ARTICLE IN PRESS

0165-1684/$ - se

doi:10.1016/j.si

�CorrespondE-mail addr

(H.-L. Yang), w

Signal Processing 88 (2008) 1191–1202

www.elsevier.com/locate/sigpro

A low-complexity adaptive antenna array code acquisition

Hua-Lung Yang, Wen-Rong Wu�

Department of Communication Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan , ROC

Received 14 November 2006; received in revised form 21 August 2007; accepted 16 November 2007

Available online 28 November 2007

Abstract

Conventionally, code acquisition with antenna array employs a correlator bank and the serial-search technique. Due to

the inherent properties of correlators and serial search, mean acquisition time of the correlator-based approach is large,

especially in strong interference environments. Recently, an adaptive-filtering approach has been applied to code

acquisition. This method simultaneously performs beamforming and code-delay estimation with a spatial and a temporal

filter. Its performance is significantly better than that of the correlator-based approach. However, its computational

complexity will be high when the code-delay uncertainty is large. In this paper, we propose a low-complexity adaptive-

filtering scheme to solve the problem. Incorporating a serial-search technique, we are able to significantly reduce the size of

the temporal filter, so does the computational complexity. We also analyze the proposed algorithm and derive closed-form

expressions for optimum solutions, mean-squared error, and mean acquisition time. Simulations show that while the

proposed system somewhat compromises the performance, the computational complexity is much lower that of the

original adaptive-filtering approach.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Code acquisition; Adaptive filter; Antenna array; DS/CDMA

1. Introduction

Direct-sequence/code-division multiple access(DS/CDMA) is a promising technique for wirelessmobile communication. It is well known that themain performance bottleneck for a CDMA systemis multiple access interference (MAI). MAI not onlyaffects data detection, but also code acquisition.Code acquisition is the first operation that a CDMAreceiver has to perform. It attempts to align thelocal code sequence and the received desired signalwith a difference less than a chip duration. After

e front matter r 2007 Elsevier B.V. All rights reserved

gpro.2007.11.008

ing author. Tel.: +886 3 5712121x54529.

esses: hualungyang.cm90g@nctu.edu.tw

rwu@faculty.nctu.edu.tw (W.-R. Wu).

successful code acquisition, other operations such aschannel estimate, code tracking, and data detectioncan follow. Thus, code acquisition is a critical taskin DS/CDMA systems.

Code acquisition for single antenna systems hasbeen extensively studied in the literature [1–11] (andreference therein). The correlator approach [1–4] ismost well known. However, the correlator is onlyoptimal for the single-user case. Its performancedegrades significantly when MAI presents [4]. Sub-space- and matrix-based methods [5,6] have beendeveloped to solve the problem. The advantage ofsubspace-based approaches is that no trainingsequences are required and the performance ismuch better than that of the correlator approach.However, these methods usually have to estimate,

.

ARTICLE IN PRESSH.-L. Yang, W.-R. Wu / Signal Processing 88 (2008) 1191–12021192

decompose, or inverse the autocorrelation matrix.This often requires high computational complexity,especially for systems with large processing gains.Recently, adaptive-filtering technique has beenintroduced to code acquisition [7–11]. It is claimedthat [7], the adaptive-filtering approach can providehigher acquisition-based capacity [3] compared tothe correlator methods. The acquisition-basedcapacity indicates the maximal number of usersthat a system can tolerate with a given acquisitionerror rate. One problem with the adaptive-filteringscheme [7] is that its computational complexitygrows linearly along with the code-delay uncertaintyrange. A low-complexity scheme was then devel-oped in [11].

It is well known that antenna arrays cansignificantly enhance acquisition performance[12–14]. In [12], each array element is equippedwith a correlator, and the correlator outputs servethe inputs of a beamformer. When directional MAIpresents, matrix inversion is required to derive thebeamformer weights [12]. In [13], an adaptivebeamformer is used to avoid the problem. However,in the presence of directional MAI, it requires a longadaptation period and the acquisition is slow. Tosolve the problem, an adaptive-filter-based arraysystem was then proposed [15,16]. This systemsimultaneously performs adaptive beamformingand code-delay estimation with a spatial and atemporal filter. The code delay can then beestimated with the peak position of the temporalfilter. It has been shown that the approachsignificantly outperforms the correlator-based sys-tem in [13].

Similar to the conventional adaptive-filteringacquisition, the computational complexity of thetemporal filter in [15] grows linearly along with theuncertain range of the code delay. When the range islarge, the computational complexity may be high. Inthis paper, we propose a low-complexity adaptivearray code acquisition scheme to solve the problem.The main idea is to divide the whole delayuncertainty range into several (delay) cells, and thensequentially search for the code delay of the desireduser among those cells. This is essentially a serial-search-technique, being able to shorten the filterlength of the temporal filter. As a result, thecomputational complexity can be reduced. As thatin [15], the proposed system employs a criterion suchthat both filters can be simultaneously adjusted by aconstrained least-mean-square (LMS) algorithm.However, the acquisition process is more involved

than that in [15]. This is because one additionaldecision has to be made before the code delay can beestimated. For each tested cell, filters are firstadapted for a period time to determine if the codedelay falls into the cell’s delay region or not. If itdoes, the spatial filter will act as a minimum mean-squared error (MMSE) beamformer and the tem-poral filter as a code-delay estimator. Thus, the codedelay can then be estimated with the peak positionof the temporal filter. If not, the next cell is testedand the process is repeated. Note that if the codedelay does not fall into the tested cell’s region, thespatial filter will act as a signal blocker with itsweighs all being zeros. This property is then used toderive an index for the cell testing. With the choiceof the number of cells, we can have an easy tradeoffbetween performance and complexity. In manycases, however, the complexity reduction is large,but the performance loss is still acceptable. In thispaper, we only consider flat-fading channels forsimplicity. It is straightforward to extend theproposed algorithm to multipath environments.Also note that with the proposed architecture, wecan also apply other types of adaptive algorithmssuch as the recursive least-squares (RLS) algorithm.In this case, the overall performance can be greatlyenhanced, but the computational complexity is alsosignificantly increased.

This paper is organized as follows. Section 2develops the proposed algorithm. Section 3 discussesissues of adaptive implementation, and Section 4carries out performance analysis. Section 5 presentssimulation results demonstrating the effectiveness ofthe proposed scheme. Finally, Section 6 drawsconclusions. Throughout this paper, we use I, k � k,and Ef�g to denote an identity matrix, a vector two-norm, and a statistical expectation operator, respec-tively. Also, let i ¼

ffiffiffiffiffiffiffi�1p

and ð�Þ�, ð�ÞT, and ð�ÞH

denote the complex conjugate, transpose, andHermitian operator, respectively.

