A New Analysis of Direct-Sequence Pseudonoise
Code Acquisition on Rayleigh Fading Channels
Wern-Ho Sheen, Member, IEEE, and Huan-Chun Wang
Abstract—Accurate performance evaluation of direct-sequence
pseudonoise code acquisition on Rayleigh fading channels is investigated in this paper. For fading channels the homogeneous Markov chain model, used to characterize the acquisition process over additive white Gaussian noise channels, is no longer valid due to the correlations between cell detections incurred by fading. Hence, the traditional direct and flow-graph approaches for performance evaluation of the code acquisition are not applicable to fading channels. In this paper, a new analysis is proposed for accurate evaluation of the acquisition performance on Rayleigh fading channels. The analysis is quite general and can include various search strategies, types of correlators, and test methods with different performance measures: probability mass function and/or moments of acquisition time. Analytical and simulation results show that the new method predicts the acquisition perfor-mance very accurately.
Index Terms—Direct sequence spread spectrum, pseudonoise
code acquisition, Rayleigh fading.
I. INTRODUCTION
P
SEUDONOISE (PN) code acquisition is one of the most challenging tasks in the design of a direct-sequence (DS) spread spectrum receiver. It has been a topic of intensive re-search for more than 20 years [1], [2]. PN code acquisition was investigated mainly for the traditional additive white Gaussian noise (AWGN) channels in the past [1]–[10]. The extension to time-variant multipath fading channels, however, has recently become more and more important because of the superiority of applying the DS spread spectrum technique to the personal and mobile communications, where the channel is modeled as a time-variant multipath fading one [11]–[15].DS code acquisition methods may 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 some others [1]–[10]. Generally speaking, the parallel search outperforms the serial search, the passive correlator outperforms the active correlator, and the variable dwell-time test outperforms the fixed dwell-time test, all at the expense of a larger system
Manuscript received June 11, 1999; revised September 17, 2001.
W.-H. Sheen is with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan R.O.C. (e-mail: whsheen@em.nctu.edu.tw).
H.-C. Wang is with the Wireless Communication Department, Computer and Communication Laboratories, Industrial Technology Research Institute, Hsinchu 310, Taiwan R.O.C.
Publisher Item Identifier S 0733-8716(01)09087-4.
complexity and/or a more complicated performance analysis. Two approaches, namely direct and flow-graph approaches, have been applied extensively for analyzing the performance of an acquisition system, probability mass function (PMF), and/or moments of the acquisition time [3], [8]–[10]. Theoretically, these two methods are only applicable to the AWGN channels because the homogeneous Markov chain modeling of the ac-quisition process, a fundamental assumption of these methods, is no longer valid for time-variant multipath fading channels due to the channel memory incurred by fading; an analysis without considering the correlations due to fading may only be considered as an approximation. Unfortunately, as to be shown, the analysis error due to no consideration on the channel correlations can be very large, especially for the acquisition system based on passive correlators. A more accurate analysis for fading channels is yet to be found.
Very recently, attempts have been given to analyze code ac-quisition operating on time-variant multipath fading channels [13]–[15].1 In these analyses, by assuming that the channel
co-herence time is smaller than the time between correct cells de-tections, the correlations incurred by fading between correct cells can be safely neglected. (Unfortunately, this approxima-tion is only justifiable for the acquisiapproxima-tion systems based on ac-tive correlators.) Hence, for this type of system, the homoge-neous Markov chain model can still be considered as valid, and the direct and flow-graph approaches can be employed for per-formance evaluation as in the AWGN channels, except that the correlations incurred by fading within the correct cell detection need to be taken into account.
In this paper a novel accurate analysis, based on direct approach, is proposed for DS code acquisition operating over Rayleigh fading channels. Thanks to the direct approach, the analysis is general enough to include various search strategies, correlators, and test methods with different performance measures: PMF and/or moments of the acquisition time.
