IT IS well-known that spread-spectrum multiple access

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1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 1, NO. 1, JANUARY Accurate DS-CDMA Bit-Error Probability Calculation in Rayleigh Fading Julian Cheng, Student Member, IEEE, Norman C. Beaulieu, Fellow, IEEE Abstract A binary direct-sequence spread-spectrum multipleaccess system with rom sequences in flat Rayleigh fading is considered. A new explicit closed-form expression is obtained for the characteristic function of the multiple-access interference signals. It is shown that the overall error rate can be expressed by a single integral whose integr is nonnegative exponentially decaying. Bit-error rates (BERs) are obtained with this expression to any desired accuracy with minimal computational complexity. The dependence of the system BER on the number of transitions in the target user signature chip sequence is explicitly derived. The results are used to examine definitively the validity of three Gaussian approximations to compare the performances of synchronous systems to asynchronous systems. Index Terms Error analysis, Rayleigh channels, spread spectrum communications. I. INTRODUCTION IT IS well-known that spread-spectrum multiple access (SSMA) has the ability to combat multipath interference, increase system capacity, improve quality of service. Much work has been reported on the calculation of the user average BER for direct-sequence code-division multiple-access (DS-CDMA) systems. Two approaches for DS-CDMA, operating on additive white-gaussian noise (AWGN) channels, have been widely reported. The first approach presumes that exact BER evaluation is intractable or numerically cumbersome, so accurate bit-error rate (BER) approximations are sought [1] [11]. Perhaps the most widely cited most widely used approximation is the so-called stard Gaussian approximation (SGA) [1] [3], [6] [10]. In the SGA, a central limit theorem (CLT) is employed to approximate the sum of the multiple-access interference (MAI) signals as an additive white-gaussian process additional to the background Gaussian noise process. The receiver design, thus, consists of a conventional single-user matched filter (correlation receiver) to detect the desired user signal. The average variance of the MAI over all possible operating conditions is used to compute the signal-to-noise ratio (SNR) at the filter (correlator) output. The SGA is widely used because it is easy Manuscript received November 20, 1999; revised February 12, 2001 March 12, 2001; accepted May 31, The editor coordinating the review of this paper approving it for publication is L. Hanzo. This work was supported in part by a postgraduate scholarship from the Natural Science Engineering Research Council of Canada (NSERC), Alberta Informatics Circle of Research Excellence (icore), NSERC Canada Research Chair (CRC) grants. The authors are with the Department of Electrical Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada ( jcheng@ee.ualberta.ca; beaulieu@ee.ualberta.ca). Publisher Item Identifier S (02) to apply; however, it is well known that performance analyses based on using the SGA often overestimate the system performance, especially when the number of users in the system is small [4]. These limitations have motivated research to improve the accuracy of the SGA. In [9], the accuracy of the SGA was improved by using the stard Hermite polynomial error correction method. In [12], the statistics of the MAI signals with rom signature sequences were extensively studied. Based on the work of [12], [4] later introduced a method termed improved Gaussian approximation (IGA). The IGA is more accurate than the SGA, especially for a small number of active users [4]. However, the IGA computation requires numerical integration multiple numerical convolutions. This method was simplified in [5] such that neither knowledge of the conditional variance distribution, nor numerical integration, nor convolution is necessary to achieve acceptable BER estimation. Thus, it is termed simplified IGA (SIGA) [5]. More recently, Morrow [11] further simplified the expression attained in [5] without significant penalty in the BER accuracy. The second approach is to perform the evaluation of the SSMA system BER without knowledge of or assumptions about the MAI distribution. Many of these techniques are based on extensions of previous studies of intersymbol interference (ISI) systems. These methods include the moment space technique [13], characteristic function method [14], method of moments [15], [16], an approximate Fourier series method [17], [18]. Generally, these techniques can achieve more accurate BER estimate than CLT-based approximations at the expense of much higher computational complexity. Fewer results exist for BERs of DS-CDMA systems operating in Rayleigh fading