A low-complexity adaptive antenna array code acquisition

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Signal Processing 88 (2008) 1191–1202

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.

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

1. Introduction

Direct-sequence/code-division multiple access (DS/CDMA) is a promising technique for wireless mobile communication. It is well known that the main performance bottleneck for a CDMA system is multiple access interference (MAI). MAI not only affects data detection, but also code acquisition. Code acquisition is the ﬁrst operation that a CDMA receiver has to perform. It attempts to align the local code sequence and the received desired signal with a difference less than a chip duration. After

successful code acquisition, other operations such as channel estimate, code tracking, and data detection can follow. Thus, code acquisition is a critical task in DS/CDMA systems.

Code acquisition for single antenna systems has been extensively studied in the literature[1–11](and reference therein). The correlator approach[1–4] is most well known. However, the correlator is only optimal for the single-user case. Its performance degrades signiﬁcantly when MAI presents[4]. Sub-space- and matrix-based methods [5,6] have been developed to solve the problem. The advantage of subspace-based approaches is that no training sequences are required and the performance is much better than that of the correlator approach. However, these methods usually have to estimate,

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decompose, or inverse the autocorrelation matrix. This often requires high computational complexity, especially for systems with large processing gains. Recently, adaptive-filtering technique has been introduced to code acquisition[7–11]. It is claimed that[7], the adaptive-filtering approach can provide higher acquisition-based capacity [3] compared to the correlator methods. The acquisition-based capacity indicates the maximal number of users that a system can tolerate with a given acquisition error rate. One problem with the adaptive-filtering scheme [7] is that its computational complexity grows linearly along with the code-delay uncertainty range. A low-complexity scheme was then devel-oped in[11].

It is well known that antenna arrays can significantly enhance acquisition performance [12–14]. In [12], each array element is equipped with a correlator, and the correlator outputs serve the inputs of a beamformer. When directional MAI presents, matrix inversion is required to derive the beamformer weights [12]. In [13], an adaptive beamformer is used to avoid the problem. However, in the presence of directional MAI, it requires a long adaptation period and the acquisition is slow. To solve the problem, an adaptive-filter-based array system was then proposed [15,16]. This system simultaneously performs adaptive beamforming and code-delay estimation with a spatial and a temporal filter. The code delay can then be estimated with the peak position of the temporal filter. It has been shown that the approach significantly outperforms the correlator-based sys-tem in[13].

Similar to the conventional adaptive-filtering acquisition, the computational complexity of the temporal filter in[15] grows linearly along with the uncertain range of the code delay. When the range is large, the computational complexity may be high. In this paper, we propose a low-complexity adaptive array code acquisition scheme to solve the problem. The main idea is to divide the whole delay uncertainty range into several (delay) cells, and then sequentially search for the code delay of the desired user among those cells. This is essentially a serial-search-technique, being able to shorten the filter length of the temporal filter. As a result, the computational complexity can be reduced. As that in[15], the proposed system employs a criterion such that both filters can be simultaneously adjusted by a constrained least-mean-square (LMS) algorithm. However, the acquisition process is more involved

than that in [15]. This is because one additional decision has to be made before the code delay can be estimated. For each tested cell, filters are first adapted for a period time to determine if the code delay falls into the cell’s delay region or not. If it does, 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 code delay can then be estimated with the peak position of the temporal filter. If not, the next cell is tested and the process is repeated. Note that if the code delay does not fall into the tested cell’s region, the spatial filter will act as a signal blocker with its weighs all being zeros. This property is then used to derive an index for the cell testing. With the choice of the number of cells, we can have an easy tradeoff between performance and complexity. In many cases, however, the complexity reduction is large, but the performance loss is still acceptable. In this paper, we only consider flat-fading channels for simplicity. It is straightforward to extend the proposed algorithm to multipath environments. Also note that with the proposed architecture, we can also apply other types of adaptive algorithms such as the recursive least-squares (RLS) algorithm. In this case, the overall performance can be greatly enhanced, but the computational complexity is also significantly increased.

This paper is organized as follows. Section 2 develops the proposed algorithm. Section 3 discusses issues of adaptive implementation, and Section 4 carries out performance analysis. Section 5 presents simulation results demonstrating the effectiveness of the proposed scheme. Finally, Section 6 draws conclusions. Throughout this paper, we use I, k k, and Efg to denote an identity matrix, a vector two-norm, and a statistical expectation operator, respec-tively. Also, let i ¼pffiffiffiffiffiffiffi1 and ðÞ, ðÞT, and ðÞH denote the complex conjugate, transpose, and Hermitian operator, respectively.

