Variable dwell-time code acquisition for direct-sequence spread ...

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 6, JUNE 2000

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Variable Dwell-Time Code Acquisition for Direct-Sequence Spread-Spectrum Systems on Time-Variant Rayleigh Fading Channels Huan-Chun Wang and Wern-Ho Sheen, Member, IEEE

Abstract—Variable dwell-time code acquisition based on multiple-dwell or sequential linear tests is investigated for direct-sequence spread-spectrum systems on time-variant Rayleigh fading channels. Not like in the conventional additive white Gaussian noise channels, the channel memory incurred by fading renders the exact analysis of the acquisition systems extremely difficult, if not impossible. In this paper, a novel method is developed to evaluate the mean acquisition time of the acquisition systems very accurately. The effects of Rayleigh fading are evaluated, and comparisons are made between double-dwell and sequential linear tests. Numerical results show that Rayleigh fading may result in 1–4-dB loss in performance, and the sequential linear test can outperform double-dwell test by a margin of 1–2 dB. The analytical results are verified by computer simulations. Index Terms—Direct-sequence spread spectrum, Rayleigh fading, variable dwell-time acquisition.

I. INTRODUCTION APID pseudonoise (PN) code acquisition is one of the most challenging tasks in the design of a direct-sequence (DS) spread-spectrum receiver [1], [2]. In the past 20 years, a vast volume of research has been conducted to devise effective acquisition systems for both the conventional additive white Gaussian noise (AWGN) and time-variant fading channels [1]–[10]. Code acquisition methods can possibly be divided into different classes according to the types of searching strategies (serial, parallel, or hybrid), correlators (passive or active), test methods (fixed or variable dwell time) and detectors (coherent or noncoherent). Variable dwell-time tests can be further classified as multiple-dwell or sequential tests.1 Generally speaking [1]–[10], parallel searching outperforms serial searching, passive correlation outperforms active correlation, all at the expense of a larger system complexity, and the methods with variable dwell-time tests outperform those with fixed dwell

R

Paper approved by D. P. Taylor, the Editor for Signal Design, Modulation and Detection of the IEEE Communications Society. Manuscript received June 30, 1997; received October 25, 1999. This paper was presented in part at the International Communications Conference (ICC’98), Atlanta, GA, June 1998. H.-C. Wang was with the Department of Electrical Engineering, National Chung Cheng University, Chia-Yi, 621 Taiwan. He is now with the Advanced Technology Center, Computer and Communications Laboratories, Industrial Technology Research Institute, Hsinchu 310, Taiwan, R.O.C. W.-H. Sheen is with the Department of Electrical Engineering, National Chung Cheng University, Chia-Yi 621, Taiwan, R.O.C. (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(00)05397-6. 1In multiple-dwell tests, the overall dwell time is variable, although the dwell time of each dwell is fixed.

time, but may suffer from lacking of an easy way for exact performance analysis [1], [2], [7]. This is especially true for the acquisition method based on the sequential test [7]. In this study, we are concerned with the noncoherent serial search acquisition based on variable dwell-time tests. Variable dwell-time code acquisition with multiple-dwell or sequential tests has been extensively studied for the conventional AWGN channels [1]–[4], [6], [7]. Mean, variance, and/or probability density function (pdf) of the acquisition time have been obtained for multiple-dwell tests [1]–[4], [8]–[10]. For sequential tests, however, exact analysis of the acquisition performance is still not available [11]–[13]. Numerical methods, computer simulations, and/or approximations are often used for the performance evaluation [13], [14]. For time-variant fading channels, due to the inherent channel memory, the performance analysis of the variable dwell-time acquisition systems becomes even more involved. In [15], an upper bound on mean acquisition time was obtained for multiple-dwell tests for both Rayleigh and Rician fading channels. In [7], a noncoherent sequential acquisition based on the noncoherent I/Q detector with continuous integration was investigated for AWGN and Rayleigh fading channels. Computer simulations were used in [7] for performance evaluation due to the difficulty of analysis. In this paper, the noncoherent variable dwell-time code acquisition with multiple-dwell or sequential linear tests is investigated for DS spread-spectrum systems on Rayleigh fading channels. A novel method is developed to evaluate the mean acquisition time of the acquisition system with the fading effect being taken into account. Active correlation is considered exclusively for its implementation simplicity. The rest of the paper is organized as follows. Section II describes the system and channel models. Section III presents the novel method for calculating the mean acquisition time for both multiple-dwell and sequential linear tests. Section IV gives some numerical results, and comparisons are made between the two tests. Finally, Section V gives our conclusions. II. SYSTEM AND CHANNEL MODELS Fig. 1 is the block diagram of the considered noncoherent serial search variable dwell-time code acquisition with active correlation.2 After the square-law detection, the signal is tested to determine if the incoming and locally generated PN codes are 2An