The remainder of this paper is organized as follows. The system model is described in Section II, where the acquisition system based on passive correlators is used as an example for analysis. Section III presents our new analytical method that can take into account the correlations incurred by fading. Section IV gives numerical results that illustrate the effects of fading correlations on the acquisition performance. Finally, the paper is concluded in Section V.
1The acquisition systems and channel models considered in [13] are more
general than that in [14] and [15]. 0733–8716/01$10.00 © 2001 IEEE
A. Channel Model
A popular channel model for the terrestrial personal and mo-bile communication environment is given by the impulse re-sponse
(1)
where
path number; path delay;
and ; Rayleigh fading and uniform random phase over , respectively.
is assumed to be fixed during the time of code acquisition because the received signal will only experience a small code Doppler shift for the terrestrial personal and mobile commu-nication environment. For the case of , the conditions for the true acquisition are more complicated to characterize for a code acquisition system. For example, assume that there are fingers in the RAKE receiver, then one may want to search the largest paths one after one; only after the largest paths have been found, the acquisition process would then be de-clared to be successful. In this paper, our main concern is how the correlations incurred by fading can be taken into account in the analysis. Hence, only the case of will be considered for simplicity. The extension to the cases with is straight-forward because of the use of direct approach.
B. Acquisition Process
As in nearly all literature, the code phase uncertainty2 of an
acquisition system is divided into cells, with a cell size to within the lock-in range of the code tracking loop used for fine code alignment. The acquisition system then searches through the code phases and determines which cells are the correct cells, according to some type of code phase correlators, search strate-gies, and test methods. As usual, the correct cells will be de-noted as the cells (hypothesis ) and the incorrect cells as the cells (hypothesis ), respectively. In practice, there may be more than one correct cell. For simplicity of presen-tation, however, only the case of one correct cell will be used as an example, although the method applies equally well to the more general case with more than one correct cell. Two types of decision errors may occur when detecting a cell, namely false dismissal of the cell and false alarm of the cells. When a false alarm occurs, the synchronization will be turned to code tracking for fine alignment. In this case, it is assumed that the false alarm can always be detected after some fixed or random time, and the synchronization will be turned back to the code
2It is well known that in addition to increase the total cell numbers, frequency
offset causes signal-to-noise ratio (SNR) degradation by the value
sin(L1fT ) sin(1f T )
whereL is the correlation length, and 1f is the frequency offset. For simple presentation, we do not include this effect in the analysis, although it can be easily done so.
Thanks to the use of direct approach, the method is applicable to various search strategies, correlators, and test methods, as will be described. For simple presentation, however, only the serial search (straight line) system based on passive correlators will be used as an example for the analysis in the following.
C. Code Acquisition Based on Passive Correlators
Fig. 1 is a code acquisition system based on passive correla-tors. is the received signal, given by
(2) where
operation of taking the real part; transmit power;
and gain and phase of the fading channel, respec-tively;
PN sequence with period ; shaping function;
chip duration;
code phase to be estimated; carrier frequency;
additive white Gaussian noise with one-side power spectral density of watts/Hz.
Define and . Then,
for Rayleigh fading and are independent and iden-tical distributions (i.i.d) Gaussian processes.
The received signal, after I/Q down conversion, chip matching, and sampling, is digitally correlated with the local PN code. The correlators’ outputs are then squared and summed to form the test variable . If is greater than a threshold, say , then the acquisition will enter a verification mode or the synchronization will be turned to code tracking. Otherwise, the search for the cell continues. Assume perfect chip synchronous sampling, then the outputs of the correlators are given by
(3)
where , , and are i.i.d.
zero-mean Gaussian variables with variance equal to . In the terrestrial mobile and personal communications, the fading rate is much smaller than (usually the symbol rate), and hence can be considered to be fixed during the duration of . Note that in Fig. 1(b) we have assumed full pe-riod correlation, i.e., the number of correlation chips is equal to . In practice, however, the number of correlation chips may be much smaller than (partial period correlation) for a large , and there will be code-self noise in the detection of cells. In this case, if the code-self noise is modeled as a Gaussian noise
Fig. 1. (a) Code acquisition based on passive correlators. (b) Passive correlator.
as is usually done in the literature [16], our method of analysis described below applies as well.