channels employing rom sequences. This paper gives a new accurate, yet tractable BER calculation solution for a binary DS-CDMA system operating in slowly fading Rayleigh channels with rom sequences. Our treatment of the subject furthers the work reported in [4], [12], [14]. Previous related work includes the following. Borth Pursley [2] studied the SNR at the output of a correlator receiver for Rician fading channels. The performance of a DS-CDMA system in a frequency nonselective Rayleigh fading channel was evaluated by Gardner Orr [3] for deterministic sequences using the SGA. In [14], Geraniotis Pursley used the characteristic function method to evaluate SSMA system performance in an AWGN channel. Later, Geraniotis [19] extended this technique to frequency nonselective selective Rician fading channels for deterministic sequences. The characteristic function method was used in studying the performance of DS-SS systems on specular multipath fading channels with multipath ISI; however, MAI was not considered in this paper /02$ IEEE

2 4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 1, NO. 1, JANUARY 2002 [20]. In [17] [18], Sunay McLane used an approximate Fourier series method to study BER performance under both frequency nonselective selective Rayleigh fading. Additionally, the system degradations due to imperfect chip phase synchronization were assessed. The contributions of this paper are as follows. We analyze a generic DS-CDMA system with Rayleigh-distributed users under both synchronous asynchronous operations for rom sequences. We examine in detail the BER dependence on the number of active users in the system for a given SNR. For asynchronous operation, we provide an explicit closed-form expression for the characteristic function of the MAI. A single integral expression is given for the overall BER the integr is shown to be positive well behaved. With this expression, we can calculate the BER to any desired accuracy with minimal computational complexity. The IGA SIGA methods are extended to a fading channel system. Numerical results obtained from our accurate method are used to assess the accuracies of the SGA, IGA, SIGA for Rayleigh fading channels. Accurate comparisons of the performances of asynchronous synchronous systems operating in Rayleigh fading are made using our new solution. The remainder of this paper is organized as follows. In Section II, we introduce the system channel models. Review of the important properties of the receiver decision statistic for rom sequences is provided in Section III for completeness. The SGA, IGA, SIGA for a Rayleigh fading channel are examined in Section IV. System performance evaluations for synchronous asynchronous operations are presented in Sections V VI, respectively. Numerical results discussions are provided in Section VII. Conclusions are drawn in Section VIII. II. SYSTEM AND CHANNEL MODELS We consider a general asynchronous binary DS-SSMA system that supports active users. More specifically, we study the system performance under the reverse link (mobiles to base station) assumption. The th transmitted signal is described by [1] where represents the transmitted signal power is the data signal, is the spreading signal, is the carrier frequency, is the carrier phase. The th user s data signal is a rom process that is a rectangular waveform, taking values from with service rate, is expressed as where, for,, otherwise. The th data bit of the th user is denoted as. Data sources are assumed uniform, i.e., (1) (2) signal can be expressed as. The spreading where is an arbitrary chip waveform that is time-limited to is the chip duration. Chip waveform is assumed to be normalized according to. The th chip of the th user is denoted, which assumes values from. All signature sequences are assumed to be rom in the following sense. Every chip polarity is determined by flipping an unbiased coin. Further justification for the rom chip sequence assumption is provided in [23]. There are chips for one data symbol the period of the signature sequence is. 1 We normalize the chip duration so that, thus,. Note that if the chip waveform is rectangular, i.e.,, the transmitted signal becomes the well-known phased-coded SS model [1]. Each signal is transmitted over a frequency-nonselective fading channel, where the user signal the interfering signals all experience mutually independent Rayleigh fading. The fading is also assumed to be slow such that coherent detection is feasible. This channel model is widely used in system design performance studies for DS-CDMA systems [6] [10], [24]. It is also a special case of the indoor wireless channel model studied in [15]. Further justification of the applicability of this channel model is given in [6]. The channel impulse response for the th transmitted signal is given by where the fading rom variables (RVs) are