A Reduced-Complexity Formulation for DS-SSMA Performance Evaluation via the Characteristic Function Method

Dandan Liu and Charles L. Despins

PCS Research Group - INRS Telecommunications (Universite du Quebec)

16, Place du Commerce, Verdun, QuCbec, Canada H3E 1H6

e-mail: liu@ and despins@inrs-telecom.uquebec.ca

Abstract - A significant body of literature has been devoted to the evaluation of error probabilities for direct-sequence spread-spectrum multiple-access (DS-SSMA) systems. This paper presents a reduced-complexity formulation for such a performance evaluation via the characteristic function method which is a powerful tool for accurate evaluation of DS-SSMA perfoimance and does not require a large number of moments of the decision statistic. In the previous version of this tech- nique, the characteristic function of the multiple-access inter- ference (MAI) component of the decision statistic (for a correlation receiver) was expressed in terms of a double inte- gral over the distributions of phase and delay. The new formu- lation generates a more compact expression for the characteristic function of the MAI, eliminates the need for numerical integration over the phase distribution (thus improving accuracy) and yields a computational complexity reduction proportional to the product of the spreading gain and the number of terms required to integrate over the phase distribution. This renders the characteristic function method more attractive for application to future wideband CDMA sys- tems and by extension, to more complex transceiver structures and time-varying channels.

I. INTRODUCTION Code-Division Multiple Access (CDMA) based on direct- sequence spread spectrum has generated increasing interest for wireless mobile and portable communications applica- tions. The wide bandwidths exploited by the radio links of these CDMA systems generally precludes a statistically sig- nificant bit error rate evaluation by pure simulation [ 11. One of the major analytical difficulties in evaluating DS-SSMA bit error rates lies in the inaccuracy [31 [41 of a standard gaussian approximation of MA1 particularly when the number of inter- ferers is small and when the spreading gain is large [4]. In [4], an improved, albeit quite complex, “conditional” gaussian approximation was shown to provide accurate results for a correlator receiver over an additive white, gaussian noise (AWGN) channel. A simplified version of this conditional gaussian approximation was later given in [5]; the latter is computed very quickly and is very accurate except [5] , once

1

1. Importance sampling can help to a certain degree [2]

again, when the number of interferers is small and when the spreading gain is large, which may occur if one employs this technique over a quasi-static fadin channel for wideband sys- tems with imperfect power contro?. In order to avoid an accu- mulation of approximations3 and to maintain reasonable computational complexity when evaluating DS-SSMA per- formance for time-varying channels and for more complex transceiver structures, it is therefore desirable to employ tech- niques which do not rely on any such gaussian assumptions.

Among the latter set of techniques, one may express the prob- ability of error in terms of a series of orthogonal polynomials (such as Hermite polynomials[6] [7]) whose coefficients involve the moments of the interference variate. However, although the moments can be computed recursively, the number of these moments required for the series to converge can be large, even more so when the interference-to-signal ratio is also large. An alternative is the use of a Gaussian quadrature rule (GQR) technique [8] which defines the unknown density function by a quadrature rule {wj, <j}j=l.N i.e. by a set of so-called weights and nodes. Using (2N+1) moments entails the solution of a set of nonlinear equations by a diagonalization of a tridiagonal matrix [8]. The GQR tech- nique has been applied to a DS-SSMA performance evalua- tion over a variety of channels (non-fading and fading) and different transceiver structures [9] [ 101 [ 111. Another possibil- ity involves using the maximum entropy method [12].

As an alternative to these “moment methods”, which all require the evaluation of a potentially large number of moments of the decision variate, one may integrate directly over the latter’s characteristic function [6] to evaluate the error probability, an approach which does not require the moments of the decision statistic. In this case, the accuracy of the technique is influenced by the need to perform multiple numerical integrals i.e. over the characteristic function itself and also over the density functions of phase and delay in terms

2. A few interferers may then dominate the MAI, on an instanta- neous basis.

3. Further approximations may then be introduced such as a union bound for systems with convolutional coding.

0-7803-2509-5195 US$4.00 0 1995 IEEE 182

of which the characteristic function in [6] is expressed. The complexity of this so-called “characteristic function” method in [6] is proportional to NKn,ng where N is the spreading gain, K is the number of users and where nz and ng are the number of terms required to numerically integrate over the density functions of phase and delily. This paper presents a reduced-complexity formulation of the characteristic function method which eliminates the need for numerical integration over the phase distribution and yields a computational com- plexity reduction proportional to the: product of the spreading gain N and the number of terms ng required to integrate over the phase distribution. The formulation is based on the numer- ical quadrature of a Laplace inversion integral [13] along a contour in the complex plane passing through a saddlepoint of the integrand. The resulting compact expression renders the characteristic function method more attractive, in terms of complexity and accuracy, for application to future wideband CDMA systems and by extension, to more complex trans- ceiver structures and time-varying channels. The technique is described, for sake of simplicity, for BPSK in section I1 of this

where R k , ] and k k , l are the continuous-time partial cross-

correlation function which are given in terms of the aperiodic cross-correlation function Ck, [15] for the kth and mth signa-

ture sequences by [ 151: R k , r ( 2 ) = C k , c ( l - N ) T c + [ c k , l ( l + l - N ) (3)

kk, l l :z ) = C k , r ( l ) T c + [ c k , c ( ‘ + l ) (4)

- ck, 1 ( l - 1 ( z - l T c )

- ‘k, 1 ( I ) 1 (‘ - l T c )

where 1 = Lz /Tc] , i.e. the integer part of z / T c , and where

the K signature sequences are periodic with period N .

