Signal Processing 82 (2002) 311–315www.elsevier.com/locate/sigpro

Fast communication

Adaptive step-size constant modulus algorithm for DS-CDMAreceivers in nonstationary environments

Peerapol Yuvapoositanona;b;1;∗, Jonathon A. Chambersa;2

aSignal and Image Processing Group, Department of Electronic and Electrical Engineering, University of Bath, Bath BA2 7AY, UKbDepartment of Electronic Engineering, Mahanakorn University of Technology, Nong-Chok, Bangkok 10530, Thailand

Received 30 July 2001

Abstract

A new adaptive step-size constant modulus algorithm for direct-sequence code division multiple access receivers ispresented for application in nonstationary channel conditions. The algorithm is derived upon the basis of adapting thestep-size to minimise the constant modulus criterion and its convergence is veri0ed analytically. Simulations show thesuperior performance of the proposed method over similar adaptive receivers and its robustness to di1erent settings of theinitial step-size. ? 2002 Elsevier Science B.V. All rights reserved.

Keywords: Constant modulus algorithm; Adaptive step-size; DS-CDMA receivers

1. Introduction

The performance and stability of a stochastic gradi-ent algorithm used within a direct-sequence code divi-sion multiple access (DS-CDMA) receiver designedto mitigate multiple access interference (MAI) are de-pendent upon the choice of an appropriate step-size.In practice, such MAI is time-varying because usersfrequently enter and exit the channel. Therefore, thechannel is far from being stationary and computationof a pre-determined step-size is di8cult. In [3], the

∗ Corresponding author.E-mail address: eeppay@bath.ac.uk (P. Yuvapoositanon).

1 This research is supported by the Mahanakorn University ofTechnology, Bangkok, Thailand.

2 With e1ect from 1st February 2002, Prof. Chambers will bewith the Centre for Digital Signal Processing Research, Divisionof Engineering, King’s College London, Strand, London WC2R2LS, UK.

adaptive step-size strategy for tracking time-varyingenvironments within DS-CDMA channels is success-fully applied to the minimum mean output energy(MOE) detector. The e1ect of multipath distortion onthe algorithm is, however, neglected in the derivationof the algorithm. This is because the MOE criterion isnot designed to equalise the channel and tends to can-cel the desired signal itself when the spreading wave-form has a mismatch [2]. In reality, the multipath dis-tortion, the cause of mismatch, must not be neglectedespecially when transmission at a high data rate is re-quired. It should also be noted that there can be asubstantial performance gap between the MOE detec-tor and the minimum mean squared error (MMSE)receiver [2], even when there is no multipath distor-tion. On the other hand, the analyses in [6] show thatthe constant modulus algorithm (CMA) receiver canperform almost as well as the Wiener (MMSE) re-ceiver provided that undesirable local minima can beavoided.

0165-1684/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.PII: S0165 -1684(01)00193 -1

312 P. Yuvapoositanon, J.A. Chambers / Signal Processing 82 (2002) 311–315

In this paper, we therefore propose a new adap-tive step-size CMA (AS-CMA) algorithm for aDS-CDMA receiver for multipath fading channels.The algorithm adaptively varies the step-size in orderto minimise the constant modulus (CM) criterion. Thealgorithm is blind in the sense that no training dataare required. Simulations have con0rmed the appli-cability of the algorithm for nonstationary multipathfading CDMA channels. The performance in termsof signal to interference plus noise ratio (SINR) ofthe proposed algorithm is shown to be superior to theexisting adaptive step-size MOE (AS-MOE) and the0xed step-size CMA receivers.

2. Signal model

Consider the real signal model of an additivewhite Gaussian noise (AWGN) K-user synchronousDS-CDMA channel, the baseband received signal isde0ned as

r(t) =∞∑

i=−∞

K∑k=1

Akbk(i)ck(t − iT ) + w(t); (1)

where Ak represents the received amplitude of the kthuser. The data bits bk(i) are independent identicallydistributed (i.i.d.) and bk(i)∈{−1;+1}. The symbolperiod is denoted by T . The spreading waveform of thekth user ck(t) is N -dimensional and has unit energyproperty, i.e., ||ck ||2 = 1. The AWGN w(t) has powerspectral density 2

w. The spreading codes ck(t) can bemodi0ed to take into account the e1ect of the channeland the pulse shaping waveform as c̃k(t)=ck(t)∗ (t)∗hk(t), where ∗ denotes convolution, (t) is the pulseshaping 0lter and hk(t) is the channel response of thekth user and need not be identical for di1erent users.The continuous-time received signal r(t) is sampledto form rn=[r(nN+Lf−1); : : : ; r(nN )]T; a length-Lf

received signal vector at the nth observation, where Lf

is the length of a receiver with tap-weight vector f .