independent, Rayleigh-distributed account for the fading channel attenuation of all signals. Each RV represents the envelope of a complex Gaussian process with unit variance in each quadrature component. The first-order probability density function (pdf) of is given by such that. Here, is the indicator function whose value is one when zero, otherwise. In (4), are the phases introduced by the fading channel are assumed uniform over independent. The RVs represent time delays account for the lack of time coordinations (asynchronism) among the transmitters as well as the channel transmission delays. These time delays are RVs assumed uniform over independent. The transmitted signal is further assumed to experience an additive background noise process, which is characterized as a zero-mean stationary white-gaussian process with two-sided power spectral 1 It has been shown in [21] that the periodic condition of the rom sequence can be removed without affecting the decision statistic presented in the sequel. Therefore, our assumption made for analytical simplicity is realistic for long codes (sequences of period much longer than the duration of data symbol) often employed in practical systems [22]. (3) (4) (5)

3 CHENG et al.: ACCURATE DS-CDMA BIT-ERROR PROBABILITY CALCULATION 5 density. Note that setting, in (4), results in an AWGN transmission model with romly phased signals. The received signal at the input of the matched filter receiver is given by In their notable work on rom sequences for the AWGN channel, Lehnert Pursley [12] simplified, conditioning on User 1 s signature sequence, as where are the partial autocorrelation functions of the chip waveform defined as where denotes convolution is assumed a uniform RV over. The average received power of the th signal is. Without loss of generality, we assume all transmitted signals have the same transmitted power set this value to two, i.e., (thus, ). Since all the aforementioned RVs are generated from different physical sources, we assume the RVs ; are mutually independent. A synchronous system model is obtained when, in (6). All other assumptions remain the same as for the asynchronous system model. III. REVIEW OF RECEIVER DECISION STATISTIC Consider using a conventional single-user matched filter to coherently demodulate the desired user signal in an asynchronous system. The average BER is the same for all users by symmetry. We assume, without loss of generality, that the target user signal has index. The decision statistic at the output of the correlator, after lowpass filtering (LPF), is given by [1] where is the unfaded amplitude of the th interfering signal can be expressed as In (7), the first, second, third term represent the desired signal component, the MAI component the noise component, respectively. In the first term, denotes the zeroth data symbol for User 1 it is a symmetric Bernoulli RV. 2 Also in the first term, is now usefully reinterpreted as the processing gain. The third term is a zero-mean Gaussian RV with variance. In the second term, is a uniform RV on representing the phase difference of the th user relative to User 1. In (8), are the continuous-time partial cross-correlation functions defined as [1] for. 2 A symmetric Bernoulli RV is defined here to be a discrete RV which takes values from f01; +1g with equal probabilities. (6) (7) (8) zero otherwise. For a rectangular chip waveform,,. Thus, (9) becomes (10) (11) (12) where is a uniform RV over, which accounts for the fractional chip displacement of the th interferer s chip relative to User 1. In (12), are symmetric Bernoulli RVs is a discrete RV that represents the sum of independent symmetric Bernoulli RVs, where equals the number of chip boundaries without transitions in User 1 s signature waveform. Similarly, represents the sum of independent symmetric Bernoulli RVs, where equals the number of chip boundaries with transitions in User 1 s signature waveform. Clearly, the marginal pdf s of are given by [12] (13) (14) Equation (9) has two-fold significance. First, it expresses the partial cross-correlation functions [see (8)] between the spreading waveforms in terms of the partial autocorrelation functions of the chip waveforms. This permits further simplification to forms such as (12), which will be used in the sequel. Second, for a target user, the simplifications also classify the total possible signature sequences into classes. As discussed later, the system average BER performance depends on the number of chip boundaries with (or without) transitions in the target user s signature sequence. Appropriate use of this fact (classification of the signature sequences) aids in determining tractable BER expressions. Our analysis will make use of a number of important results which are repeated here for completeness. The proofs further discussions of these results are given in [4] [12].