2. Proposed low-complexity code acquisition

Consider that there are K users in a mobile celland each user is given an aperiodic pseudo-noise(PN) code sequence with a period much longer thana symbol period. The transmitted signal of the kthuser in baseband can be expressed as

xkðtÞ ¼X1

j¼�1

dkðjÞXU�1l¼0

ck;jðlÞpðt� lT c � jUT cÞ, (1)

ARTICLE IN PRESS

+ +

Constrained LMS

[ ( )]q Ht nw 1( )tx n qM−

[ ( )]q Hs nw

( )nr

Find peak-weightlocation

go to code tracking loop

advance code-phase by chipstMqZ ζ≥

No

Yes

+-

Fig. 1. System diagram of the proposed system, where

Zq ¼ kwqs ðNÞk

2, q ¼ 0; . . . ;Q� 1.

H.-L. Yang, W.-R. Wu / Signal Processing 88 (2008) 1191–1202 1193

k ¼ 1; . . . ;K , where dkðjÞ is the jth BPSK symbol ofthe kth user, ck;jðlÞ the lth chip of the spreadingsignal for dkðjÞ, pðtÞ a unit-amplitude rectangularchip-pulse with a chip-duration Tc, and U thenumber of chips in a symbol. At the receiver, auniformly linear array with M sensors is placed andthe element spacing is assumed to be half awavelength of the carrier. Then, the chip-ratesampled received signal vector in baseband can berepresented as

rðnÞ ¼XK

k¼1

akakxkðn� tkÞ expð�iykÞ þ gðnÞ, (2)

where code delays tk, k ¼ 1; . . . ;K are assumedto be integers between ½0;UÞ, and gðnÞ is anM � 1, complex, and zero-mean Gaussian noisevector with a covariance matrix s2ZI. Also, ak, ak,and yk stand for the steering vector, the ampli-tude, and the carrier-phase offset, associatedwith the kth user, respectively. Note that yk isuniformly distributed over ½�p; pÞ and ak isgiven by ak ¼ ½1; expð�ip sinfkÞ; . . . ; expð�ipðM�1ÞsinfkÞ�

T, where fk denotes the direction-of-arrival(DoA) of the kth user’s signal. Without loss ofgenerality, the first user is seen as the desired user.We also assume that d1ðjÞ ¼ 1 during the acquisi-tion period.

As described, the whole delay uncertainty U isdivided into cells. Let Q ¼ dU=M te, where Mt

denotes the filter length of the temporal filter.Among these Q cells, the actual code delay only fallsinto the delay region of a certain cell. Let the cellwhose delay region includes the desired code delaybe the inphase cell and others be outphase cells.Thus, we have one inphase cell and Q� 1 outphasecells.

Fig. 1 illustrates the block diagram of theproposed system. As seen, the spatial filter wq

s

combines M array outputs into a single output,where q ¼ 0; . . . ;Q� 1 denotes the cell index. Thetemporal filter w

qt uses x1ðn� qM tÞ as its input

signal and the spatial filter output as its referencesignal, where x1ðnÞ is the desired user’s PN sequence[since d1ðjÞ ¼ 1]. As far as an inphase cell isconcerned, the system is the same as that in [16].From Fig. 1, we can see that the spatial filter can actlike a beamformer to reject interference, while thetemporal filter can be a code-delay estimator. Inother words, the optimum temporal filter will have aunique peak weight whose location corresponds tothe code delay [16]. However, for the outphase cells,

there is no correlation between the input and thereference signals. The optimum spatial filter willbecome a signal blocker (all weights are zeros).Using the characteristic, we propose to perform celldetection with the magnitude of the spatial filterweights. If kwq0

s k2 exceeds a preset threshold, then

the q0th cell is considered as the inphase cell. Oncethe inphase cell is identified, the peak weight in

wq0

t can be located. Let the peak weight be wq0

t;D,

where wq0

t;Dwith 0pDoMt denotes the (Dþ 1)th

element of wq0

t . Then, the code delay can be

estimated with t1 ¼ q0M t þ D.As shown, the difference between these two-filter

outputs forms the error signal from which we canperform minimization. The cost function to mini-mize is the same as that in [16]. For each cell, we let

Jq ¼ Efj½wqt �HxqðnÞ � ½wq

s �HrðnÞj2g, (3)

q ¼ 0; . . . ;Q� 1, where wqs9½w

qs;o; . . . ;w

qs;M�1�

T,

wqt9½w

qt;0; . . . ;w

qt;M t�1

�T, and xqðnÞ9½x1ðn� qM tÞ;

x1ðn� qM t � 1Þ; . . . ;x1ðn� qM t �Mtþ 1Þ�T. From

(3), it is simple to observe that a minimum Jq (which

is zero) occurs at wqt ¼ 0 and wq

s ¼ 0, and this is an

undesired trivial solution. To avoid that, we have tomake a constraint on the solution. Here, we pose aunit-norm constraint, i.e.,

kwqt k

29½wqt �Hw

qt ¼ 1; q ¼ 0; . . . ;Q� 1. (4)

Thus, minimization of (3) turns out to be aconstrained optimization problem. We use theLagrange multiplier method [17] to transform theconstrained optimization problem into an uncon-strained one. From (3) and (4), we have an

ARTICLE IN PRESSH.-L. Yang, W.-R. Wu / Signal Processing 88 (2008) 1191–12021194

equivalent cost function as

Jq ¼ Efj½wqt �HxqðnÞ � ½wq

s �HrðnÞj2g þ xq

f1� ½wqt �Hw

qt g

¼ ½wqt �HRq

xwqt þ ½w

qt �HKqwq

s þ ½wqs �H½Kq�Hw

qt

þ ½wqs �HRrw

qs þ xq

f1� ½wqt �Hw

qt g, ð5Þ

where

KqðM t�MÞ9� EfxqðnÞrHðnÞg, (6)

RrðM�MÞ9EfrðnÞrHðnÞg,

RqxðMt�M tÞ

9EfxqðnÞ½xqðnÞ�Hg,

and xq denotes the Lagrange multiplier for the qthcell. Differentiating (5) with respect to ½wq

s �� and

½wqt �� and setting the results to be zero-vectors, we

obtain

qJq

q½wqs �� ¼ ½K

q�Hwqt þ Rrw

qs ¼ 0, (7)

qJq

q½wqt �� ¼ Rq

xwqt þ Kqwq

s � xqwqt ¼ 0. (8)