Under , by using the property that the out-of-phase cor-relation of a PN sequence can be approximated as zero for full period correlation, one has
(4)
which is a zero-mean Gaussian variable with variance equal to (5) where
(6) is the received signal to noise density ratio. Define
. It can be shown that the correlation
coef-ficient of and is
o.w.
(7)
For a large enough , . Hence
(8) are well approximated as independent variables under , as in the AWGN channels. This is not the case under , however, due to the channel memory incurred by fading. In this case, we have
(9)
where is the correlation coefficient of and ,
and is a positive integer. This correlation between cells renders the homogeneous Markov chain model, a fundamental assumption of the direct and flow-graph approaches for the per-formance analysis on AWGN channels, invalid. A new method that can take this correlation into account needs to be devised for performance analysis. From (7), it is also easy to see that the detections between and cells are approximately in-dependent.
III. NEWANALYSIS
In this section, the acquisition system with no verification mode will be used as an example to illustrate the new analytical method. Some performance examples with a verification mode are given in Section IV.
As in [3], the acquisition process can be represented by a signal flow graph. For example, the signal flow graph for the straight-line search is shown in Fig. 2(a), where is the prob-ability that the th cell is the starting cell, is the gain characterizing the detection of an cell, and and are the gains for the correct detection and false dis-missal of the cell, respectively, with to denote the fact that the detection of the cell at present time depends on pre-vious cells detections. Let be the time of doing th de-tection of the cell. Then, for the th detection of the cell, . That is, the th detection of the cell at the time depends on the cell detections at the times of . Recall that the detections of cells are inde-pendent of one another.
For an acquisition system with no verification mode (10) (11) (12) where
probability of false alarm;
(in ) penalty time assumed to be constant; and probabilities of the correct detection and
false dismissal, respectively.
The unit time in (10)–(12) is . Fig. 2(a) can be further sim-plified as in Fig. 2(b). At this point, one might attempt to apply Mason’s formula to obtain the generating function of the acquisition time , as in the AWGN case. (In effect, contains all the information about the PMF of the acquisition time .) Unfortunately, Mason’s formula does not apply in a
fading channel case because and are now
func-tions of time. A new method is required for the performance evaluation.
B. Calculation of , , and
From definition and (8), it is easy to see that under , are i.i.d. central chi-square variables, and
(13) On the other hand, however, the evaluation of and becomes much more involved due to the correlations incurred
by fading. Since , we will only consider .
Assume that for the th detection of the cell is of in-terest. From the definition
(14) Unlike the case, now are correlated central chi-square distributed variables, and, to our best of knowledge, there is still no easy way for the evaluation of (14). In the following, a novel method is proposed to evaluate to the desired accuracy.
Fig. 2. (a) Signal flow diagram for the straight-line search. (b) Simplified signal flow diagram.
To begin with, the following definitions are necessary (also see [15]).
Definition 1 [17]: A symmetric random matrix of dimen-sion is said to have a Wishart distribution with degrees of freedom and parameter , if can be written as
(15)
where , , are independently
dis-tributed normal -vectors with zero mean and covariance matrix , where denotes the operation of transpose. The matrix is called the Wishart matrix.
Definition 2 [17]: Let be the Wishart matrix with the dis-tribution . The joint distribution of the diagonal ele-ments of given by
(16)
is called the -variate central chi-square distribution with de-grees of freedom and parameter , denoted by . With these definitions, we have the following theorem [15], [18].