4 6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 1, NO. 1, JANUARY 2002 Fact 1: The MAI terms conditioned on are independent. Fact 2: The RVs,,, conditioned on are independent. Fact 3: The RVs,,, conditioned on are zero-mean. Based on above facts, the following can further be shown [4]. Fact 4: The MAI terms are uncorrelated (unconditionally). In (19), is zero-mean Gaussian with unit variance, i.e.,. This is true since is Rayleigh-distributed, is uniform over,, are independent [26, p. 146]. However, is not Gaussian, as shown in Section VI. Therefore, by the same arguments as in [4], we state that the distribution of the MAI is approximately Gaussian, conditioned on,,, where. The conditional variance of the MAI, denoted as, is given by [4] IV. CENTRAL LIMIT THEOREM APPROXIMATIONS A. Stard Gaussian Approximation In the SGA, as used in AWGN, a CLT is invoked to approximate the MAI process as Gaussian [1]. The SGA is widely used [1] [3], [6] [10] because of its simplicity. However, the SGA is also known to seriously overestimate system performance (or underestimate the BER values) for small values of [4] in AWGN channels. The average variance of the MAI is obtained by averaging over,,. For, the average variance is given by (15) Averaging across the interferers Rayleigh fading amplitudes, (15) becomes The average BER given by (16) for each user is then approximated (17) where. Averaging over in (17) with respect to the Rayleigh distribution (5) using the integral identity [24, p. 101, eq. 3.61], we approximate the average BER in Rayleigh fading using the SGA as (18) (20) Unlike the SGA, which assumes an average variance value for the MAI or the first moment of, the IGA exploits knowledge of the distribution or all moments of. The BER, given,is approximated by (21) The evaluation of the pdf for requires finding the conditional variance density function for each interferer, numerically evaluating a -fold convolution, taking the expectation with respect to. Finally, two additional numerical integrations are required to obtain the overall average BER. It was shown in [4] that, consequently, the BER for an AWGN channel obtained from the IGA is significantly more accurate than the BER obtained from the SGA especially for small values of. Holtzman [5] further simplified the calculation of (21) by using only the first second moments of, provided certain prior analyzes are done. The inaccuracies of Holtzman s simplification have been shown to be minor [5]. For our problem, using Holtzman s method assuming ( is the energy per bit), the overall BER can be shown to be B. Improved Gaussian Approximation The IGA is also a technique based on a CLT. For an AWGN channel,,, the IGA approximates the MAI as Gaussian, conditioned on,, for large, but finite values of [4], where. Here, we extend the IGA to the flat Rayleigh fading channel. Denoting the MAI as, from (7) we have where are given by (22a) (19) (22b)

5 CHENG et al.: ACCURATE DS-CDMA BIT-ERROR PROBABILITY CALCULATION 7 (22c) where (22a) is valid provided. It should be emphasized that (22a) (22c) are valid for any values of, though in our derivation, we have assumed. Alternatively, (22a) (22c) can be derived by taking account of the distribution of the received power. The instantaneous received power for the th user is or, where is a chi-square RV with two degrees of freedom unit variance. In [25], Holtzman s method is extended by including the first second moments for the received signal power. Using, as well as [25, p , eqs. C.96 C.98], we again obtain (22a) (22c). V. SYNCHRONOUS BER ANALYSIS In this section, we consider the BER calculation for a synchronous system, i.e., in (6). The BER for a synchronous system operating in flat Rayleigh fading is known. However, the present derivation both serves completeness clarifies the derivation understing of the asynchronous results to follow. The results will also be used later to compare the performances of asynchronous synchronous systems. We first assume all signature sequences are deterministic. The output of the matched filter, after LPF, for User 1 is given by [24] of implies the independence of. Since a sum of independent Gaussian RVs has a Gaussian distribution, it follows that is a Gaussian RV with zero-mean variance. By symmetry using the independence of, one has, we obtain the BER for a Rayleigh- Averaging over the pdf of faded user as (25) (26) The BER expression (26) is the same as the expression derived in [24, eq ] as expected. Physically, from (26), one sees that the interferers act like additional independent Gaussian background noise. This is because the MAI on the flat Rayleigh fading channel (inclusion of the modulation on the carriers) has a Gaussian first-order distribution assuming synchronous transmission. Importantly, this implies that the optimum receiver (that does not perform user-interference cancellation) is a correlator detector. In Section VI, it is shown that this is not the case for asynchronous transmission. For uniform rom signature sequences, E [24] (23) where is a zero-mean Gaussian RV with variance, is the signal component, the interference term is given by (24a) (24b) (24c) VI. ASYNCHRONOUS BER ANALYSIS (27) In this section, we will use the characteristic function