Performance Evaluation for Band-limited DS-CDMA Systems Based on Simplified Improved Gaussian Approximation

Transcription

1 1204 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 7, JULY 2003 Performance Evaluation for Band-limited DS-CDMA Systems Based on Simplified Improved Gaussian Approximation Guozhen Zang and Cong Ling, Student Member, IEEE Abstract The standard Gaussian approximation (SGA) for error analysis of direct-sequence code-division multiple-access (DS-CDMA) systems is very optimistic in many cases. Improved Gaussian approximation (IGA) is a technique that produces accurate error probabilities, but is still computationally intensive. Simplified IGA (SIGA) has complexity similar to that of SGA and, at the same time, provides sufficient accuracy. In this paper, we consider SIGA for DS-CDMA systems employing random sequences in a band-limited scenario. The validity of IGA for band-limited systems is established in a rigorous mathematical sense. Then a key parameter in SIGA is derived via a frequency-domain approach. Applications to a number of typical chip waveforms, including the popular sinc and raised-cosine pulses, are investigated. Performance comparison with IGA-based lower and upper bounds shows that SIGA yields very accurate probability of error. Index Terms Band-limited signals, code-division multiple access (CDMA), error analysis, frequency domain analysis, Gaussian distributions. I. INTRODUCTION THE evaluation of bit-error probabilities for direct-sequence code-division multiple-access (DS-CDMA) systems is a problem of long interest. Over the past few decades, substantial research has been devoted to approximations [1], bounds [2], [3] or exact calculations [4], [5] for the performance of DS-CDMA. Early work [1] treated the multiple-access interference (MAI) as a Gaussian random variable, and used the signal-to-noise ratio (SNR) as a single figure of merit in performance evaluation. The Gaussian approximation (GA) of this method was based on intuitive application of the central limit theorem. However, it is generally not accurate enough, since the density of MAI decays at a much slower rate than a Gaussian density for large arguments [3]. It is well known that the standard Gaussian approximation (SGA) substantially underestimates the error probability when the number of users is small and when the spreading factor is large. Accurate analysis was developed by Geraniotis et al. [4], [5] Paper approved by M. Chiani, the Editor for Wireless Communication of the IEEE Communications Society. Manuscript received October 2, 2001; revised May 29, 2002 and September 28, This work was supported by the National Science Foundation of China under Grant G. Zang is with the Nanjing Institute of Communications Engineering, Nanjing , China ( zgz78@sina.com). C. Ling was with the Nanjing Institute of Communications Engineering, Nanjing , China. He is now with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore ( cling@ieee.org). Digital Object Identifier /TCOMM (see also [3]), which relied on numerical integration of the characteristic function of MAI. Unfortunately, the characteristic-function approach is computationally intensive, especially as the sequence length gets large. To reduce the computational complexity, Morrow and Lehnert [6] proposed an improved Gaussian approximation (IGA) technique, which was derived from the fact that MAI is conditionally Gaussian as the sequence period tends to infinity. Based on this observation, Holtzman [7] introduced a simple, yet sufficiently accurate approximation to calculate the error probabilities by using an expansion of a function of random variables in differences. Holtzman s simplified IGA (SIGA) has received much attention since its publication. It was extended to unequal-power interfering users [8], to arbitrary time-limited chip waveforms [9], to generalized quadriphase DS-CDMA [10], to multistage interference cancellation [11], and to multicode CDMA [12]. All these papers on SIGA assumed that the chip waveform is time limited to, where is the chip duration, and that the system occupies an infinite bandwidth so that the chip waveform experiences no distortion during transmission. A practical DS-CDMA system, in contrast, always involves bandlimitation filtering to restrict out-of-band radiation. For example, both the IS-95 [13] and IS-2000 [14] standards limit the bandwidth of radio frequency (RF) signals within about Hz, while wideband CDMA (WCDMA) employs square-root raised cosine (sqrt-rc) pulse shaping with a rolloff factor of Bandlimitation filtering causes the chip waveforms to disperse over the time axis and overlap one another, which would violate the assumption of the above-referenced papers. Nearly no paper addressed SIGA in an infinite chip duration scenario before this paper was submitted. The difficulty lies in calculation of the second-order moment of the conditional variance of MAI [7], which would contain an infinite number of terms. In this paper, this problem is solved via a frequency-domain approach. We obtain a simple formula of the second-order moment for use in SIGA. We notice that SGA for band-limited CDMA has been investigated by many authors (see, e.g., [15] [18]). The derivation of Viterbi [15], [16] is, by far, the most straightforward. Nonetheless, we need to follow the method in [13] and [18] to derive SIGA in a band-limited scenario. Interestingly, a sudden surge of analysis for band-limited DS-CDMA [26] [30] occurred after the submission of this paper. In particular, we learned of the work by Yoon [26] which considered a similar problem by applying the Poisson sum formula. The final error-probability expressions obtained by /03$ IEEE