Since Rr is a full rank matrix, its matrix inversionexists. From (7), we have

wqs ¼ �R

�1r ½K

q�Hwqt . (9)

Substituting (9) into (8), we have

fRqx � KqR�1r ½K

q�Hgwqt � xqw

qt ¼ 0. (10)

It is simple to observe that the solution of xq in(10) denotes the eigenvalue of Rq

x � KqR�1r ½Kq�H,

while wqt is the corresponding eigenvector. Note that

an eigenvector wqt satisfies (4) automatically. Once

wqt is derived, w

qs can be found using (9). Multiplying

(10) with ½wqt �H, we obtain

xq¼ ½w

qt �HfRq

x � KqR�1r ½Kq�Hgw

qt . (11)

Substituting (9) into (5) and using (11), we have

Jq ¼ ½wqt �HRq

xwqt � ½w

qt �HfKqR�1r ½K

q�Hgwqt

� ½wqt �HfKqR�1r ½K

q�Hgwqt

þ ½wqt �HfKqR�1r ½K

q�Hgwqt

¼ ½wqt �HfRq

x � KqR�1r ½Kq�Hgw

qt

¼ xq, ð12Þ

which is identical to (11) exactly. Let solutions to(7)–(8), which are optimum weights, be wq

s;o and wqt;o,

and the corresponding minimum value of (12) beJ

qmin. We then conclude that J

qmin is equal to the

minimum eigenvalue Rqx � KqR�1r ½K

q�H and wqt;o is

the corresponding eigenvector. Substituting wqt;o into

(9), we can then obtain wqs;o.

To simplify notations, we rewrite (2) as

rðnÞ ¼XK

k¼1

akakxkðn� tkÞ expð�iykÞ þ gðnÞ ð13Þ

¼ a expð�iyÞxðn� nM t � DÞ

þXK

k¼2

akakxkðn� tkÞ expð�iykÞ þ gðnÞ, ð14Þ

where we let a1 ¼ a, a1 ¼ 1, y1 ¼ y, x1ðnÞ ¼ xðnÞ,and t1 ¼ nM t þ D. It is simple to see that theinphase cell is the cell that q ¼ n.

We first consider the scenario of the inphase cell.As mentioned, the proposed system is just the sameas that in [16]. From [16], we can have

xnmin ¼ Jnmin ¼ 1� aHR�1r a, (15)

wnt;o ¼ ½0; . . . ; 0|fflfflfflffl{zfflfflfflffl}

D

; 1; 0; . . . ; 0�T expðicÞ, (16)

wns;o ¼ R�1r a expð�i½y� c�Þ, (17)

where c is an arbitrary angle. From (16) and (17),we can see that both filters do not have uniquesolutions. This is not surprising since we only posethe magnitude constraint. Also, note that wn

s;o is justthe conventional MMSE beamformer (R�1r a).

Now, let us consider the remaining Q� 1 out-phase cells. Since qan, we have Kq ¼ 0 [see (6)].Then, (5) becomes

Jq ¼ ½wqs �HRrw

qs þ ½w

qt �HRq

xwqt þ xq

f1� ½wqt �Hw

qt g,

(18)

qan, where Rqx ¼ I (the long-code assumption).

Also, (7)–(8) become

qJq

q½wqs �� ¼ Rrw

qs ¼ 0, (19)

qJq

q½wqt �� ¼ w

qt � xqw

qt ¼ 0. (20)

From (19), we have wqs;o ¼ 0, since Rr is a full-

rank matrix. The spatial filter will block all signalfrom entering the temporal filter. From (20), wecan see xq

min ¼ Jqmin ¼ 1, and there is no unique

solution for wqt;o either. Any vector satisfies the unit-

norm constraint can serve as an optimum solution.

3. Adaptive implementation and convergence analysis

In Section 2, we have proposed a low-complexitycode acquisition system modifying the system in

ARTICLE IN PRESSH.-L. Yang, W.-R. Wu / Signal Processing 88 (2008) 1191–1202 1195

[16]. Optimum weights of the filters are derived withthe eigen-decomposition technique. However, therequired computational complexity of eigen-decom-position is on the order of OðM3

t Þ. In addition, thematrix inversion of Rr is required in (10). Toalleviate these problems, we propose to use anadaptive algorithm to derive the optimum filterweights. The adaptive algorithm we consider is theLMS algorithm which is well known for itssimplicity and robustness. As shown, we have aunit-norm constraint on the temporal filter. Apply-ing this constraint, we then obtain a constrainedLMS algorithm. In what follows, we will describethe algorithm and examine related issues such as thestep size bound and steady-state mean-squarederror (MSE). Besides, we also analyze the outputSINR of beamformer [see gðnÞ in Fig. 1] for aninphase cell.

3.1. Constrained LMS and convergence issue

Rewriting (3), we have

JqðnÞ ¼ ½wqvðnÞ�

HRqvw

qvðnÞ; q ¼ 0; . . . ;Q� 1, (21)

where

wqvðnÞ9½½w

qt ðnÞ�

T; ½wqs ðnÞ�

T�T,

vqðnÞ9½½xqðnÞ�T;�rTðnÞ�T,

and

Rqv9EfvqðnÞ½vqðnÞ�Hg.

The gradient of (21) is by

qJqðnÞ

q½wqvðnÞ�

� ¼ Rqvw

qvðnÞ. (22)

Using (22), we can apply a gradient decentalgorithm to obtain the optimum solution, denotedas wq

v;o. However, Rqv needs to be estimated. The

simplest estimate of Rqv is to use instantaneous value

from vqðnÞ½vqðnÞ�H and this yields a stochasticgradient decent algorithm, called the LMS algo-rithm [17]. We then can have the filter adaptation as

wqvðnþ 1Þ ¼ wq

vðnÞ þ mf�vqðnÞ½vqðnÞ�HwqvðnÞg, (23)

where m is the step size controlling the convergencerate. Recall that we have the constraint kw

qt ðnÞk ¼ 1.