method to compute the average BER under asynchronous transmission conditions in flat Rayleigh fading. It will be shown in Section VII that the average BER can be obtained with this method to any desired accuracy. We first examine the statistics of each interferer. From (19), we have, where is a zero-mean unit-variance Gaussian RV is defined in (12). Thus, given, is a Gaussian RV with zero mean conditional variance. This implies through (12) that given,,,,, is Gaussian the conditional pdf for follows as In (24a), we have defined the full-period, cross-correlation coefficient between the th User 1 s signature sequences as. We set. Note that, as before, is a zero-mean, unit-variance Gaussian RV. Since takes values from with equal probability independently of the values of, is also a zero-mean unit-variance Gaussian RV. Now, note that the independence (28) Since may take negative values, a modulus operation is required in (28). Averaging over,,, (which is equiv-

6 8 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 1, NO. 1, JANUARY 2002 Fig. 1. Conditional characteristic function for each interferer 8 (!) with N =31 B =30. alent to averaging over all interferers spreading sequences data sequences, see [12, eq. 12]) yields Note that to obtain (29a), Fact 2 has been used, i.e., given are independent. It is clear from (29a) that the pdf of given is not Gaussian, though the functional form is a weighted summation of Gaussian-like terms. We postpone averaging here since they appear in the denominators of the exponential function arguments giving an intractable integral. The characteristic function of,given,is (29a) where (29b) (29c) (29d) (29e) The s now appear in the numerators of the exponential function arguments averaging can be carried out to give

7 CHENG et al.: ACCURATE DS-CDMA BIT-ERROR PROBABILITY CALCULATION 9 Fig. 2. Integr of (39) with K =5, N =31, B =15for received SNR = 20 db. from (31) that is real. This implies is symmetric about. The function in (32) has an interesting geometric interpretation. To see this, for, we write (31) where (32) The detailed derivation of is given in the Appendix; the other terms are derived in a similar manner. Observe also (33) where lies between, is the first-order derivative of the function, is the third-order derivative of the function. From (33), it is clear that, for, the function is just the central difference derivative approximation (scaled by a constant factor) of the derivative of the stard function evaluated at. For, the second term in (33) vanishes for all values of this approximation

8 10 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 1, NO. 1, JANUARY 2002 Fig. 3. Integr of (40) with K =5, N =31, B =30. becomes an exact scaling of the derivative of the function evaluated at. Using the fact that the s given are independent, we have the characteristic function for the total interference term,given,as (34) Let, where is a Gaussian RV representing the background noise, is the total other-user interference given, is the total disturbance given. Since the other-user interference background noise are independent, we have We use the Fourier inversion formula for the real integral [27, p. 40, eq. 3 22] to find the distribution function of,, which is to be used to calculate the BER, as (36) The conditional BER for our target user can be expressed, by symmetry, as (35) (37)

9 CHENG et al.: ACCURATE DS-CDMA BIT-ERROR PROBABILITY CALCULATION 11 Fig. 4. System BER performance without background noise for N =7. Averaging over the pdf of [28, eq ], using the integral identity (38) all signature sequences romly assigned by a base station for each service request is (41) we have When the effect of the background noise is negligible,, so (39) becomes (39) (40) Equations (34), (39), (31) [(or (34), (40) (31) for the noiseless case] give the average BER experienced by a target user with a signature sequence that has a given value of. The average BER for all users or for one target user averaged over An interesting result seen in (41) is that the average BER over all signature sequences can be obtained by averaging over classes of sequences rather than over possible rom sequences. Consequently, the order of the computational complexity in the number of user sequences is reduced from exponential to linear. This conclusion was stated previously in [12]. VII. NUMERICAL RESULTS AND DISCUSSIONS In previous analyses of SSMA system performance based on the characteristic function method, the characteristic function of each interferer is given as iterated integrals [14], [18] with two to three levels of numerical integration. At best, it can be expressed as a single integral for a binary phase shift keying (BPSK) system with rectangular chip waveform for the AWGN channel, but the integr is not well behaved [14, eq. 14]. In (31), we present an explicit closed-form expression for the characteristic function involving only the exponential function the function. It can be shown that the expression requires computation summation of at most terms. Since the error function is widely available in many scientific software tools such as Matlab Maple, this expression can be readily programmed for direct evaluation. A typical plot of is illustrated in Fig. 1 for.

