

c. w?" = H * P F ~ (4 2 until I I W ~ + ' ) - W, II / I I W $ ) ~ ~ ~ < <

Table 3.2: SBCbfACI

In Fig. 3.7 we compare the MSE for the LSRCMA and SBChI.-\CI but using WmYLS as initialization in both cases, and the same simulation parameters as in section 3.2. It

is seen that the two are virtually identical. When channel identification is incorporated

into the SBChIXCI than there will be a much greater difference. as section 3.1 wili show.

Looking at how W, is calculated using pseudo training symbols in (3.51) it is obvious

.3.4. Simulation Ezamples

Comparison of two LS-type Aigorithrns 10' - 1 I - LS with seudo trainin

LS regukrired with 1-2 i

l o O 7

L u cn I

IO-' 7

1 t t

5 10 7 5 Number of Training Symbols (Nt)

Figure 3.7: Comparison of LSRCZVIA with SBCSIACI using same initialization

that the best choice for p is one. This is something which Kuzminskiy lias used in

simulations [li] .[lS] but vas not understood until now.

Finally. in Fig. 3.8 tve compare the performance of the SBChlhCI for the three

values of a used in Fig. 3.6. We see that semi-blind channel estimation is needed if the

SBCbIACI is going to work.

Simulation Exarnples

This section compares the MSE and probability of error performance of the simple LS es-

timator w ~ , ~ ~ with the LSRCMA and SBCMACI. FVe compare the difference in .LISE vs.

Xt (number of training symbols) performance as the following parameters are changed: L

(channel length), d l (number of users), -4 (amount of spatial diversity). and .Vb (number

3.4. Simulation Ezarnples

Performance of SBCMACI for Different alpha 10' . I t 1 I 1 I . . 1 - SBCMACI: alphad.0

-r- SSCMACI: alpha4.5 : - - i SBCMACI: alphaS.0 .

Figure 3.8: Performance of SBCXIACI for Different n

of data symbols). The probability of error vs. SNR is compared for different Yb. The

following parameters remain the same for al1 simulations: .V = 16. C = 10? p = 1.

d = 0'. and a = 3. When simulating an asynchronous system. the delays for inter-

fering users were randomly picked between O and iV - 1 chips. Channel coefficients.

Lm-I {tlk[l]},=, . are randomly generated for each user (m = 1,. % -. . JI). and for each an-

tenna element (a = 1. - . . . -4) from a complex Gaussian distribution of unit variance and

zero mean. The channel coefficients do not change during the transmission of ,Vb data

symbols. Tables 3.3: and 3.4 list the values of the parameters used in the experiments.

In al1 simulations with the SBCblACI it is assumed that the dimensionality of the sig-

nal space is k n o m exactly. This is justified when the SNR is above 5 dB given the

performance of the XIDL for rank estimation (as discussed in section 2.4.2).

3.4. Sinidation Exumples

Table 3.3: Simulation Pararneters for SBCiLIACI

Figure 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16

A 1 p 1 SNR 1 Async. 1

Table 3.4: Simulation Parameters for LSRCMA and LS

3.4.1 Cornparison for Different Channel Lengt hs

L 2.3 3.4 2 4 2 2 2 3

The simulation parameters are .Vb = 100. -4 = 2? SNR = 10 dB. and .CI = S. In Fig. 3.9

is the bISE comparison for a synchronous system, with L = 2 and L = 3. In these cases

we used p = 1 for the LSRC'VIA and p = 2 for the SBCMACI. Figure 3.10 are the results

for an asynchronous system, with L = 3 and L = 4, and p = 3. p = 2 for the SBCKACI

and LSRCh1.A respectively. Results are averaged over 100 independent trials. In both

cases it is obvious that the SBCMACI greatly outperforms the LSRCMA and LS.

N 16 16 16 16 16 16 16 16

3.4.2 Dependence on Number of Data Symbols (ATb)

Ab1 8 8

Both semi-blind algorithms d l improve as the amount of data increases' due to the

influence of the blind component. The LS algorithm obviously stays the same. In Fig.

