Pseudonoise code tracking loop for a CDMA system with imperfect power control

INTERNATIONAL JOURNAL OF COMMUNICATION SYSTEMSInt. J. Commun. Syst. 2001; 14:419}430 (DOI: 10.1002/dac.482)

Pseudonoise code tracking loop for a CDMA systemwith imperfect power control

Jin Young Kim*��

Access Network Group, Central Research and Development Center, SK Telecom, Korea

SUMMARY

In this paper, performance of a pseudonoise (PN) code tracking loop is analysed and simulated fora direct-sequence/code-division-multiple-access (DS/CDMA) system with imperfect power control ina multipath fading channel. A non-coherent "rst-order delay-locked loop (DLL) is considered as a PN codetracking loop. Power control error is modelled as a log-normally distributed random variable. Themultipath fading channel is modelled as a two-ray Rayleigh fading model that is typically applied to landmobile communication environments. The tracking performance of DLL is evaluated in terms of trackingjitter and mean-time-to-lose-lock (MTLL). From the simulation results, it is shown that for smallerdiscriminator o!set, tracking jitter performance is improved while MTLL performance is degraded. It isshown that large power control error and heavy shadowing substantially degrade the PN trackingperformance. Therefore, an accurate power control scheme is very essential to compensate for fading andshadowing for a DS/CDMA system. The extension of analysis to higher-order loop models is verystraightforward. The analysis in this paper can be applied to design the PN code tracking loop fora DS/CDMA system. Copyright � 2001 John Wiley & Sons, Ltd.

KEY WORDS: code tracking; CDMA; power control

1. INTRODUCTION

Direct-sequence/code-division-multiple-access (DS/CDMA) is a promising technique for mobileand personal communication systems because of anti-interference, random access capability, andincreased capacity, etc. To exploit the advantages of a DS/CDMA technique, a CDMA receivermust synchronize a locally generated pseudonoise (PN) code with the incoming one. Thesynchronization of a DS/CDMA system consists of two steps: acquisition (coarse alignment) andtracking ("ne alignment) [1}5]. PN tracking is necessary to detect the timing with maximumenergy. PN code tracking in a multipath fading channel is a critical issue in the design ofDS/CDMA receivers since a service disconnection may be caused by the failure of code tracking.

* Correspondence to: Jin Young Kim, Access Network Group, 8th Floor, Central Research and Development Center, SKTelecom, 9-1 Sunae-dong, Pandang-gu, Sungnam City, Kyunggi-do 463-020, Korea.

� E-mail: [email protected]

Received 5 January 2000Copyright � 2001 John Wiley & Sons, Ltd. Accepted 5 September 2000

This situation results in reducing the channel utilization, which corresponds to decrease in thecapacity of mobile communication system.

Conventionally, a non-coherent tracking loop is preferred because it is insensitive to datamodulation and presumes a reliable carrier tracking prior to PN code synchronization [1,2].Delay-locked loop (DLL) and tau-dither loop (TDL) are extensively considered in the literature[1}8]. Although TDL overcomes gain imbalance problem of DLL, it su!ers from degradation innoise performance. The DLL and the TDL can be applied to the coherent and non-coherentdetection schemes. In most of the CDMA applications, the non-coherent tracking loops are moreoften employed than the coherent loops because, spread signal-to-noise ratio (SNR) is typicallytoo low to achieve carrier recovery before code synchronization [1,9]. In an AWGN channel,linear and non-linear approaches have been employed to analyse tracking performance [9,10].A linear and a non-linear approaches are adequate for a high-loop SNR and for a low-loop SNR,respectively. For multipath fading channels, the non-linear approach is more appropriate espe-cially for a weak specular component. Renewal process approach (RPA) has been proposed for anexact non-linear analysis [9,10].

Power control is a major design criterion in a DS/CDMA system for two reasons: (1) to makethe received power less dependent on fading and shadowing and (2) to combat near}far problemand co-channel interference. A number of power control schemes have been proposed tominimize the e!ects of fading, shadowing, and near}far problems [11,12]. In a power controlscheme, the transmitted power from all mobile stations is adjusted such that all signals arereceived with equal power at the base station. In most of the previous researches on the CDMA,perfect power control has been achieved by employing the adaptive power control. However, the"eld test on a CDMA performance demonstrates that a power control error determined bylognormal distribution is more realistic in the practical case. The e!ect of imperfect power controlon PN acquisition performance has already been considered [13]. However, the impact of powercontrol error and shadowing on PN tracking performance has not been taken into account so far.