10 12 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 1, NO. 1, JANUARY 2002 Fig. 5. System BER performance without background noise for N =31. Observe that has a Gaussian-like shape that decays exponentially in for increasing frequency (the function can be well approximated as exponential in for large values of arguments). Note that (31) implies that the pdf of the MAI has support over ; this is in contrast to the the AWGN channel case in which the support of the MAI pdf is finite over, where is some constant. Consequently, the BER bounding technique described in [12] is not applicable to Rayleigh fading channels. The conditional BER expressions in (39) (40) are important new results. Known error rate expressions are given in a form similar to (36), but with an integr that contains a singularity point, moreover, is oscillatory with a varying period (dependent on ). This is a very undesirable feature for numerical integration. A series expansion method was suggested in [14] to enhance the accuracy; however, large numbers of terms are required to obtain a good approximation the speed of convergence depends on certain parameters. The integr functions in (39) (40) are extremely well behaved in the sense that they are smooth, strictly nonnegative, have no oscillations, decay exponentially with increasing. This is clearly seen from typical integr functions plotted in Fig. 2 for the case of a system with background noise in Fig. 3 for a case without background noise. Hence, simple numerical integration techniques can be employed to obtain the BER to any desired accuracy. In this paper, all numerical results were obtained using the composite Simpson s rule. Compared to the approximate Fourier series method [18], our new integral solution also provides greater dynamic range for the average error rate. This is because the series given in [18] is an alternating series. Fig. 4 shows numerical results for the conditional BER with three values of,,,. A short sequence period of no background noise are assumed. For, User 1 s signature sequence has no transitions at any chip boundary. This corresponds to a narrow-b signal subject to wide-b interferences; hence, it yields the worst BER performance. On the other h, for, User 1 s signature sequence has transitions at every chip boundary, consequently, highest spreading gain; this results in the best BER performance. However, the unfavorable autocorrelation properties of such sequences may introduce difficulties for chip synchronization [11]. These two extreme case performances provide upper lower bounds for the BER experienced by any user in the system for a single particular transmission using a particular signature sequence. User sequences having the mean value of, i.e.,, have transitions at half of the chip boundaries of User 1. As expected, the BER lies between the BERs of the cases. Fig. 4 shows the accurate BER computed using the characteristic function method described in Section VI, as well as the BER obtained by the SGA. Several interesting observations can be made. As shown, the accurate BER averaging over all possible values of s is well approximated by the average BER obtained by assuming the mean value of. These two numbers agree to two to three significant figures for many cases [29]. We also observe from Fig. 4 that the SGA provides excellent approximation to the accurate BER computed via the charac-

11 CHENG et al.: ACCURATE DS-CDMA BIT-ERROR PROBABILITY CALCULATION 13 Fig. 6. System BER performance with background noise for N =31. teristic function method this is true even for the case of a single interferer. We observe, from our numerical results, that the BER estimates based on the SGA consistently overestimate the accurate BER. This is in contrast to the situation for AWGN channels where the SGA underestimates the BER [4], [5]. Fig. 5 examines a system with larger processing gain. Similar observations conclusions can be drawn. Comparing Fig. 4 with Fig. 5, we see that the SGA is a better approximation for than. This is expected in consideration of a CLT. These tests on the validity of the SGA suggest there is no substantial benefit to using the IGA or SIGA rather than the SGA for our fading channel system. Indeed, our numerical results for show that, in most cases, the improvements in the BER accuracy are in the third significant digits for the SIGA [29]. Our definition of synchronous operation in Section II implies perfect chip alignment, but no phase