2 ZANG AND LING: PERFORMANCE EVALUATION FOR BAND-LIMITED DS-CDMA SYSTEMS 1205 where indicates the integer portion of, is the real chip waveform given by the inverse Fourier transform of, and and are random variables representing time delays and carrier phases that are independent and uniformly distributed on and, respectively. User 1 is considered to be the signal of interest. We set and to zero, respectively, to model perfect code and carrier tracking. The received signal is defined as Fig. 1. Asynchronous DS-CDMA system model. Yoon are essentially the same as ours. However, the heuristic approach to derive the results and the associated performance comparison are unique to this paper. In addition, we provide a rigorous proof for the validity of IGA for strictly band-limited systems that was previously unavailable, thereby putting the analysis of this paper, as well as [25] [30], on a solid theoretic basis. Cho et al. [25], [30] applied the Taylor series expansion to the function inherent in IGA, thereby yielding another error-rate expression different from SIGA. Besides, Yoon presented the exact performance evaluation for band-limited systems whose complexity is exponential in the spreading factor, as well as the reduced-complexity IGA, both via the characteristic-function method [28]. This paper is organized as follows. In Section II, we introduce the system model of filtered DS-CDMA with random sequences. Section III is devoted to SIGA for filtered DS-CDMA. Applications to several typical chip waveforms are described in Section IV. Section V presents numerical results and comparison with performance bounds derived from IGA. II. SYSTEM MODEL The asynchronous binary DS-CDMA system model is represented by Fig. 1. There are users accessing the channel simultaneously, each of which employs a transmitter and receiver of the form illustrated in the figure. The th user generates a data signal where is the data bit, and for and 0, elsewhere. The spreading sequence, generated at rate, is of infinite period. This is an appropriate model for systems employing very long sequences, such as IS-95. The spreading factor is given by. Let both and be modeled as sequences of independent and identically distributed (i.i.d.) random variables taking values on with equal probability. The impulse modulator generates an impulse each seconds. The impulses are shaped by a baseband filter of the transfer function, multiplied by an amplitude constant, and RF-modulated at carrier frequency. At the receiver end, the signal contributed by the th user is given by (1) (2) where is the desired signal, is the MAI due to the additional users, given by, and is the additive white Gaussian noise (AWGN) of double-sided power spectrum density. After down converting, the received signal is passed through a matched filter, whose output is sampled at rate, despread with, and summed to yield the final decision statistic, where where is the signal power, is the sampled noise at the output of the matched filter, and is the continuous-time auto-correlation function of the chip waveform, defined as for a real chip pulse. To normalize the chip waveform, we assume the energy constraint. In addition, is assumed to satisfy. In the signal term, the components for will generally produce interchip interference (ICI). For sake of simplicity, we select a square-root Nyquist pulse for so that is free of ICI. Thus, we can write the signal term as If does not satisfy the Nyquist criterion, the approach of [29] should be followed to take the effect of ICI into account, but nonetheless, the analytic results obtained in this paper might serve as a lower bound to the actual performance. The noise is independent from one sample to another, with the same (3) (4) (5) (6)

3 1206 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 7, JULY 2003 variance. Because of the aperiodic random sequence model, the data bit in (4) can be absorbed into for the purpose of performance evaluation. Hence, the interference due to the th user can be rewritten as (7) Note that the receiver has an infinite observation interval, due to the infinite duration of. This is quite different from previous papers that considered only two partial auto-correlation functions of the chip waveform [1] [9]. III. ERROR PROBABILITIES A. Justification of IGA Before employing SIGA or IGA, we should show that the MAI is asymptotically Gaussian when certain operating conditions are given. For full-response chip waveforms that are time limited to, it was shown [6] that this is true when MAI is conditioned on chip delays, random phases, and another parameter, the number of chip boundaries within a data-bit duration where transitions occur. Holtzman s SIGA [7] was based on those conditions. Later, Torrieri [19] relaxed the conditions to eliminate the necessity of. He proved the central limit theorem is applicable as long as chip delays and carrier phases are fixed. This conclusion was also drawn in [10] by analyzing the characteristic function of MAI as tends to infinity. Unlike the case of full-response chip waveforms, the interference in (7) of a filtered system consists of an infinite number of terms. Therefore, it is necessary to reexamine the validity of IGA for. Since the terms contain the same random sequence,, the terms are not statistically independent, even if and are given. Moreover, [19, Lemma], crucial to proving the independence for in the full-response case, breaks down for the current problem. At this point, we have to invoke the central limit theorem for dependent random variables. The independence assumption of the standard central limit theorem can be relaxed in many ways. In particular, IGA can be easily justified if the chip waveform is time limited to. This is a good approximation for band-limited chip waveforms as long as is sufficiently large. In fact, chip waveforms are necessarily time limited in digital hardware realization, e.g., the IS-95 pulse is time limited to [13]. A central limit theorem for -dependent sequences [23] ensures the validity of IGA for such chip waveforms. More precisely, a sequence is said to be -dependent if and are independent whenever. Because is time limited to for chip waveforms time limited to, it is seen from (7) that and do not have the same elements of if. Hence, they are statistically independent for fixed and. In the terminology of -dependency, is -dependent. Nevertheless, the theory of -dependent sequences is insufficient to justify IGA for strictly band-limited systems. We have to resort to other mathematical tools. An excellent overview of central limit theorems for various dependent random variables is available in [24]. The conventional method to check the validity of central limit theorems involves the use of mixing conditions, but the authors were unable to verify the decay-rate condition in [23]. Fortunately, by resorting to a theorem from [24] based on the Martingale theory, we managed to prove the following result in Appendix I. Proposition 1: For DS-CDMA systems employing square-root Nyquist pulses and random sequences, if and are given, then as, the normalized MAI due to the th user converges in distribution to a Gaussian random variable with zero mean and variance The necessary conditions of Proposition 1 are that and is piecewise smooth. The series in (8) converges for many non-nyquist pulses, as well. However, the condition of convergence is unclear. The proposition is valid for band-limited or nonband-limited systems. For band-limited systems, the rate of convergence to the Gaussian law is probably slower than that for full-response chip waveforms. As demonstrated in [24], the rate of convergence for many types of dependent sequences is on the order of, rather than, in the i.i.d. case. B. Moments of To calculate the moments of, we adopt a frequency-domain representation of (8) given in [13] where we exploit the even symmetry of waveforms. The expectation known. By making use of the fact that (8) (9) for real chip is well, the

4 ZANG AND LING: PERFORMANCE EVALUATION FOR BAND-LIMITED DS-CDMA SYSTEMS 1207 second-order moment can be written as Observing that the MAI tends to be Gaussian when and are given, IGA improves the accuracy significantly [6] by taking advantage of the distribution of (14) SGA ignores the specific distribution of and replaces it by its mean. Finding the distribution of is not easy, as it involves convolutions of the conditional density function of the interference. By applying a theory from perturbation analysis, Holtzman [7] approximated the above average by an expansion in differences of as a function of (10) The derivation of this paper depends heavily on the following property: (15). (11) From (11), we have for, and zero, otherwise. Thus, we find Generally, this expression is accurate if does not deviate too much from its mean. We now return to our problem. The mean of is given by (16) (12) while the variance is given by Owing to the even symmetry of, (12) can be rewritten as (13) This series converges to a finite value if the necessary conditions of Proposition 1 are satisfied. To see this, bearing in mind that the infinite sum in (13) is nothing but, and recalling from (8) that for a Nyquist pulse, we immediately have. C. Improved Gaussian Approximation All the Gaussian approximation techniques can be derived from the distribution of the MAI variance, where [6]. may be expressed as and is a function of the random variables and. Let and denote the mean and standard deviation of. (17) where we have assumed the interference arising from distinct users to be mutually independent. An alternative to SIGA was proposed by Cho et al. [25], [30] which is based on the Taylor series expansion of (14) and requires the calculation of high-order moments of. Without making any approximation, Yoon was able to evaluate IGA (14) exactly and more efficiently than [6] by the characteristic-function method [28], but a very long time is still needed to compute the low bit-error rate.

