IVE SYNCHRONIZATION FOR SPREAD SPECTRUM SYSTEMS

Mohamed G. El-Tarhuni and Asrar U. Sheikh PCS Research Laboratory, Department of Systems & Computer Engineering

Carleton University 1 125 Colonel By Drive Ottawa, Canada K1S 5B6

email: Sheikh@sce.carleton.ca Abstract-- Code acquisition in direct sequence spread- spectrum systems must occur before data demodulation can start. In this paper, a new hybrid approach to code acquisition is proposed by using an adaptive filter. The proposed system tests several phases concurrently and, hence, offers fast acquisition with small hardware requirements. Moreover, the same system could be employed in code tracking.

I. INTRODUCTION

Direct sequence spread spectrum (DS-SS) systems offer several advantages such as interference rejection, multipath diversity, multiple access capability, and accurate ranging. Recently, greater interest has been devoted to use DS-SS systems in many mobile radio applications. However, to take advantage of SS systems, code synchronization must be established by the receiver. This is usually accomplished in two steps: acquisition and tracking [ 11. In this paper, only code acquisition is considered.

Acquisition is the process by which the two codes are coarsely aligned such that their phase offset is less than one chip period. Previous work in the acquisition problem concentrates mostly on correlation techniques where the system searches for acquisition either serially [2][3] or in parallel [4][5][6]. Hybrid systems were also developed [7]. Another approach to acquisition is based on sequential estimation schemes 181. In this paper, we will present a new scheme for code synchronization based on adaptive filtering applications.

The proposed system provides fast acquisition performance as well as code tracking capability; thus, a separate code tracking circuit is not required. However, in this paper, only code acquisition is considered. Expressions for the mean acquisition time, detection probability, and false alarm probability are derived, and numerical results are presented. These are followed by a discussion. The paper is organized as follows: the structure of the proposed acquisition system is described in section I1 and the analysis of the system is presented in section 111. Section IV shows some numerical results and the paper is concluded in section V.

11. ACQUISITION SYSTEM DESCRIBTION

The proposed acquisition system is based on an adaptive finite impulse response (FIR) filter that processes the received SS signal in order to extract the delay offset between the PN code in the incoming signal and a locally generated replica of the same code. The filter has M coefficients (taps) which are adjusted by using the least-mean-square algorithm (LMS) to minimize the mean squared error between the filter output and a desired reference signal. The block diagram of the system is shown in Fig. 1 .

A standard model is used for the received DS-SS signal given by [ 1][9]

Y ( t ) = f i b ( t - z T,) a ( t - z T,) cos ( 2 ?fCt + c p ) ( 1 )

where P is the power of the received signal, b(t) is the information data, a(t) is the PN spreading signal, f,, and cp are the frequency and random phase of the carrier, respectively, T, is the chip duration, and z is the random time delay that must be estimated by the synchronizer which is assumed to be an integer random variable over the code period L. The receiver’s thermal noise, nP(t), is modeled as an Additive White Gaussian Noise (AWGN). The spreading waveform, a(t) , is given by

+ I ? ( t ) P

a ( t ) = C a ( 2 ) I - I (t-iT,) T i

where a (i) E { -1, 1 } with equal probability, and I I T is a rectangular shaping pulse with unit amplitude and duration 7: Assume. as in [7][9], that carrier synchronization is obtained prior to acquisition, no data exist during acquisition, and, without loss of generality, the received signal power IS

normalized to unity. The equivalent baseband signal, x(t), is given by

x ( t ) = a ( t - 7 T J + n ( t ) ( 3 )

where n(t) is a zero mean AWGN with two-sided power spectral density iVd2. The baseband signal, x(t), is passed through a chip matched filter whose output is sampled at the chip rate to form the adaptive filter input sequence x(n). The filter output is

v(n) = w ‘ ( n ) . ( n ) (4)

where w‘(n)=[wo(n) wl(n) ..... w~. l (n) ] is the tap-weight vector of the filter coefficients at the nth time instant; subscripts O , l , ..., M- 1 denote the tap number. Also, x‘(n)=[x(n) x(n- 1) ..... x(n-M+l)] is the filter input vector at the nth instant which consists of the present sample and past M-1 samples of the received signal. The superscript “T’ denotes vector transposition operation.

