Accurate DS-CDMA Bit-Error Probability Calculation in ... - CiteSeerX

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Accurate DS-CDMA Bit-Error Probability Calculation in Rayleigh Fading Julian Cheng, Student Member, IEEE, and Norman C. Beaulieu, Fellow, IEEE

Abstract—A binary direct-sequence spread-spectrum multipleaccess system with random 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 integrand is nonnegative and 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 and to compare the performances of synchronous systems to asynchronous systems. Index Terms—Error analysis, Rayleigh channels, spread spectrum communications.

I. INTRODUCTION

I

T IS well-known that spread-spectrum multiple access (SSMA) has the ability to combat multipath interference, increase system capacity, and 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 and most widely used approximation is the so-called standard 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 and March 12, 2001; accepted May 31, 2001. The editor coordinating the review of this paper and approving it for publication is L. Hanzo. This work was supported in part by a postgraduate scholarship from the Natural Science and Engineering Research Council of Canada (NSERC), Alberta Informatics Circle of Research Excellence (iCore), NSERC and Canada Research Chair (CRC) grants. The authors are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 1536-1276(02)00187-3.

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 standard Hermite polynomial error correction method. In [12], the statistics of the MAI signals with random 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 and 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], and 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 and employing random 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 random sequences. Our treatment of the subject furthers the work reported in [4], [12], and [14]. Previous related work includes the following. Borth and 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 and Orr [3] for deterministic sequences using the SGA. In [14], Geraniotis and Pursley used the characteristic function method to evaluate SSMA system performance in an AWGN channel. Later, Geraniotis [19] extended this technique to frequency nonselective and 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

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[20]. In [17] and [18], Sunay and McLane used an approximate Fourier series method to study BER performance under both frequency nonselective and selective Rayleigh fading. Additionally, the system degradations due to imperfect chip and 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 and asynchronous operations for random 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 and the integrand is shown to be positive and well behaved. With this expression, we can calculate the BER to any desired accuracy and with minimal computational complexity. The IGA and 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, and SIGA for Rayleigh fading channels. Accurate comparisons of the performances of asynchronous and 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 and channel models. Review of the important properties of the receiver decision statistic for random sequences is provided in Section III for completeness. The SGA, IGA, and SIGA for a Rayleigh fading channel are examined in Section IV. System performance evaluations for synchronous and asynchronous operations are presented in Sections V and VI, respectively. Numerical results and 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 active users. More specifically, we system that supports study the system performance under the reverse link (mobiles to base station) assumption. The th transmitted signal is described by [1]

. The spreading signal

can be expressed as (3)

is an arbitrary chip waveform that is time-limited to and 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 . All signature sequences are assumed from to be random in the following sense. Every chip polarity is determined by flipping an unbiased coin. Further justification for the random chip sequence assumption is provided in [23]. There are chips for one data symbol and the period of the signature sequence is .1 We normalize the chip duration so that and, thus, . Note that if the chip waveform is rectangular, , the transmitted signal i.e., becomes the well-known phased-coded SS model [1]. Each signal is transmitted over a frequency-nonselective fading channel, where the user signal and 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 and 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

(4) are indepenwhere the fading random variables (RVs) dent, Rayleigh-distributed and account for the fading channel atrepresents the envelope of a tenuation of all signals. Each RV complex Gaussian process with unit variance in each quadrature component. The first-order probability density function (pdf) of is given by (5)

(2)

. Here, is the indicator function such that and zero, otherwise. In (4), whose value is one when are the phases introduced by the fading channel and and independent. The RVs are assumed uniform over represent time delays and account for the lack of time coordinations (asynchronism) among the transmitters as well as the channel transmission delays. These time delays are RVs and independent. The transmitted assumed uniform over signal is further assumed to experience an additive background , which is characterized as a zero-mean stanoise process tionary white-Gaussian process with two-sided power spectral

, for , and , where otherwise. The th data bit of the th user is denoted as . Data sources are assumed uniform, i.e.,

1It has been shown in [21] that the periodic condition of the random 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].