2. Proposed low-complexity code acquisition Consider that there are K users in a mobile cell and each user is given an aperiodic pseudo-noise (PN) code sequence with a period much longer than a symbol period. The transmitted signal of the kth user in baseband can be expressed as

xkðtÞ ¼ X1 j¼1 dkðjÞ X U 1 l¼0 ck;jðlÞpðt lTcjUTcÞ, (1)

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k ¼ 1; . . . ; K, where dkðjÞ is the jth BPSK symbol of

the kth user, ck;jðlÞ the lth chip of the spreading

signal for dkðjÞ, pðtÞ a unit-amplitude rectangular

chip-pulse with a chip-duration Tc, and U the

number of chips in a symbol. At the receiver, a uniformly linear array with M sensors is placed and the element spacing is assumed to be half a wavelength of the carrier. Then, the chip-rate sampled received signal vector in baseband can be represented as

rðnÞ ¼X

K

k¼1

akakxkðn tkÞexpðiykÞ þgðnÞ, (2)

where code delays tk, k ¼ 1;. . . ; K are assumed

to be integers between ½0; U Þ, and gðnÞ is an M 1, complex, and zero-mean Gaussian noise vector with a covariance matrix s2

ZI. Also, ak, ak,

and yk stand for the steering vector, the

ampli-tude, and the carrier-phase offset, associated with the kth user, respectively. Note that yk is

uniformly distributed over ½p; pÞ and ak is

given by ak¼ ½1; expðip sin fkÞ; . . . ; expðipðM 1Þ

sin f_kÞT, where f_k denotes the direction-of-arrival (DoA) of the kth user’s signal. Without loss of generality, 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 is divided into cells. Let Q ¼ dU =Mte, where Mt

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

Fig. 1 illustrates the block diagram of the proposed 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. The temporal ﬁlter wqt uses x1ðn qMtÞ as its input

signal and the spatial ﬁlter output as its reference signal, where x1ðnÞ is the desired user’s PN sequence

[since d1ðjÞ ¼ 1]. As far as an inphase cell is

concerned, the system is the same as that in [16]. FromFig. 1, we can see that the spatial filter can act like a beamformer to reject interference, while the temporal filter can be a code-delay estimator. In other words, the optimum temporal filter will have a unique peak weight whose location corresponds to the code delay[16]. However, for the outphase cells,

there is no correlation between the input and the reference signals. The optimum spatial ﬁlter will become a signal blocker (all weights are zeros). Using the characteristic, we propose to perform cell detection with the magnitude of the spatial ﬁlter weights. If kwq0

sk2 exceeds a preset threshold, then

the q0_{th cell is considered as the inphase cell. Once}

the inphase cell is identiﬁed, the peak weight in wq_t0 can be located. Let the peak weight be wq0

t; ^D,

where wq0

t; ^D with 0p ^DoMt denotes the ( ^D þ 1)th

element of wq_t0. Then, the code delay can be estimated with ^t1 ¼q0Mtþ ^D.

As shown, the difference between these two-ﬁlter outputs forms the error signal from which we can perform minimization. The cost function to mini-mize is the same as that in[16]. For each cell, we let

¯Jq _¼_Efj½wq tHxqðnÞ ½wqsHrðnÞj2g, (3) q ¼ 0; . . . ; Q 1, where wq_s9½wq_s;o; . . . ; wq_s;M1T, wq_t9½wq_t;0; . . . ; wq_t;Mt1T, and xq_ðnÞ9½x 1ðn qMtÞ; x1ðn qMt1Þ;. . . ; x1ðn qMtMtþ1ÞT. From

(3), it is simple to observe that a minimum ¯Jq(which is zero) occurs at wq_t ¼0 and wq_s ¼0, and this is an undesired trivial solution. To avoid that, we have to make a constraint on the solution. Here, we pose a unit-norm constraint, i.e.,

kwq_tk29½wq_tHwq_t ¼1; q ¼ 0; . . . ; Q 1. (4) Thus, minimization of (3) turns out to be a constrained optimization problem. We use the Lagrange multiplier method [17] to transform the constrained optimization problem into an uncon-strained one. From (3) and (4), we have an

+ +

Constrained LMS [ q( )]H t n w x n qM1( − t) [ q( )]H s n w ( )n r Find peak-weight

location _{tracking loop} go to code advance code-phase by chipsM_t q Z ≥ζ No Yes +

-Fig. 1. System diagram of the proposed system, where Zq_{¼ kw}q

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equivalent cost function as Jq¼Efj½wq_tHxqðnÞ ½wq_sHrðnÞj2g þxqf1 ½wq_tHwq_tg ¼ ½wq_tHRq_xwq_t þ ½wq_tHKqwq_sþ ½wq_sH½KqHwq_t þ ½wq_sHRrwqsþx q_{f1 ½w}q tHw q tg, ð5Þ where Kq_ðMtMÞ9 EfxqðnÞrHðnÞg, (6) RrðMMÞ9EfrðnÞrHðnÞg, Rq_xðMt_Mt_Þ9EfxqðnÞ½xqðnÞHg,