equivalent digital implementation may be employed in practice.

0090–6778/00$10.00 © 2000 IEEE

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Fig. 1. Typical structure for code acquisition systems with a square-law detector.

aligned to within the lock-in range of the code tracking loop. If they do, then a sync hit is declared, and the synchronization goes to code tracking. Otherwise, the local PN code is moved to a new phase for test. Since the phase movement is often done in a discrete step, the phase uncertainty, denoted as , can be divided into cells with a size equal to the adjustment step. Denote as the hypothesis , and the incorthe correct cells in the . Then, the PN code acquisition rect ones as the hypothesis process is a serial of simple binary hypothesis testing. Two types of errors, namely false alarm and missed detection, may occur. is accepted when is False alarm denotes the event that is accepted when true, and missed detection denotes that is true. For presentation simplicity, only one correct cell in the will be considered, although our analysis is applicable to the case with multiple correct cells, as well. In addition, exact align. ment will be assumed under In addition to AWGN, the transmitted signal is also affected by frequency-nonselective Rayleigh fading. Hence, the received is given by3 signal

is the normalized code phase error, is the chip duration, local code, of spreading function defined by

is the code phase of the is the autocorrelation

if otherwise

(4)

and are the baseband uncorrelated for a large , and zero-mean Gaussian processes with the variance of (5) In (2), the code-self noise is safely neglected since in practical applications. After sampling, we have (6) where (7)

(1) is a spreading function with where is the carrier power, is the incoming code phase, is the carrier raperiod and are the in-phase and quadradian frequency, is ture components of the fading channel, respectively, and watts per AWGN with one-sided power spectrum density of and are uncorhertz. For Rayleigh fading channels, related zero-mean Gaussian processes with equal variance . is The channel is assumed to be wide-sense stationary, and and . statistically independent to For the applications of interest, for example, PCS (personal communication services) and land mobile radio, the fading rate of the of the channel is much smaller than the bandwidth (ideal) bandpass filter. The bandwidth will be assumed as , where is the symbol rate. Hence, the output of the bandpass filter is given by

(2) where (3) 3The effects of data modulation and frequency offsets will not be considered, although they can be easily incorporated in the analysis as in [1].

, and under , respectively, where . As is evident, the samples are highly correlated due to the inherent memory of the fading channel, and under are independent, identical distributed (i.i.d.) random variables if . In the squeal, under because under of the assumption of perfect alignment.

under

A. Sequential Linear Test It is well known that for i.i.d. samples, the Wald’s sequential probability ratio test (SPRT) is optimum in the sense that given the probabilities of false alarm and missed detection, the average sample number (ASN) of any test is equal to or larger than that of SPRT [11], [12]. Unfortunately, as of Fig. 1 just shown, for fading channels, the samples , and for correlated samples, are highly correlated under there is no optimum test being established for general pdf yet, although the generalized SPRT (GSPRT) has been shown are dependent to be optimum when the distribution of Gaussian [17], [18]. Further, SPRT and/or GSPRT are often too complex to be implemented for correlated samples since and are required the joint pdf’s of samples both under to form the likelihood ratio [12], [17], [18]. In this part, an easy-to-implement sequential test, called sequential linear test

WANG AND SHEEN: VARIABLE DWELL-TIME CODE ACQUISITION FOR DS SPREAD-SPECTRUM SYSTEMS

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Fig. 2. Sequential linear test.

