Variable Step Size Affine Projection Algorithm for Adaptive Multiuser DS-CDMA MMSE Receiver

Wireless Personal Communications (2007) 42:107–130DOI 10.1007/s11277-006-9169-8 c© Springer 2006

Variable Step Size Affine Projection Algorithm for AdaptiveMultiuser DS-CDMA MMSE Receiver

ADITYA TRIVEDI1 and DALJIT KUMAR MEHRA2

1Department of Electronics Engineering, Madhav Institute of Technology and Science, Gwalior, Madhya Pradesh474 005, IndiaE-mail: [email protected]

2Department of Electronics and Computer Engineering, Indian Institute of Technology, Roorke 247667, IndiaE-mail: [email protected]

Abstract. In this paper, we consider the use of affine projection algorithm (APA) for interference suppression indirect sequence code-division multiple-access (DS-CDMA) system. We first derive the multiuser fixed step-sizeAPA (FSS-APA) algorithm. The computational complexity offered by the APA algorithm is linear in terms of thenumber of taps with additional terms of O(L2) and a matrix inversion of dimension L , where L is known as theorder of the filter. The value of L is chosen very small as compared to the number of filter-taps NT . We next proposea novel variable step-size APA (VSS-APA) algorithm, which further improves the performance of the FSS-APAalgorithm with very small increase in computational complexity as compared to the FSS-APA. It is demonstratedthat the performance of the APA based minimum mean-square error (MMSE) receivers is far superior to that ofthe normalized least-mean-square (NLMS) based receivers. Though, the recursive-least-square (RLS) algorithmbased adaptive receivers offer better performance but at the cost of much higher computational complexity.

Keywords: DS-CDMA systems, MMSE receivers, adaptive algorithms, interference suppression

1. Introduction

Direct sequence code-division multiple-access (DS-CDMA) offers advantage over other mul-tiple access methods like time-division multiple-access (TDMA) and frequency-divisionmultiple-access (FDMA) in terms of multipath resistance, inherent frequency diversity, inter-ference rejection, and the potential use of advanced antenna and receiver structures [1]. Thecapacity of wireless CDMA systems is limited by the interference generated by the trans-mission from other mobiles known as multiple access interference (MAI). It also results innear-far problem when the interfering signal is stronger than the intended signal. The problemof MAI becomes severe in asynchronous systems, where orthogonality between spreadingcodes cannot be maintained, unlike in synchronous systems. As the number of interferersincreases, MAI becomes substantial, causing degradation in system performance [2]. Anotherlimitation on the performance of CDMA systems is imposed by multipath fading, which alsointroduces intersymbol interference (ISI) due to multipath propagation [3].

The conventional matched filter (MF) receiver correlates the received signal with the signa-ture sequence of the desired user, ignoring the existence of multiple access interference (MAI)[2]. One common strategy to deal with the near-far problem in the conventional receivers isto use power control which causes wastage of precious bandwidth [4].

108 A. Trivedi and D.K. Mehra

Another strategy to overcome MAI, ISI and near-far problem is to use multiuser detection(MUD). Here, information about multiple users is used jointly to detect each individual user [2].MUD techniques provide significant performance improvement over the conventional detector.The optimal multiuser detector as proposed by Verdu [5] in 1986, is based on maximum-like-lihood sequence detection (MLSD). The optimum MLSD receiver requires information aboutthe number of users, signal amplitude, and transmission delay for all the multipath of allthe users and their signature sequences. It results in excessive computational complexity forobtaining a priori knowledge of all these parameters, which increases exponentially with thenumber of users. Since optimum MUD is computationally expensive, a number of subopti-mum MUD techniques have been explored, including minimum mean-square error (MMSE)receivers, with less computational complexity (where the complexity of the receiver structuregrows linearly with the number of interferers), providing substantial improvement over theconventional detector. Although, these linear detectors do not achieve minimum bit-error rate,they satisfy alternative optimization criteria based on performance indices such as asymptoticefficiency and near-far resistance [6].

The MMSE detector tries to minimize mean-square error (MSE) between the transmittedand the estimated bits. The estimated bits are obtained by using linear transformation, whichinvolves calculation of the inverse of correlation matrix [7], taking into consideration thenoise present in the system. Both the MMSE and decorrelator receivers are optimally near-farresistant [8].

In 1990, Xie et al. [9] first proposed the MMSE DS-CDMA receiver for additive whiteGaussian noise (AWGN) channel. The linear transformations were chosen to minimize eithera MSE or a weighted squared error performance criterion. However, the receiver requiresthe knowledge of the users’ parameters and since it is not adaptive it also needs to performcomputationally complex matrix inversion.

In order to overcome the problem of parameter estimation for implementing these receiversand to avoid the need for matrix inversion, adaptive receivers like adaptive MMSE receiversfor multiuser DS-CDMA detection have been proposed by many researchers [4, 10, 11] amongothers. In adaptive MMSE receivers, we only need to send training data sequence for eachactive user. The training sequence is used for the initial adaptation of the receiver. After thetraining phase ends; adaptation during actual data transmission occurs in decision-directedmode. The adaptive MMSE receiver directly processes samples of the received signal at chip(or fraction of a chip) interval, and operates like an adaptive equalizer, suppressing the inter-ference from the other users [12]. In [10], an N -tap MMSE receiver is proposed, where N isthe processing gain. In 1995, Miller [4] proposed an adaptive receiver for combating near-farproblem. The receiver is a CMF followed by an adaptive equalizer structure to perform de-spreading operation. The receiver is shown to be immune to the near-far problem. Traininganalysis of adaptive MMSE receiver is given in [11], where it is shown that MMSE receiver cantolerate a 30–40 dB near-far situation ( interferer’s power is 30–40 dB more than the desireduser’s power) without excessively long convergence time.

The key issues involved in the adaptive implementation of a filter are rate of convergence,computational complexity, and misadjustment error of the adaptive algorithm. The perfor-mance of the least-mean-square (LMS) and the normalized LMS (NLMS) adaptive algorithmssuffer from slow convergence in near-far situation and/or when the number of users is large[13], although the computational complexity involved is small, of the order of O (NT ), whereNT is number of filter taps. The performance of the recursive-least-square (RLS) algorithmis superior in comparison but at the cost of excessive computational complexity

(O

(N 2

T

)).

Variable Step Size APA for Adaptive Multiuser DS-CDMA MMSE Receiver 109

Computational complexity of the RLS algorithm becomes prohibitive, when the number ofNT is large. Therefore, it is desirable for an adaptive algorithm to have small computationalcomplexity as compared to the RLS algorithm with reasonable performance.