100 100

Async. N E' N Y N ?J 3 Y

-4 2 2 2 2

1,3 2 2 2

13 8 8

5, 10 10 10

p 2 3 3 3 2 2 2 2

100, 150 100, 150

100 100

100. 125 135

SNR 10 10 10 10 10 10

0-10 0-10

MSE Companson for Increasing L: Synchmnous

W Co 5

2 4 6 8 10 12 14 16 Number of Tralning Syrnbols

Figure 3.9: M E Comparison for Increasing L: Synchronous

MSE Cornparison for Increasing L: Asynchronous L 1 L 1 1 1 L

+ LSRCMA: L=3 + LS: L=4 - - LSRCMA: L=4 1

I 1 1 1 1 1 t

2 4 6 8 10 12 14 16 Number of Training Symbols

Figure 3.10: MSE Comparison for Increasing L: Asynchronous

3.4. Simulation Examples

MSE Cornparison for Increasing Number of Data Symbols 1 I 1 I I 1 1 - SBCMACI: Nb=tW ' - - SBCMACI: Nb=l5O * - LSRCMA: Nb-t O0

+ LSRCMA: Nb=lSO '

-A- LS: Nb=tW

\ \ \ \

t 5 . - - - - 5 - - - - - - - - - - _ -

- - a _ - - - - - - - . 4

1 1 1 1 1 I 1

2 4 6 8 IO 12 14 Nurnber ot Training Syrnbols

Figure 3.11: MSE Cornparison for Increasing Number of Data Symbols: Synchronous

3.11 the LISE performance is compared for iVb = 100 and :\ib = 150. The simulation

parameters are -4 = 2, ;CI = 12. SNR = 10 dB. and L = 2. p = 2 for the SBCMACI.

and p = 1 for the LSRCMA. Resiilts have been averaged over 100 independent trials.

The system is synchronous. It is seen that both semi-blind algorithms do show an im-

provement for increasing !Vb. Performance of the LSRChf-4 is very poor for this range

of training s p b o l s because the initialization from the LS estimator is poor (the dimen-

sionality of the signal space being M ( p + L - 1) = 24 > 16. Figure 3.12 is a coniparison

for an asynchronous system. In this case L = 4, = 8. and other parameters the same

as before. We used p = 2 for the LSRCMA and p = 3 for the SBCMACI.

3.4. Simulation Ezarnp les

MSE Comparison for Increasing Number of Data Symbols: Asynchronous

- - - - _ - - - - - - - - _ _ _ - - - - - - _

2 4 6 8 t O 12 14 16 Number of Training Symbols

Figure 3.12: Comparison for Increasing 'iumber of Data Symbols: .-\synchronous

3.4.3 Dependence on Amount of Spatial Diversity (A)

-4 controls the amount of spatial diversitu. As .4 increases the SISE should decrease.

The required nurnber of training symbols should not increase with increasing -4. this is

because increasing .A effects the row space of G,. but the number of training symbols is

related to the column space. In Fig. 3.13 is a comparison for -4 = 1 and -4 = 3. The

system is synchronous, and the simulation parameters are AI = 8, SNR = 10 dBt and

L = 2. We used ;Vb = 100. p = 2 for the SBCblACI and ivb = 150, p = 1 for the

3.4.4 Dependence on Number of Users ( M )

As the number of users. M. increases, so should the required number of training symbols.

In Fig. 3.14 is a cornparison for A-1 = 5 and A1 = 10. The systern is synchronous. with

3.4. Simulation Exumples

2 4 6 8 10 12 14 Number of Trainmg Symbots

Figure 3.13: MSE Cornparison for Increasing Spatial Diversity

the following parameters L = 2, -4 = Y' SNR = 10 dB. We used ;\ = 100. p = 2 for the

SBCbl.-\CI and & = 150. p = 1 for the LSRCXIA.

3.4.5 Probability of Error Cornparison

The calculation of the probability of error was done by averaging over 2000 indepen-

dent trials. Fig. 3-15 is a ccmparison for a synchronous system. and Fig. 3.L6 for an

asynchronous system. The simulation parameters are -4 = 2. JI = 10, L = 2 for the

synchronous system and L = 3 for the asynchronous system. We used = 3. p = 2

for the SBCMACI. and .Vt = 20, p = t for the LSRChI.4 and LS. Again. we see the

dramatic improvernent of SBCMACI over LSRCiLIA and LS.