(1) represents the transmitted signal power is the where is the spreading signal, is the carrier fredata signal, is the carrier phase. The th user’s data signal quency, and is a random process that is a rectangular waveform, taking values from with service rate , and is expressed as

CHENG et al.: ACCURATE DS-CDMA BIT-ERROR PROBABILITY CALCULATION

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density . Note that setting , in (4), results in an AWGN transmission model with randomly phased signals. at the input of the matched filter The received signal receiver is given by

In their notable work on random sequences for the AWGN , conditioning channel, Lehnert and Pursley [12] simplified on User 1’s signature sequence, as

where and are the partial autocorrelation functions of the chip waveform defined as (10)

(6) where denotes convolution and is assumed a uniform RV over . The average received power . Without loss of genof the th signal is erality, we assume all transmitted signals have the same transmitted power and set this value to two, i.e., (thus, ). Since all the aforementioned RVs are generated from different physical sources, we assume the

and (11) and zero otherwise. For a rectangular chip waveform , , and . Thus, (9) becomes

are

(12)

mutually independent. A synchronous system model is obtained when , in (6). All other assumptions remain the same as for the asynchronous system model.

is a uniform RV over , which accounts for the where fractional chip displacement of the th interferer’s chip relative and are symmetric Bernoulli RVs and to User 1. In (12), 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. represents the sum of independent symmetric Similarly, Bernoulli RVs, where equals the number of chip boundaries with transitions in User 1’s signature waveform. Clearly, and the marginal pdf’s of and are given by [12]

RVs

;

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 and . The target user signal has index decision statistic at the output of the correlator, after lowpass filtering (LPF), is given by [1]

(13) (7) and where is the unfaded amplitude of the th interfering signal and can be expressed as (8) In (7), the first, second, and third term represent the desired signal component, the MAI component and the noise component, respectively. In the first term, denotes the zeroth data symbol for User 1 and it is a symmetric Bernoulli RV.2 Also is now usefully reinterpreted as the proin the first term, is a zero-mean Gaussian RV cessing gain. The third term with variance . In the second term, is a unirepresenting the phase difference of the th form RV on and are the conuser relative to User 1. In (8), tinuous-time partial cross-correlation functions defined as [1] and for . 2A symmetric Bernoulli RV is defined here to be a discrete RV which takes values from ; with equal probabilities.

f01 +1g

(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 signature sequences into classes. As the total possible 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 and further discussions of these results are given in [4] and [12].

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Fact 1: The MAI terms conditioned on are independent. , , and conditioned on are Fact 2: The RVs , independent. , , and conditioned on are Fact 3: The RVs , zero-mean. Based on above facts, the following can further be shown [4]. Fact 4: The MAI terms are uncorrelated (unconditionally).

is zero-mean Gaussian with unit variIn (19), . This is true since is Rayleigh-disance, i.e., , and , are independent tributed, is uniform over [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 , , and , where . The conditional variance of the MAI, denoted as , is given by [4]

IV. CENTRAL LIMIT THEOREM APPROXIMATIONS A. Standard 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 [4] in underestimate the BER values) for small values of AWGN channels. The average variance of the MAI is obtained , the by averaging over , , and . For average variance is given by (15)

(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

Averaging across the interferers’ Rayleigh fading amplitudes, (15) becomes (21) (16) The average BER given by

for each user is then approximated

(17)

where . Averaging over in (17) with respect to the Rayleigh distribution (5) and using the integral identity [24, p. 101, eq. 3.61], we approximate the average BER in Rayleigh fading using the SGA as

The evaluation of the pdf for requires finding the conditional variance density function for each interferer, numerically -fold convolution, and taking the expectaevaluating a tion 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 and 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, ( is using Holtzman’s method and assuming the energy per bit), the overall BER can be shown to be

(18)

B. Improved Gaussian Approximation The IGA is also a technique based on a CLT. For an AWGN , , the IGA approximates channel, for large, the MAI as Gaussian, conditioned on , , and but finite values of [4], where and . Here, we extend the IGA to the flat Rayleigh fading channel. Denoting the MAI as , from (7) we have (19)

(22a)

where

and

are given by

(22b)