The adaptive filter coefficients are adjusted by using the LMS algorithm to minimize the mean squared error (MSE) between the filter output, y(n) , and a desired response, d(n) , which is chosen as the locally generated despreading sequence at a certain phase,i.e., d ( n ) = a ( n - ?) ; ? can assume an integer value between 0 and L- I . The time uncertainty region to be searched for the actual delay T is divided into q = r L / W cells, where rzl is the smallest integer greater than or equals to z. Note that, correct acquisition can only be obtained under one cell which we call the in-phase hypothesis

0-7803-3157-5196 $5.00 0 1996 IEEE 170

H I , while all other cells, satisfy the out-of-phase hypothesis I f o ,

may cause a false alarm error. Starting with the first cell, the system checks the MSE for convergence towards the minimum mean square error (MMSE), which is obtained by the optimum Weiner filter [lo], for a fixed period of time. If convergence is detected, the delay offset between the incoming code and the locally generated code is assumed to be within the time spanned by the filter which is M chips. The delay is estimated from the number of cells already tested and the tap number at which the peak value of the filter taps is located. On the other hand, if convergence is not detected, the phase of the reference sequence is updated by Mchips and the next cell is tested.

The optimum tap-weight vector for the system under consideration can be shown to be [ 1 I ]

SNR' -~ ; H , and j = i - T SNR< + 1

0 ; H , and j # i - T or H ,

( 5 )

for j=0,1,2, ..., M-1. The SNR, is the signal-to-noise ratio per chip. Therefore, under HI, the filter coefficients are peaked at the tap number which equals to the delay offsel between the two codes; while, under H,, the filter actually turns itself off since it can not minimize the MSE because of the lack of correlation between the incoming signal and its reference signal.

The LMS algorithm is a stochastic gradient-based algorithm which is used to search for the minimum of the mean squared error. The algorithm is used to update the filter tapis according to the following formula [IO]

w ( n + l ) = ~ ( n ) + p e ( n ) x ( n ) (6)

e ( n > = d ( n ) - Y ( n )

where p is the step size which controls the convergence speed and the steady state excess MSE [lO][12], and e(n) is the error signal. Since the MSE, which is an ensemble average of the square of the error signal, is not readily avadable at the receiver to test for convergence, the time average of the squared error over a window of S samples is used instead. This decision statistic is defined as

S (7)

I s s A = - E e2 ( n >

n: I

After an adaptation period, A, is checked for convergence K,, times and if B or inore tests, out of Kmax. indicate convergence, i.e., A, 5 q where q is a preset threshold value, the system assumes that the range of phases under test are within the time span of filter; however, if the number of threshold crossings is less than B, the phase of the local sequence is advanced by M chips and the adaptation process is repeated.

111. ACQUISITION TIME ANALYSIS

The acquisition process is modeled as a Markov process which can be described by a state transition diagram with q states. Each state represents M phase shifts of the code

sequence. Two extra states, denoted as ACQ and FA, are used to indicate the acquisition and false alarm conditions, respectively. For each state under test, the filter coefficients are adapted for a time period of T, and assume that T,=K,ST,., where KO is an integer. After the adaptation period, the short term time average of the squared error is tested for convergence for a maximum of K,,, tests. If B tests or more indicate convergence in the MMSE sense, acquisition is declared and tracking is initiated. This decision is either correct with probability Pi, if the correct cell, assumed to be sq. was under test and the peak of the tap-weight vector was located at the correct phase offset a , which is defined as tap number ^t - T. or false with probability PFj or PF". The probability PFj denotes the in-phase false alarm probability which occurs when .sy is under test while the tap weight peak is not located at tap number a at the Bth threshold crossing. The other probability. PFo, represents the out-of-phase false alarm probability which occurs when a state other than sq is being tested when B threshold crossings were detected.

A false alarm will be detected by the tracking mode of operation after wasting a time of Tp, which represents the false alarm penalty time of Tp=KpSTc where Kp is an integer. Lipon clearing the false alarm error, the acquisition process is restarted from the state next to the one at which the false alarm has occurred. On the other hand, a miss of acquisition occurs if the system fails to detect B threshold crossings of the short term time average while sq is under test. This occurs with probability

A portion of the state transition diagram that represents the acquisition system is shown in Fig. 2a. There is only one absorbing state which is the detection state ACQ. Figure 2a can be simplified to Fig. 2b.The transfer function between the different states are defined as

PM= I -PD-P,.