5 1208 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 7, JULY 2003 IV. APPLICATIONS Expressions derived in Section III are applicable to partial response as well as full-response chip waveforms. We apply the SIGA to a number of typical chip waveforms in this section. For convenience, we assume a perfectly power-controlled system, so that,, unless otherwise stated. A. Rectangular Pulse The first example is the conventional rectangular chip waveform. It helps to validate the correctness of the frequency-domain approach, though SIGA for this kind of chip waveform has been derived in literature. The transfer function of a rectangular chip waveform over the interval is given by where is well known [16] (18). The mean of the MAI variance Fig. 2. Chip waveform of IS-95 and IS (19) while the second moment is given by (20) This series converges fairly rapidly because Fig. 3. Transfer function H (f ) of chip waveforms band-limited to [01=2T ; 1=2T ]. This means that the terms in the sum of (20) decays as, so that considerable accuracy can be obtained by truncating at. Using a numerical integration, we obtain. Substituting it into (17) we arrive at B. Band-limited Pulses: If the bandwidth of the chip waveform is limited to,, extremely compact representation is obtained. Since and its shifted versions in frequency by a multiple of do not overlap, those terms for in (13) vanish. Therefore, (16) and (17) reduce to Though this is less than Holtzman s original result [7] (21) (22) where proper normalization of and was used to render them disappear, Morrow [20] showed there is little loss in accuracy by retaining only the second-order term, which is exactly the same as ours. (23) This class has practical applications, such as the IS-95 system, as well as IS Fig. 2 illustrates the IS-95 chip waveform, truncated to duration of (delayed by for causality in realization) [13]. The IS-95 pulse is approximately a sinc function, and consequently, has a flat in-band spectrum for. There exist other techniques to generate band-limited chip waveforms. The transfer functions of two such waveforms are plotted in Fig. 3. The main lobe

6 ZANG AND LING: PERFORMANCE EVALUATION FOR BAND-LIMITED DS-CDMA SYSTEMS 1209 TABLE I MEAN AND VARIANCE OF 9 FOR VARIOUS KINDS OF CHIP WAVEFORMS square wave (MLSQ) pulse is the main lobe of a sinc function in frequency; 1/2 MLSQ is half the main lobe of a square wave [21]. For visual ease, Fig. 3 is normalized to, but energy constraint is retained in all calculations. The mean and variance of for these three kinds of chip waveforms are summarized in Table I. Note that the later two do not conform to the bound, for they are not Nyquist pulses. One can see that the sinc pulse has the least mean and variance of, while the MLSQ pulse has the greatest. The performance of 1/2 MLSQ is close to that of the sinc waveform. Generally, the less flat the spectrum is, the higher the mean and variance of are. The optimality of the flat-spectrum pulse with respect to the mean of MAI variance is well known [16], given the band-limited constraint. It can be seen from (23) that it is optimal with respect to the variance of, as well. This is expected to yield low error probabilities, because (15) grows with in regions of interest. On the other hand, the sinc pulse attains the bound with equality, thereby being the worst among the class of Nyquist pulses in terms of this quantity. There is a little more that is worthy to be discussed about this class of band-limited pulses. It is seen from (10) that will be zero if the term is absent. It corresponds to a system with perfectly synchronized carrier phases, or a system employing BPSK modulation but quadrature phase-shift keying (QPSK) spreading, such as the IS-95 system. QPSK spreading renders the variance of MAI to be unaffected by carrier phases [13], [16]. In this case, rather surprisingly, has no influence on and, and as a result, low probability of error. The price, of course, is a wider bandwidth. V. NUMERICAL EXAMPLES In this section, we report the performance curves and compare SIGA to the lower and upper bounds derived from IGA [19]. The error probability (14) may be alternatively expressed by where (25) (26) is the conditional error probability for given and. Define as the conditional error probability for given. Lower and upper bounds on can be developed for bandlimited systems in a fashion similar to [19]. First, it is evident from (9) that Thus, an upper bound on, where is given by (27) (24) It implies that from an IGA perspective, the error probabilities for such a CDMA system with perfect delay tracking for interfering users are equal to that without delay tracking for interfering users. It also implies that IGA and SIGA for such a strictly band-limited CDMA system are exactly the same as SGA. That is, SGA is already accurate for IS-95-type systems, as long as the MAI is modeled as a conditionally Gaussian random variable. C. Generalized Band-limited Pulses The calculation of error probabilities for generalized bandlimited chip waveforms is illustrated by the sqrt-rc pulse. The RF bandwidth of the sqrt-rc pulses is, where is the rolloff factor, making overlap with two of its shifted versions. The mean and variance of for this waveform is included in Table I. When,wehave and. Comparing to the sinc waveform, we find produces lower values of (28) Second, by using Jensen s inequality [19], a lower bound may be obtained, 1 where (29) If the Jensen inequality is further used for, then we discover the interesting fact that SGA is a lower bound to, thereby also a lower bound to IGA. 1 A sufficient condition allowing application of Jensen s inequality is that the instantaneous signal-to-interference-plus-noise ratio is greater than or equal to 3=2 1:8 db [19].