alignment. The BERs under synchronous operation are plotted in Figs We observe that the BER under synchronous operation is no better than the worst case BER (upper bound for ) under asynchronous operation. The poor error rate performance under synchronous operation for our fading channel system agrees with previous findings for the AWGN channel [4]. Fig. 6 shows the accurate BER results the SGA for a system operating with different background noise levels for. We observe that for all values of SNR, the SGA provides excellent approximation to the accurate BER computed via the characteristic function method, even for a system with a single interfering signal. VIII. CONCLUSION An accurate analytical solution for the BER of a DS-CDMA system operating over Rayleigh fading channels has been derived. Also, a new closed-form expression was provided for the characteristic function of the inerfering signals. In contrast to previous analytical solutions, the new solution is of very moderate complexity, requiring a single numerical integration of an exponentially decaying positve integr for any number of system users. The solution is rapid for small moderate values of processing gain. Any arbitrary degree of accuracy in the results can be achieved by using stard techniques of numerical integration. The new solution has been applied to assess the validities of three Gaussian BER approximations for Rayleigh fading to compare the BER performances of asynchronous systems to synchronous systems in Rayleigh fading. APPENDIX In this appendix, we show the details in obtaining in (31). For,, integration of in the first exponential term in (30) gives

12 14 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 1, NO. 1, JANUARY 2002 where the last equality follows from the fact. Define in (32) follows immediately. For,. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their comments, which helped to improve the presentation of this paper. REFERENCES [1] M. B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication Part I: System analysis, IEEE Trans. Commun., vol. COM-25, pp , Aug [2] D. E. Borth M. B. Pursley, Analysis of direct-sequence spread spectrum multiple-access communication over Rician fading channels, IEEE Trans. Commun., vol. COM-27, pp , Oct [3] C. S. Gardner J. A. Orr, Fading effects on the performance of a spread spectrum multiple-access communication system, IEEE Trans. Commun., vol. COM-27, pp , Jan [4] R. K. Morrow J. S. Lehnert, Bit-to-bit error dependence in slotted DS/SSMA packet systems with rom signature sequences, IEEE Trans. Commun., vol. 37, pp , Oct [5] J. M. Holtzman, A simple, accurate method to calculate spread-spectrum multiple-access error probabilities, IEEE Trans. Commun., vol. 40, pp , Mar [6] L. B. Milstein, T. S. Rappaport, R. Barghouti, Performance evaluation for cellular CDMA, IEEE J. Select. Areas Commun., vol. 10, pp , May [7] A. M. Monk, M. Davis, L. B. Milstein, C. W. Helstrom, A noisewhitening approach to multiple-access noise rejection Part 1: Theory background, IEEE J. Select. Areas Commun., vol. 12, pp , [8] O. K. Tonguz M. M. Wang, Cellular CDMA networks impaired by Rayleigh fading: System performance with power control, IEEE Trans. Veh. Technol., vol. 43, pp , Aug [9] B. Q. Long, J. D. Hu, P. Zhang, Method to improve Gaussian approximation accuracy for calculation of spread-spectrum multiple-access error probabilities, IEE Electron. Lett., vol. 31, pp , [10] K. L. Cheah, S. W. Oh, K. H. Li, Efficient performance analysis of asynchronous cellular CDMA over Rayleigh-fading channels, IEEE Commun. Lett., vol. 1, pp , May [11] R. K. Morrow, Accurate CDMA BER calculations with low computational complexity, IEEE Trans. Commun., vol. 46, pp , Nov [12] J. S. Lehnert M. B. Pursley, Error probabilities for binary directsequence spread spectrum communications with rom signature sequences, IEEE Trans. Commun., vol. COM-35, pp , Jan [13] K. Yao, Error probability of asynchronous spread spectrum multipleaccess communication systems, IEEE Trans. Commun., vol. COM-25, pp , Aug [14] E. A. Geraniotis M. B. Pursley, Error probability for direct-sequence spread-spectrum multiple-access communications Part 2: Approximations, IEEE Trans. Commun., vol. COM-30, pp , May [15] M. Kavehrad, Performance of nondiversity receivers for spread spectrum in indoor wireless communication, AT& T Tech. J., vol. 64, pp , July Aug [16] K. -T. Wu S. -A. Tsaur, Error performance for diversity DS-SSMA communication in fading