Generally, frequency-domain adaptive digital filters have advantages over time-domainimplementations [14] by providing faster convergence rate, comparable to the performanceof the RLS algorithm but with lower computational complexity. However, frequency domainadaptive filtering is not directly applicable to DS-CDMA receivers, since the input signal issampled at the chip rate while its output is sampled at the bit rate, i.e., the shifting propertyof the input data does not exist. Further, these methods introduce medium to large processingdelays and often lack tracking capability [15]. Since the shifting property of the input datadoes not exist, fast versions of RLS algorithm (such as fast transversal filter (FTF) and fastKalman algorithms) are also not applicable in this case.

Complexity reduction can be achieved by using block adaptive filtering algorithms such asblock LMS (BLMS) [16], in which the filter coefficients are adjusted once per output block,while maintaining equivalent convergence properties. Hassan and Mehra [17] proposed theuse of block adaptive algorithms for interference suppression in DS-CDMA systems. It wasshown that the block algorithm is near-far resistant and moderate block size achieves reason-able performance while introducing small processing delay. A generalization of the NLMSalgorithm, known as affine projection algorithm (APA), was proposed by Ozeki and Umeda[18]. Sankaran and Beex [19] analyzed the convergence behaviors of the APA filter.

It is well known that for any algorithm with a fixed value of the algorithm parameters(for example step-size parameter in LMS and NLMS algorithms) for tap weight adaptation,there exists a trade-off between the attainable convergence speed and the misadjustment val-ues. When the step-size is adjusted to obtain faster convergence, the misadjustment becomeslarger, and vice versa. Consequently, if we adaptively control the step size so that it remainslarge in the early stages of filter convergence and becomes smaller as the convergence pro-ceeds, both fast convergence and low estimation error could be obtained. Different adaptivestep-size control algorithms have been proposed and studied based on this concept.

Kwong et al. [20] proposed a variable step-size LMS algorithm in which the step size iscontrolled in relation to the error signal power. In [21], step-size is chosen, so that it tries toreduce the squared error at each instant. In [22], step-size changes as per the sign of successivesamples of the error gradient. In [23], step-size variation depends upon the cross correlationbetween the input signal and the error signal. Aboulnasr and Mayyas [24] use the square of atime-averaging estimate of the auto-correlation of the successive error samples, thus mitigat-ing the adverse effect of the measurement noise in finding the minimum misadjustment error.The performance of this algorithm [24] is significantly better than the fixed step size LMSalgorithm and other variable step-size algorithms proposed in [20–23].

In this paper, we first propose to use regularized fixed step-size affine projection algorithm(FSS-APA) for the adaptive implementation of multiuser detector (based on MMSE crite-ria. The FSS-APA based Adaptive MMSE receivers were first considered by the author in[25]. Next, we propose a novel variable step-size affine projection algorithm (VSS-APA) formultiuser detection, which further improves the performance of the FSS-APA based adaptiveMMSE receivers. The performance of both the algorithms is found to be superior to the NLMSand the block or PRA (partial rank algorithm) algorithms in terms of convergence characteris-tics, with significantly less computational complexity as compared to the RLS algorithm. Thecomputational complexity offered by the APA algorithms is linear in the length of transversalfilter with additional terms of the order of O(L2) and an inversion of L-by-L matrix, where


L is the order of filter, which is considerably smaller than the length of the filter NT . In theAPA, weight adaptation is done on the basis of multiple, delayed input signal vectors, whereasthe NLMS algorithms updates the weights on the basis of a single input vector. The APAconverges faster than the NLMS algorithm, as more input vectors are used. Also with morenumber of input vectors, the convergence rate improves, whereas the rate of improvementitself decreases. The APA is better alternative than the NLMS algorithm in applications wherethe input signal is highly correlated [19], as in multiuser detection of DS-CDMA systems.

To obtain faster convergence and lower misadjustment error, we employ the VSS-APAin place of the FSS-APA. The advantage of the VSS-APA algorithm was first discussed bythe author in [26]. In the VSS-APA, the step size is varied proportional to the square of theautocorrelation between adjacent error blocks, whose concept is derived based on the criterionas proposed in [24]

The DS-CDMA signal model is introduced in Section 2. Receiver architecture, whichprocesses samples from chip-matched filter (CMF) outputs, is also included in this section.MMSE detection is described in Section 3. In Section 4 FSS-APA as used for DS-CDMAmultiuser detection is derived. The VSS-APA is discussed in Section 5. Simulation results,illustrating the APA algorithms’ performance and their discussions are provided in Section 6.Conclusions drawn are given in Section 7.

2. Signal Model and Receiver Structure

In a DS-CDMA system having K active users, the kth active user’s base-band (low-passequivalent) transmission can be written as,

xk(t) =∑

i

Akbk(i)sk (t − iT − νk) (1)

where bk(i) is the symbol transmitted during i th interval, iT ≤ t ≤ (i + 1) T . T is the sym-bol duration and bk(i) ∈ {−1,+1} .Ak and νk represent the kth users amplitude and delay,respectively. Delay νk can be expressed as νk = (dk + δk) Tc, where dk is aninteger between0 and N − 1 (N is the processing gain) and δk lies between [0, 1). We consider short codeCDMA system, where same signature sequence is employed for each symbol interval. sk(t)is the signature waveform associated with the kth user, which may be written as,

sk(t) =N∑

j=1

ak( j)ψ (t − jTc) (2)

where Tc is the chip interval and the processing gain N = T/Tc.ψ (t) is the chip waveform

and ak( j) ∈{±1/

√N

}is normalized so that sk (t) has unit energy and duration T .