Finally, Tables 3.5 and 3.6 compare the probability of error of the blind rnethod in

[44] with the SBCkI.AC1. This is the same blind algorithm that was mentioned in section

3.4. Simulation Examples

MSE Cornparisan for lncreasing Numôer of Users (M)

2 4 6 8 t O 12 14 Number of Training Symbols

Figure 3.14: 'VISE Comparison for Increasing Number of Csen

O 1 2 3 4 5 6 7 8 9 1 O SNR (dB)

Figure 3.15: Probability of Error Comparison: Synchronous

3.5. Multistage Extension

Figure 3.16: Probability of Error Cornparison: .-\synchronous

1.3. The systeni is synchronous. with -4 = 2. JI = 8. .V = 16. L = 2 and p = 2. It

is seen that by incorporating a small number of training symbols. in this case 5. ive are

able to achieve a practical probability of error for a small data burst. where the blind

method would fail.

Table 3.5: Probability of Error for Blind Method

3.5 Multistage Extension

*

When operating in a cellular environment the base station has knowledge of the spreading

waveforms and training symbols for all intra-ceIl users. Hence. if there are M users.

Blind Method, Yb = 200 Blind Method. Sb = 100 SNR i BER

2 SNR BER

4 0.091 0.0164

3 2 0.101

5 0.0851

3 0.098

-4 0.0098 - 0.0125

5 0.0113

3.5. Multistuge Extension 56

Table 3.6: Probability of Error for SBCMACI

we assume that JIh- 5 A1 are intra-ce11 and so the LSRChI.4 or SBCBL.-\CI can be

used to recover these signals independent of each other. in some cases this strategv

will be successful for al1 users. however. in other cases one user might be successfully

recovered but another user rnight not (if that user experienced a deep fade for example).

If knowledge of the spreading codes and training symbols for these intra-ce11 users is

exploited, then we can improve performance by jointly processing each user's signal. In

this section we present a simple multistage algorithm for doing this which consists of the

bllowing steps:

'

i. Attempt to recover the signal for the mth user (rn = 1 + JIK) with either the

LSRCbIA or SBCSI.XI. This is the first stage.

SBCMACI, iVb = 100, Xt = 1

2. If successfuul~ and other user% signals were unsuccessful, subtract the rnth user's

signal (and ot her successful users) from what was ini tially received. hopefully re-

ducing the interference to a point at which the remaining users are successful (to

be reprocessed in the nert stage).

SBCMACI. !Vb = 100, -\il = 5

3. Attempt to recover the signals for the remaining users. Continue subtracting suc-

cessful users until no users are left or there are none successful for that stage (in

this case we have failed to recover the signals for al1 intra-ce11 users).

SNR BER

This kind of approach has been used in the past when working with the C51A [35].

[ ~ 9 ] . It exploits the fact that the CMX h a local minima: in one stage a user might be

trapped in a local minima. but in the next stage: given the fact that the interference

4 0.0396

3 0.0498

SNR BER

O 0.0131

5 0.0290

2 0.061

1 0.0064

3 - 0.0031

3 0.0013

3.5. Multistage Extension 57

is less. the algorithm might achieve a global minimum. In [19] the following simple

argument is used to explain the usefulness of a multistage structure:

Assume that the probability of global convergence for a given user in the first stage

is p l . Then, since fading is assurned independent for each user, the probability of global

convergence for al1 .CIK users in the first stage is (pl)"". However. the probability of

global convergence for at least one user is:

If at l e s t one user can be recovered in the first stage then it is reaçonable to assume

that the probability of global convergence in the nest stage. PL> is greater than pl because

there is less interference.

Since 2 (pl)a'*". by relaving the condition of global convergence for al1 tisers to

global convergence for a t least one user the multistage approach successively improws

the chances of recovering al1 user's signals.

This can be seen from Table 3.7. The simulated system is as follows: = 100.