In this paper, the performance of non-coherent "rst-order DLL is analysed and simulated fora DS/CDMA system with imperfect power control in a multipath fading channel. The multipathfading channel is modelled as a two-ray Rayleigh fading model that is typically applied to landmobile communication environment. The performance of DLL is evaluated in terms of trackingjitter and mean-time-to-lose-lock (MTLL) by the method of RPA. In a DS/CDMA system, thesignal can be degraded or distorted by multipath fading, shadowing, imperfect power control andmultiple access interference (MAI), etc. For e!ective and reliable PN tracking, the combined e!ectof above factors is considered in the analysis of PN tracking performance. The imperfection ofpower control system is well modelled by the logarithmic standard deviation of the lognormalpower distribution of the received signal.

The rest of the paper is organized as follows: In Section 2, PN tracking model is described in thetwo-ray multipath fading model. In Section 3, the PN tracking performance of DLL is in terms oftracking jitter and MTLL. In Section 4, simulation results are presented and, "nally, theconclusions are drawn in Section 5.

2. SYSTEM MODEL

In a non-coherent "rst-order DLL shown in Figure 1, the received signal is "rst correlated withlocally generated early and late PN sequences, that is, c (t!�L

�#�¹

�) and c(t!�L

�!�¹

�). Then

420 J. Y. KIM

Copyright � 2001 John Wiley & Sons, Ltd. Int. J. Commun. Syst. 2001; 14:419}430

Figure 1. Block diagram of PN code tracking loop.

the error signal is obtained by band-pass "ltering, squaring, and di!erencing the correlatoroutputs. The loop is closed by low-pass "ltering of the error signal and the output of the loop"lter is used to drive an voltage control clock (VCC). The output of VCC is used to correct thecode phase error of the local PN code. The parameter � (0(�(1) is called the early}latediscriminator o+set. The acquisition unit continually adjusts the phase of the local code until theincoming and local are aligned, so that the code phase error is within the permissible range(��

, ��

).For a PN tracking loop, this permissible range is usually the range of the discriminator

characteristic or S-curve. The code phase error is de"ned as the normalized phase di!erencebetween the incoming and local codes and given by

� (t)"[�

�(t)!�L

�(t)]

¹�

(1)

where ��

(t) and �L�

(t) are the phases of the incoming and local codes, respectively, and ¹�

is a chipduration. In this case, there is a probability, P

�(�), that the lock detector in a synchronization

block mistakenly triggers reacquisition process due to adverse channel environments.

3. PERFORMANCE ANALYSIS

In the analysis of PN tracking performance, the followings are assumed: (1) data sequence,spreading sequence, and AWGN are mutually independent, (2) tracking boundaries are absorb-ing, (3) individual tracking process is statistically identical, (4) both reverse and forwardlinks su!er from identical shadowing, and (5) mobile user estimates signal strength bymeasuring pilot signal and controls its transmission power. For the multipath fading channel, thetwo-ray fading model is employed. It is widely used to represent the dispersive component offading, and describes the multipath propagation in terms of primary and dominant interferingrays.

421PSEUDONOISE CODE TRACKING LOOP


3.1. DS/CDMA signal modelling

The kth user's transmitted signal of a DS/CDMA system with BPSK modulation is given by

s�(t)"�2P

�a�(t)c

�(t) cos (2� f

�t#�

�) (2)

where P�

is signal power, a�(t) is data sequence with bit duration ¹, c

�(t) is spreading sequence

with chip duration ¹�, f

�is carrier frequency, and �

�is carrier phase. The data and spreading

sequences are given by

a�(t)"

��

��

d��

p�

(t!i¹ ) (3)

c�(t)"

��

��

c��

� (t! i¹�) (4)

where a��

3�#1,!1� is a data sequence, c��

3�#1,!1� is a spreading sequence, p�

(t) is bitwaveform, and � (t) is chip waveform. Let the reference user be user 1 without loss of generality.