Theorem: If a covariance matrix (positive def-inite) can be factorized as
(17)
where , is the identity matrix
with dimension , and is a matrix
with rank , then the PDF of the distribution is given by
(18)
where the expectation is taken with respect to the -distributed Wishart matrix , and
o.w. (19) is the noncentral chi-square distribution with degrees of freedom and the noncentrality parameter , where the th order modified Bessel function of the first kind. From Definition 2 and (8), it is easy to see that
under is distributed as with the covariance matrix given by (20) where o.w. (21) Furthermore, let (22) It follows that (positive semidefinite) is real
symmetric. Assuming that rank , let
be the positive eigenvalues of ,3
then the matrix in (17) can be obtained as
(23)
where and , and
is the orthonormal eigenvector associated with . Hence, by the described theorem, given , the random variables
, are independent with the conditional pdf
(24) Recall that has the Wishart distribution , is the
th row of the matrix , and [from (19)]
(25)
3In practice, the matrixW 6W 0 I can always be approximated as having
a rankd, if = 0; j > d.
With (24), now (14) becomes
(26) It is well known that
(27) where is the Marcum generalized function. Hence
(28)
Many methods can be employed for the evaluation of the expec-tation in (28) [19]. The method based on the Monte Carlo inte-gration will be used for the numerical examples in Section IV.
C. Direct Approach
In the direct approach [8], the search strategy is characterized by a functional, say , where denotes that the true ac-quisition is accomplished at the th detection of the cell, and
is the starting cell. For example, using Fig. 2
(29) for the straight line search. Given , then the generating function of the acquisition time is given by
(30)
where is the generating function of conditioned and . Define to be the set of
that lead to the true acquisition, conditioned on and . Then
(31)
where is the term in associated with the
time sequence . For the cases with different search strategies, i.e., more than one cell and/or more than one path, the only difference in the analysis is to use different functionals ; the method has a broad range of applicability in this aspect.
Two observations are important regarding the calculation of (30). First, the infinite sum must be approximated by just in-cluding to . (In our numerical examples, the
max-imal .) Second, given the sets
in the evaluation of that reduces the complexity of this method very significantly. That is, we have
(32)
Of course, for each term in (32) only the with those are needed in the evaluation of the associated and
.
IV. NUMERICALEXAMPLES
The Jakes’ two-dimensional (2-D) isotropic scattering channel model and simulation method [20] are adopted in the following numerical examples. In the model, the correlation function of the in-phase and quadrature-phase fading compo-nents is given by
(33) where is the maximum Doppler shift, and is the ze-roth-order Bessel function of the first kind. Three fading rates,
namely , , and are
considered, where . As mentioned, the Monte
Carlo integration is used to evaluate the expectation over the Wishart distribution required in (28), although the other integration algorithms [19] may also be used. It has been found that for all the numerical results. Also, less than 10 samples are needed for the Monte Carlo integration to obtain the desired accuracy. For simplicity, only the worst case, that is
and , is used to illustrate the
acqui-sition performance. and in all results.
In practice, an active correlator with threshold crossing is usu-ally employed as a lock detector during tracking. It is common that the correlation time is taken to be around ten times of that used in the code acquisition to ensure the detection of in-lock or out-of-lock conditions. In our examples, the correlation time (penalty time) of the lock detector is assumed to be 12 times that used in code acquisition.
Figs. 3–5 are the results for the acquisition system with no verification mode. Fig. 3 shows an example PMF of the acquisi-tion time under different fading rates. ( , normalized to ). Also shown is the PMF (approximation) obtained with the traditional direct approach without considering the cor-relations incurred by fading. As is evident, the discrepancy be-tween the true and approximate PMF is very large. In Fig. 3, the largest peak is located at , which means that the event with and no false alarm has the largest probability. The
second peak is located at , which is
asso-ciated with the event that and one false alarm. Note that there is a total of 62 of this type of event.
Fig. 4 shows the performance of mean acquisition time as a function of the threshold under various fading rates. Clearly,
Fig. 3. Probability mass function of acquisition time with no verification mode.