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

s

and ½wq_t and setting the results to be zero-vectors, we obtain qJq q½wqs ¼ ½KqHwq_t þRrwqs ¼0, (7) qJq q½wqt ¼Rq_xwq_t þKqwq_sxqwq_t ¼0. (8) Since Rr is a full rank matrix, its matrix inversion

exists. From (7), we have

wq_s ¼ R1_r ½KqHwq_t. (9) Substituting (9) into (8), we have

fRq_xKqR1_r ½KqHgwq_t xqwq_t ¼0. (10) It is simple to observe that the solution of xq in (10) denotes the eigenvalue of Rq_xKqR1_r ½KqH, while wqt is the corresponding eigenvector. Note that

an eigenvector wqt satisﬁes (4) automatically. Once

wq_t is derived, wq_s can be found using (9). Multiplying (10) with ½wqtH, we obtain

xq¼ ½wq_tHfRq_xKqR1_r ½KqHgwq_t. (11) Substituting (9) into (5) and using (11), we have Jq¼ ½wq_tHRq_xwq_t ½w_tqHfKqR1_r ½KqHgwq_t

½wq_tHfKqR1_r ½KqHgwq_t þ ½wq_tHfKqR1_r ½KqHgwq_t ¼ ½wq_tHfRq_xKqR1_r ½KqHgwq_t

¼xq, ð12Þ

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

s;oand w q t;o,

and the corresponding minimum value of (12) be Jq_min. We then conclude that Jq_min is equal to the minimum eigenvalue Rq_xKqR_r1½KqH and wq_t;o is the corresponding eigenvector. Substituting wq_t;ointo (9), we can then obtain wq

s;o.

To simplify notations, we rewrite (2) as rðnÞ ¼ X K k¼1 akakxkðn tkÞexpðiykÞ þgðnÞ ð13Þ ¼a expðiyÞxðn nMtDÞ þX K k¼2 akakxkðn tkÞexpðiykÞ þgðnÞ, ð14Þ where we let a1¼a, a1¼1, y1¼y, x1ðnÞ ¼ xðnÞ,

and t1¼nMtþD. It is simple to see that the

inphase 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 same as that in [16]. From[16], we can have

xn_min¼Jn min ¼1 aHR1r a, (15) wn_t;o¼ ½0;. . . ; 0 |fflfflfflffl{zfflfflfflffl} D ; 1; 0; . . . ; 0TexpðicÞ, (16) wn_s;o ¼R1_r a expði½y cÞ, (17) where c is an arbitrary angle. From (16) and (17), we can see that both filters do not have unique solutions. This is not surprising since we only pose the magnitude constraint. Also, note that wn

s;ois just

the conventional MMSE beamformer (R1_r a). Now, let us consider the remaining Q 1 out-phase cells. Since qan, we have Kq_¼_{0 [see (6)].}

Then, (5) becomes Jq¼ ½wq_sHRrwqsþ ½w q tHRqxw q t þxqf1 ½w q tHw q tg, (18) qan, where Rq

x¼I (the long-code assumption).

Also, (7)–(8) become qJq q½wqs ¼Rrwqs ¼0, (19) qJq q½wqt ¼wq_t xqwq_t ¼0. (20) From (19), we have wq

s;o¼0, since Rr is a

full-rank matrix. The spatial filter will block all signal from entering the temporal filter. From (20), we can see xq_min ¼Jq_min¼1, and there is no unique solution for wqt;oeither. 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-complexity code acquisition system modifying the system in

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[16]. Optimum weights of the ﬁlters are derived with the eigen-decomposition technique. However, the required computational complexity of eigen-decom-position is on the order of OðM3_tÞ. In addition, the matrix inversion of Rr is required in (10). To

alleviate these problems, we propose to use an adaptive algorithm to derive the optimum ﬁlter weights. The adaptive algorithm we consider is the LMS algorithm which is well known for its simplicity and robustness. As shown, we have a unit-norm constraint on the temporal ﬁlter. Apply-ing this constraint, we then obtain a constrained LMS algorithm. In what follows, we will describe the algorithm and examine related issues such as the step size bound and steady-state mean-squared error (MSE). Besides, we also analyze the output SINR of beamformer [see gðnÞ in Fig. 1] for an inphase cell.

3.1. Constrained LMS and convergence issue Rewriting (3), we have ¯Jq_{ðnÞ ¼ ½w}q vðnÞHRqvwqvðnÞ; q ¼ 0; . . . ; Q 1, (21) where wq_vðnÞ9½½wq_tðnÞT; ½wq_sðnÞTT, vqðnÞ9½½xqðnÞT; rTðnÞT, and Rq_v9EfvqðnÞ½vqðnÞHg. The gradient of (21) is by q ¯JqðnÞ q½wqvðnÞ ¼Rq_vwq_vðnÞ. (22)

Using (22), we can apply a gradient decent algorithm to obtain the optimum solution, denoted as wq

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

simplest estimate of Rq_vis to use instantaneous value from vqðnÞ½vqðnÞH and this yields a stochastic gradient decent algorithm, called the LMS algo-rithm[17]. We then can have the ﬁlter adaptation as wq_vðn þ 1Þ ¼ wq_vðnÞ þ mfvqðnÞ½vqðnÞHwq_vðnÞg, (23) where m is the step size controlling the convergence rate. Recall that we have the constraint kwqtðnÞk ¼ 1.