The properties of the asynchronous DS-SSMA system under consideration irnply that the collection of variables bk , zk and

paper but can be easily extended [14] to QPSK and other forms of four-phase signaling (e.g. MSK and OQPSK). In sec- ( p k , ( 2 I k 2 IC) , is a set of mutually independent random

tions 111 and Iv, the accuracy of the technique is evaluated by variables. Under fairly general conditions, zk and ( p k are comparing its results with those of previous methods.

independent and uniformly distributed on the intervals [ 0, T]

and [0,2x] , respectively. b;k) is a sequence of independent 11. PERFORMANCE EVALUATION TECHNIQUE

A. System Model data bits for each user k and Pr( bjk’ = + 1) = for each k

andj . The probability of bit error can be written in terms of the decision vaiate z,,r in The DS-SSMA system model for IK users employed in this

paper is described j,n [15]. The decision statistic for the detec- simp,y as:

( 5 ) tion of binary, asynchronous, phase-coded SSMA signals via a P , = Pr(Zour ( 1 ) < O l b i l ) = + I ) correlation receiver is [15] [ 161

B. The Characteristic Function Method + 11 (1)

(1) T -bo ( I ) + T& 5 Ik, 1 ( b k , z k ! ’ P k )

II -_. k = 2 v An efficient algorithm for using the characteristic function Zout =

desired signal MA1

where is zero-mean, white, Gaussian noise (AWGN) with variance equal to NoT/4 and wher,e ~ ~ 1 2 is the value of its

double-sided spectral density function. Ps denotes the power

of the desired signal and Pj denotes the power of each jam-

mer. The equal jammer power assumption is made for nota- tional simplicity and for convenience in presenting numerical results. T is the data bit duration. The vector

AWGN method to compute the probability of error, as in (3, has been proposed [13] in the context of a static binary symmetric channel. This technique was later applied [ 171 to the perform- ance evaluation of FD-TDMA cellular radio links on an indoor fading radio channel. The method involves the numeri- cal quadrature of a L,aplace inversion integral which is approximated via the trapezoidal rule as [ 13 J

U

bk = (b<$, b iK) ,) represents a pair of consecutive data bits 1 where G ( s ) := - R e [ s - ’ g ( s ) ] , and where g(s) denotes of the kth signal and zk and vk represent the time delay (mod- Tc

ulo r ) and the phase angle (modulo 2n ), respectively, of the kth signal relative to the first.

the moment-generating function4 (mgf> of the variate z,,, in

eq. (1) and where U is large enough such that G (so +juAy)

The function Ik , which appears in (1) represents the normal-

ized multiple-access interference (MAI) due to the kth signal. 4. The mgf of a random variable X is simply E[e-SX], while the

characteristic function is E [ e - J ~ X ] ,

183

is negligible for U > U . The numerical quadrature is per- T K

= i b j ' O [ s & ( Rk, + k k , 1 (')))'' formed on a straight vertical path through the integrand's

s- g (s) single positive saddlepoint so (found through a sim-

ple iterative procedure) on the Re(s) axis 1131 using only the mgf of the decision variate and without explicit knowledge of its probability density function. This mgf is given by:

(7)

k = 2 1

0

k = 2 0 g ($1 = E [exp (-sZour) I = gq (SI g t (SI g,(s)

where where I o ( ) is the modified Bessel function of order 0. By

defining the functions %+ (m, k ) and %- (m, k ) as g Y (s) = E [ exp( - s g T b L 1 ) ) ] = exp( -SE] (8) T

are respectively the mgfs of the signal and noise components of the decision variate. Two different cases can be considered in terms of the mgf of the multiple-access interference (MAI) component: (1) BER with a given phase ( p k and delay zk , (2) average BER over all ( b , z, cp) . 1) deterministic Dhase and delay

k = 2

and by using a result from [18], (11) can be simply rewritten as

K M s 2 m p m 1

g g ( S ) = T r J L 2 % + ( m A

'2l-I c + 2 w % k )

k = 2 m = 0 8 ~ ( m ! ) (13) K M s 2 m p m

1

k = 2 m = 0 8 ( m ! )

2m m s P .