Fig. 4. Mean acquisition time versus threshold under different fading rates (no verification mode).
the traditional analysis may underestimate the mean acquisition time by a large margin depending on thresholds and fading rates. As expected, the analysis error is larger for a smaller fading rate because a smaller means that the correlation due to fading will extend longer. Note that the mean acquisition time obtained with the traditional analysis is independent of fading rate. Also shown in the figure is the simulation mean acquisition time. As seen, the analytical and simulation results agree very well for all the range of thresholds. Simulations are done with 10 sam-ples to obtain the desired accuracy. In Fig. 3 we do not have simulation results because it is too time-consuming to be really conducted.
Fig. 5 shows the minimum mean acquisition time (those ob-tained with optimum thresholds) versus under various fading rates. The optimum thresholds are obtained from Fig. 4, and, as seen, they depend on Doppler rate, SNRs, penalty time, and some other parameters. More than an order of analysis errors are observed with the traditional analysis, depending on the fading
Fig. 5. Minimum mean acquisition time versus the received chip energy to noise ratio under different fading rates (no verification mode).
Fig. 6. Minimum mean acquisition time versus the received chip energy to noise ratio under different fading rates (coincidence detector).
rates and SNRs. In addition, a smaller minimum mean acquisi-tion time is obtained with a fast fading.
Figs. 6–8 are the results for the acquisition systems with a verification mode. The coincidence detector proposed in [3] is employed here. That is, after the first threshold is exceeded, then an active correlator is activated and the test variable is com-pared to the second threshold . If is exceeded for out of times, a successful code acquisition will be declared. Oth-erwise, the search for the cell continues. The detailed anal-ysis with a verification mode can be found in [15].
Fig. 6 shows the minimum mean acquisition time versus under various fading rates. Again, the traditional analysis may underestimate the true value by a large margin, depending on the fading rates and SNRs. Also, the analytical and simulation results agree very well in the figure. Figs. 7 and 8 are example mean acquisition times as a function of ( ) with ( ) being fixed, respectively. Similar results are observed in these two figures.
Fig. 7. Mean acquisition time versus the first thresholdV under different fading rates (coincidence detector).
Fig. 8. Mean acquisition time versus the second thresholdV under different fading rates (coincidence detector).
Finally, we note that the calculation of in (32) is not as complicated as it appears to be. For example, it takes only few minutes to complete a point in Figs. 4–8 (in a commercially available work station with 10 samples for Monte Carlo inte-grations) instead of about one day by using simulation with 10 samples; the analytical method has a tremendous advantage in computation time over simulations. In addition, it is very dif-ficult to use simulations to obtain accurate PMF of the acqui-sition time as the one shown in Fig. 3, especially to simulate with enough accuracy the rare events with probabilities smaller than 10 .
V. CONCLUSION
In this paper, a novel analytical method is proposed to ana-lyze direct-sequence pseudonoise code acquisition systems over Rayleigh fading channels. The method is unique in that the channel memory incurred by fading between different cell de-tections can be taken into account. Thanks to the use of a
di-ferent performance measures: probability mass function and/or moments of acquisition time. Numerical results show that the new method predicts the acquisition performance very accu-rately, and the traditional analysis may suffer from a large anal-ysis error, especially for the acquisition systems based on pas-sive correlators.
REFERENCES
[1] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread
Spec-trum Communications, MD: Computer Science Press, 1985, vol. III.
[2] R. L. Peterson, R. E. Ziemer, and D. E. Borth, Introduction to Spread
Spectrum Communications. Englewood Cliffs, NJ: Prentice-Hall, 1995.
[3] A. Polydoros and C. L. Weber, “A unified approach to serial search spread-spectrum code acquisition—Part I: General theory,” IEEE Trans.
Commun., vol. COM-32, pp. 542–549, May 1984.
[4] , “A unified approach to serial search spread-spectrum code acqui-sition—Part II: A matched filter receiver,” IEEE Trans. Commun., vol. COM-32, pp. 550–560, May 1984.