(8) H , ( Z ) = PDZ'"" + K J ST'

H , ( Z ) = ( 1 - PD -

where the parameters are defined as follows: Z is the unit delay operator, S is the short term time average window, Kd is the number of tests required to detect E threshold crossings when sq is under test, B I Kd < K,,,, K f is the number of tests at

which a false alarm occurs, B 5 K < K m U x , and Kp is the .f -

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with small SNR, is the noncentral x2 with S degrees of freedom

penalty time caused by a false alarm. From the simplified diagram, it is possible to obtain the generating function as

PA cQ( 1 The mean acquisition time is given by [2][3]

7 2 where <- = Jss , and I, ( ) is the jth order modified Bessel (17)

function of the first kind. The pdf of A, under H , is the same

as that given by (24) but with a different value for o', obtained

from (20) and with 5' = 1 .

Using (16) into (1 7) yields after some algebra 1

E U a c q } = - c [ I + ( 4 - 1) (2 -P , ) /2 1 ( K ' + K 1 PD "'718)

The probability that As is less than the threshold is simply + [PFi+(4-1)P,o(2-PD)/21Kp

where both Kd and Kfwere assumed to be fixed at a value of K,, (worst case scenario). P,,,, = l f l s j H i ( 9 ) d 9 ; i = 0 , 1 (25)

At steady state, i.e., after the filter taps converge to their - m

optimum value, the error signal is Gaussian with a mean value given by [ 111 and the detection and false alarm probabilities are given by

+ 1 : H ,

; H , large SNR, (19)

(27) ; H , small SNR(

P,, = Pi/ ( 1 - Pt)P,

and a variance o f

J S S (1 -J,7,, ; H , small SNRc where P E is the probability of making an error about the

coefficients' peak position. Pc can be determined by

approximating the filter taps, at steady state, as independent identically distributed Gaussian random variables' to get [ 1 I ]

where Jss is the steady state MSE produced by the LMS algorithm given by [ I 11

where 2SNRc-p(SNRc+ 1)

(22) = 2SNRc-p(SNRc+ 1) ( M i - 1)

Referring to (7) it can be shown that the probability density function (pdf) of the decision statistic A, under H I with large

SNR, is the central x' with S degrees of freedom given by

where Q is the Q-function defining the area under the tail o f a Gaussian random variable, the mean, m a , is the optimum

Wiener solution given by (5), and the variance, oLequals to

(30)

9 [ 1 I I [ 121 -___ & p 2 - I e 2 u : / s P I 2

; 920 (23) o: = 1 +SNRc fhsI H , = S (+s) 2s'2 r ( s / 2 )

where r ( ) is the gamma function. The pdf of A, under H I 1, It was shown in [ 131, under some general assumptions, that the filter taps are actually independent Gaussian random variables.

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IV. NUMERICAL RESULTS AND DISCUSSION

As an application for the previous analysis, we consider a DS-SS system with a PN code of length L=127. Different values for the number of adaptive filter taps are considered: M=4, IO, 16, 22 and 28. The averaging window for the error signal is set to S=10. The parameters B and KO,, are fixed at 3 and 10 respectively. The adaptation period is approximated by the steady state time of the LMS algorithm which is defined as [ 121 K, = 2/yhu,, where huv is the average eigenvalue of the autocorrelation matrix of the filter input. The false alarm penalty time is taken to be Kp=q(Ku+K,,).

In calculating the detection and false alarm probabilities numerical integration of (25) and (29) is used, then substituting into (26), (27), and (28). The effect of the LA4S step size parameter and the threshold value on these probabilities is shown in Figs. 3 ,4 and 5, for a SNR, of -5dB and i%16 taps. It is clear that an inappropriate choice of these par,ameters may result in a large degradation on these probabilities. Thus, optimization of the LMS step size and the threshold value, for each value of SNR,, is necessary to obtain good performance. The LMS step size and the threshold value are numerically selected to minimize the mean acquisition time.

Figure 6 shows the optimized mean acquisitiion time as a function of the SNR, with the number of filter taps as a parameter. The mean acquisition time is reduced as the number of taps increases because of the reduction in the nuimber of cells to be searched for acquisition, i.e. increased parallelism. However, a diminishing gain is observed due to the larger adaptation period required by the LMS algorithm to converge to the optimum solution.