8m ( m ! ) where M is large enough such that -2%+ ( m , k ) is negli-

0 0 >

184

According to [6], this yields a computational complexity pro- portional to 2N(K-lI)n,npu where nz , n$ and nu are respec- tively the number of points required for numerical evaluation of the integrals over [O,T,], over [0,7c/2] and over the charac- teristic function itself. However, as shown by (13), the compu- tational complexity of the new method is proportional to only 2M(K-l)nzn,,, i.e. independent of N and such that the com- plexity reduction is n,+N/M. As M need not be very large [ 181 in practice to provide good accuracy (e.g. M < 30), the com- plexity reduction will be quite significant particularly for wideband systems with large spreading gain N. If one pays closer attention to the details of the computation, it should be noted that this complexity evaluation (relative to what is given in [6]) did not account for the calculation of the 0k,, functions

in the previous version [6] and of the Rk, (z) and 8 , (7)

functions of the new version, as required in (12) and given by (3) and (4). Once this has been considered, the actual com- plexities are 4N2(K-l)nTn,pu for the previous version and 8MN(K-l)n,nU. for the new version, yielding a complexity reduction of nmN/2M.

' i .

x 4 H Walsh: 15 4

I

Eb/No= 8dB I phase=2.802657

delay=O 711213

4 : L , .

L E :

IV. RESULTS In this section we apply the characteristic-function method described in section 1I.B to the evaluation of the probability of error for the binary DS-SSMA system. All of the numerical results presented in this paper are based on a BPSK DS- SSMA system with a rectangular chip waveform.

A. Deterministic phase, delay and sequences First, we evaluate the system with fixed phase ( p k and delay

zk and we compare the results obtained by the new version of

the characteristic function method (with fixed delay and phase) against the results produced by two other different methods: (1) in Pursley's method [$I, the BER for given b, ( p k and zk can be calculated by

where a = m& and the Q-function is defined by

ca

dx [18]. The results of the bit-by-bit

Y simulations were obtained over 100000 information bits.

- 5 1 A I -10 0 10 20 30 40

JSR (dB) lO-20

Fig. 1 BER of BPSK DS-SSMA (fixed phase and delay).

Table 1 : BER of BPSK. DS-SSMA systems with different methods (Eb/l\lo=SdB,N=32,Walsh codes) and fixed phase

and delay.

char. char. Pursley Pursley func. method method method (K=2) (K=3)

(K=2) (K=3)

JSR func. (dB) method

-10 6.07515 6.07515 6.01484 6.01484 (x10-3)

0 7.20067 7.20067 6.57236 6.57236 (~10-3)

10 2.1480'7 2.14807 1.29671 1.29671 ( ~ 1 0 - 2 )

15 7.01141 7.01141 3.33456 3.33456 (~10-2)

20 2.04231 2.04231 1.15520 1.15520 (~10-1)

30 2.52977 2.52977 2.55354 2.55354 (~10-1)

The results are shown in Fig. 1 and in Table 1. There are 2 users in Fig. 1 and the processing gain N is 32. All curves are generated with the same signature sequences (4th and 15th sequences of a Walsh code set), as well as with the same delay

( zk =0.711213) and phase ( ( p k =2.802657). The dashed lines

show the results, from the new version of the characteristic function method. The circles show the results from a bit-by- bit simulation. The results from Pursley's method are overlap- ping with the dashed lines. It can be noted that the characteris- tic function method's results match exactly those of Pursley's method.

B. Average BER over ]phase and delay for deterministic

The accuracy of the new and previous versions of the charac- teristic function method can be compared against the results of the previous section (fixed phase and delay) averaged by

sequences

185

numerically generating 100000 sets of phase and delay (“sta- tistically averaged BER’)). In order to compare with the numerical results quoted in [6], the same N=31, m-sequences used in [6] are also employed with the new method. The results, for equal-power users, are presented in Fig. 2.

N=3 1, m-sequence

2 EbNo (dB)

Fig. 2 Average BER of BPSK DS-SSMA

Table 2: Average BER for BPSK DS-SSMA systems (K=3, N=3 1, m-sequences)

Char. fct. statistically Char. fct. Eb/No (new averaged (results in (dB) version) BER [61)

8 8.47558 8.46249 8.4870 (~10-4 )

9 3.13337 3.12905 3.1340 ( ~ 1 0 - 4 )

10 1.06713 1.06387 1.0760 ( ~ 1 0 - 4 )

11 3.40283 3.39345 3.4680 ( ~ 1 0 - 5 )

12 1.04015 1.03994 1.0810 ( ~ 1 0 - 5 )

13 3.12726 3.13590 3.3470 (~10-6 )

14 9.51967 9.47199 10.480 ( ~ 1 0 - 7 )

The four curves with dashed lines represent the average BER calculated by the new version of the characteristic function method where the number of users (K) is 2,3,4 and 6, in order of increasing BER. The circles represent the results for the statistically averaged BER and the results quoted from [6] are represented by stars. The three sets of results are found to be in good agreement. Further insight into the accuracy of the two versions of the characteristic function method is obtained by comparing, for K=3, the actual exact numerical values against the statistically averaged BER. The new version’s results are seen to be closer to the statistically averaged BER which is a consequence of the elimination of the numerical integration with respect to the random phase variable.

V. CONCLUSION This paper has presented a new formulation for DS-SSMA performance evaluation via the characteristic function method. This new version yields a computational complexity reduction proportional to the spreading gain which makes it particularly attractive for future wideband CDMA systems. It also eliminates the need for numerical integration with respect to a random phase variable thus improving accuracy.

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