3. Adaptive step-size CMA algorithm

The stochastic gradient adaptive gain strategy is in-troduced in [1] for the nonblind adaptive system totrack nonstationary environments. We adopt the ordi-nary di1erential equation (ODE) approach suggestedin [1] for the derivation of the adaptive step-size CMAalgorithm. The CM criterion for real signals is given

by JCM = E(z2n − �)2, where zn = f Tn rn is the outputof the receiver. The dispersion constant � is equal tounity for binary phase shift keying (BPSK) signals.By minimising JCM with respect to fn at a particularstep-size � or f �n and taking the instantaneous gradi-ent, the 0xed-� CMA receiver weight update equationis given by

f �n+1 = f�n − �(z2n − �)znrn; (2)

where zn = (f �n )Trn. Let �n and fn be the actual se-quences of step-sizes and estimates of the receivertap-weights, we arrive at

fn+1 = fn − �n(z2n − �)znrn; (3)

where �n is the time-varying step-size. By minimisingJCM w.r.t. �n,

14@E(z2n − �)2

@�n=

14@T(z2n − �)2

@fn

@fn@�n

= (z2n − �)znrTn Yn; (4)

where Yn represents the derivative (@fn=@�)|�=�n asde0ned in [3]. We arrive at the update of the step-size

�n+1 = [�n − �(z2n − �)znrTn Yn]�+�−; (5)

where [ · ]�+�− denotes truncation to the limits of therange [�−; �+] and � denotes the adaptation rate of thestep-size �n with �¿ 0. From (3), the update equationof Yn is given by

Yn+1 = [I − �n(3z2n − �)rnrTn ]Yn − (z2n − �)znrn: (6)

In Appendices A and B the convergence propertiesof (5) and (6) are investigated. Unlike the AS-MOEalgorithm of [3], AS-CMA is an inherently uncon-strained optimisation scheme. Therefore, rn does notneed to be replaced by the projection rn orthogonalto the spreading code ck = [ck(Lf − 1); : : : ; ck(0)]T

for the kth desired user. Eqs. (3), (5) and (6) con-stitute the new adaptive step-size CMA algorithm forreal signals. It is straightforward to extend these tothe complex case, but we retain the real version forconsistency with [2,3].

4. Numerical results

We considered a synchronous DS-CDMA systemwith spreading gain 31 (N = 31). Without loss of

P. Yuvapoositanon, J.A. Chambers / Signal Processing 82 (2002) 311–315 313

0 500 1000 15000

5

10

15

20

25

30

Number of symbols

SIN

R(d

B)

AS-MOEAS-CMAfixed- µ CMA

Fig. 1. The SINR performance of AS-MOE, 0xed-� CMA andAS-CMA receivers in the single-path channel.

generality, the 0rst user was the desired user withunity power. The background noise was zero meanAWGN with SNR = 20 dB referenced to the desireduser. Six 20 dB MAI users were in the channel attime zero. At time n = 500, another four 30 dB MAIusers were added to the channel. At time n = 1000,two of the 20 dB MAI users and three of the 30 dBMAI users exited the channel. We considered two testcases: single-path and multipath fading scenarios. Theinitial step-sizes of the AS-CMA and AS-MOE algo-rithms were set at �0 = 10−4. The step-size of the0xed-� CMA receiver was 10−4. For both AS-MOEand AS-CMA, the adaptation rates � were 5 × 10−8.The upper and lower step-size limits �+; �− of bothalgorithms were set at 10−2 and zero, respectively.For the case of no-multipath fading, the SIR plots areshown in Fig. 1. For the inclusion of four near-farMAI users at time n= 500, AS-MOE is able to trackthe channel variation but with noticeable degradationin SINR level while the 0xed-� CMA algorithm takesmore time to reach its asymptotic level. The AS-CMAalgorithm successfully tracks the abrupt changes withhigh SINR upon convergence.In the second test case, 0ve-ray multipath fading

channels were assumed for all users where the last fourrays were uniformly distributed in delay over [0; 5Tc),where Tc is the chip interval, with = 0:3 (Fig. 2).After time n= 1000, the SINR of the AS-MOE algo-rithm drops dramatically due to the mismatch causedby multipath. On the other hand, AS-CMA shows su-

0 500 1000 1500-5

0

5

10

15

20

25

Number of symbols

SIN

R(d

B)

AS-MOEAS-CMAfixed- µ CMA

Fig. 2. The SINR performance of AS-MOE, 0xed-� CMA andAS-CMA receivers in the 0ve-ray multipath channel.

0 500 1000 15000

0.5

1

1.5x 10

-3

Number of symbols

µ AS

CM

A

Fig. 3. The trajectories of step-sizes �n for AS-CMA inthe single-path channel when initial step-sizes �0 are: (i)(− −) 10−5; (ii) (−�−) 10−4; (iii) (−♦−) 10−3.

perior abilities both in tracking of abrupt changes andalso in equalising the channel. The 0xed-� CMA al-gorithm lacks such ability unless its step-size was ap-propriately chosen. Fig. 3 shows three trajectories ofdi1erent initialised step-sizes for the AS-CMA algo-rithm for the single-path case. With the 100-fold vari-ation in �0 = {10−5; 10−4; 10−3}, all trajectories stillshow an approximately identical behaviour in conver-gence even after the occurrence of the sudden changesin the channel. The results suggest the robustness ofthe algorithm to initialisation of step-sizes.