7 1210 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 7, JULY 2003 Fig. 4. Comparison of IGA, SIGA, and SGA for the sinc waveform, for N = 64, K =4. Fig. 6. Comparison of IGA, SIGA, and SGA for the sqrt-rc waveform, for N =64, K =4, and =1. Fig. 5. Comparison of IGA, SIGA, and SGA for the MLSQ waveform, for N =64, K =3. For chip waveforms band-limited to,, the lower bound coincides with the upper bound, again because does not overlap,. In Fig. 4, the error probabilities of SIGA, IGA, and SGA are plotted against for the sinc waveform for,. Fig. 5 compares the error probabilities for the MLSQ waveform for,. It is apparent that SIGA produces error probabilities that are very close to IGA, but SGA is optimistic by many orders in the probability of error at high SNRs. Fig. 6 demonstrates the lower and upper bounds versus SIGA for the sqrt-rc waveform with,,.itis seen that SIGA lies between the lower and upper bounds, but SGA is much lower. It was observed in numerical evaluation that SIGA is quite accurate with respect to the IGA-based bounds, except when. The probability of error predicted by SIGA is a little pessimistic in this case. The scenario of is of limited interest, however, because SIGA can be computed only at low SNRs. Since our analysis is accurate for small user numbers Fig. 7. Comparison of SGA and SIGA for a coded CDMA system, for N = 64, K =7, with 6 db power imbalance. such as and, it will become more accurate as gets larger. Therefore, SIGA derived in this paper may be used with confidence in most circumstances. The performance of a coded CDMA system with a rate-1/2, 16-state (23, 35) convolutional code is illustrated in Fig. 7, which is subject to near far effect with 6 db power imbalance, i.e.,,. It is assumed that and remain constant in the duration of a codeword. The error probabilities are evaluated by means of the union bound where is the free distance, is the information weight spectrum of the code, and is the pairwise error probability for two codewords at distance. may be calculated in accordance with (15) or other methods. The sum of the union bound is truncated to the first ten terms in evaluation. It is seen from Fig. 7 that with coding, SIGA still is much better than SGA. Finally, our analysis can be extended to CDMA systems with distinct spreading and despreading chip waveforms [21],

8 ZANG AND LING: PERFORMANCE EVALUATION FOR BAND-LIMITED DS-CDMA SYSTEMS 1211 Theorem 1: Let denote a stationary and ergodic sequence with,, and put. Suppose that is a sub- field of and, and put and If and as, then converges in distribution to the standard normal law. We take and. Then,,.Wehave Fig. 8. Comparison of SGA and SIGA for the noise-whitening receiver, for N = 64, K =5. [22] straightforwardly. For interested readers, we describe SIGA under such circumstance, using an example of the noise-whitening receiver [21] in Appendix II. Fig. 8 illustrates error probabilities of SGA and SIGA for the sinc, MLSQ, and 1/2 MLSQ waveforms. VI. CONCLUSIONS This paper dealt with performance evaluation for practical DS-CDMA systems employing band-limitation filtering. We gave a strong justification for IGA of the individual MAI, based on the central limit theorems for dependent random variables. The variance of was derived via a frequency-domain approach. Applications to a number of typical chip waveforms, including sinc and sqrt-rc pulses that are popular in existing and upcoming cellular CDMA systems, were considered. Performance comparison with lower and upper bounds derived from IGA showed that SIGA yields very accurate error performance. Although the IS-95 waveform is not exactly a sinc pulse [13], the actual performance is expected to be very close to that obtained using a sinc model. We also demonstrated that SIGA and IGA are the same for IS-95-type systems employing BPSK modulation and QPSK spreading. Another interesting result discovered in the paper is that SGA is practically a lower bound to IGA. When When, the second expectation vanishes, so that When, both expectations vanish, so that. Hence APPENDIX I A. Proof of Proposition 1 Proof: To validate the convergence to Gaussian distribution for fixed and, we purposely extend the range of in (7) to so that an infinite sequence may be defined. It is not difficult to show that is stationary and ergodic. Everything starts from a probability space with ergodic measure-preserving transformation, where is the sample space, is a field, and is the probability measure. Let denote the space of mean-square summable random sequences. The proof is achieved via an application of [24, Th. 5.3]. with (30) (31)