The received signal for the AWGN channel can be written as,

r (t) =K∑

k=1

xk(t)+ n (t) (3)

where n (t) is white Gaussian noise with power spectral density No/2.We consider the adaptive implementation of linear MMSE receiver employing CMF at

the front end. In this architecture, perfect estimation of the time delay of individual user is


assumed and one CMF is employed per user. The received signal is passed through a CMFand sampled at the chip rate then the output of the CMF at the nth chip sample correspondingto the i th bit of the kth user is given by

rnk (i) =

iT +(n+1)Tc+υk∫

iT +nTc+υk

r (t) ψ (t − iT − nTc − νk) dt (4)

If we take chip rate samples at the output of CMF corresponding to each user, then received

signal vector r j (i)

(=

(r1

j (i), r2j (i), . . . ., r

Nj (i)

)T)

corresponding to j th user’s CMF, during

i th symbol duration, can be written as [27],

r j (i) = A j b j (i)p+j +

K∑

k=1k �= j

Ak(bk(i)p

+k + bk (i − 1)p−

k

) + n(i) (5)

where n(i) is the N -by-1 vector of circularly symmetric complex noise samples during i th,symbol duration, assumed to be white Gaussian with variance σ 2. Because the users are asyn-chronous, each interferer contributes two interfering vectors. The vectors p+

k and p−k contain

the j th user’s CMF output samples within the time window spanned by r j (i) in response tothe inputs sk (t − iT − νk) and sk (t − (i − 1) T − νk), respectively, where j �= k, i.e., nthelement of the vector p+

k can be given as (for rectangular chip waveform) [10],

p+k (n) = (1 − δk) ak (n − dk) χn≥dk + δkak (n − dk − 1) χn≥dk+1 (6a)

where χA is the indicator function for the set A. Similarly, nth element of the vector p−k is

given as,

p−k (n) = (1 − δk) ak (n + N − dk) χn≤dk−1 + δkak (n + N − dk − 1) χn≤dk (6b)

The CMF is assumed to be synchronized to the j th user, so that p+j is the spreading code for

user j , and p−j contains only zeros.

For this receiver structure, which we call as fully connected (FC) architecture [25], wecollect N samples per CMF corresponding to each user, then for K users, we aggregate KNsamples. The sampled outputs of each CMF are shared amongst users in this architecture. Letr(i) be the overall data vector of length KN obtained by stacking the N samples of each user’sCMF during i th symbol interval,

r(i) =(

r1(i)T , r2(i)

T , . . . , rK (i)T)T

(7)

The FC multiuser detector architecture is shown in Figure 1 for two users’ case. For obtain-ing synchronism between all detected data symbols of all users, we can delay the signal atthe input to the kth user’s CMF by cT − νk , where c is the smallest integer that results inpositive delay. Since νk ∈ [0, T ) for k = 1, 2, . . . .., K , value of c may be taken as unity. Itmay be noted that in what we call as non-connected (NC) architecture r1(i), r2(i), . . . , rK (i)are not stacked and each user is detected from the samples collected by its CMF, i.e., thereis no cross coupling as shown in Figure 1. In the FC architecture by making available thesamples corresponding to other users, parallel interference cancellation takes place [28].


∑

∑

Figure 1. Fully connected multiuser detector for two users.

The FC architecture offers diversity advantage by combining energies of the desired userfrom the samples collected by other users’ CMF output.

Let w j,q [n] be the nth tap (filter coefficient) of j th user’s CMF output going to qth user’ssummer, where 1 ≤ j, q ≤ K , and 1 ≤ n ≤ N . All ws are adapted according to MMSE cri-teria. The output signal-to-interference and noise ratio (SINR) of an MMSE detector dependson its ability to capture the received energy of the desired user while suppressing MAI withoutunduly amplifying the noise. Now we define K N length filter coefficients for the kth user ini th symbol duration as,

wk(i) =(

wk,1(i)T ,wk,2(i)

T , . . . ,wk,K (i)T)T

(8)

where the N -by-1 length vector, w j,q(i) = (w j,q [1], w j,q [2], . . . , w j,q [N ])T .

In the FC architecture, if w j,q = 0, when j �= q, then this architecture reduces to the NCarchitecture, as proposed in [10, 29].

3. MMSE detection

The receiver is supplied with a desired response b(i) during training phase. After the trainingmode, the adaptive filter switches to the decision directed mode. The NT -by-1 vector of thetap inputs at time i is denoted by r(i) and the corresponding estimates of the desired responsesat the filter output sampled at interval T corresponding to K users is given as,

b̂(i) = WH (i)r(i) (9)

where b̂(i) =(

b̂1(i), b̂2(i), . . . , b̂K (i))T

. W(i) is NT -by-K matrix of filter taps, in

which each column represents the taps corresponding to each user’s filter i.e., W(i) =(w1,w2, . . . ,wK ).


An MMSE detector tries to minimize the cost function J , the mean-square-error, defined as,

J = E{

eH (i)e(i)}, where e(i) = b(i)− b̂(i) (10)

Therefore, J = E∥∥∥b(i)− b̂(i)

∥∥∥

2

= E∥∥∥b(i)− WH (i)r(i)

∥∥∥

2

= E

((b(i)− WH (i)r(i)

)H (b − WH (i)r(i)

))

= E(

bH (i)b(i)− 2Re(

bH (i)WH (i)r(i))

+ rH (i)W(i)WH (i)r(i))

(11)

where E is expectation operation. The optimal MMSE solution may be obtained by taking thederivative of the right hand side of the above equation and setting it equal to zero. Assumingthe independence between the expectation operation and differentiation operation, we get theoptimum solution for matrix of filter taps as,

Wo = R−1V (12)

in which R is NT -by-NT correlation matrix of the input vector, r(i), i.e.,

R = E{

r(i)rH (i)}

(13)

and V is NT -by-K multiuser cross-correlation matrix,

V = E{

r(i)bH (i)}

(14)

i.e., V may be obtained by grouping K cross-correlation column vectors corresponding toeach user, as V = (v1, v2, . . . , vK ). The Eq. (12) is known as Wiener–Hopf equation.

In fact, minimization of J is equivalent to minimization of MSE of individual user and itis equivalent to running K independent filters, one corresponding to each user. Computationalcomplexity of the adaptive algorithms may be reduced if the filters corresponding to each usershare the same data vector.

For MMSE receivers, the residual interference can be modeled as Gaussian random variable[6, 30]. In this case, the theoretical (analytical) Pe corresponding to a user can be expressedin term of its output SINR [4, 12, 28] as,

Pe = E(

Q(√

SINR))

(15)

where Q (x) = 1√2π

∞∫

x

e−t2/2dt (16)

and expectation is carried out for different relative delays and signature sequences. The numberof users supported by any particular receiver architecture and adaptive algorithm depend uponthe output SINR. As the number of users increase, residual MSE increases, which results in thedegradation of the overall performance of the receiver. In fact, output SINR is the key param-eter that governs the performance of the MMSE receiver in CDMA system [30]. Output SINR


is plotted as a function of number of users for different receiver architectures and adaptivealgorithms. Output SINR in dB is defined as follows,

SINR (dB) = 10 log10σ 2

b − Jmin

Jmin(17)

where Jmin is the minimum MSE and σ 2b is the variance of the input symbols. The output SINR

is directly dependent upon the residual MSE (Eq. (17)), and hence, Pe is related to residualMSE (Eq. (15)). The upper bound on the Pe of the DS-CDMA receiver is given by the socalled matched filter bound in which interference caused from the other users is assumed asGaussian noise. The lower (single-user) bound is evaluated by considering the thermal noiseonly (no MAI and ISI).