In most of the previous researches on the PN tracking loop, perfect power control (P�"P,

(1)k)K) has been generally assumed by employing adaptive power control [1}5]. However,the "eld measurements on CDMA performance demonstrate that a power control error deter-mined by lognormal distribution is more realistic in the practical operating environment. Theprobability density function (p.d.f ) of received signal power is modelled by

f (P�)"

1

�2��P�

exp�!lnP

�2�

�� (5)

where ��

is the logarithmic variance and P�

is assumed to have logarithmic mean of zero.The major factors a!ecting the received signal power are the distance and shadowing loss. The

received power at the base station for the kth user is given by

P�"P

d��

�10�� (6)

where P

is a constant which depends on the parameters of transmitter and receiver, d�

is distancebetween the kth mobile user and base station, is path loss exponent, and

�is a random variable

corresponding to shadowing and power control error which is lognormally distributed withstandard deviation of �

�and �

�dB for shadowing and power control error, respectively.

Then, the received signal at the receiver in a two-ray fading model is given by

r (t)"��

�2P��

g��

a�(t!�

��)c

�(t!�

��) cos (2� f

�t#�

��)

#

��

�2P��

g��

a�(t!�

��!�

��)c

�(t!�

��!�

��) cos (2�f

�t#�

��)#n (t) (7)

where ��"�

��!�

��is delay di!erence between the "rst and second paths, g

�� is a path gain

with Rayleigh distribution, ��

is propagation delay of the "rst path of the kth user, ��

is carrier

422 J. Y. KIM


phase o!set of the "rst path, ��

"��

!2� f��

, and n (t) is AWGN with power spectral densityN

/2. The �

��and the �

��(i"2,2, K ) are independent identically distributed (i.i.d.) random

variables uniformly distributed in [0, 2�], and ��

(i"2,2,K) are i.i.d. random variables.Therefore, �

��is the time delay to be tracked by the tracking loop. The decision statistic of

reference user is given by

Z�"D#I

�#I

(8)

where D is the sum of desired user's component and self-interference, I�

is the MAI component,and I

is AWGN component. The D is composed of reference user component and self-

interference component, and is given by

D"�2P��

g��

cos ��

�

c�

(t!��

) a�

(t!��

)a�

(t) dt

#�2P��

g��

cos ��

�

c�

(t!��

!��

)a�

(t!��

!��

)a�

(t) dt (9a)

"

��2P

��g��

cos ��

(¹�!��

�� )G#�2P

��

�� cos �

��

��

�� (9b)

"D�#D

(9c)

where D�

is desired user component, D

is self-interference component due to chip timing error,�"d

��c��

c��

, and ��"d

��c��

c��

(i"1,2, G!1) are i.i.d. random variables with$1,and G"¹/¹

�is processing gain.

The MAI component is given by

I�"

��

��

�2P��

g��

cos ��

�

c�(t!�

�� )a

�(t!�

�� )a

�(t) dt (10)

The AWGN component I

is a Gaussian random variable with zero mean and variance N¹/4,

and given by

I"2 �

�

n (t)a�(t) cos (2� f

�t) dt (11)

Since the p.d.f. of MAI and self-interference is di$cult to obtain, the DS/CDMA systemanalysis has been evaluated using approximations [14]. For a large number of users, the p.d.f. isusually assumed to approximate Gaussian process with a variance equal to the sum of variancesof individual interfering users. The e!ect of AWGN, MAI, and fading is incorporated into theanalysis of PN tracking performance simply by increasing the variance of interference. When thenumber of users is large enough (typically, larger than 10), the p.d.f. of MAI is assumed toapproximate Gaussian process, with a variance equal to the sum of variances of individualinterferers and of the AWGN. For MAI modelling in this paper, Gaussian approximation isemployed. The variances of I

�and I

are derived. We adopt the results for variances of Reference

[14] to "nd the PN tracking performance.