This constraint can be easily satisﬁed if normal-ization is performed on wq_tðnÞ at every iteration. The overall adaptation procedure is given as

eqðnÞ ¼ ½wq_vðnÞHvqðnÞ, (24) HqðnÞ ¼ diag 1 kwq_tðnÞk; . . . ; 1 kwq_tðnÞk |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Mt ; 1; . . . ; 1 |fflfflfflffl{zfflfflfflffl} M 8 > > > < > > > : 9 > > > = > > > ; , (25) wq_vðn þ 1Þ ¼ HqðnÞwq_vðnÞ mvqðnÞ½eqðnÞ, (26) n ¼ 0; 1; . . . ; N 1; q ¼ 0; . . . ; Q 1, where diagfg denotes a diagonal matrix consisting of the argu-ments that it includes, and N the iteration number for each cell. As we can see, HqðnÞ normalizes wq_tðnÞ at every iteration. After training, we have to detect the inphase cell first. To do that, we propose to compare kwq

sðNÞk2 with a preset threshold. If

kwq

sðNÞk2 is larger than the threshold, the cell is

deemed as the inphase cell. Then, the peak location of wqtðNÞ is located and the code delay is estimated.

Otherwise, we go to the next cell and start the process all over again. To guarantee convergence, m has to be selected properly. Here, we perform the mean convergence analysis to derive a step size bound. Subtracting wq

v;o¼ ½½w q

t;oT; ½wqs;o

TT _from

both sides of (26), we have

Dwq_vðn þ 1Þ ¼ Dwq_vðnÞ þ ½HqðnÞ Iwq_vðnÞ mvqðnÞf½wq_vðnÞHvqðnÞg ¼Dwq_vðnÞ þ ½HqðnÞ Iwq_vðnÞ mvqðnÞ½vqðnÞH½Dwq_vðnÞ þ wq_v;o ¼ fI mvqðnÞ½vqðnÞHgDwq_vðnÞ þ ½HqðnÞ Iwq_vðnÞ mvqðnÞ½eq_oðnÞ_, ð27Þ where eq oðnÞ9½wqv;o H_vq_{ðnÞ and Dw}q vðnÞ9wqvðnÞ wqv;o:

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

EfDwq_vðn þ 1Þg ¼ ½I mRq_vEfDwq_vðnÞg

þ ½EfHqðnÞg IEfwq_vðnÞg. ð28Þ Let Kq¼diagflq_v;1; . . . ; lq_v;MtþMg with lq_v;j being an eigenvalue of Rq_v, and Uq be a matrix consisting of the eigenvectors of Rq_v. Multiplying (28) with ½UqH and letting gq_{ðnÞ ¼ ½U}qH_EfDwq

vðnÞg, we obtain

gqðn þ 1Þ ¼ ½I mKqgqðnÞ

þ ½UqH½EfHqðnÞg IEfwq_vðnÞg. ð29Þ Since wq_tðnÞ is normalized at every iteration and the step size is usually small, it is reasonable to assume that HqðnÞ I and the second term in the right-hand

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side of (29) can be ignored. Iterating (29), we obtain gqðnÞ ¼ ½I mKqngqð0Þ. (30) Thus, for (29) to converge, the following condition must be satisﬁed:

0omo 2

lq_v;max, (31)

where lq_v;maxdenotes the maximum eigenvalue of Rq_v. This result is the same as that of the conventional LMS algorithm[17]. From (30), we can also see that gqð1Þ ¼0. In other words, Efwq_vðnÞg ¼ wq_v;o, when n ! 1.

Note that while the conventional LMS algorithm requires 2ðMtþMÞ multiplications per iteration,

the constrained LMS algorithm developed here needs extra Mt multiplications for calculation of

kwq_tðnÞk and extra Mt divisions for normalization

[see (26)].

3.2. Steady-state MSE analysis

We now derive the steady-state MSE of the constrained LMS algorithm. Invoking the direct-averaging method [17] and using (27), we can write the correlation matrix of the tap-weight error vector as

Pqðn þ 1Þ9EfDwq_vðn þ 1Þ½Dwq_vðn þ 1ÞHg ¼ ½I mRq_vPqðnÞ½I mRq_v þm2Jq_minRq_v

þEf½HqðnÞ Iwq_vðnÞ

½wq_vðnÞH½HqðnÞ Ig. ð32Þ As stated, wq_tðnÞ is normalized at every iteration and the step size is usually small. Thus, HqðnÞ I and the last term in the right-hand side of (32) can be ignored. Let ¯PqðnÞ9½UqHPqðnÞUq and observe that ½UqHRq_vUq¼Kq. Pre-multiplying and post-multi-plying both sides of (32) with ½UqH and Uq, respectively, we have

¯

Pqðn þ 1Þ ¼ ½I mKq ¯PqðnÞ½I mKq þm2Jq_minKq. (33) Let the jth element on the diagonal of ¯PqðnÞ be ¯pq_jðnÞ. Then, ¯pq_jðn þ 1Þ ¼ ð1 mlq_v;jÞ2¯pq_jðnÞ þ m2_Jq minl q v;j, (34) j ¼ 1; . . . ; MtþM. When n ! 1, ¯pqjðn þ 1Þ ¯pq_jðnÞ. From (34), we derive ¯pq_jð1Þ ¼ mJ q min 2 mlq_v;j. (35)