CHENG et al.: ACCURATE DS-CDMA BIT-ERROR PROBABILITY CALCULATION

and

(22c) . It should be where (22a) is valid provided emphasized that (22a)–(22c) are valid for any values of and , though in our derivation, we have assumed and . Alternatively, (22a)–(22c) can be derived by taking account of the distribution of the received power . The instantaneous or , where received power for the th user is is a chi-square RV with two degrees of freedom and unit variance. In [25], Holtzman’s method is extended by including the first and second moments for the received signal power. Using and , as well as [25, p. 588–589, eqs. C.96–C.98], we again obtain (22a)–(22c). V. SYNCHRONOUS BER ANALYSIS In this section, we consider the BER calculation for a synin (6). The BER chronous system, i.e., for a synchronous system operating in flat Rayleigh fading is known. However, the present derivation both serves completeness and clarifies the derivation and understanding of the asynchronous results to follow. The results will also be used later to compare the performances of asynchronous and synchronous systems. We first assume all signature sequences are determin. The output of the matched filter, after LPF, for istic and User 1 is given by [24]

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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 and variance . By symmetry and using the independence of and , one has (25)

Averaging over the pdf of faded user as

, we obtain the BER for a Rayleigh-

(26)

The BER expression (26) is the same as the expression derived in [24, eq. 3.135] 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 random signature sequences, E [24] and (27)

(23) where

is a zero-mean Gaussian RV with variance , is the signal component , and the interference term is given by

(24a)

(24b)

(24c) In (24a), we have defined the full-period, cross-correlation between the th and User 1’s signature coefficient sequences as . We set and . Note that, as before, is a zero-mean, unit-variance Gaussian RV. Since takes values from with equal probability , is also a zero-mean independently of the values of unit-variance Gaussian RV. Now, note that the independence

VI. ASYNCHRONOUS BER ANALYSIS 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 , where is interferer . From (19), we have a zero-mean unit-variance Gaussian RV and is defined in , is a Gaussian RV with zero mean and (12). Thus, given . This implies through (12) conditional variance , , , , is Gaussian and the condithat given , tional pdf for follows as

(28) Since may take negative values, a modulus operation is required in (28). Averaging over , , , (which is equiv-

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Fig. 1. Conditional characteristic function for each interferer 8

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(!) with N = 31 and B = 30.

alent to averaging over all interferers’ spreading sequences and data sequences, see [12, eq. 12]) yields

and 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 and is not Gaussian, though the functional form is a weighted summation of “Gaussian-like” terms. We postpone here since they appear in the denominators of the averaging exponential function arguments giving an intractable integral. and , is The characteristic function of , given

(29a) where (29b) (29c) (29d) (29e)

The ’s now appear in the numerators of the exponential function arguments and averaging can be carried out to give

CHENG et al.: ACCURATE DS-CDMA BIT-ERROR PROBABILITY CALCULATION

Fig. 2.

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Integrand of (39) with K = 5, N = 31, and B = 15 for received SNR = 20 dB.

is real. This implies is symfrom (31) that . The function in (32) has an intermetric about esting geometric interpretation. To see this, for , we write

(31) where

(33)

lies between and , is the first-order derivative of the function, and is the third-order derivafunction. From (33), it is clear that, for , the tive of the (32) function is just the central difference derivative approximation (scaled by a constant factor) of the derivative of the stanThe detailed derivation of is given in the Appendix; dard function evaluated at . For , the second term the other terms are derived in a similar manner. Observe also in (33) vanishes for all values of and and this approximation where

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Fig. 3. Integrand of (40) with K = 5, N = 31, and 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

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) (34)

The conditional BER for our target user can be expressed, by symmetry, as

, where is a Gaussian RV representing Let is the total other-user interference the background noise, is the total disturbance given . Since the given , and other-user interference and background noise are independent, we have

(35)

(37)

CHENG et al.: ACCURATE DS-CDMA BIT-ERROR PROBABILITY CALCULATION

Fig. 4.

System BER performance without background noise for

Averaging over the pdf of [28, eq. 3.952–1]

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N = 7.

, and using the integral identity

all signature sequences randomly assigned by a base station for each service request is

(38)

(41)

we have 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 random 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]. (39) VII. NUMERICAL RESULTS AND DISCUSSIONS When the effect of the background noise is negligible, and , so (39) becomes

(40) Equations (34), (39), and (31) [(or (34), (40) and (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

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 integrand 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 and the function. It can be shown that the expression reterms. quires computation and summation of at most Since the error function is widely available in many scientific software tools such as Matlab and Maple, this expression can be readily programmed for direct evaluation. A typical plot of is illustrated in Fig. 1 for and .