For a large PN sequence period N, autocorrelation function is approximated over the interval[!N/2, N/2) by the ideal autocorrelation

R�()"�

1!� � , � �)1

0, otherwise(12)

3.2. PN tracking performance

The error signal is given by

e(t)"��

H�(0) [P

��g��

v��

(�, 0)#P��

g��

v��

(�,��/¹

�)

#2P��

P��

g��

H�(�

��)

H�(0)

v��

(�, ��/¹

�) cos (�

��!�

��)]#n

�(�, t ) (13)

where

H�()"�

�

��

S��

( f ) �H�( f ) � exp ( j2� f) df (14)

v��

(�, )"R��

(�!�#)!R��

(�#�#) (15)

v��

(�, )"[R��

(�!�)R��

(�!�#)![R��

(�#�)R��

(�#�#)] (16)

In (13), the S��

( f ) is power spectrum density of data sequence a�(t), H

�( f ) is transfer function of

band-pass "lter (BPF). To describe the dynamic behaviour of DLL, the stochastic di!erentialequation is given by

de (t)

dt"f

�!G

��

[P��

H�(0)C (�)#n

�(�, t)] l (t) (17)

where G

is VCC gain, l (t) is impulse response of loop "lter, and f�

is Doppler shift.The overall discriminator characteristic is shown in Figure 2, and is given by

C (�)"C�(�)#C

�(�) (18)

where C�(�) is desired discriminator component and C

�(�) is interference discriminator component

due to the e!ect of the second path. The overall discriminator characteristic in a two-raymultipath fading channel is shown in Figure 2. When there is a multipath e!ect, the linear regionnear �"0 may not exist. In this case, the non-linear analysis is more appropriate than the linearapproach for analysing the tracking performance.

Let p(��g) be the stationary p.d.f. of the tracking jitter conditioned on the channel impulseresponse g. In a RPA, the p (��g) satis"es the following Fokker}Planck equation with boundaryconditions p (�

��g)"p (�

��g)"0.

��

z��

(�)p (��g)

P�

(�) �!1

2

� �z(�)p (��g)

P�

(�) �"� (�)�N t �g

(19)

424 J. Y. KIM


Figure 2. Discriminator characteristics in a multipath fading channel.

where � (�) is the p.d.f. of the initial code phase error and depends on acquisition process,

P�

(�) "1!P�

(�) (20)

z��

"!G

��

H�(0)C(�)#

1

4

dz

(�)d�

(21)

z

(�)"G �

�

��

R�

(�, ) d (22)

and �N t�g is conditional MTLL.The tracking jitter and MTLL are obtained by [7]

��"�g

��g p(g) dg (23)

�N�"�g

�N t �g p(g) dg (24)

where ��g is the conditional tracking jitter [7], and p(g) is the joint p.d.f. of the channel impulseresponse.

4. SIMULATION RESULTS

Some simulation results are presented in this section. For simulation examples, the binarynon-return-to-zero (NRZ) pulse for a chip pulse shape, path loss exponent "4, chiprate"1.2288 Mcps (IS-95 CDMA system speci"cation), processing gain"128, carrier frequencyf�"900 MHz, the number of multipath"2 (i.e. two-ray multipath fading model), and Doppler



Figure 3. Tracking jitter vs discriminator o!set for power control error.

frequency f�"f

�v/c"30 Hz (c is speed of light and v is vehicle speed) are assumed. The ideal

BPF is used with a cuto! frequency of R("1/¹ ). For simplicity of analysis, the P

�(�) is assumed

to be

P�

(�)"�1!�, ��)1

0, otherwise(25)

In the simulations of the power control scheme, (1) power control step size of 1 dB is used, (2)feedback power control scheme is employed with power control feedback delay (the delay fromwhen power control command is sent to when the power is adjusted). It is also assumed that thereis a very accurate acquisition unit in a DS/CDMA synchronization block, and boundaryabsorption occurs at �(t)"$1, where �(t) is the phase di!erence between the incoming and localPN sequences normalized by chip duration. Since the closed-form expressions for Equations (23)and (24) do not exist, they are computed by numerical method using Gauss-quadrature integra-tion.

In Figure 3, tracking jitter vs discriminator o!set is shown for the various values of powercontrol errors with standard deviation of shadowing �

�"5 dB and the number of user K"15. It

is shown that the tracking jitter becomes larger as the discriminator o!set. It is also shown thatpower control error substantially degrades the tracking jitter performance. As the power controlerror becomes larger, the performance degradation becomes higher.

In Figure 4, MTLL vs discriminator o!set is shown for the various values of power controlerrors with �

�"5 dB and K"15. It is shown that MTLL becomes larger as the discriminator

o!set. It is shown that the MTLL performance is also severely degraded by power control error.The degree of MTLL performance degradation becomes higher as the power control error.