[5] Y. T. Su, “Rapid code acquisition algorithms employing PN matched filters,” IEEE Trans. Commun., vol. COM-36, pp. 724–733, June 1988. [6] U. Cheng, “Performance of a class of parallel spread spectrum code acquisition schemes in the presence of data modulation,” IEEE Trans.
Commun., vol. COM-36, pp. 596–604, May 1988.
[7] V. M. Jovanovic and E. S. Sousa, “Analysis of noncoherent correlation in DS/BPSK spread spectrum acquisition,” IEEE Trans. Commun., vol. 43, pp. 565–573, Feb./Mar./Apr. 1995.
[8] H. Meyr and G. Poltzer, “Performance analysis for general PN spread spectrum acquisition techniques,” IEEE Trans. Commun., vol. COM-31, pp. 1317–1319, Dec. 1983.
[9] V. M. Jovanovic, “Analysis of strategies for serial-search spread spectrum code acquisition-direct approach,” IEEE Trans. Commun., vol. COM-36, pp. 1208–1220, Nov. 1988.
[10] G. E. Corazza, “On the MAX/TC criterion for code acquisition and its application to DS-SSMA systems,” IEEE Trans. Commun., vol. 44, pp. 1173–1182, Sept. 1996.
[11] E. Sourour and S. C. Gupta, “Direct-sequence spread-spectrum parallel acquisition in a fading mobile channel,” IEEE Trans. Commun., vol. 38, pp. 992–998, July 1990.
[12] S. Tantaratana, A. W. Lam, and P. J. Vincent, “Noncoherent sequential acquisition of PN sequences for DS/SS communication with/without channel fading,” IEEE Trans. Commun., vol. 43, pp. 1738–1746, Feb./Mar./Apr. 1995.
[13] G. E. Corazza and A. Polydoros, “Code acquisition in CDMA cellular mobile networks—Part I: Theory,” in Proc. IEEE ISSSTA’98, Sun City, South Africa, Sept. 2–4, 1998, pp. 454–458.
nonselective multipath fading channels,” Wireless Network, vol. 5, pp. 11–21, 1999.
[15] H-C. Wang and W-H. Sheen, “Variable dwell-time code acquisition for direct sequence spread spectrum systems on time-variant Rayleigh fading channels,” IEEE Trans. Commun., vol. 48, pp. 1037–1046, June 2000.
[16] W. Zhuang, “Composite GPS receiver modeling, simulation and appli-cations,” Ph.D. dissertation, Univ. New Brunswick, Canada, 1992. [17] N. C. Giri, Multivariate Statistical Analysis. New York: Dekker, 1996. [18] T. Royen, “On some central and noncentral multivariate chi-square
dis-tributions,” Statistica Sininca, vol. 5, pp. 373–397, 1995.
[19] A. R. Krommer and C. W. Ueberhuber, Numerical Integration. New York: Springer-Verlag, 1994.
[20] W. C. Jakes, Jr., Microwave Mobile Communications. New York: Wiley, 1974.
[21] H-C. Wang, “Pseudo-noise code acquisition for direct sequence spread spectrum systems on multipath fading channels,” Ph.D. dissertation, Na-tional Chung Cheng Univ., Taiwan, July 1999.
Wern-Ho Sheen (S’89–M’91) received the B.S.
degree from the National Taiwan University of Science and 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 Telecommunica-tion Laboratories, Taiwan, R.O.C, and from 1993 to 2001 he was with the Department of Electrical Engi-neering, National Chung Cheng University, Taiwan, R.O.C. Since 2001, he has been with the Department of Communication Engi-neering, National Chiao Tung 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.
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, and 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, respec-tively.
Since October 1999, he has been with Computer and Communications Labo-ratories, Industrial Technology Research Institute, where he is mainly involved in the projects of wireless communications. His current research interests in-clude communication theory, spread-spectrum communications, software de-fined radio, and personal and mobile communications.