Both Pe and residual MSE depend upon the input SNR (signal-to-noise ratio), which isdefined in dB as follows,

SNR (dB) = 10 log10σ 2

b

σ 2 (18)

where σ 2 is the variance of AWGN noise.

4. FSS-APA Algorithm

In the multiuser adaptive algorithm, adaptation of the filter weights for all the users take placesimultaneously. However, the multiuser adaptive algorithms can be implemented only in thearchitectures where there is a common data vector for all the users as is the case with the FCarchitecture [13]. In the NC architecture single user adaptive algorithms has to be used, whereadaptation for each user has to be performed separately.

The affine projection adaptive filter may be derived based on multiple constraints optimi-zation technique. We will first derive the FSS-APA algorithm for single user adaptation, whichmay be used for the NC architecture. Subsequently, we will extend it to multiuser FSS-APAalgorithm, which may be used for the FC architecture.

4.1. S i ng l e U s e r F S S - A PA A l g orithm

Let rk(i) be the received signal vector, which consists of samples of CMF outputs correspond-ing to the kth user in the i th symbol duration. In APA filter of order L , we try to predict the Lmost recent outputs [31], bk(i), bk (i − 1) , . . . .., bk (i − L + 1) from the most recent L inputreceived signal vector rk(i), rk (i − 1) , . . . ., rk (i − L + 1), where,

rk(i) =(

r1k (i), r

2k (i), . . . ., r

NTk (i)

)T(19)

In the above equation rnk (i) is the kth user’s nth sample in the i th observation interval as given

in Eq. (4).


Based on minimum disturbance criteria to reduce noise amplification, we minimize thesquared Euclidean norm of the change in the tap-weight vector [32],

δwk (i + 1) = wk (i + 1)− wk(i) (20)

subjected to the set of L constraints, where L is known as the order of the APA filter,

bk (i − j) = wHk (i + 1) rk (i − j) , for j = 0, 1, 2, . . . L − 1 (21)

where kth user’s NT -by-1 tap-weight vector is given as,

wk (i + 1) =(w1

k (i + 1) , w2k (i + 1) , . . . , wNT

k (i + 1))T

(22)

Following the method of Lagrange multipliers [32] with multiple constraints, we definethe cost function for updating the tap weight vector as,

Ck(i) = ‖wk (i + 1)− wk(i)‖2 +L−1∑

j=0

Re(λ∗

k, j

(bk (i − j)− wH

k (i + 1) rk (i − j)))

(23)

In the above equations λk, j s are the Lagrange multipliers pertaining to multiple constraintscorresponding to the kth user.

Next to present the derivation in a compact manner, we define the following:Let Ak be L-by-NT data matrix, whose Hermitian transpose is defined as,

AHk (i) = (rk(i), rk (i − 1) , . . . , rk (i − L + 1)) (24)

L-by-1 desired response vector for kth user, whose Hermitian transpose is given as,

bHk (i) = (bk(i), bk (i − 1) , . . . , bk (i − L + 1)) (25)

and an L-by-1 Lagrange vector, whose Hermitian transpose is defined as

λHk = (

λk,0, λk,1, . . . , λk,L−1)

(26)

using Eqs. (24), (25), and (26) into Eq. (23), we can write the cost function as follows,

Ck(i) = ‖wk (i + 1)− wk(i)‖2 + Re(

bk(i)− Ak(i)wk (i + 1)H λk

)(27)

Differentiating the cost function Ck(i) with respect to complex valued weight vectorwk(i + 1), we have,

∂Ck(i)

∂w∗k (i + 1)

= ∂

∂w∗k (i + 1)

×(‖wk (i + 1)− wk(i)‖2 + Re

(bk(i)− Ak(i)wk (i + 1)H λk

))(28)

Now using the rules of differentiation with respect to a complex-valued vector [32, 33], wecan write the conjugate derivative of weight vector as,

∂Ck(i)

∂w∗k (i + 1)

= 2 (wk (i + 1)− wk(i))− AHk (i)λk (29)


Setting this derivative equal to zero and using Eq. (20), we have,

δwk (i + 1) = 1

2AH

k (i)λk (30)

From Eqs. (21), (24) and (25), we may write for the kth user’s desired constraint vector as,

bk(i) = Ak(i)wk (i + 1) (31)

Now premultiply both the sides of the Eq. (30) by Ak(i) and using Eq. (20), we have,

Ak(i)δwk (i + 1) = 1

2Ak(i)AH

k (i)λk

⇒ Ak(i) (wk (i + 1)− wk(i)) = 1

2Ak(i)AH

k (i)λk (32)

Using Eqs. (31) and Eq. (32) and rearranging terms, we can write,

bk(i) = Ak(i)wk(i)+ 1

2Ak(i)AH

k (i)λk (33)

The difference between bk(i) and Ak(i)wk(i) at iteration i is known as L-by-1 a priorierror vector, i.e.,

ek(i) = bk(i)− Ak(i)wk(i) (34)

With the help of Eqs. (33) and (34), we can solve for the Lagrange vector λk ,

λk = 2(

Ak(i)AHk (i)

)−1ek(i) (35)

Putting the value of λk into Eq. (30), we have,

δwk (i + 1) = AHk (i)

(Ak(i)AH

k (i))−1

ek(i) (36)

Now, we can introduce step-size parameter µ into the Eq. (36) for controlling the changein the weight vector from one iteration to the next as follows,

δwk (i + 1) = µAHk (i)

(Ak(i)AH

k (i))−1

ek(i) (37)

using Eq. (20), we may write the weight update equation for the APA filter as follows,

wk (i + 1) = wk(i)+ µAHk (i)

(Ak(i)AH

k (i))−1

ek(i) (38)

It may be observed from Eq. (38) that, the weight adaptation of the APA filter involvesmatrix inversion of dimension L (L-by-L matrix). Since the order of the APA filter L is usu-ally small as compared to the filter length, the computations involved are also small. Next, weobtain the limits for µ, within which the APA will remain stable. We assume that the physicalmechanism for generating the desired response bk(i) is governed by the multiple regressionmodel as given below [32],


bk(i) = wRrk(i)+ n(i) (39)

where wR is model’s unknown parameter vector and n(i) is additive disturbance (noise). Thedifference between wR and wk(i) is known as weight-error vector εk(i),