[19], is investigated. As will be seen, the test neglects the correlations between samples and hence is not an optimum test. Nevertheless, it can provide a significant performance improvement over double-dwell tests as will be illustrated in Section IV. Fig. 2 is the block diagram of a sequential linear test. As seen, the test compares the running sum

(8) , where is the pdf to the two thresholds . If , then no decision will of under if be made, and the sampling continues. Otherwise, accept , and accept if . From (6), the pdf’s of under and are given by

(9) o.w.

Fig. 3. Multiple-dwell test.

B. Multiple-Dwell Test Fig. 3 is a typical structure of the -dwell test, where is the sample number of each dwell. The samples of different dwells are nonoverlapped, and the strategy of immediate rejection is employed for the search and lock controlling. That is, if

and (11) fails to exceed the threshold , then the for some will be accepted only cell will be rejected immediately, and for all . if exceeds the corresponding threshold o.w. (10) respectively. As is evident in (8), the correlations between samples have been neglected, hence it is not an optimum test. Nevertheless, it is a good approximation to the optimum test when the signal-to-noise-ratio (SNR) becomes small and is much easier to implement than SPRT and GSPRT. It is noted that the effects become less promiof correlations between samples under nent for the low SNR cases. In a sequential test, the fundamental question is as follows: does the test reach a decision in finite steps? For SPRT with i.i.d. samples, the answer is “yes” as proved by Wald in [11]. In Appendix A, we show that the sequential linear test of (8) indeed reaches a decision in finite steps with probability one.

III. PERFORMANCE ANALYSIS The performance of PN code acquisition is often characterized in terms of the mean, variance, or pdf of the acquisition time [1]–[4], [8]–[10]. Here, only the mean acquisition time will be used for performance evaluation and comparisons. It was shown that insofar as the data error rate is concerned, the mean acquisition time is the single important parameter to characterize the acquisition performance [4]. A. Mean Acquisition Time The mean acquisition time of an acquisition system is often evaluated by using a transfer function approach based on a Markov chain modeling of the code acquisition process. For active correlation, the Markov chain model is (approximately)

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valid for fading channels [20]. Hence, the expression of mean acquisition time that was obtained for the conventional AWGN channel is also valid for the channel considered here. The expression is given by (for straight line search) [1], [2]

where

(22) (12) is the cell number of the time uncertainty, i.e., is the average dwell time under is the average , and is the detection probability. As dwell time under is needed in order will be seen, the false alarm probability to calculate . Hence, our aim of performance analysis is to oband for the acquisition systems of interest. tain

where

, and

B.

1) Evaluation of

and (25) Note that the value given in the original paper [(30)] [21] is not correct. and are obtained, is calculated by After

of Sequential Linear Test and

(23) (24)

: Define (13)

(26) Using (9) and (10), then

is given by (14)

where (15)

(16) is the signal-to-noise power ratio (SNR), and symbol duration. With this notation, (8) becomes

is the

(17) Under

are i.i.d. variables with the pdf o.w.

(18)

is the average penalty time when a false alarm occurs. where and : Since the samples and 2) Evaluation of hence under are highly correlated, the evaluation of and is much more involved than that of and . Under , where is the ASN. Hence, in the following, and . we are mainly concerned with the evaluation of The first step toward the evaluation is to obtain the joint pdf for . To this aim, the following of definitions are useful. Definition 1 [23]: A symmetric random matrix of dimenis said to have a Wishart distribution with sion degrees of freedom and parameter if can be written as (27) are independently where distributed Gaussian -vectors with zero mean and covariance matrix , and denotes the matrix transpose. The matrix is called a Wishart matrix. Definition 2: Let be the Wishart matrix with the distribu. The joint distribution of the diagonal elements tion of given by (28)

where (19) are i.i.d., and can be Since the random variables obtained exactly by solving Albert’s equations as in [21]. The results are as follows [21]: (20) and

(21)

is called the -variate central chi-square distribution with degrees of freedom and parameter , denoted by . With these definitions, we have the following theorem. covariance matrix (positive Theorem 1: If a definite) can be factorized as (29) and where is a matrix with rank , then the pdf of the distribution is given by