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

exð1Þ. From [17], we then have

Jq_exð1Þ ¼ X MtþM j¼1 ¯pq_jð1Þlq_v;j¼Jq_min X MtþM j¼1 mlq_v;j 2 mlq_v;j. (36) Denote the steady-state MSE of the LMS adapta-tion as Jq_ss. Finally, we have

Jq_ss¼Jq_minþJq_exð1Þ. (37)

3.3. Output SINR at beamformer for an inphase cell Now, let us analyze the output SINR of the beamformer. We consider the inphase cell (q ¼ n), and the output is by

gðnÞ9½wn_sðnÞHrðnÞ ð38Þ

¼ ½wn_sðnÞHða expðiyÞxðn nMtDÞ þ zðnÞÞ,

ð39Þ where zðnÞ consists of MAI and noise. Using (17), we can ﬁnd the output SINR of the optimum beamformer, denoted as SINRo, as

SINRo¼ ½wn_s;oHRAwns;o ½wn s;o H_R zwns;o ð40Þ ¼ a H_R1 r RAR1r a aH_R1 r RzR1r a , ð41Þ

where RA¼aaH and Rz9EfzðnÞzHðnÞg. Since we

use adaptive ﬁlter-weights to approximate the optimum weights, we have to include the excess MSE in the SINR calculation. Thus, we can rewrite (41) as SINRo¼ aHR1_r RAR1r a aH_R1 r RzR1r a þ Jnexð1Þ , (42) where Jn_exð1Þ is from (36). 4. Performance analysis

The performance of acquisition is generally measured with the mean acquisition time, which is the averaged time for correct acquisition. The mean acquisition time of the proposed system is a function of the probability of false alarm, the probability of missing (denoted as PM), and the

probability of correct acquisition (denoted as PD).

In this section, we will ﬁrst derive these probabilities and then calculate the mean acquisition time.

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4.1. Mean acquisition time

As mentioned, the proposed scheme performs sequential cell testing. Since there are Q possible cells, there are Q possible states in the system. Label these states as fs0; . . . ; sQ1g in the circular state

diagram [1], as shown in Fig. 2. In the ﬁgure, the state labeled as ACQ indicates the state of correct acquisition. That labeled as FA is the state of false alarm. Using this diagram, we can evaluate the averaged time reaching the ACQ state, i.e., the mean acquisition time. Without loss of generality, we assume sQ1 being the state of an inphase cell, and

thus it is connected to the ACQ state. Also, let Zq_9kwq

sðNÞk2. As described in Section 2, the

optimum wq

s;o, for outphase cells are all-zero vectors,

and the corresponding Zq _{should be small. On the}

other hand, Zq of the inphase cell should be large. Using this property, we set a threshold z for the detection of the inphase cell. Thus, the acquisition problem can be seen as a hypothesis testing problem. Note that correct acquisition means that the inphase cell is correctly detected and at the same time the optimum peak-weight location ðDÞ is also correctly estimated. There are two types of false alarm. We name the false alarm occurring in an inphase cell as an inphase false alarm, which means that the inphase cell is correctly detected but the peak location is not (i.e., ^DaD), and the false alarm occurring in an outphase cell as an outphase false alarm.

FromFig. 2, we can see that the transfer function (TF) between sQ1 and ACQ can be expressed as

HbðzÞ ¼ PDzNþ1 [1,8,9], where N þ 1 denotes the

time for iteration and cell detection, and z the unit-delay operator. The probability of missing, PM, is

deﬁned as the probability of ZQ1oz. Thus, if only

the missing is considered, the TF between sQ1 and

s0 can be expressed as HaðzÞ ¼ PMzNþ1. The

probability of the inphase false alarm, denoted as PFi, is equal to PFi¼1 PDPM. On the other

hand, if only the inphase false alarm is considered, the TF between sQ1 and FA can be expressed

HfðzÞ ¼ PFizNþ1. Note the system has to stay Tp

chips once it enters the FA state. The quantity Tpis

generally referred to as the penalty time[1]. The TF between the input and the output of the FA state can be described as HdðzÞ ¼ zTp. Thus, we can have

the TF between sQ1 and s0 as HgðzÞ9HaðzÞþ

HfðzÞHdðzÞ. The TF between any sq, q ¼ 0; 1;. . . ;

Q 2, and the FA state will be HcðzÞ ¼ PFozNþ1,

where PFois 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, is

not considered, then the TF between these two consecutive states can be described as HeðzÞ ¼

ð1 PFoÞzNþ1. Thus, we can have the TF between

any two consecutive states, sqand sqþ1, q ¼ 0; 1;. . . ;

Q 2, as HhðzÞ9HeðzÞ þ HcðzÞHdðzÞ.