The improvement in the mean acquisition lime for the proposed system over straight serial-search schemaes (with 112- chip increments) is roughly twice the number of filler taps used, i.e. 2M. Other hybrid acquisition schemes, which test 2M cells at a time, have similar improvement, however, 2M correla.tors and decision devices are needed. In contrast, tlhe proposed system requires a single M-tap adaptive filter to attain such gain. Moreover, it is possible to use the same systern in tracking the SS signal and, thus, eliminating the need for a separate code tracking loop; this is currently under investigation [ 111.

V. CONCLUSIONS

A new approach to DS-SS code acquisition is developed in which the code acquisition problem is considered as an adaptive filtering application. Analysis for the mean acquisition tim, detection, and false alarm probabilities are presented. The proposed system has faster acquisition performance than straight serial-search schemes, and less hardware requirement than hybrid techniques. It is noticed that increasing the adaptive filter length, i.e. the number of taps, reduces the mean acquisition time owing to the increased parallelism. The LMS algorithm is used to adapt the filter taps due to its simplicity, however, other adaptation algorithms could be employed.

REFERENCES

search spread spectrum code acquisition-part I: General Theory,” IEEE Trans. Commun., vol. COM-32, No. 5, pp. 542- 549, May 1984. [3] J. Holmes and C. Chen, “Acquisition time performance of PN spread spectrum systems,” JEEE Trans. commun., vol. COM-2.5, pp.778-783, Aug. 1977. 1[4] L. Milstein, J. Gevargiz, and P. Das, “Rapid acquisition for direct sequence spread-spectrum communication using parallel :SAW convolvers,” IEEE Trans. Commun., vol. COM-33, No.

1-51 Y. Su, “Rapid code acquisition algorithms employing PN matched filters,” IEEE Trans. C o m u n . , vol. COM-36, No. 6. pp. 724-733, June1988. 161 E. Sourour and S. Gupta, “Direct sequence spread spectrum parallel acquisition in a fading mobile channel,” 39th IEEE IVehic. Tech. Conf., San Francisco, California, pp. 774-779, May 1989. 171 C. Baum and V. Veeravalli, “Hybrid acquisition schemes for direct sequence CDMA systems,” Inter. Commun. Conf.. pp. 1433-1437, New Orleans, LA, May 1-5, 1994 [ 81 R. Ward, “Acquisition of pseudonoise signals by sequential estimation,” E E E Trans. Commun. Tech., vol. COM- 13, No. 4, pp. 475483, Dec. 1965. [9] K. Chawla and D. Sanvate, “Acquisition of PN sequences in chip synchronous DSISS systems using a random sequence model and the SPRT,” IEEE Trans. Commun., vol. COM-42, No. 6, pp. 2325-2332, June 1994. [IO] S. Haykin, Adaptive Filter Theory, 2nd. ed., Prentice-Hall, New Jersey, 199 I . [ 111 M. G. El-Tarhuni, Application of Adaptive Filtering to Direct-Sequence Spread-Spectrum Code Synchronization, 1’h.D. Thesis proposal, Department of Systems & Computer Engineering, Carleton university, Canada, Jan. 1996. [I21 B. Widrow and S. Stearns, Adaptive Signal Processing, Prentice-Hall, New Jersey, 198.5. [13] N. Bershad, and L. Qu, “On the probability density function of the LMS adaptive filter weights,” IEEE Trans. Acoust., Speech. Signal Processing, vol. ASSP-37, No. I , pp. 413-56, Jan. 1989.

7. pp. 593-600, July 1985.

algorithm m

chips

Fig. 1 ]Block diagram for the acquisition system [ I ] M. Simon, et al., Spread Spectrum Communications, vol. I- 111, Computer science press, Rockville, Maryland, 1985. [2] A. Polydoros and C. Weber, “A unified approach to serial

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Fig. 2 Flow graph model for the acquisition system

1 2 3 4 5 6 Step size parameter, p IO-?

10-4

Fig. 3 Probability of detection.

3 3 ,,------" 0.8 ; / 1

lo-'ui 1'5 ' I 2 2'5 3'5 4 4'5 5 5 5 6

Step size parameter, p

Fig. 4 In-phase false alarm probability.

SNR, = -5dB 1 M= 16 taps S=lOchips 1

* f 0 108,LL i- -,_- J_-I -A 1 - - - L A--

i s 2 2 s 3 3s 4 4 5 5 5 5 6 % i o 3 Step size parameter. p

Fig 5 Out-of-phase false alarm probability.

L=127 i

I -15 -10 -5 0

Signal-to-noise ratio per chip, dB

Fig. 6 Optimum mean acquisition time perfonnance.

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