(30)

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where the expectation is taken with respect to the -distributed Wishart matrix

Recall that is the th row of the matrix is an matrix, and is defined as in (31). From (38) and (14), it are independent with the pdf follows that given

o.w. (31)

(39)

degrees of is the noncentral chi-square distribution with , and the freedom and the noncentrality parameter th-order modified Bessel function of the first kind. Proof: Use Definition 1.1, (2.8a), and take the derivative on both sides of (3.3) in [22]. Q.E.D. From Definition 2 and (6), it is easy to see that under is distributed as with the covariance matrix given by

where is defined as in (19). As will be seen, it is this property that enables us to evaluate of conditional independence of and very accurately. and , we define the following: To evaluate (40) (41) and

(32) (42)

where (33) and

is the correlation coefficient of and ], and

[also of

(34)

(43)

) is not lin-

since the events that the true acquisition is achieved at the test is given by of sample are disjoint for different ’s. Further,

o.w. because Note that early dependent. Let

(or

is the probability that the true acquisition is achieved at the test of the th sample, and the probability that the test has not , i.e., been terminated after the test of the th sample under more than samples are required to make the decision. Thereis given by fore, by definition, the detection probability

(35) is a positive semidefinite real then it follows that symmetric matrix and can be factorized as

(44)

(36) where is positive semidefinite befor some matrix . cause it itself is a covariance matrix. Let be the nonnegligible eigenvalues of and thus is approximated as of rank .4 The significance of this approximation will become clear later. With this approximation, then the matrix is given by

and where is the orthonormal eigenvector associated with by Theorem 1, given , the random variable independent with the conditional pdf

approximation would be possible, if 

=

 0; j > m.

is the probability that the test is terminated, i.e., a decision is made, at the test of the first sample, and

(37)

(46)

, and . Hence, are

is the probability that the test is terminated at the test of the th sample. As a result, and can be evaluated to any accuracy, if and are known for all . In Appendix A, it is and converge to zero in the rate bounded by shown that with given in (A-14). Hence, in practice, and can be approximated by truncating the summations in (43) and (44) to the th term which gives the desired accuracy. and due to Unfortunately, it is not esay to evaluate fact that are highly correlated random variables. In the following, based on (38) and (30), a method is developed to eval-

(38) 4The

(45)

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uate and very accurately. The basic idea is as follows. Given a which gives the desired accuracy in (43) and (44), directly as in (40)–(42), instead of evaluating and and , given , is evaluated first, and then and can be obtained with (47)

for (49), (50), and (52). A numerical integration method needs to be used for their calculations. C.

and

of Multiple-Dwell Test

and : Recall that under 1) Evaluation of are independent variables with ples

and

, the sam-

(53) o.w.

(48) Recall that the operation of expectation is done with respect to , the random matrix with the Wishart distribution and is an matrix and is the rank of the matrix . In addition, given are independent with the pdf given in (39). Because of this, the evaluation and will be much easier as will be shown shortly. Note that in this method, most of the computational complexity is due to the fact that all the calculations need to be performed for each outcome of . Therefore, a small will be a very desirable property so as to keep the complexity low. This explains why we to by need to approximate the rank of the matrix just keeping the largest eigenvalues in (37). The value in order to obtain the desired accuracy depends on fading rate and the value , which in turn depends on SNR and thresholds and . Generally speaking, a larger and/or a larger fading rate will and hence increase the complexity of this demand a larger method. In our numerical examples with the considered fading and are good enough. rates and SNR’s, From (17), (39), (41), and (42), we have (49) and (50) is defined as in (51), shown at the bottom of the where page. In addition, it is straightforward to show (52), shown at the bottom of the page. No closed-form expressions are available

Hence, we have [1], [2], [4] (54) and the ASN (55) with

(56) where (57), shown at the bottom of the next page, is the pdf of . After the random variable of (11). In (55), and are obtained, is calculated as (26). and : Under where 2) Evaluation of is the ASN. Hence, our next step is to evaluate and . are correlated, and are given by Since the samples [1], [2], [4] (58)