channels, Proc. Inst. Elect. Eng. Commun., vol. 141, pp , Oct [17] M. O. Sunay P. J. McLane, Calculating error probabilities for DS CDMA systems: When not to use the Gaussian approximation, in Proc. IEEE GLOBECOM, London, U.K., Nov. 1996, pp [18], Probability of error for diversity combining in DS-CDMA systems with synchronization errors, Eur. Trans. Telecommun., vol. 9, pp , Sept. Oct [19] E. A. Geraniotis, Direct-sequence spread-spectrum multiple-access communications over nonselective frequency-selective Rician fading channels, IEEE Trans. Commun., vol. COM-34, pp , Aug [20] E. A. Geraniotis M. B. Pursley, Performance of coherent direct-sequence spread-spectrum communications over specular multipath fading channels, IEEE Trans. Commun., vol. COM-33, pp , June [21] M. O. Sunay, Performance analysis of direct sequence code division multiple access systems, Ph.D. dissertation, Queen s University, Kingston, ON, Canada, [22] Cellular System-Base Station Compatibility Stard for Dual-Mode Wideb Spread Spectrum Cellular System IS-95A, Telecommunications Industry Association, TIA/EIA, Washington, DC, [23] E. Geraniotis B. G. Ghaffari, Performance of binary quaternary direct sequence spread-spectrum multiple-access systems with rom signature sequences, IEEE Trans. Commun., vol. 39, pp , May [24] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, [25] T. S. Rappaport, Wireless Communications: Principles Practice. Piscataway, NJ: IEEE Press, [26] A. Papoulis, Probability, Rom Variables, Stochastic Processes, 3rd ed. New York: McGraw-Hill, [27], The Fourier Integral its Applications. New York: McGraw- Hill, [28] I. S. Gradshteyn I. M. Ryzhik, Table of Integrals, Series Products. New York: Academic, [29] J. Cheng, Performance Analyses of Digital Communication Systems With Fading Interference, Ph.D. dissertation, to be published. Julian Cheng (S 96) received the B.Eng. degree (First Class) in electrical engineering from the University of Victoria, Victoria, BC, Canada in 1995 the M.Sc. (Eng.) degree in mathematics engineering from Queen s University, Kingston, ON, Canada, in He is currently working toward the Ph.D. degree in electrical computer engineering at the University of Alberta, Edmonton, AB, Canada. His current research interests include digital communications over fading channels statistical signal processing for wireless applications. Mr. Cheng received a postgraduate scholarship from the Natural Sciences Engineering Research Council of Canada (NSERC).

13 CHENG et al.: ACCURATE DS-CDMA BIT-ERROR PROBABILITY CALCULATION 15 Norman C. Beaulieu (S 82 M 86 SM 89 F 99) received the B.A.Sc. (honors), M.A.Sc., Ph.D. degrees in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 1980, 1983, 1986, respectively. He was a Queen s National Scholar Assistant Professor with the Department of Electrical Engineering, Queen s University, Kingston, ON, Canada, from September 1986 to June 1988, an Associate Professor from July 1988 to June 1993, a Professor from July 1993 to August In September 2000, he became the icore Research Chair in Broadb Wireless Communications, University of Alberta, Edmonton, AB, Canada, in January 2001, the Canada Research Chair in Broadb Wireless Communications. His current research interests include broadb digital communications systems, fading channel modeling simulation, interference prediction cancellation, decision-feedback equalization. Dr. Beaulieu is a Member of the IEEE Communication Theory Committee served as its Representative to the Technical Program Committee of the 1991 International Conference on Communications as Co-Representative to the Technical Program Committee of the 1993 International Conference on Communications the 1996 International Conference on Communications. He was General Chair of the Sixth Communication Theory Mini-Conference in association with GLOBECOM 97 Co-Chair of the Canadian Workshop on Information Theory He has been an Editor of Wireless Communication Theory of the IEEE TRANSACTIONS ON COMMUNICATIONS since January 1992, an Associate Editor for Wireless Communication Theory of the IEEE COMMUNICATIONS LETTERS since November 1996, Editor-in-Chief of the IEEE TRANSACTIONS ON COMMUNICATIONS since January 2000, on the Editorial Board of the PROCEEDINGS OF THE IEEE since November He received the Natural Science Engineering Research Council (NSERC) E. W. R. Steacie Memorial Fellowship in 1999 the University of British Columbia Special University Prize in Applied Science in 1980.