SNR = 10 dB. .V = 16. L = 3. A = 2. .\.I = 8 For bISBCIIhCI (Multi-SBCMACI) and

the same parameters except L = 2 for the iLILSRChL.4. The system is synchronous. tvith

results being averaged over 100 independent trials. .As .LiK increases for a given .Vt the

estirnated probability of global convergence increases. and similar behavior is observed for

increasing Xt for a fked J.[h.. Hence. this multistage algorithm can reduce the required

number of training synibols.

As another example, Fig. 3.17 shows the improvement in MSE performance for the

'rILSRChI.A for MK = 1< SiK = -4. and i& = JI = 8. The other simulation parameten

are :Vb = 100. SNR = 10 dBt N = 16, L = 2 and -4 = 2. The -stem is synchronous.

Results have been averaged over 100 independent trials.

3.5. hfultistage Extension

Table 3.7: Probability of global convergence for multistage algont hm

MSE lmprovement for MLSRCMA: M=B r 1 I 1 1 1 1 I - LS: Mk t l

MLSRCMA: Mk=l - - MLSRCMA: Mk=4 + MLSRCMA: Mk=B

2 4 6 8 1 O 12 14 16 18 20 Nurnber of Tra~ning Syrnbols

Figure 3.17: MLSRCM .A Performance Improvement

The criterion for signal recovery is distance from the source alphabet:

Note that r,(k) has been replaced with îL(k) to indicate the received signal vector for

the lth stage. In this case W, may represent the space-time weight vector calculated

using the LSRCMA. SBCMACI, or LS.

3.5. Multistage Eztension

Stage #1

Stage #2

Stage #3

User #l User #2 User #3 User #4

Figure 3.18: Multistage Erample

When rl, < q for some q > O (Le. 0.05) then

is used to estimate the data bits for the mth user. after which the signal is respread.

reconvolved with an estimate of the propagation channel, and then subtracted from rt.

Consider. for esample. the scenario depicted in Figure 3.18 (Alh- = 4. J I = 8). In the

first stage the signal for user #3 can be subtracted so that in stage 2 users 1 and 4

are acquired and finally. after the 3rd stage. al1 intra-ce11 users have been successfully

dernodulated. On the s-axis is shown 4 for each user for each stage where applicable. In

this case q = 0.05. which mas also used to obtain the results in Table 3.7 and Fig. 3.17.

Assume that in the Zth stage the signal for the rnth user finalty satisfies (3.55). L e

t hen perform the following steps:

1. Gse (3.56) to estirnate b,[k].

3.6. Sumrnary and Comrnents

2. Form an estirnate of the spread-spectrurn signal:

3. Estimate p,:

1 Pm = arg min -l&, - ~ 8 ~ 1 1 ~ + apHirimp

P -4iV~v~

where i!vb and k~~ are defined in section 3.3.1 (with l a simple generalization for

the received vector a t the !th stage). If implementing the ULSRCIIA then set

CY = O. othenvise for the MSBCMACI a good choice is n = 3 (see section 3.3.1).

4. Cornpute the received multipath signal contributed by the mth user:

Steps 1-4 need to be repeated depending on the number of signals that satisfy (3.55)

for the 1 th stage. Assuming that K users were demodulated then:

where d coritains the indices of the K users.

3.6 Summary and Comments

In this chapter we have introduced a number of different semi-blind dgorithms. One

class. the LSRCMA. perform interference suppression Nithout channel estimation. The

other. SBCSIACI. hILSRCblAt and MSBCbIACI, need channel estimation either as part

3.6. Summary and Comments 61

of the initialization (SBCMACI), or at an intermediate point (MLSRChlA). or both

('VISBCMACI) . Simulation results st rongly suggest that the use of channel estimation is

very effective at reducing the required number of training symbols compared with the

LSRCMLIA. The use of channel estimation. honrever: increases the coniputational burden.

The computational cornplexity of the LSRCMA is on the order of (.-l.Vp)2. but the

computational complexity of the SBCSIACI is on the order of ( - 4 - V j ~ ) ~ because we need

an eigen-decomposition of the autocorrelation matriv for channel estimation.

Chapter 4

Conclusion

This chapter provides concluding remarks and comrnents on future work.