9 1212 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 7, JULY 2003 provided that the above series converges to a finite value. It is seen that the series, when viewed as a function of,is even, periodic with period. As such, it can be expressed in a Fourier series,if and is piecewise smooth [26, Prop. 1]. As the Dirichlet condition is satisfied, the Fourier series converges. Further, since is real valued and has even symmetry, we have (32) where the third equality follows from the property of a square-root Nyquist pulse. Hence, the series is bounded. Since we assume a random sequence model, and are statistically independent for. Then, it can be observed from (7) that and are uncorrelated, given and. Hence, the variance of is, while the variance of is given by which is clearly independent of. Therefore, the condition that as is satisfied trivially. In accordance with Theorem 1, converges to the Gaussian distribution unless. The case only happens when or, but this leads. Combining the two cases, we arrive at the conclusion that, conditioned on and, converges to the Gaussian distribution with variance shown in (8). It is easily checked that the above proof process will no longer hold if contains either or. In other words, it is indeed necessary to condition on and for the validity of IGA. APPENDIX II A. Noise-Whitening Receiver For chip waveforms whose spectra are not flat across the given bandwidth, it is possible to maximize SNR of the decision variable by inserting a noise-whitening filter ahead of the matched filter. Combining the two filters results in a receive filter having the dispreading waveform distinct from the spreading waveform [21] (33) where denotes the transmit filter. The parameters of SIGA for chip-rate samples are modified as follows. 1) Signal component:. 2) Noise variance:. 3) Mean of :. 4) Second moment of :. SIGA is obtained by substituting these parameters into (15) (17). ACKNOWLEDGMENT The authors wish to thank the Editor for his suggestion on the organization of this paper. They are also grateful to the anonymous reviewers for their constructive comments and bringing [25] to the authors attention. 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] M. B. Pursley, D. V. Sarwate, and W. E. Stark, Error probability for direct-sequence spread-spectrum multiple-access communications Part I: Upper and lower bounds, IEEE Trans. Commun., vol. COM-30, pp , May [3] J. S. Lehnert and M. B. Pursley, Error probabilities for binary directsequence spread-spectrum communications with random signature sequences, IEEE Trans. Commun., vol. COM-35, pp , Jan [4] E. A. Geraniotis and M. B. Pursley, Error probability for direct-sequence spread-spectrum multiple-access communications Part II: Approximations, IEEE Trans. Commun., vol. COM-30, pp , May [5] E. A. Geraniotis and B. Ghaffari, Performance of binary and quaternary direct-sequence spread-spectrum multiple-access systems with random signature sequences, IEEE Trans. Commun., vol. 39, pp , May [6] R. K. Morrow and J. S. Lehnert, Bit-to-bit error dependence in slotted DS/SSMA packet systems with random signature sequences, IEEE Trans. Commun., vol. 37, pp , Oct [7] J. M. Holtzman, A simple, accurate method to calculate spread-spectrum multiple-access error probabilities, IEEE Trans. Commun., vol. 40, pp , Mar [8] T. S. Rappaport, Wireless Communications: Principles and Practice. New York: IEEE Press, 1996, pp [9] H. H. Nguyen and E. Shwedyk, On error probabilities of DS-CDMA systems with arbitrary chip waveforms, IEEE Commun. Lett., vol. 5, pp , Mar [10] T. M. Lok and J. S. Lehnert, Error probabilities for generalized quadriphase DS/SSMA communication system with random signature sequences, IEEE Trans. Commun., vol. 44, pp , July [11] R. M. Buehrer and B. D. Woerner, Analysis of adaptive multistage interference cancellation for CDMA using an improved Gaussian approximation, IEEE Trans. Commun., vol. 44, pp , Oct [12] S. J. Lee, T. S. Kim, and D. K. Sung, Bit-error probabilities of multicode direct-sequence spread-spectrum multiple-access systems, IEEE Trans. Commun., vol. 49, pp , Jan [13] J. S. Lee and L. E. Miller, CDMA Systems Engineering Handbook. Norwood, MA: Artech House, 1998, ch. 1 and 7. [14] Physical Layer Standard for CDMA2000 Spread Spectrum Systems,, TIA/EIA/IS (Ballot Version). [15] A. J. Viterbi, Very low rate convolutional codes for maximum theoretical performance of spread-spectrum multiple-access channels, IEEE J. Select. Areas Commun., vol. 8, pp , May 1990.