(51)

(52)

WANG AND SHEEN: VARIABLE DWELL-TIME CODE ACQUISITION FOR DS SPREAD-SPECTRUM SYSTEMS

and and

and

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is the Marcum generalized can be obtained directly by

function. Finally,

(59)

(67) and

respectively. In (58),

(68) (60) Again, the expectation is taken with respect to the -distribution.

and

(61)

IV. NUMERICAL RESULTS

in (38). Then, from (38) and (31), Let , the random variable of (11) are given independent with the conditional pdf [24], shown in (62) at the bottom of page, where

The Jakes’ two-dimensional isotropic scattering channel model [25] is adopted in the following numerical examples. Also, the simulation method given in [25] is employed in all our simulations (with 10 simulation samples). For this model, the correlation function of the in-phase and quadrature-phase fading components is given by

(63)

(69)

with given

, and

. As a result,

(64) and

(65) where

(66)

is the maximum Doppler shift, and is the zewhere roth-order Bessel function of the first kind. Two fading rates, and , have been connamely sidered. For multiple-dwell systems, only the results for the double-dwell test are presented. It has been known that for multiple dwell tests, the most significant improvement is obtained by going from single dwell to double dwell [3], [15]. For simplicity, Monte Carlo integration is used to evaluate the expectarequired in (47), tion over the Wishart distribution (48), (67), and (68), although the other integration algorithms [28]–[30] may be more efficient. It has been found that for being as large as 1000, and for most cases can provide a good result. Also, less than 10 samples are used for Monte Carlo integration to obtain the desired accuracy. Fig. 4 compares the simulation and analytical results for sequential linear test with various threshold values and . The threshold values shown in the figure have been nor. Since and can be obtained exactly malized by as in (20) and (21), respectively, only the comparisons for and are given here. As evident, the simulation and analytical results agree very well for a large range of SNR’s. Comparisons

(57) o.w.

(62) o.w.

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Fig. 4. Comparison of simulation and analytical results for sequential linear test.

Fig. 6. Comparison of sample numbers for sequential linear test, double-dwell test, and fixed sample size test under .

Fig. 5. Comparison of sample numbers for sequential linear test, double-dwell . test, and fixed sample size test under

H

Fig. 7. Example mean acquisition time performance of variable-dwell time acquisition systems.

between simulation and analytical results for double-dwell tests are to be shown in Fig. 7. Figs. 5 and 6 compare the ASN to reach a specific perforand for the fixed sample size (FSS), doublemance of and , dwell (DW), and sequential linear tests (SLT), under , that is (0.9,10 ) respectively. Two typical pairs of , it is shown and (0.65,10 ), are used as examples. Under that double dwell outperforms the fixed sample size by a large pairs, and the sequential margin for both considered linear test outperforms double dwell by a smaller margin. Actually, for the pair (0.9,10 ), the difference of the two tests , as shown in Fig. becomes small for high SNR’s. Under 6, sequential linear test outperforms double dwell by a large pairs, but double dwell margin for both considered

is outperformed by the fixed sample size test. This is because instead of is minimized for a given for double-dwell test, in order to obtain a smaller mean acquisition time. Recall that in the phase uncertainty to be searched through, there cell, and all others are cells. is only one Fig. 7 shows an example minimum mean acquisition time for the acquisition systems using double-dwell and sequential linear tests. The system parameters are as follows. , and hence is the period of the PN code, , and . It has been known that for double-dwell and sequential tests, the minimum mean acquisition time is quite insensitive to the value of penalty time [3], are shown here. [15]. Hence, only the results for The mean acquisition time is calculated by the expression (12),