Using the TFs derived above, we now can redraw the diagram in Fig. 2 as that in Fig. 3. In what follows, we use Fig. 3 to calculate the mean acquisition time. We deﬁne the probability of correct acquisition starting from time zero and ending at time n as PACQðnÞ. Then, its z-transform is

given by PACQðzÞ ¼

X1 n¼0

PACQðnÞzn, (43)

which can be the generating function of acquisition time. Denote the mean acquisition time as Tacqand

it can be derived from[1] Tacq9

d

dzPACQðzÞjz¼1. (44)

Note that the unit of (44) is chip. Assuming that we can start searching at any state in fs0; . . . ; sQ1gwith

FA ACQ 1 Q s− 2 Q s− s0 1 s 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+1 D : Hd (z) = zT p A B C C D D D E E F

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

ACQ S_Q−1 S₀ Hb (z) Hg (z) Hh (z) Hh (z) S_Q−2 Q−2

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equal probability 1=Q, we rewrite (43) as PACQðzÞ ¼ 1 Q X Q1 q¼0 Pq;ACQðzÞ ð45Þ ¼ 1 QHbðzÞ X Q1 q¼0 Pq;Q1ðzÞ, ð46Þ

where Pq;ACQðzÞ denotes the TF between sq and

ACQ states, and Pq;Q1ðzÞ the TF between sq and

sQ1. UsingFig. 3, we can have

Pq;Q1ðzÞ ¼

HQ1q_h ðzÞ 1 HgðzÞHQ1h ðzÞ

. (47)

Substituting (47) into (46), we obtain PACQðzÞ ¼ 1 Q HbðzÞ 1 HgðzÞHQ1h ðzÞ X Q1 q¼0 HQ1q_h ðzÞ ð48Þ ¼ 1 Q HbðzÞ½1 HQ_hðzÞ ½1 HgðzÞHQ1h ðzÞ½1 HhðzÞ . ð49Þ Using (49) in (44), we ﬁnally obtain

Tacq¼ 1 PD 1 þ ðQ 1Þ2 PD 2 ðN þ 1Þ þ PFiþ ðQ 1ÞPFo 2 PD 2 Tp . ð50Þ

Observing (50), we ﬁnd that a large PFoand a small

PDcan enlarge Tacqsigniﬁcantly. It should be noted

that PFo is more harmful to Tacq than PFi. This is

because there are Q 1 outphase cells and only one inphase cell. For an ideal situation that PD¼1 and

PFi¼PFo¼0, we have

Tacq;LB¼

Q þ 1

2 ðN þ 1Þ, (51)

which can serve as the lower bound of (50). 4.2. Probabilities derivation

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

sðNÞ has a mean vector of

wq_s;o and wq_tðNÞ has a mean vector wq_t;o. We denote their covariance matrices as Cq_s9Ef½wq

sðNÞ wqs;o ½wq sðNÞ wqs;oHg and C q t9Ef½w q tðNÞ w q t;o½w q tðNÞ

wq_t;oHg, respectively. As a common practice, the step

size is usually small. Thus, we can use the Taylor expansion to expand 1=ð2 mlq_v;jÞ in (35) with respect to mlq_v;j¼0. Then, we can derive

1 2 mlq_v;j¼ 1 2þ 1 4ml q v;jþ . (52)

From (33), it can be seen that the matrix ¯PqðnÞ will become diagonal as n ! 1. Using this property and truncating the terms higher than the ﬁrst-order in (52), we then have ¯ PqðnÞ ¼m 2J q minI þ m2 4 J q minK q . (53)

Pre-multiplying and post-multiplying both sides of (53) with Uq _{and ½U}qH_{, we obtain}

PqðnÞ ¼m 2J q minI þ m2 4 J q minR q v. (54)

Note that the MtMt upper-left submatrix of

PqðnÞ corresponds to Cq_t and the M M lower-right submatrix of that can be Cq_s. Thus, we can write Cq_s ¼Jq_min m 2I þ m2 4 Rr mJ q min 2 I, (55) Cq_t ¼Jq_min m 2þ m2 4 I mJ q min 2 I, (56)

where Jq_minis the MMSE evaluated in Section 2. For notational clarity, we let s2

19mJnmin=2 and

s2 09mJ

q

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

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

Let us calculate PDnow. Since wqs;o¼0 for qan,

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

PFo¼ Z 1 z 1 sM 0 2M=2GðM=2Þ bM=21exp b 2s2 0 db, (57) where GðÞ stands for the gamma function[19], and z is usually selected on some level to prevent a large PFo (e.g. PFo¼0:01). Let PC be the probability of

correct inphase cell detection. Then, PC9 Z 1 z 1 2s2 1 b s2 ðM2Þ=4 exp s 2_þ_b 2s2 1 IM=21 p sffiffiffib s2 1 db, ð58Þ

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where s2_9kwn

s;ok2 and IM=21ðÞ the ðM=2 1Þth

order modiﬁed Bessel function of the ﬁrst kind[19]. Next, we evaluate the probability of ^D ¼ D, say PD.