εk(i) = wR − wk(i) (40)

and εk (i + 1) = wR − wk (i + 1) (41)

From Eqs. (38), (40), and (41), we have,

εk (i + 1) = εk(i)− µAHk (i)

(Ak(i)AH

k (i))−1

ek(i) (42)

Using the definition of mean-square deviation(Dk(i) = E

(‖εk(i)‖2)), we can write,

Dk (i + 1)− Dk(i) = µ2 E

(ek(i)

(Ak(i)AH

k (i))−1

ek(i)

)

− 2µE

{Re

(ξ

Hu (i)

(Ak(i)AH

k (i))−1

ek(i)

)}(43)

where ξu(i) is known as undisturbed error vector and is defined as,

ξu(i) = Ak(i) (wR − wk(i)) (44)

It can be observed that Dk(i) decreases monotonically with increase in iterations, providedthat the step-size parameter µ satisfies the condition,

0 < µ <2E

{Re

(ξH

u (i)(Ak(i)AH

k (i))−1

ek(i))}

E(

eHk (i)

(Ak(i)AH

k (i))−1

ek(i)) (45)

This is the stability condition for the APA filter.In case of noisy data and highly correlated input, the L-by-L matrix product Ak(i)AH

k (i)may be ill conditioned and its inversion may pose numerical difficulty. To overcome this prob-lem, we use regularization by adding the term δI to the matrix product Ak(i)AH

k (i), where δ isa small positive constant known as regularization parameter and I is L-by-L identity matrix.

The Eq. (38) may be modified to include the regularization parameter as,

wk (i + 1) = wk(i)+ µAHk (i)

(Ak(i)AH

k (i)+ δI)−1

ek(i) (46)

This is the weight update equation for the single user APA adaptive algorithm for CDMA sys-tems, which is used to evaluate the performance of the NC architecture. In the NC architecturefor detecting more than one user, the algorithm needs to run separately for each of the users.

4.2. M ul t i us e r F S S - A PA A l g orithm

We next extend the single-user FSS-APA algorithm to multiuser FSS-APA algorithm. Mul-tiuser adaptive algorithms give advantage in terms of saving computations if the data vec-tor is common for all the users [13]. For the FC architecture data vector r(i) is commonfor all the users so that multiuser adaptation can take place. In the FC architecture, r(i)is obtained by stacking chip rate sampled outputs corresponding to each of the user, i.e.,


r(i) = ((r1(i))T , (r2(i))T , . . . ., (rK (i))T

)T, this results in NT-by-1 length vector where

NT = K N .In multiuser FSS-APA algorithm, similar to single-user FSS-APA algorithm, we try to

predict the most recent output vectors b(i),b (i − 1) , . . . ,b (i − L + 1) from the most recentinputs r(i), r (i − 1) , . . . , r (i − L + 1), where b(i) = (b1(i), b2(i), . . . , bK (i))T .

Multiuser tap weight matrix W (i + 1) of dimension NT -by-K is obtained by stacking Kusers’ NT -by-1 weight vector as,

W (i + 1) = (w1 (i + 1) ,w2 (i + 1) , . . . ,wK (i + 1)) (47)

We define L-by-K desired response matrix consisting of K users’ L recent symbols as,

B(i) = (b(i),b (i − 1) , . . . ,b (i − L + 1))T (48)

The estimate of b(i) at i th iteration can be given as,

b̂(i) = WH (i)r(i) (49)

TheL-by-K error matrix E(i) may be given as,

E(i) = B(i)− A(i)W(i)

i.e., E(i) = (e1(i), e2(i), . . . , eK (i)) (50)

In the above equation A(i) is L-by-NT data matrix, whose Hermitian transpose is given asAH (i) = (r(i), r (i − 1) , . . . , r (i − L + 1)). Grouping the K users’ tap weight vectors, wecan write the multiuser weight update equation as,

W (i + 1) = W(i)+ µAH (i)(

A(i)AH (i)+ δI)−1

E(i) (51)

In the above equation µ and δ are step-size and regularization parameters, respectively,as for single user APA algorithm. Note that for L = 1, the algorithm reduces to the NLMSalgorithm, which can be considered as a special case of APA [19]. If we let L = NT andthe matrix inversion is replaced with the exponentially weighted, time averaged correlationmatrix, the algorithm then become equivalent to the RLS algorithm.

In block or PRA, to reduce the complexity of the APA, weights are updated once for everyblock of length L [19], [17]. However, the convergence performance of the block algorithmis inferior to that of the APA as will be shown in the simulation results. Moreover, block algo-rithm’s computational complexity is bursty, so depending upon the speed of the implementedtechnology, there is often a delay in the generation of the error vector [34].

The multiuser APA algorithm is summarized in Table 1.

5. Multiuser VSS-APA Algorithm

The performance of the FSS-APA based DS-CDMA receivers can be further improved byemploying variable step-size APA (VSS-APA) in place of FSS-APA. In VSS-APA, large step-size is used in the initial stages of adaptation to speed up the convergence rate. When thealgorithm reaches its steady state, then a small step-size is used to achieve a low level ofmisadjustment. So the overall performance can be improved as compared to the FSS-APAalgorithm.


Table 1. Summary of the multiuser APA algorithm

NT =number of filter taps

Parameters L =filter order (number of multiple constraints)

µ = step-size parameter

r(i) = NT -by-1 tap-input vector at iteration i .

Given data b(i) = (b1(i), b2(i), . . . , bK (i))T desired response at iteration i

Initialize W (0) = 0. For i = 0, 1, 2, . . ...,compute

AH (i) = (r(i), r (i − 1) , . . . , r (i − L + 1))

Computations B(i) = (b(i),b (i − 1) , . . . ,b (i − L + 1))T

E(i) = B(i)− A(i)W(i)

W (i + 1) = W(i)+ µAH (i)(A(i)AH (i)+ δI

)−1 E(i)

Several criteria can be used for adjusting the step-size parameter. We adjust the step-sizeparameter according to the square of the time-averaged estimate of the autocorrelation of con-secutive error vectors ek(i) and ek (i − 1). This criterion is similar to one, which was proposedin [24] for the LMS algorithm. We modify it to be used in the multiuser FSS-APA to obtainmultiuser VSS-APA.