H

WANG AND SHEEN: VARIABLE DWELL-TIME CODE ACQUISITION FOR DS SPREAD-SPECTRUM SYSTEMS

which is valid for the straight line search strategy. The minimum value of (12) is obtained by optimizing the system parameters of thresholds and dwell times (for double dwell). The optimization is performed with the differential evolution global optimization algorithm proposed in [26] and [27]. The algorithm has been found very robust and converges within 100 steps for all the considered cases. Results for both and have been shown. As seen, the fading rate does not make much difference on the mean acquisition time performance, and the sequential linear test outperforms the double-dwell test by a margin of 1–2 dB. (It was shown in [15] that depending on the penalty time, a 1–4-dB improvement can be obtained by double-dwell test over the single-dwell one.) The minimum mean acquisition time of SPRT for the conventional AWGN channel is also shown in the figure for comparison. As is evident, fading may result in a 1–4-dB loss in performance for the sequential linear test for the SNR’s of interest. Similar degradation caused by Rayleigh fading was found in [15] for double-dwell systems. In Fig. 7, some computer simulation results are also given for double-dwell systems. The simulation and analytical results agree very well as can be seen in the figure.

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and it follows that (A-5) . The proof is done by substituting (A-5) into for all (A-2). Q.E.D. Theorem A-1: The sequential linear test of (8) reaches a decision in finite steps with probability one. Proof: Let denote the sample number when the sequen, we tial linear test of (8) reaches a decision. Clearly, for have (A-6) Hence, it is sufficient to show that

(A-7) 1)

Case: From (17) and (18), we have

V. CONCLUSIONS Noncoherent serial search PN code acquisition based on multiple-dwell or sequential linear tests is investigated for direct-sequence spread-spectrum systems on Rayleigh fading channels. A novel analytical method is proposed to obtain the test performance in terms of the detection and false alarm probabilities and ASN’s, which in turn are used to obtain the mean acquisition time of the acquisition system. The effects of fading are evaluated, and comparisons are made between double-dwell and sequential linear tests. Example results show that fading may result in a 1–4-dB loss in performance and the sequential linear test can outperform the double-dwell test with a margin of 1–2 dB.

(A-8) is given in (18). Recall that

where

for

, hence

(A-9) Substituting (A-9) repeatedly into (A-8), then

APPENDIX A In this appendix, we will show that the sequential linear test of (8) reaches a decision in finite steps with probability one. To proceed, the following lemma is needed. be defined as in (31), then Lemma A-1: Let

(A-10) . and (A-7) follows by letting Case: Given , from (17) and (39), we have 2)

(A-1) . for all Proof: From [22, eq. (2.8a)] (A-11) is

where (A-2)

for

given in (39). Again, , it follows that

since

where (A-3) By integrating (A-3) by part repeatedly, we have (A-4) (A-12)

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Substituting (A-1) into the right-hand side of (A-12), we have

(A-13) for all , where (A-14) Further, by substituting (A-13) repeatedly into (A-11), we have for all

(A-15) and then

(A-16) Finally, (A-7) is obtained by letting

in (A-16). Q.E.D.

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Huan-Chun Wang was born in Hvalien, Taiwan, R.O.C., on January 14, 1969. He received the B.S.E.E. degree from the Chung Yuan Christian University, Taiwan, R.O.C., in 1992, the M.S.E.E. and Ph.D. degrees from the National Chung Cheng University, Chia-Yi, Taiwan, R.O.C., in 1994 and 1999, respectively. Since October 1999, he has been with the Advanced Technology Center at Computer and Communications Laboratories, Industrial Technology Research Institute, where he is mainly involved in the projects of multimedia mobile communications. His current research interests include communication theory, spread-spectrum communications, and personal and mobile communications.

Wern-Ho Sheen (S’89–M’91) received the B.S. degree from the National Taiwan Institute of Technology, Taiwan, R.O.C., in 1982, the M.S. degree from the National Chiao Tung University, Taiwan, R.O.C., in 1984, and the Ph.D. degree from the Georgia Institute of Technology, Atlanta, in 1991. From 1984 to 1993, he was with Telecommunication Laboratories, Taiwan, where he was mainly involved in the projects of personal communications and basic rate ISDN. Since 1993, he has been with the Department of Electrical Engineering, National Chung Cheng University, Taiwan, R.O.C., where he is currently a Professor. His research interests include adaptive signal processing, spread-spectrum communications, and personal and mobile radio systems.