Let Yj¼ jwnt;jðNÞj

2_{for j ¼ 0;}_{. . . ; M}

t1. When N is

large enough, YD has a noncentral chi-square

distribution with two degrees of freedom, whereas Yj, jaD, has a chi-square distribution. The

corresponding probability density functions can be shown as p_YDðyÞ ¼ 1 2s2 1 exp jw n t;o;Dj2þy 2s2 1 ! I0 ffiffiffiy p jwnt;o;Dj s2 1 , (59) p_YjðyÞ ¼ 1 2s2 1 exp y 2s2 1 ; y40; jaD, (60)

where wn_t;o;D denotes the ðD þ 1Þth element in wn_t;o with jwn_t;o;Dj2¼1. With (59)–(60), PD is given by

PD¼ PrðYjoYDÞ ð61Þ ¼ Z 1 0 Z b 0 p_Yjðb0Þdb0 Mt1 p_YDðbÞ db; jaD, ð62Þ where the i.i.d. property has been applied in (62). Finally, we can have PD¼PCPD, PM¼1 PC,

PFi¼1 PDPM. Then, (50) can be evaluated.

5. Simulation results

To demonstrate the effectiveness of the proposed system, we report some simulation results in this section. First, we set common parameters used in simulations as follows: U ¼ 256, M ¼ 8, K ¼ 8, Tp¼100U chips, s2Z ¼1, m ¼ 3 103, wqsð0Þ ¼ 0, and wqtð0Þ ¼ ð1= ffiffiffiffiffiffiffi Mt p Þ½1;. . . ; 1Tfor q ¼ 0;. . . ; Q 1. Also, for convenience, the DoAs are fixed to be ff_kgK_k¼1¼ 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 adaptive system. This includes the MSE convergence of the system, the SINR convergence behavior of the spatial filter, and the weight convergence of the temporal filter. All experimental results are derived from an average of 400 trials. In these experiments, we let Mt¼8, the array input SINR be 10 dB

(a ¼ 1), and the powers of jammers be equal.Fig. 4 shows the MSE convergence curve for the proposed system with the inphase cell (q ¼ n). It can be seen that 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 same figure, we also show the MSE with an outphase cell (qan). It is apparent that the experimental MSE is more fluctuating. This is because that Jq_min, qan, is much greater than Jn_min making the corresponding excess MSE larger. Fig. 5 illustrates the SINR convergence curve for the beamformer output [see gðnÞ in (38)]. The SINR starts from 11 dB and eventually reaches the optimum value 4.18 dB. The theoretical value is derived from (42) and is shown with the horizontal line in the figure. We omit the results for qan, in which the experimental SINR is around 10 dB. This indicates that the spatial filter cannot suppress interference for outphase cells. From the figure, we conclude that the adaptive

0 500 1000 1500 2000 0 0.5 1.0 1.5 Iteration MSE Experimental, q = ν Theoretical, q = ν Experimental, q ≠ ν Theoretical, q ≠ ν

Fig. 4. Convergence curve for MSE when m ¼ 3 103 _and

Mt¼8. Theoretical value is from (37).

0 500 1000 1500 2000 −12 −10 −8 −6 −4 −2 0 2 4 6 SINR dB Experimental Theoretical Iteration

Fig. 5. Convergence curve for SINR of gðnÞ at an inphase cell, m ¼ 3 103_{and M}

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spatial filter can effectively suppress interference when it operates in the inphase cell.Fig. 6presents several experimental beam patterns calculated from wn_sðNÞ. Here, we let N ¼ 2000. The optimum beam pattern, derived from (17), is also shown. Note that the arrow signs indicate the signal DoAs and only the DoA of the desired user, f₁, is labeled. As seen, the spatial filter, acting as a beamformer, can steer the main beam to the incident direction f₁ and put nullities in the directions of interference. The convergence behavior of the temporal filter weights is shown in Fig. 7. We can see that the tap weight whose indices correspond to the code delay, jwn

t;DðnÞj2, converges to unity, while other

weights, jwn

t;jðnÞj2, jaD, converge to a very small

value (only one weight is shown in the ﬁgure). InFigs. 8 and 9, we can see that the theoretical results calculated using derived formulas all match the simulated ones very well. Then, we calculate the mean acquisition time of the proposed system. Before that, we have to evaluate related probabil-ities. Fig. 8shows the comparison of experimental and theoretical PFo(versus z). Here, the array input

SINR is set as 10 dB. As we can see, PFodecreases

rapidly as the threshold increases. The experimental results with Mt¼8 match the theoretical results [in

(57)] better than those with Mt¼16. We also see

that the experimental results for N ¼ 1000 and N ¼ 2000 are close.Fig. 9shows the similar comparison for PC. Here, experiment and theoretical results

agree very well for N ¼ 2000. However, they agree poorly for N ¼ 1000. This is because the spatial

ﬁlter has not converged with the given number of iterations, and Zn tends to be smaller than the threshold. This behavior is different from that in PFocalculation. FromFigs. 8 and 9, we can see that

the best z is around 0.07. Using this value, PFo can

be close to 0 and PCto 1. The theoretical value of

PDis usually very close to 1. With 104trials, we ﬁnd

PD¼1 (N ¼ 2000, Mt ¼8 or Mt¼16). As a result,

we can let PCPD. Since the interference is mainly

suppressed by the spatial ﬁlter, PFi is close to 0.