In Figure 5, tracking jitter vs discriminator o!set is shown for the various values of powercontrol error and shadowing deviations with K"15. It is shown that the shadowing a!ects thetracking jitter performance in the whole range of discriminator o!set.

In Figure 6, MTLL robustness vs discriminator o!set is shown for the various values of powercontrol error and shadowing deviations with K"15. It is shown that the MTLL performance isalso degraded due to shadowing in the whole range of discriminator o!set.

426 J. Y. KIM


Figure 4. MTLL vs discriminator o!set for power control error.

Figure 5. Tracking jitter vs discriminator o!set for shadowing deviations.

Figure 6. MTLL vs discriminator o!set for shadowing deviations.



Figure 7. Tracking jitter vs discriminator o!set for varying number of users.

Figure 8. MTLL vs discriminator o!set for varying number of users.

To assess the e!ect of the MAI coming from the interfering users, the additional simulationresults are presented. In Figure 7, tracking jitter vs discriminator o!set is shown for varying thenumber of users with �

�"5 dB. As evidenced in the "gure, the tracking jitter performance is

degraded by degrees with the number of interfering users in the system. The e!ect of MAI ontracking jitter is more signi"cant for larger discriminator o!set than for smaller discriminatoro!set. These results have also been con"rmed in previous research [15}17]. In Figure 8, MTLL vsdiscriminator is shown for varying number of users with �

�"5 dB. It is also found that the MAI

signi"cantly degrades the MTLL performance.

5. CONCLUSIONS

The PN code tracking performance of the non-coherent "rst-order DLL was evaluated fora DS/CDMA system with imperfect power control in a multipath fading channel. The fading

428 J. Y. KIM


channel was modelled as a two-ray Rayleigh fading channel. The performance of DLL wasevaluated in terms of tracking jitter and MTLL. It is also shown that large power control errorsubstantially degrades the PN tracking performance. Therefore, it is concluded that a veryaccurate power control scheme is required to compensate for fading and shadowing. In thispaper, the IS-95 CDMA system parameters are mainly employed. In the future simulations, thewideband-CDMA system parameters will be considered for applications to IMT-2000 system.The extension of analysis to higher-order loop models is very straightforward. The considerationsin this paper can be applied to the design of a PN code tracking loop for a reverse link ofa DS/CDMA system.

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perturbation approach. IEEE Journal on Selected Areas in Communications 1990; 8:809}817.7. Sheen WH, Stuber GL. E!ects of multipath fading on delay-locked loops for spread spectrum systems. IEEE

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Systems 1974; 2:415}424.9. Lindsey WC, Meyr H. Complete statistical description of the phase error processes generated by correlative tracking

systems. IEEE ¹ransactions on Information ¹heory 1977; 23:194}202.10. Meyr H. Nonlinear analysis of correlative tracking systems using renewal process theory. IEEE ¹ransactions on

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AUTHOR'S BIOGRAPHY

Jin Young Kim received BSc, MSc, and PhD degrees from the School of ElectricalEngineering, Seoul National University, Seoul, Korea, in 1991, 1993, and 1998,respectively. From 1995 to 1997, he has conducted research in various areas ofwireless communication systems and networks with special attention on CDMAsystems at the Wireless Communications Laboratory, Institute of New Media andCommunications (INMC) of Seoul National University. He was a researcher at theInter-university Semiconductor Research Center (ISRC) of Seoul National Universityin 1998. He was Postdoctoral Research Fellow at the Information Sciences andSystems Research Group, Department of Electrical Engineering, Princeton Univer-sity, NJ, U.S.A. from August 1998 to February 2000. He is now with Central Researchand Development Center, SK Telecom, Korea. He has been selected as a recipient ofan overseas postdoctoral research fellowship program from KOSEF (Korea Science

and Engineering Foundation) in 1998. His research interests include spread-spectrum communications withemphasis on PN acquisition and tracking, for applications to mobile and satellite communication systems,broadcasting technology, adaptive antenna array, packet radio network, optical communication, synchroni-zation, channel coding, multimedia transmission technique, wireless internet, wireless ATM, detection andestimation, and information theory. He is listed in Marquis's Who's Who in the World 2001 Edition. DrKim is now a member of IEEE, IEE, IEICE, IEEK, and KICS.

430 J. Y. KIM


Pseudonoise code tracking loop for a CDMA system with imperfect power control

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