4.1 Final Remarks

We have presented three algorithms for semi-blind interference suppression in DS-CDhl.4

systems. The LSRCXIA is similar to diagonal loading, except that the diagonal matris

has been replaced by the time-averaged autocorrelation m a t m . The SBCSIXCI uses

channel identification as a precursor to equalization. I t is an enhancement to LSRChLA

since it uses cyclostationarity and constant-modulus knowledge. Also. it is specific to

DS-CDMA systerns. It cannot be generalized include the TDMA framework considered

in O t her semi- blind algorit hms [17], (1 $1.

It has been observed that the number of training symbols needed by the LSRChlA

is approximately ( p + L - 1). this being the dimensionality of the signal space. For

the SBCS.1-KI, however. it is possible to do much better since channel estimation needs

on the order of [(q, + l)/'i] training symbols. this being the minimum to ensure that

XE;XF* has full rank. Since (q, + 1) < - (L - l)M we see that the minimum nurnber

4.2. Fz~ture Work 63

of training symbols needed by the SBCMACI is L - 1: a nurnber much smdler than

M ( p + L - 1).

Intuitively, it makes sense that channel estimation should cause such a big improve-

ment. The cross-correlation vector c m only be calculated using training syrnbols. so in

comparison to the autocorrelation matrix, which can be calculated using al1 data sym-

bols. i t is the area we need to improve the most. The LSRCMA attempts to accomplish

this by creating pseudo training symbols that are calculated using the CbIA. This can

be an effective strategy, but it does not take advantage of the spread-spectrum benefits

providecl by DS-CD-VlA systems. For every symbol there are Y chips, so if estimation

can be done on the chip level. through channel estimation. it should be better. In a

general mu1 t iuser SDhI.4 syst ern [42] semi-blind subspace-based channel estimation as a

precursor for equalization is difficult due to the presence of a rnatris ambiguity factor (sec

section 2.4). but for DS-CDb1.A systems knowledge of the spreading code removes this

ambiguity. The SBCLLACI is specific to DS-CDMA systerns and takes full advantage of

its bencfits.

Finally. we presented a decision-aided multistage algorit hm and showed how it can

be used to further reduce the required number of training. The multistage algorithm

incorporates system and signal information by using the fact that basestations will know

the spreading codes and training symbols for al1 intra-ce11 users.

4.2 Future Work

Synchronization is an important topic that has not been addressed here. Throughout

this work it has been assumed that the desired user is synchronized with the basestation.

In practice. so-called code-acquisition circuits are used to set the timing. in near/far

situations these circuits will suffer from the same problems as the traditional correlation-

4.2. Future Work 64

type detectors. Synchronization for the 5GiI and ISI infected system we consider in this

work has not received a lot of interest. It is interesting to see how serni-blind algorit hms

can be applied for this task.

-4 bet ter analysis of the semi-blind channel estimation technique is required. Exactly

how a should change depending on the size of the data burst and the number of training

symbols is not well understood. In this work a value for a was picked based on a few

experiniental results. but perhaps it is possible to derive an equation describing the best

choice for o (in the MSE sense). This seems prornising since the estimator derived using

cr has a closed form expression (see eq. (3.47)).

In the derivation of the LSRCblA it was argued that the particular Bussgang nonlin-

earity that was chosen was a good choice since it concentrated on estimating the phase of

the transmitted symbol. with the magnitude being known to be 1. This tvill work well for

BPSK. 4-QAM. 4-PSK and other alphabets that have constant modulus. For alphabets

that are not constant modulus. such as 16-$.W. perhaps a better nonlinearity esists.

What exactly it might be is an open question.

In this work we have incorporated system and signal information to reduce the number

of training syrnbolç. It would be beneficial to see how systems can be changed to suit the

needs of semi-blind algorithms. For example. the required number of training s p b o l s

depends on the amount of spatial diversity, .A. In cases where the fading is high the

amount of spatial diversity may not be enougli. By changing the transmission protocol to

incorporate retransmissions it is possible to increase A tvithout adding antenna elements.

Exactly how such a divetsity scheme would work. and how throughput will change is an

interesting question to answer.

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