Error matrix E(i), which consists of K users’ L length column error vectors each, i.e.,E(i) = (e1(i), e2(i), . . . , eK (i)), where ek(i) = (ek(i), ek (i − 1) , . . . , ek (i − L + 1))T .We next define operation ⊗ as,

E(i)⊗ E (i − 1) =(

e1(i)T e1 (i − 1) , e2(i)

T e2 (i − 1) , . . . , eK (i)T eK (i − 1)

)(52)

resulting in 1-by-K row vector.In the proposed algorithm, time average estimate of autocorrelation between successive

error vectors for K users is given by K -by-1 column vector as,

Z(i) = βZ (i − 1)+ (I − β) {E(i)⊗ E (i − 1)}T (53)

where β is a diagonal matrix with elements β1, β2, . . . , βK , corresponding to each user. βsare positive constants (0 < β < 1) which govern the quality of estimation [24]. In the initialstages of adaptation, the square of the autocorrelation estimate between successive error vec-tors is high resulting in large step-size, causing fast convergence. For near optimum solution,the error autocorrelation approaches zero, resulting in a smaller step size so that final misad-justment error is minimized. The proposed step size update for multiuser detection is givenby

µ (i + 1) = αµ(i)+ γZ2(i) (54)

where µ(i) = (µ1(i), µ2(i), . . . , µK (i))T . By using different step-size parameters for differ-ent users, we have the option of selecting different step-size parameters for the users, dependingon their SNR and mobility. γ = diag (γ1, γ2, . . . , γK ) with all γ > 0 and 0 < α < 1.γ con-trols the rate of convergence and final misadjustment error. βs and γ s are chosen dependingupon the nonstationarity and the SNR of the concerned user. For example, in the stationaryenvironment β is usually taken as close to but smaller than unity. And for the nonstationaryenvironment a smaller value of β may be used. Further, α is usually chosen same for differentenvironment and SNR as in [24]. We keep the value of α same for all the users. The upper


limit of the step-size is set to µmax, which is near the point of instability of the algorithm andlower limit µmin is chosen as trade-off between desired level of misadjustment and trackingcapability of the algorithm. Further, Eq. (51) gets modified to,

W (i + 1) = W(i)+{

AH (i).(


E(i)}

µ (i + 1) (55)

In the above equation, we convert the step-size column vector, as obtained from Eq. (54), into a K -by-K diagonal matrix.

We now summarize the additional operations performed in the VSS-APA as compared tothe FSS-APA.

(i) Calculate autocorrelation of the error vectors corresponding to K users,

Z(i) = βZ (i − 1)+ (I − β) {E(i)⊗ E (i − 1)}T

(ii) update the step-size parameter

µ (i + 1) = αµ(i)+ γZ2(i)

(iii) adapt the weights with the updated step-size parameter

W (i + 1) = W(i)+{

AH (i)(


E(i)}

µ (i + 1)

Computational complexity of the multiuser FSS-APA, and the multiuser VSS-APA is givenin Table 2. For comparison sake, computational complexity involved in the multiuser RLS algo-rithm is also reproduced in the Table 2. Computational complexity is given in terms of numberof operations required per user per iteration.

Gaussian elimination method utilizing LU decomposition (factorization) is used for obtain-ing the inverse of L-by-L matrix

(A(i)AH (i)+ δI

), which has a computational complexity

of L3/3 + L2 − L/3 multiplication/division and L3/3 + L2/2 − 5L/6 addition/subtraction[35].

It may be noted from the Table 2 that, the VSS-APA involves, additional (L+5) multi-plication/division and (L + 2) addition/subtraction operations as compared to the FSS-APA.Thus, the increase in the computational complexity is very small while using the VSS-APA in

Table 2. Computational complexity comparison of the multiuser FSS- APA, multiuser VSS-APA,and multiuser RLS algorithms

Algorithm Computational complexity

Addition/subtraction Multiplication/division

RLS3N 2

TK + 2NT − NT

K3N 2

TK + 2NT + 2NT

K + 1K

FSS-APAL2 NT

K + L (2NT + L − 1)

+ L3

3K + L2

2K − 5L6K

L2 NTK + L (2NT + L + 1)

+ L3

3K + L2

K + 2L3K

VSS-APAL2 NT

K + L (2NT + L)

+ L3

3K + L2

2K − 5L6K + 2

L2 NTK + L (2NT + L + 2)

+ L3

3K + L2

K + 2L3K + 5

Note that for obtaining the computational complexity for single user adaptation algorithm, asused in the NC architecture, choose K = 1 in the Table above.


place of the FSS-APA. However, as will be shown in the simulation results, the performanceimprovement is substantial.

For comparing the computational complexity involved in the RLS, VSS-APA, and FSS-APA algorithms, we use filter length NT = 186, when number of users K = 6. Orderof APA filters L is taken as four. The multiplication operations involved in the multi-user RLS algorithm are 17,732. Whereas, the multiplication operations involved in theVSS-APA and the FSS-APA are 2020 and 2011, respectively. It clearly demonstrates thecomputational advantage of using APA algorithms, which will further increase with theincrease in the number of filter taps as is the case of multiple receive antennas receiverstructures [13].

6. Simulation Results

In all the simulations, asynchronous transmission is assumed. In asynchronous system, theusers’ delays νks are uniformly distributed between zero and T . In the NC architecture, it isassumed that the receiver is synchronized with the desired user. In the FC architecture, it isassumed that the CMF corresponding to a particular user is synchronized with that user. Thefilter length for the NC architecture is N (processing gain), whereas for the FC architecture,it is K N . K is the total number of users in the system. Gold sequences of length 31 are usedfor the simulation. In each independent trial, signature sequence are picked up randomly anddelays corresponding to different users are generated uniformly between the interval [0, T ).In the implementation of the RLS algorithm, value of the forgetting factor is chosen as 0.995for all the simulations. Optimum step-size for the LMS and NLMS algorithm and regular-ization parameter for the RLS algorithm are chosen by conducting a number of simulationexperiments so that minimum MSE is obtained at the end of specified number of iterations. Inthe implementation of the VSS-APA, we have taken the value of α = 0.97 and regularizationparameter value δ = 0.01 in all the simulations. Values of other parameters are obtained fromthe trial and error method. In particular, values of βs vary from around 0.9–0.99 and that ofγ s around 0.1 in different situations.

Convergence rate of the different adaptive algorithms is compared by plotting residualMSE (in dB) as a function of number of training bits transmitted. In calculating the probabilityof BER, step size and regularization parameter is adjusted at each input SNR. Probability ofBER is calculated, at the end of 500 iterations, using Eq. (15), since it has been verified that,both the theoretical (analytical) results given by Eq. (15) and the results obtained from theMonte-Carlo simulations closely match each other. AWGN channel is considered. Ensembleaveraging over more than 100 independent trials is performed and their mean is plotted in theresults.