This constraint can be easily satisfied if normal-ization is performed on w

qt ðnÞ at every iteration. The

overall adaptation procedure is given as

eqðnÞ ¼ ½wqvðnÞ�

HvqðnÞ, (24)

HqðnÞ ¼ diag1

kwqt ðnÞk

; . . . ;1

kwqt ðnÞk|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

M t

; 1; . . . ; 1|fflfflfflffl{zfflfflfflffl}M

8>>><>>>:

9>>>=>>>;,

(25)

wqvðnþ 1Þ ¼ HqðnÞwq

vðnÞ � mvqðnÞ½eqðnÞ��, (26)

n ¼ 0; 1; . . . ;N � 1; q ¼ 0; . . . ;Q� 1, where diagf�gdenotes a diagonal matrix consisting of the argu-ments that it includes, and N the iteration numberfor each cell. As we can see, HqðnÞ normalizes w

qt ðnÞ

at every iteration. After training, we have to detectthe inphase cell first. To do that, we propose tocompare kwq

s ðNÞk2 with a preset threshold. If

kwqs ðNÞk

2 is larger than the threshold, the cell isdeemed as the inphase cell. Then, the peak locationof w

qt ðNÞ is located and the code delay is estimated.

Otherwise, we go to the next cell and start theprocess all over again. To guarantee convergence, mhas to be selected properly. Here, we perform themean convergence analysis to derive a step sizebound. Subtracting wq

v;o ¼ ½½wqt;o�

T; ½wqs;o�

T�T fromboth sides of (26), we have

Dwqvðnþ 1Þ ¼ Dwq

vðnÞ þ ½HqðnÞ � I�wq

vðnÞ

� mvqðnÞf½wqvðnÞ�

HvqðnÞg�

¼ DwqvðnÞ þ ½H

qðnÞ � I�wqvðnÞ

� mvqðnÞ½vqðnÞ�H½DwqvðnÞ þ wq

v;o�

¼ fI� mvqðnÞ½vqðnÞ�HgDwqvðnÞ

þ ½HqðnÞ � I�wqvðnÞ � mvqðnÞ½eq

oðnÞ��,

ð27Þ

where eqoðnÞ9½w

qv;o�

HvqðnÞ and DwqvðnÞ9wq

vðnÞ � wqv;o:

Taking the statistical expectation of (27), applyingthe direct-averaging method [17], and using theorthogonality principle, we then have

EfDwqvðnþ 1Þg ¼ ½I� mRq

v �EfDwqvðnÞg

þ ½EfHqðnÞg � I�EfwqvðnÞg. ð28Þ

Let Kq¼ diagflq

v;1; . . . ; lqv;M tþMg with lq

v;j being aneigenvalue of Rq

v , and Uq be a matrix consisting ofthe eigenvectors of Rq

v . Multiplying (28) with ½Uq�H

and letting gqðnÞ ¼ ½Uq�HEfDwqvðnÞg, we obtain

gqðnþ 1Þ ¼ ½I� mKq�gqðnÞ

þ ½Uq�H½EfHqðnÞg � I�EfwqvðnÞg. ð29Þ

Since wqt ðnÞ is normalized at every iteration and the

step size is usually small, it is reasonable to assumethat HqðnÞ � I and the second term in the right-hand

ARTICLE IN PRESSH.-L. Yang, W.-R. Wu / Signal Processing 88 (2008) 1191–12021196

side of (29) can be ignored. Iterating (29), we obtain

gqðnÞ ¼ ½I� mKq�ngqð0Þ. (30)

Thus, for (29) to converge, the following conditionmust be satisfied:

0omo2

lqv;max

, (31)

where lqv;max denotes the maximum eigenvalue of Rq

v .This result is the same as that of the conventionalLMS algorithm [17]. From (30), we can also see thatgqð1Þ ¼ 0. In other words, Efwq

vðnÞg ¼ wqv;o, when

n!1.Note that while the conventional LMS algorithm

requires 2ðMt þMÞ multiplications per iteration,the constrained LMS algorithm developed hereneeds extra M t multiplications for calculation ofkw

qt ðnÞk and extra Mt divisions for normalization

[see (26)].

3.2. Steady-state MSE analysis

We now derive the steady-state MSE of theconstrained LMS algorithm. Invoking the direct-averaging method [17] and using (27), we canwrite the correlation matrix of the tap-weight errorvector as

Pqðnþ 1Þ9EfDwqvðnþ 1Þ½Dwq

vðnþ 1Þ�Hg

¼ ½I� mRqv �P

qðnÞ½I� mRqv � þ m2J

qminR

qv

þ Ef½HqðnÞ � I�wqvðnÞ

�½wqvðnÞ�

H½HqðnÞ � I�g. ð32Þ

As stated, wqt ðnÞ is normalized at every iteration and

the step size is usually small. Thus, HqðnÞ � I andthe last term in the right-hand side of (32) can beignored. Let P

qðnÞ9½Uq�HPqðnÞUq and observe that

½Uq�HRqvU

q ¼ Kq. Pre-multiplying and post-multi-plying both sides of (32) with ½Uq�H and Uq,respectively, we have

Pqðnþ 1Þ ¼ ½I� mKq�PqðnÞ½I� mKq

� þ m2JqminK

q.

(33)

Let the jth element on the diagonal of PqðnÞ be p

qj ðnÞ.

Then,

pqj ðnþ 1Þ ¼ ð1� mlq

v;jÞ2p

qj ðnÞ þ m2Jq

minlqv;j , (34)

j ¼ 1; . . . ;M t þM. When n!1, pqj ðnþ 1Þ �

pqj ðnÞ. From (34), we derive

pqj ð1Þ ¼

mJqmin

2� mlqv;j

. (35)

The additional MSE due to the use of the LMSalgorithm is generally referred to as the excess MSE,denoted as Jq

exð1Þ. From [17], we then have

Jqexð1Þ ¼

XM tþM

j¼1

pqj ð1Þl

qv;j ¼ J

qmin

XM tþM

j¼1

mlqv;j

2� mlqv;j

.

(36)

Denote the steady-state MSE of the LMS adapta-tion as Jq

ss. Finally, we have

Jqss ¼ J

qmin þ Jq

exð1Þ. (37)

3.3. Output SINR at beamformer for an inphase cell

Now, let us analyze the output SINR of thebeamformer. We consider the inphase cell (q ¼ n),and the output is by

gðnÞ9½wnsðnÞ�

HrðnÞ ð38Þ

¼ ½wnsðnÞ�

Hða expð�iyÞxðn� nM t � DÞ þ zðnÞÞ,

ð39Þ

where zðnÞ consists of MAI and noise. Using (17),we can find the output SINR of the optimumbeamformer, denoted as SINRo, as

SINRo ¼½wn

s;o�HRAw

ns;o

½wns;o�

HRzwns;o

ð40Þ

¼aHR�1r RAR

�1r a

aHR�1r RzR�1r a

, ð41Þ

where RA ¼ aaH and Rz9EfzðnÞzHðnÞg. Since weuse adaptive filter-weights to approximate theoptimum weights, we have to include the excessMSE in the SINR calculation. Thus, we can rewrite(41) as

SINRo ¼aHR�1r RAR

�1r a

aHR�1r RzR�1r aþ Jn

exð1Þ, (42)

where Jnexð1Þ is from (36).