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Fig. 5.

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System BER performance without background noise for

N = 31.

Observe that has a Gaussian-like shape that decays for increasing frequency (the funcexponentially in for large tion can be well approximated as exponential in values of arguments). Note that (31) implies that the pdf of the ; this is in contrast to the the MAI has support over AWGN channel case in which the support of the MAI pdf is fi, where is some constant. Consequently, the nite over BER bounding technique described in [12] is not applicable to Rayleigh fading channels. The conditional BER expressions in (39) and (40) are important new results. Known error rate expressions are given in a form similar to (36), but with an integrand that contains a singularity point and, 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 and the speed of convergence depends on certain parameters. The integrand functions in (39) and (40) are extremely well behaved in the sense that they are smooth, strictly nonnegative, have no oscillations, and decay exponentially with increasing . This is clearly seen from typical integrand functions plotted in Fig. 2 for the case of a system with background noise and 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 , , and three values of , . A short sequence period of and no background , User 1’s signature sequence noise are assumed. For has no transitions at any chip boundary. This corresponds to a narrow-band signal subject to wide-band interferences; hence, it yields the worst BER performance. On the other hand, for , User 1’s signature sequence has transitions at every chip boundary and, 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 and lower bounds for the BER experienced by any user in the system for a single particular transmission using a particular signature sequence. User sequences , have transitions at having the mean value of , i.e., half of the chip boundaries of User 1. As expected, the BER lies and cases. between the BERs of the 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-

CHENG et al.: ACCURATE DS-CDMA BIT-ERROR PROBABILITY CALCULATION

Fig. 6.

System BER performance with background noise for

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N = 31.

teristic function method and 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 . Similar examines a system with larger processing gain observations and 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. 4 and 5. We observe that the BER under synchronous operation is no better than the ) under asynchronous worst case BER (upper bound for 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 and 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 integrand for any number of system users. The solution is rapid for small and moderate values of processing gain. Any arbitrary degree of accuracy in the results can be achieved by using standard techniques of numerical integration. The new solution has been applied to assess the validities of three Gaussian BER approximations for Rayleigh fading and 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

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where the last equality follows from the fact . in (32) and follows immediately. For Define , .

ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their comments, which helped to improve the presentation of this paper.

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Julian Cheng (S’96) received the B.Eng. degree (First Class) in electrical engineering from the University of Victoria, Victoria, BC, Canada in 1995 and the M.Sc. (Eng.) degree in mathematics and engineering from Queen’s University, Kingston, ON, Canada, in 1997. He is currently working toward the Ph.D. degree in electrical and computer engineering at the University of Alberta, Edmonton, AB, Canada. His current research interests include digital communications over fading channels and statistical signal processing for wireless applications. Mr. Cheng received a postgraduate scholarship from the Natural Sciences and Engineering Research Council of Canada (NSERC).

CHENG et al.: ACCURATE DS-CDMA BIT-ERROR PROBABILITY CALCULATION

Norman C. Beaulieu (S’82–M’86–SM’89–F’99) received the B.A.Sc. (honors), M.A.Sc., and Ph.D. degrees in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 1980, 1983, and 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, and a Professor from July 1993 to August 2000. In September 2000, he became the iCORE Research Chair in Broadband Wireless Communications, University of Alberta, Edmonton, AB, Canada, and in January 2001, the Canada Research Chair in Broadband Wireless Communications. His current research interests include broadband digital communications systems, fading channel modeling and simulation, interference prediction and cancellation, and decision-feedback equalization. Dr. Beaulieu is a Member of the IEEE Communication Theory Committee and served as its Representative to the Technical Program Committee of the 1991 International Conference on Communications and as Co-Representative to the Technical Program Committee of the 1993 International Conference on Communications and the 1996 International Conference on Communications. He was General Chair of the Sixth Communication Theory Mini-Conference in association with GLOBECOM’97 and Co-Chair of the Canadian Workshop on Information Theory 1999. 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, and on the Editorial Board of the PROCEEDINGS OF THE IEEE since November 2000. He received the Natural Science and Engineering Research Council (NSERC) E. W. R. Steacie Memorial Fellowship in 1999 and the University of British Columbia Special University Prize in Applied Science in 1980.

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