Substituting derived experimental probabilities into (50), we can then calculate the mean acquisition time. Fig. 10 shows the result. In the ﬁgure, lower bounds derived from (51), are also shown. It is simple to see that if z is too small, PFowill be large,

−1.5 −1.0 0 1.0 1.5 10−3 10−2 10−1 100 Beam−pattern Experimental Theoretical φ₁ −0.5 DoA in radian 0.5

Fig. 6. Experimental and theoretical beam patterns for an inphase cell (N ¼ 2000, Mt¼8, and M ¼ 8). Arrow signs

indicate the DoAs associated with all users. The labeled one, f₁, is the DoA of the desired user.

200 400 600 800 1000 1200 1400 1600 1800 2000 10−4 10−3 10−2 10−1 100 Magnitude−squared tap−weights Iteration Experimental, |w_{t, Δ}(n)|2 Theoretical, |w_t,_Δ(n)|2 Experimental, |w_{t, j}(n)|2, j ≠ Δ

Fig. 7. Convergence curve for squared temporal ﬁlter weights jwn

t;jðnÞj2 for j ¼ 0;. . . ; Mt1 and Mt¼8. Theoretical value is

obtained from (16). 0.02 0.04 0.06 0.08 0.12 0.14 0 0.05 0.10 0.15 0.20 0.25 PFo Experimental, M_t = 16, N = 2000 Experimental, Mt =16, N = 1000 Experimental, Mt =8, N = 2000 Experimental, Mt = 8, N = 1000 Theoretical 0.10 Threshold ζ

Fig. 8. Experimental and theoretical probabilities of outphase false alarm PFoversus threshold z.

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

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

Finally, we conduct performance comparison for the correlator-based scheme in [13], the adaptive array system in [16], and the proposed system. We let U ¼ 256, s2

Z¼1, a ¼ 1, and the powers of all

jammers be equal. As addressed in[13], the derived theoretical threshold is not accurate enough to guarantee that a designated probability of false alarm (set as 0.01 here) can be achieved. Thus, we experimentally search for the threshold, processing

period (for adaptation), and step size that gives the minimal mean acquisition time (for each array input SINR). To ensure a fair comparison, we also search for an optimum set fm; N; zg that provides the optimum performance for the proposed system (Mt¼8 or 16). Here, PFo is set as 0:01. Similarly,

for the system in[16], the performance is optimized over fm; Ng.Fig. 11shows the performance compar-ison for these systems in various SINRs. From the figure, we first can see that the correlator-based system has the worst performance. This is because the beamformer training cannot be accomplished in a short processing period, especially in serious MAI environments. The system in [16] exhibits the best performance. It can outperform the correlator-based system by an order of magnitude. Comparing to that in [16], the proposed system somewhat compromises the performance. However, its com-putational complexity is much lower. For example, with Mt¼8, the temporal filter size is just₃₂1 of that

in[16]. We also can see that for the proposed system with Mt ¼8 performs slightly worse than that with

Mt¼16. We can expect that the larger the Mt, the

smaller the performance loss. Thus, we can have an easy tradeoff between performance and computa-tional complexity.

6. Conclusions

In this paper, we proposed a low-complexity adaptive array code acquisition scheme, especially being suited to large-delay channel environments. Applying the serial-search technique, we can greatly reduce the temporal ﬁlter size, so does the

0.02 0.04 0.06 0.08 0.12 0.14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 Threshold _ζ P C Experimental, M_t = 16, N = 2000 Experimental, Mt = 16, N = 1000 Experimental, M_t = 8, N = 2000 Experimental, M_t = 8, N = 1000 Theoretical 0.9 0.10

Fig. 9. Experimental and theoretical probabilities of PCversus

threshold z. 0.02 0.04 0.06 0.08 0.10 0.12 0.14 104 105 106 Threshold _ζ

Mean acquisition time

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

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

103 104 105 106

SINR (dB)

Mean acquisition time

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

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computational complexity. The proposed scheme also allows an easy tradeoff between performance and computational complexity. With the special designed structure, the proposed system is able to suppress MAI and estimate code delay simulta-neously. It can outperform the conventional correlator-based system. We also analyze the con-vergence behavior and the mean acquisition time of the proposed scheme, and derive related closed-form expressions. Simulations verify that theoretical and experimental results agree well. In this paper, we only consider the ﬂat-fading yet integer chip-delay channels. With minor modiﬁcations, the proposed system can be easily extended to multi-path yet fractional chip-delay channels [16]. This issue 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. Stro¨m, An efﬁcient code-timing estimator for DS-CDMA signals, IEEE Trans. Signal Process. 45 (January 1997) 82–89.

[6] E.G. Stro¨m, 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.,