In Figure 2, we plot the convergence characteristics of the FC architecture, employingthe FSS-APA algorithm. We consider four users system and near-far situation with the threeinterferers at 10 dB power advantage relative to the desired user, which is kept at 20 dB inputSNR. In this example, we have chosen three different APA filters having orders, L = 2, 4,and 8. For a fair comparison of the convergence performance, we keep the same residual MSEfor all the situations. As is evident from the Figure, order 2 APA filter converges in about 240iterations, and order 4 APA filter converges in as many as 170–180 iterations. Using order8 APA filter give very small improvement over the order 4 APA filter. It may be concludedthat with the increase in order of the APA filter, convergence rate improves, however the


Figure 2. Convergence characteristics of FSS-APA of order 2,4, and 8 for FC architecture in AWGN channel withK = 4 (users). Input SNR of the desired user is 20 dB. interferers are at 10 dB power advantage relative to thedesired user.

rate at which improvement takes place start decreasing. With the increase in the order of theAPA filter, computational complexity starts growing O(L2) with an additional term of matrixinversion of dimension L . We choose order 4 APA filter as a good compromise between thecomputational complexity and performance. In all our simulations involving either FSS-APAor VSS-APA filter, we will use order 4 APA.

The convergence characteristics of the NLMS, FSS-block, and the RLS algorithms is com-pared with the FSS-APA in Figure 3. We consider four users system with the FC architecture.Input SNR of the desired user is 20 dB with the three interferers at 10 dB power advantage rel-ative to the desired user. As may be seen from the Figure, the NLMS algorithm’s convergenceperformance is worst as expected. The block algorithm [17] converges in about 220 iterations.We have used length 4 block algorithm for fair comparison with the FSS-APA. The FSS-APAoutperforms both the FSS-block and the NLMS algorithms and reaches its steady state valuein about 170–180 iterations. The RLS algorithm converges in about 70–80 iterations. Though,the RLS algorithm gives the best performance, but at much higher computational complexityas mentioned earlier.

It may be concluded that considering the performance of APA and its computational com-plexity, it may be preferred over the RLS algorithm in situations where the number of filtertaps to be adapted is large.

In Figure 4, convergence characteristics of the RLS, FSS-APA, VSS-APA, FSS-block [17],and VSS-block algorithms are plotted for the FC architecture with four users. Three interferersare at 10 dB power advantage relative to the desired user, which is kept at 20 dB input SNR.As may be seen from the Figure, the RLS algorithm converges in about 70–80 iterations. TheVSS-APA converges in about 120–130 iterations as compared to 170–180 iterations takenby the FSS-APA. It clearly shows the superior performance offered by the VSS-APA by just


Figure 3. Convergence characteristics of NLMS, FSS-block, FSS-APA (order 4), and RLS algorithms for FCarchitecture in AWGN channel with K = 4 (users). Input SNR of the desired user is 20 dB. interferers are at 10 dBpower advantage relative to the desired user.

Figure 4. Convergence characteristics of FSS-block, FSS-APA, VSS-block, VSS-APA, and RLS algorithms forFC architecture in AWGN channel with K = 4 (users). Input SNR of the desired user is 20 dB. interferers are at10 dB power advantage relative to the desired user.

having L + 5, i.e., nine more multiplication operations as compared to the FSS-APA. Bothfixed step-size and variable step-size block algorithms are implemented in the simulation. Weimplemented the VSS-block algorithm by using the same step-size parameter update equation


Figure 5. Near-far resistance performance of FC architecture using FSS-APA and VSS-APA algorithms in AWGNchannel with K = 2 (users). Input SNR of the desired useris 20 dB interferer’s power is varied from 0 to 20 dBrelative of the desired user’s power.

as used for the VSS-APA (Eq. (54)). It may be noted that length 4 block algorithms are usedin the simulation. As may be observed from the Figure that, the performance of both theblock algorithms is somewhat inferior to that of their APA counterparts. In particular, theFSS-block algorithm converges in about 220 iterations, which is 40–50 iterations more thanthe FSS-APA. In fact the VSS-block algorithm give almost the same performance as that ofthe FSS-APA, both converged in about 170–180 iterations, whereas the VSS-APA convergesin about 120–130 iterations.

Near-far resistance of the FC MMSE detector is demonstrated in Figure 5 using both theVSS-APA and FSS-APA. We have taken two users system, in which the interferer’s power isincreased from 0 to 20 dB in the increment of 4 dB relative to the desired user’s power, whichis kept fixed at 20 dB. Output SINR is measured at the end of 500 iterations by adjusting theparameters of both the algorithms to provide minimum MSE. The output SINR of the desireduser degrades by about 0.1 and 0.4 dB using the VSS-APA and FSS-APA algorithms respec-tively for the FC architecture, when the interferer’s power is 20 dB more than the desired user’spower. It may be concluded that, though both the algorithms offer good near-far resistance,but the performance of the VSS-APA based MMSE receiver is better than the FSS-APA basedreceiver.

In Figure 6, we plot the output SINR of the desired user as a function of the number ofusers for the FC architecture using the RLS, VSS-APA, FSS-APA, and NLMS algorithms.All the interferers are at 10 dB power advantage relative to the desired user. Output SINRof the desired user is calculated at the end of 1000 iterations. We may observe that outputSINR increases when the number of users increases up to eight users; this is because in FCarchitecture number of filter taps increases in direct proportion to the number of users (K N )and more useful energy is collected. But for more number of users, MAI becomes dominant


Figure 6. Output SINR of FC architecture as a function of number of users using RLS,VSS-APA,FSS-APA, andNLMS algorithms in AWGN channel. Input SNR of the desired user is 20 dB. Interferers are at 10 dB advantagerelative to the desired user.

and also tap adaptation take longer time, resulting in the performance degradation. When thenumber of users is 16, the output SINR of the desired user is 21.5, 17, 15, and 10.5 dB using theRLS, VSS-APA, FSS-APA, and NLMS algorithms, respectively. If the desired output SINRis 15 dB then the FSS-APA can accommodate 16 users, whereas the NLMS algorithm canaccommodate only 13 users. It may be concluded that, the output SINR obtained using boththe VSS-APA and FSS-APA is intermediate to the RLS and the NLMS algorithms, with theVSS-APA clearly outperforming the FSS-APA.