4. Performance analysis

The performance of acquisition is generallymeasured with the mean acquisition time, which isthe averaged time for correct acquisition. The meanacquisition time of the proposed system is afunction of the probability of false alarm, theprobability of missing (denoted as PM), and theprobability of correct acquisition (denoted as PD).In this section, we will first derive these probabilitiesand then calculate the mean acquisition time.

ARTICLE IN PRESSH.-L. Yang, W.-R. Wu / Signal Processing 88 (2008) 1191–1202 1197

4.1. Mean acquisition time

As mentioned, the proposed scheme performssequential cell testing. Since there are Q possiblecells, there are Q possible states in the system. Labelthese states as fs0; . . . ; sQ�1g in the circular statediagram [1], as shown in Fig. 2. In the figure, thestate labeled as ACQ indicates the state of correctacquisition. That labeled as FA is the state of falsealarm. Using this diagram, we can evaluate theaveraged time reaching the ACQ state, i.e., the meanacquisition time. Without loss of generality, weassume sQ�1 being the state of an inphase cell, andthus it is connected to the ACQ state. Also, letZq9kwq

s ðNÞk2. As described in Section 2, the

optimum wqs;o, for outphase cells are all-zero vectors,

and the corresponding Zq should be small. On theother hand, Zq of the inphase cell should be large.Using this property, we set a threshold z for thedetection of the inphase cell. Thus, the acquisitionproblem can be seen as a hypothesis testing problem.Note that correct acquisition means that the inphasecell is correctly detected and at the same time theoptimum peak-weight location ðDÞ is also correctlyestimated. There are two types of false alarm. Wename the false alarm occurring in an inphase cell asan inphase false alarm, which means that the inphasecell is correctly detected but the peak location is not(i.e., DaD), and the false alarm occurring in anoutphase cell as an outphase false alarm.

From Fig. 2, we can see that the transfer function(TF) between sQ�1 and ACQ can be expressed asHbðzÞ ¼ PDz

Nþ1 [1,8,9], where N þ 1 denotes thetime for iteration and cell detection, and z the unit-delay operator. The probability of missing, PM, isdefined as the probability of ZQ�1oz. Thus, if only

FA

ACQ

1Qs −

2Qs −0s

1s

A : Ha (z) = PM zN+1

B : Hb (z) = PD zN+1

C : Hc (z) = PFo zN+1

F : Hf (z) = PFi zN+1

E : He (z) = (1 – PFo) zN+1D : Hd (z) = zTp

AB

CC

D

DD

E

E

F

Fig. 2. Circular state diagram of the proposed system.

the missing is considered, the TF between sQ�1 ands0 can be expressed as HaðzÞ ¼ PMzNþ1. Theprobability of the inphase false alarm, denoted asPFi, is equal to PFi ¼ 1� PD � PM. On the otherhand, if only the inphase false alarm is considered,the TF between sQ�1 and FA can be expressedHf ðzÞ ¼ PFiz

Nþ1. Note the system has to stay Tp

chips once it enters the FA state. The quantity Tp isgenerally referred to as the penalty time [1]. The TFbetween the input and the output of the FA statecan be described as HdðzÞ ¼ zTp . Thus, we can havethe TF between sQ�1 and s0 as HgðzÞ9HaðzÞþ

Hf ðzÞHdðzÞ. The TF between any sq, q ¼ 0; 1; . . . ;Q� 2, and the FA state will be HcðzÞ ¼ PFoz

Nþ1,where PFo is the probability of outphase false alarm.

If the outphase false alarm between two con-secutive states, sq and sqþ1, q ¼ 0; 1; . . . ;Q� 2, isnot considered, then the TF between these twoconsecutive states can be described as HeðzÞ ¼

ð1� PFoÞzNþ1. Thus, we can have the TF between

any two consecutive states, sq and sqþ1, q ¼ 0; 1; . . . ;Q� 2, as HhðzÞ9HeðzÞ þHcðzÞHdðzÞ.

Using the TFs derived above, we now can redrawthe diagram in Fig. 2 as that in Fig. 3. In whatfollows, we use Fig. 3 to calculate the meanacquisition time. We define the probability ofcorrect acquisition starting from time zero andending at time n as PACQðnÞ. Then, its z-transform isgiven by

PACQðzÞ ¼X1n¼0

PACQðnÞzn, (43)

which can be the generating function of acquisitiontime. Denote the mean acquisition time as Tacq andit can be derived from [1]

Tacq9d

dzPACQðzÞjz¼1. (44)

Note that the unit of (44) is chip. Assuming that wecan start searching at any state in fs0; . . . ; sQ�1g with

ACQ

SQ−1S0

Hb (z)

Hg (z)Hh (z)

Hh (z)

SQ−2

Q−2

Fig. 3. Simplified state diagram.

ARTICLE IN PRESSH.-L. Yang, W.-R. Wu / Signal Processing 88 (2008) 1191–12021198

equal probability 1=Q, we rewrite (43) as

PACQðzÞ ¼1

Q

XQ�1q¼0

Pq;ACQðzÞ ð45Þ

¼1

QHbðzÞ

XQ�1q¼0

Pq;Q�1ðzÞ, ð46Þ

where Pq;ACQðzÞ denotes the TF between sq andACQ states, and Pq;Q�1ðzÞ the TF between sq andsQ�1. Using Fig. 3, we can have

Pq;Q�1ðzÞ ¼H

Q�1�qh ðzÞ

1�HgðzÞHQ�1h ðzÞ

. (47)

Substituting (47) into (46), we obtain

PACQðzÞ ¼1

Q

HbðzÞ

1�HgðzÞHQ�1h ðzÞ

XQ�1q¼0

HQ�1�qh ðzÞ

ð48Þ

¼1

Q

HbðzÞ½1�HQh ðzÞ�

½1�HgðzÞHQ�1h ðzÞ�½1�HhðzÞ�

. ð49Þ

Using (49) in (44), we finally obtain

Tacq ¼1

PD1þ ðQ� 1Þ

2� PD

2

� �ðN þ 1Þ

�

þ PFi þ ðQ� 1ÞPFo2� PD

2

� �Tp

�. ð50Þ

Observing (50), we find that a large PFo and a smallPD can enlarge Tacq significantly. It should be notedthat PFo is more harmful to Tacq than PFi. This isbecause there are Q� 1 outphase cells and only oneinphase cell. For an ideal situation that PD ¼ 1 andPFi ¼ PFo ¼ 0, we have

Tacq;LB ¼Qþ 1

2ðN þ 1Þ, (51)

which can serve as the lower bound of (50).