10 ZANG AND LING: PERFORMANCE EVALUATION FOR BAND-LIMITED DS-CDMA SYSTEMS 1213 [16], CDMA: Principles of Spread Spectrum Communication. New York: Addison-Wesley, 1995, ch. 2. [17] J. E. Salt and S. Kumar, Effects of filtering on the performance of QPSK and MSK modulation in D-S spread spectrum systems using RAKE receivers, IEEE J. Select. Areas Commun., vol. 12, pp , May [18] Y. Asano, Y. Daido, and J. M. Holtzman, Performance evaluation for band-limited DS-CDMA communication system, in Proc. IEEE 43rd Vehicular Technology Conf., Secaucus, NJ, May 1993, pp [19] D. J. Torrieri, Performance of direct-sequence system with long pseudonoise sequences, IEEE J. Select. Areas Commun., vol. 10, pp , May [20] R. K. Morrow, Accurate CDMA BER calculations with low computational complexity, IEEE Trans. Commun., vol. 46, pp , Nov [21] A. M. Monk, M. Davis, L. B. Milstein, and C. H. Helstrom, A noisewhitening approach to multiple access noise rejection Part I: Theory and background, IEEE J. Select. Areas Commun., vol. 12, pp , June [22] Y. Huang and T. S. Ng, Capacity enhancement of band-limited DS-CDMA system using weighted dispreading function, IEEE Trans. Commun., vol. 47, pp , Aug [23] P. Billingsley, Probability and Measure, 2nd ed. New York: Wiley, 1986, pp [24] P. Hall and C. C. Heyde, Martingale Limit Theory and Its Application. New York: Academic, [25] J. H. Cho, Y. K. Jeong, and J. S. Lehnert, A closed-form BER expression for band-limited DS/SSMA communications, in Proc. IEEE Military Communications Conf., Los Angeles, CA, Oct , 2000, pp [26] J. H. Cho and J. S. Lehnert, An optimal signal design for band-limited asynchronous DS-CDMA communications, IEEE Trans. Inform. Theory, vol. 48, pp , May [27] Y. C. Yoon, A simple and accurate method of probability of bit error analysis for asynchronous band-limited DS-CDMA systems, IEEE Trans. Commun., vol. 50, pp , Apr [28], An improved Gaussian approximation for probability of bit-error analysis of asynchronous band-limited DS-CDMA systems with BPSK spreading, IEEE Trans. Wireless Commun., vol. 1, pp , July [29] W. Gao, J. H. Cho, and J. S. Lehnert, Chip waveform design for DS/SSMA systems with aperiodic random spreading sequences, IEEE Trans. Wireless Commun., vol. 1, pp , Jan [30] J. H. Cho, Y. K. Jeong, and J. S. Lehnert, Performance of band-limited DS/SSMA communications, IEEE Trans. Commun., vol. 50, pp , July Guozhen Zang was born in Henan Province, China, in She received the B.S. and M.S. degrees in electrical engineering from the Nanjing Institute of Communications Engineering, Nanjing, China, in 1999 and 2002, respectively, where she is currently working toward the Ph.D. degree. Her research interests include spread spectrum communications and satellite communications. Cong Ling (A 01 S 02) was born in Anhui Province, China, in He received the B.S. and M.S. degrees in electrical engineering from the Nanjing Institute of Communications Engineering, Nanjing, China, in 1995 and 1997, respectively. He is currently working toward the Ph.D. degree at the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. From 1998 to 2001, he was a Lecturer at the Nanjing Institute of Communications Engineering. His research interests are in the general area of wireless communications, with emphasis on spread spectrum, coding and iterative processing techniques, and space time communication. Mr. Ling was awarded the Singapore Millennium Scholarship in He is a Student Member of the IEEE Communications Society and IEEE Information Theory Society.