The probability of the error performances of the APA based FC MMSE receivers are con-sidered in Figure 7. The probability of BER is calculated after 500 iterations. Parameters ofthe FSS-APA and VSS-APA algorithms are adjusted after conducting a number of simulationexperiments in such a manner to get minimum MSE at the end of 500 iterations, which alsoresults in minimum BER. Four users case is considered in near-far situation, in which theinterferers are at 10 dB power advantage relative to the desired user. As may be seen fromthe Figure that the RLS, VSS-APA and FSS-APA algorithms give nearly the same perfor-mance. There is a gain of approximately 0.6 dB as compared to the NLMS algorithm at 10−3

probability of error.In Figure 8, we plot the probability of error performance for the eight users case employing

the FC architecture, with all the interferers at 10 dB power advantage relative to the desireduser. We use both the VSS-APA and FSS-APA algorithms and for comparison, performance ofthe RLS and NLMS algorithms is also plotted. Probability of error is calculated after 500 itera-tions. It may be seen from the Figure that at 10−3 probability of error, the SNR required by theRLS, VSS-APA, FSS-APA, and NLMS algorithms are 7.5, 8.5, 9.2, and 10.5 dB, respectively.The VSS-APA is having 2 dB advantage relative to the NLMS algorithm and is inferior by 1 dBas compared to the RLS algorithm. Comparing with Fig. 7, it is obvious that as the number


Figure 7. Probability of error performance of FC architecture using NLMS, APA, VSS-APA, and RLS algorithmsin AWGN channel with k = 4 (users). Interferers are at 10 dB power advantage relative to the desired user.

Figure 8. Probability of error performance of FC architecture using NLMS, FSS-APA, VSS-APA, and RLS algo-rithms in AWGN channel with k = 8 (users). Interferers are at 10 dB power advantage relative to the desireduser.

of users starts increasing; the RLS algorithm provides superior performance as compared tothe APA based filters. However, the computational complexity of the RLS algorithm is muchhigher as discussed earlier.


Figure 9. Convergence characteristics of NLMS, FSS-APA, VSS-APA, and RLS algorithms for FC architecturewith K = 4 (users) and single antenna in 10 paths, exponential PDP fading channel. Input SNR of the desired useris 20 dB. Interferers are at 10 dB power advantage.

In Figure 9, we plot the convergence characteristics of the RLS, VSS-APA, FSS-APA,and NLMS algorithms for the FC architecture for 10 paths, exponential PDP channel withK = 4 (users). We assume a quasi-static Rayleigh fading channel model. In the exponentialpower delay profile considered, a highly frequency-selective tapped delay line channel modelis assumed, where multipath arrive at Tc interval length following the arrival of the first path,i.e., τk,l = τk,1 + (l − 1) Tc, where l = 2, . . . . . . .., L P . This situation arises where the delayprofile is more or less continuous [36]. In this situation multipath spread is taken to be approx.2.2µ s., causing a multipath span of 9Tc, which spreads ISI up to about 1/3rd of symbol.Input SNR of the desired user is kept at 20 dB and near-far situation is assumed with the threeinterferers at 10 dB power advantage relative to the desired user. As may be observed fromthe figure that the RLS, VSS-APA, and FSS-APA algorithms converge in about 270, 340,and 470 iterations, respectively. The NLMS algorithm’s convergence performance is worstand it takes about 850–900 iterations to converge. It may be concluded, that for fading multi-path channels also, the VSS-APA algorithm’s performance is nearly comparable to that of thecomputationally expensive RLS algorithm.

A comparison in the computational complexity offered by the multiuser VSS-APA andmultiuser RLS algorithms for fading channels with multiple receive antennas is in order. Weconsider four users FC architecture system with four receive antennas. Number of filter tapsNT equals to 992(= 2K N M) for length 31 sequences, where M is the number of receiveantennas and filter length is doubled to capture the whole of the energy of the desired symbolbecause of multipath. From Table 2, number of multiplication/division operations required forthe multiuser RLS algorithm are 7,40,528, whereas multiuser VSS-APA algorithm requiresonly 11,943 multiplication/division operations, a difference of a factor of 62.


7. Conclusions

In this paper, we first proposed the use of APA algorithm for interference suppression inDS-CDMA system and derived the multiuser FSS-APA algorithm. The computational com-plexity offered by the APA algorithm is linear in terms of number of taps with additionalterms of O(L2) and a matrix inversion of dimension L , where L is known as the order ofthe filter. L is taken very small as compared to the number of taps of the filter NT . In thesimulation, we take the value of L , as small as four to obtain substantial improvement in theperformance as compared to the NLMS algorithm. We next proposed a novel variable step-size APA (VSS-APA) algorithm, which further improves the performance of the FSS-APAalgorithm, in which the computational complexity is increased by just (L + 5) multiplicationand (L + 2) addition operations as compared to the FSS-APA. By extensive simulations, itwas shown that the performance of the APA based MMSE receivers is far superior to that ofthe NLMS based receivers in terms of residual MSE, number of users accommodated, andprobability of error performance. Though, the RLS algorithm based adaptive receivers offerbetter performance but it involves much higher computational complexity of O(N 2

T ).

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Aditya Trivedi is working as Reader in the Department of Electronics and Computer Sci-ence and Engineering with the Madhav Institute of Technology and Science, Gwalior, M.P.,India. He received his bachelor degree (Electronics) from the same institute in 1987. He didhis M.Tech. (Communication Systems) from IIT Kanpur in 1997. He obtained his doctoratefrom IIT Roorkee in 2004 in the area of wireless communication engineering. His teach-ing and research interest include Digital communication, CDMA systems, Signal processing,and Networking. He has published 10 papers in various national and international journalsconferences.

D.K. Mehra was born in Amritsar, India, on May 25, 1946. He received the B.E. and M.E.degree in electrical communication engineering from the Indian Institute of Science, Bangalor,India in 1968 and 1970, respectively, and the Ph.D. degree from the Indian Institute of Tech-nology, Kanpur, India in 1978. In 1975, he joined the Electronic and Computer EngineeringDepartment of Indian Institute of Technology Roorkee (Previously University of Roorkee),India, where he became a Professor in 1987. His main techning and research interests are inthe area of adaptive signal processing techniques and their applications for interference sup-pression in CDMA systems, high mobility OFDM systems and digital communication overfading dispersive channels.

Variable Step Size Affine Projection Algorithm for Adaptive Multiuser DS-CDMA MMSE Receiver

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