4.2. Probabilities derivation

Since Zq is random, we have to characterize itsstatistical properties. It is mentioned in [18] thatwhen an adaptive filter approaches the steady state,its weights have a Gaussian distribution. From theanalysis in the previous section, we see that in thesteady state (N !1), wq

s ðNÞ has a mean vector ofwqs;o and w

qt ðNÞ has a mean vector w

qt;o. We denote

their covariance matrices as Cqs9Ef½wq

s ðNÞ � wqs;o�

½wqs ðNÞ � wq

s;o�Hg and C

qt9Ef½w

qt ðNÞ � w

qt;o�½w

qt ðNÞ�

wqt;o�

Hg, respectively. As a common practice, the step

size is usually small. Thus, we can use the Taylorexpansion to expand 1=ð2� mlq

v;jÞ in (35) withrespect to mlq

v;j ¼ 0. Then, we can derive

1

2� mlqv;j

¼1

2þ

1

4mlq

v;j þ � � � . (52)

From (33), it can be seen that the matrix PqðnÞ will

become diagonal as n!1. Using this propertyand truncating the terms higher than the first-orderin (52), we then have

PqðnÞ ¼

m2

JqminIþ

m2

4J

qminK

q. (53)

Pre-multiplying and post-multiplying both sides of(53) with Uq and ½Uq�H, we obtain

PqðnÞ ¼m2

JqminIþ

m2

4J

qminR

qv . (54)

Note that the M t �Mt upper-left submatrix ofPqðnÞ corresponds to C

qt and the M �M lower-right

submatrix of that can be Cqs . Thus, we can write

Cqs ¼ J

qmin

m2Iþ

m2

4Rr

� ��

mJqmin

2I, (55)

Cqt ¼ J

qmin

m2þ

m2

4

� �I �

mJqmin

2I, (56)

where Jqmin is the MMSE evaluated in Section 2. For

notational clarity, we let s219mJnmin=2 and

s209mJqmin=2, qan. From (55)–(56), we can see that

these filter weights are approximately independentand identically distributed (i.i.d.).

Let us calculate PD now. Since wqs;o ¼ 0 for qan,

we find that Zq, qan is chi-square distributed withM degrees of freedom, while Zn is noncentral chi-square distributed with M degrees of freedom.Thus, the probability of PFo is given by

PFo ¼

Z 1z

1

sM0 2M=2GðM=2Þ

bM=2�1 exp �b2s20

� db,

(57)

where Gð�Þ stands for the gamma function [19], and zis usually selected on some level to prevent a largePFo (e.g. PFo ¼ 0:01). Let PC be the probability ofcorrect inphase cell detection. Then,

PC9Z 1z

1

2s21

bs2

� ðM�2Þ=4exp �

s2 þ b2s21

� IM=2�1

�ffiffiffib

p s

s21

� db, ð58Þ

ARTICLE IN PRESS

0 500 1000 1500 2000

0

0.5

1.0

1.5

Iteration

MS

E

Experimental, q = νTheoretical, q = νExperimental, q ≠ ν

Theoretical, q ≠ ν

Fig. 4. Convergence curve for MSE when m ¼ 3� 10�3 and

M t ¼ 8. Theoretical value is from (37).

0 500 1000 1500 2000−12

−10

−8

−6

−4

−2

0

2

4

6

SIN

R d

B

ExperimentalTheoretical

Iteration

Fig. 5. Convergence curve for SINR of gðnÞ at an inphase cell,

m ¼ 3� 10�3 and M t ¼ 8. Theoretical value is from (42).

H.-L. Yang, W.-R. Wu / Signal Processing 88 (2008) 1191–1202 1199

where s29kwns;ok

2 and IM=2�1ð�Þ the ðM=2� 1Þthorder modified Bessel function of the first kind [19].Next, we evaluate the probability of D ¼ D, say PD.Let Y j ¼ jw

nt;jðNÞj

2 for j ¼ 0; . . . ;Mt � 1. When N islarge enough, YD has a noncentral chi-squaredistribution with two degrees of freedom, whereasY j , jaD, has a chi-square distribution. Thecorresponding probability density functions can beshown as

pYDðyÞ ¼

1

2s21exp �

jwnt;o;Dj

2 þ y

2s21

!I0

ffiffiffiyp jwn

t;o;Dj

s21

� ,

(59)

pY jðyÞ ¼

1

2s21exp �

y

2s21

� ; y40; jaD, (60)

where wnt;o;D denotes the ðDþ 1Þth element in wn

t;o

with jwnt;o;Dj

2 ¼ 1. With (59)–(60), PD is given by

PD ¼ PrðY joYDÞ ð61Þ

¼

Z 10

Z b

0

pY jðb0Þdb0

� �M t�1

pYDðbÞdb; jaD,

ð62Þ

where the i.i.d. property has been applied in (62).Finally, we can have PD ¼ PCPD, PM ¼ 1� PC,PFi ¼ 1� PD � PM. Then, (50) can be evaluated.

5. Simulation results

To demonstrate the effectiveness of the proposedsystem, we report some simulation results in thissection. First, we set common parameters used insimulations as follows: U ¼ 256, M ¼ 8, K ¼ 8,

Tp ¼ 100U chips, s2Z ¼ 1, m ¼ 3� 10�3, wqs ð0Þ ¼ 0,

and wqt ð0Þ ¼ ð1=

ffiffiffiffiffiffiffiMt

pÞ½1; . . . ; 1�T for q¼ 0; . . . ;Q�1.