314 P. Yuvapoositanon, J.A. Chambers / Signal Processing 82 (2002) 311–315

5. Conclusion

A new adaptive step-size CMA algorithm is pro-posed for a DS-CDMA receiver. The algorithm isdesigned to adapt the step-size in order to minimisethe CM criterion. Simulation shows the e1ectivenessof the algorithm in both interference cancellation ina time-varying environment and channel equalisationfor a multipath fading channel. Robustness against dif-ferent initial settings of the step-size is also shown.

Appendix A

Following Appendix A of [5], we consider theboundedness of the Lyapunov function |Yn|2 of theAS-CMA algorithm. As shown in [6], the locationsof the CM local minima are close to those for Wienerreceivers and are approximately in the directions ofthe Wiener receivers. We therefore con0ne the anal-ysis to within a compact region B with boundary@B in the tap-weight parameter space where localminima of CM receivers exist, and we do likewisein Appendix B. With the assumption of local conver-gence and that the term (z2n − �) within the gradientat large time will be small, we arrive at

En|Yn+1|2 − |Yn|2

=En|Yn+1 − Yn|2 + 2En[Y Tn Yn+1 − |Yn|2]

+En| − �n(3z2n − �)rnrTn Yn + (z2n − �)znrn|2

=− 2�nEn(3z2n − �)Y Tn rnr

Tn Yn

+2EnY Tn rn(z

2n − �)zn

+En�2n(3z

2n − �)2 Y T

n rnrTn rnr

Tn Yn

−En2�n(3z2n − �)(z2n − �)znY Tn rn|rn|2

+En(z2n − �)2z2n |rn|2

=− 2�nEn(3z2n − �)Y Tn rnr

Tn Yn + 2EnY T

n rn(z2n

− �)zn + O(4�2n)|Yn|2 + O(1) + O(|fn|2):

(A.1)

An approach to analysing the convergence of (A.1)based upon a 0rst-order perturbed test function canthereby be followed in the same manner as in [5].

With the assumption of small �+, the bound on Yn isdescribed by

lim supn

E|Yn|26O(1=�2−): (A.2)

Appendix B

We need to derive the convergence theorems by0rst rewriting (5) as

�n+1 = �n − �(z2n − �)znrTn Yn + �Qn; (B.1)

where Qn is a sequence of real-valued random vari-ables and functions as a reNection term to return�n−�(z2n −�)znrTn Yn to the constraint set [�−; �+] [3].We then interpolate �n to create the continuous-timerepresentation ��(t) = �n; 0¡t ∈ [n�; n� + �) andde0ne the interpolation for the reNection term asQ�(t) = �

∑t=�−1j=0 Qj; t¿ 0. To prove the conver-

gence theorem we conduct the following steps. The0rst is to prove that {��(·)} is tight, it is requiredthat the uniform integrability of {(z2n − �)znrTn Yn} beproved. Since by the Cauchy–Schwarz inequality,E|(z2n − �)znrTn Yn|26KE|Yn|2 for some K , then ittranslates into proving that E|Yn|2 ¡∞. From (A.2),the moment bound of E|Yn|2 =O(1=�2

−) suggests that{(z2n − �)znrTn Yn} and hence {Q�(·)} for n¡∞, andsmall �¿ 0 are uniformly integrable. Therefore, thetightness of {��(·); Q�(·)} is satis0ed.

In view of Theorem 5:1 of [5] and Theorem 4:1 of[3], with the tightness of {��(·); Q�(·)} inD2(−∞;∞)with any weakly convergent subsequence which haslimit {�(·); Q(·)}, there is a continuous functiong(·) such that for each m¿ 0 as n → ∞ for each�∈ [�−; �+]

1n

n+m∑j=m

Em(z2j − �)zjrTj Y�j ⇒ g(�); (B.2)

where g(�) = 14 (@E(z

2n − �)2=@�) and Y �

j denotes@f �j =@�. Due to the absolute continuity of Q(·),Q(t) = Q0 +

∫ t0 q(u) du and the limit {�(·); Q(·)} is

a solution of

d�dt

= g(�(t)) + q(t) �(0) = �0: (B.3)

In order to con0rm the validity of (B.3), use the weakconvergence technique de0ned in [4] to show that the

P. Yuvapoositanon, J.A. Chambers / Signal Processing 82 (2002) 311–315 315

weak limit satis0es

�(t + s)− �(t) =∫ t+s

tg(u) du+

∫ t+s

sq(u) du:

(B.4)

Hence, the validity of (B.3) is con0rmed.

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