Also, for convenience, the DoAs are fixed to be

ffkgKk¼1¼ f0:41; 0:56; 0:78;�0:20;�0:52;�1:12; 1:12;

0:94g (radians) in all simulations.In the first set of simulations, we examine the

convergence behaviors of the proposed adaptivesystem. This includes the MSE convergence ofthe system, the SINR convergence behavior of thespatial filter, and the weight convergence of thetemporal filter. All experimental results are derivedfrom an average of 400 trials. In these experiments,we let M t ¼ 8, the array input SINR be �10 dB(a ¼ 1), and the powers of jammers be equal. Fig. 4shows the MSE convergence curve for the proposedsystem with the inphase cell (q ¼ n). It can be seenthat the steady-state MSE value approaches to the

theoretical value 0.28 around n ¼ 1300. The theore-tical MSE value is calculated from (37). In the samefigure, we also show the MSE with an outphase cell(qan). It is apparent that the experimental MSE ismore fluctuating. This is because that J

qmin, qan, is

much greater than Jnmin making the corresponding

excess MSE larger. Fig. 5 illustrates the SINRconvergence curve for the beamformer output [seegðnÞ in (38)]. The SINR starts from �11 dB andeventually reaches the optimum value 4.18 dB. Thetheoretical value is derived from (42) and is shownwith the horizontal line in the figure. We omit theresults for qan, in which the experimental SINR isaround �10 dB. This indicates that the spatial filtercannot suppress interference for outphase cells.From the figure, we conclude that the adaptive

ARTICLE IN PRESS

200 400 600 800 10001200140016001800200010−4

10−3

10−2

10−1

100

Mag

nitu

de−s

quar

ed ta

p−w

eigh

ts

Iteration

Experimental, |wt, Δ(n)|2

Theoretical, |wt, Δ(n)|2

Experimental, |wt, j(n)|2, j ≠ Δ

Fig. 7. Convergence curve for squared temporal filter weights

jwnt;jðnÞj

2 for j ¼ 0; . . . ;M t � 1 and M t ¼ 8. Theoretical value is

obtained from (16).

0

0.05

0.10

0.15

0.20

0.25P

Fo

Experimental, Mt = 16, N = 2000Experimental, Mt =16, N = 1000Experimental, Mt =8, N = 2000Experimental, Mt = 8, N = 1000Theoretical

H.-L. Yang, W.-R. Wu / Signal Processing 88 (2008) 1191–12021200

spatial filter can effectively suppress interferencewhen it operates in the inphase cell. Fig. 6 presentsseveral experimental beam patterns calculated fromwnsðNÞ. Here, we let N ¼ 2000. The optimum beam

pattern, derived from (17), is also shown. Note thatthe arrow signs indicate the signal DoAs and onlythe DoA of the desired user, f1, is labeled. As seen,the spatial filter, acting as a beamformer, cansteer the main beam to the incident direction f1

and put nullities in the directions of interference.The convergence behavior of the temporal filterweights is shown in Fig. 7. We can see that thetap weight whose indices correspond to the codedelay, jwn

t;DðnÞj2, converges to unity, while other

weights, jwnt;jðnÞj

2, jaD, converge to a very smallvalue (only one weight is shown in the figure).

In Figs. 8 and 9, we can see that the theoreticalresults calculated using derived formulas all matchthe simulated ones very well. Then, we calculate themean acquisition time of the proposed system.Before that, we have to evaluate related probabil-ities. Fig. 8 shows the comparison of experimentaland theoretical PFo (versus z). Here, the array inputSINR is set as �10 dB. As we can see, PFo decreasesrapidly as the threshold increases. The experimentalresults with M t ¼ 8 match the theoretical results [in(57)] better than those with M t ¼ 16. We also seethat the experimental results for N ¼ 1000 and N ¼

2000 are close. Fig. 9 shows the similar comparisonfor PC. Here, experiment and theoretical resultsagree very well for N ¼ 2000. However, they agreepoorly for N ¼ 1000. This is because the spatial

−1.5 −1.0 0 1.0 1.510−3

10−2

10−1

100

Bea

m−p

atte

rn

ExperimentalTheoretical

φ1

−0.5DoA in radian

0.5

Fig. 6. Experimental and theoretical beam patterns for an

inphase cell (N ¼ 2000, M t ¼ 8, and M ¼ 8). Arrow signs

indicate the DoAs associated with all users. The labeled one,

f1, is the DoA of the desired user.

0.02 0.04 0.06 0.08 0.12 0.140.10

Threshold ζ

Fig. 8. Experimental and theoretical probabilities of outphase

false alarm PFo versus threshold z.

filter has not converged with the given number ofiterations, and Zn tends to be smaller than thethreshold. This behavior is different from that inPFo calculation. From Figs. 8 and 9, we can see thatthe best z is around 0.07. Using this value, PFo canbe close to 0 and PC to 1. The theoretical value ofPD is usually very close to 1. With 104 trials, we findPD ¼ 1 (N ¼ 2000, M t ¼ 8 or Mt ¼ 16). As a result,we can let PC � PD. Since the interference is mainlysuppressed by the spatial filter, PFi is close to 0.Substituting derived experimental probabilities into(50), we can then calculate the mean acquisitiontime. Fig. 10 shows the result. In the figure, lowerbounds derived from (51), are also shown. It issimple to see that if z is too small, PFo will be large,

ARTICLE IN PRESS

0.02 0.04 0.06 0.08 0.12 0.140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.0

Threshold ζ

PC

Experimental, Mt = 16, N = 2000Experimental, Mt = 16, N = 1000Experimental, Mt = 8, N = 2000Experimental, Mt = 8, N = 1000Theoretical

0.9

0.10

Fig. 9. Experimental and theoretical probabilities of PC versus

threshold z.

0.02 0.04 0.06 0.08 0.10 0.12 0.14

104

105

106

Threshold ζ

Mea

n ac

quis

ition

tim

e

Mt = 16, N = 2000Mt = 16, N = 2000, Lower boundMt = 8, N = 2000Mt = 8, N = 2000, Lower boundMt = 8, N = 1000Mt = 8, N = 1000, Lower bound

Fig. 10. Experimental mean acquisition time (chips) versus z.

103

104

105

106

SINR (dB)

Mea

n ac

quis

ition

tim

e

Proposed, Mt = 16 Proposed, Mt = 8 System in [16]

Fig. 11. Mean acquisition time comparison for U ¼ 256.

H.-L. Yang, W.-R. Wu / Signal Processing 88 (2008) 1191–1202 1201

leading to large mean acquisition time. On thecontrary, if z is too large, PM being equal to 1� PC

will be large, leading to large mean acquisition timealso. From the figure, we can observe that z can bechosen in a wide range of value such that meanacquisition times can approach lower bounds.

Finally, we conduct performance comparison forthe correlator-based scheme in [13], the adaptivearray system in [16], and the proposed system. Welet U ¼ 256, s2Z ¼ 1, a ¼ 1, and the powers of alljammers be equal. As addressed in [13], the derivedtheoretical threshold is not accurate enough toguarantee that a designated probability of falsealarm (set as 0.01 here) can be achieved. Thus, weexperimentally search for the threshold, processing

period (for adaptation), and step size that gives theminimal mean acquisition time (for each array inputSINR). To ensure a fair comparison, we also searchfor an optimum set fm;N; zg that provides theoptimum performance for the proposed system(Mt ¼ 8 or 16). Here, PFo is set as 0:01. Similarly,for the system in [16], the performance is optimizedover fm;Ng. Fig. 11 shows the performance compar-ison for these systems in various SINRs. From thefigure, we first can see that the correlator-basedsystem has the worst performance. This is becausethe beamformer training cannot be accomplished ina short processing period, especially in serious MAIenvironments. The system in [16] exhibits the bestperformance. It can outperform the correlator-based system by an order of magnitude. Comparingto that in [16], the proposed system somewhatcompromises the performance. However, its com-putational complexity is much lower. For example,with Mt ¼ 8, the temporal filter size is just 1

32of that

in [16]. We also can see that for the proposed systemwith M t ¼ 8 performs slightly worse than that withM t ¼ 16. We can expect that the larger the M t, thesmaller the performance loss. Thus, we can have aneasy tradeoff between performance and computa-tional complexity.

6. Conclusions

In this paper, we proposed a low-complexityadaptive array code acquisition scheme, especiallybeing suited to large-delay channel environments.Applying the serial-search technique, we cangreatly reduce the temporal filter size, so does the

ARTICLE IN PRESSH.-L. Yang, W.-R. Wu / Signal Processing 88 (2008) 1191–12021202

computational complexity. The proposed schemealso allows an easy tradeoff between performanceand computational complexity. With the specialdesigned structure, the proposed system is able tosuppress MAI and estimate code delay simulta-neously. It can outperform the conventionalcorrelator-based system. We also analyze the con-vergence behavior and the mean acquisition time ofthe proposed scheme, and derive related closed-form expressions. Simulations verify that theoreticaland experimental results agree well. In this paper,we only consider the flat-fading yet integer chip-delay channels. With minor modifications, theproposed system can be easily extended to multi-path yet fractional chip-delay channels [16]. Thisissue may serve as a topic for further research.

References

[1] A. Polydoros, C. Weber, A unified approach to serial search

spread-spectrum code acquisition—part I and II, IEEE

Trans. Commun. 32 (May 1984) 542–560.

[2] A.J. Viterbi, Principle of Spread Spectrum Communications,

Addison-Wesley, New York, 1995.

[3] U. Madhow, M.B. Pursley, Acquisition in direct-sequence

spread-spectrum communication networks: an asymptotic ana-

lysis, IEEE Trans. Inform. Theory 39 (3) (May 1993) 903–912.

[4] T.K. Moon, R.T. Short, C.K. Rushforth, Average acquisi-

tion time for SSMA channels, in: IEEE Military Commu-

nication Conference, 1991, pp. 1042–1046.

[5] D. Zheng, J. Li, S.L. Miller, E.G. Strom, An efficient code-

timing estimator for DS-CDMA signals, IEEE Trans. Signal

Process. 45 (January 1997) 82–89.

[6] E.G. Strom, et al., Propagation delay estimation in

asynchronous direct-sequence code-sequence multiple access

systems, IEEE Trans. Commun. 44 (January 1996) 84–93.

[7] M.G. El-Tarhuni, A.U. Sheikh, An adaptive filtering PN

code acquisition scheme with improved acquisition based

capacity in DS/CDMA, in: 9th IEEE International Sympo-

sium on Personal, Indoor, and Mobile Radio Communica-

tions, vol. 3, 1998, pp. 1486–1490.

[8] M.G. El-Tarhuni, A.U. Sheikh, Adaptive synchronization

for spread spectrum systems, in: IEEE Vehicular Technology

Conference, vol. 1, April 1996, pp. 170–174.

[9] M.G. El-Tarhuni, Application of adaptive filtering to direct-

sequence spread-spectrum code synchronization, Ph.D.

Dissertation, Department of System and Computer Engi-

neering, Carleton University, Canada, 1997.

[10] R.F. Smith, S.L. Miller, Acquisition performance of an

adaptive receiver for DS-CDMA, IEEE Trans. Commun. 47

(9) (September 1999).

[11] H.L. Yang, W.R. Wu, Multirate adaptive filtering for DS/

CDMA code acquisition, in: IEEE International Symposium

on Signal Processing and Information Technology, Decem-

ber 2003, pp. 363–366.

[12] Y. Zhang, L. Zhang, G. Liao, PN code acquisition and

beamforming weight acquisition for DS-CDMA systems

with adaptive array, in: 14th IEEE International Symposium

on Personal, Indoor, and Mobile Radio Communications,

vol. 2, 2003, pp. 1385–1389.

[13] B. Wang, H.M. Kwon, PN code acquisition using smart

antenna for spread-spectrum wireless communications—

part I, IEEE Trans. Veh. Technol. 52 (1) (January 2003)

142–149.

[14] S. Kim, Approximate maximum likelihood approach for

code acquisition in DS-CDMA systems with multiple

antennas, IEICE Trans. Commun. E 88-B (3) (March

2005) 1054–1065.

[15] H.L. Yang, W.R. Wu, A novel adaptive code acquisition

using antenna array for DS/CDMA systems, in: IEEE

International Workshop on Antenna Technology: Small

Antennas and Novel Metamaterials, March 2005.

[16] H.L. Yang, W.R. Wu, A novel adaptive antenna array for

DS/CDMA code acquisition, IEEE Trans. Signal Process.

55 (9) (2007) 4567–4580.

[17] S. Haykin, Adaptive Filter Theory, 3rd ed., Prentice-Hall,

New Jersey, 1996.

[18] N.J. Bershad, L.Z. Qu, On the probability density function

of the LMS adaptive filter weights, IEEE Trans. Acoust.

Speech Signal Process. 37 (1) (January 1989) 43–56.

[19] J.G. Proakis, Digital Communications, 4th ed., McGraw-

Hill, New York, 2000.