Mc-cdma performance evaluation over a multipath ... - IEEE Xplore

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 8, AUGUST 2001

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MC-CDMA Performance Evaluation Over a Multipath Fading Channel Using the Characteristic Function Method Besma Smida, Charles L. Despins, and Gilles Y. Delisle

Abstract—This letter presents an efficient and accurate performance evaluation method for binary MC-CDMA systems with deterministic signature sequences. This method is based on a formulation of the characteristic function, and does not resort to any assumption on the statistical or spectral behavior of the interference. The accuracy of the Gaussian approximation method is also considered. Index Terms—Characteristic function method, MC-CDMA, multipath fading channel, performance analysis.

I. INTRODUCTION IRECT-SEQUENCE spread-spectrum (DS-SS) code-division multiple-access (CDMA) signals have a wide bandwidth (BW) and may be subject to frequency-selective multipath fading. It is well known that without coding or diversity, the performance of DS-CDMA can degrade rapidly when the number of users increases. Another problem occurs in high data applications if the channel delay spread exceeds the symbol duration. In this case, the conventional DS-SS is subject to severe intersymbol interference (ISI) and is practically not usable [1]. A technique of reducing the symbol rate is essential in this case. Multicarrier (MC) systems proposed in [3] are one way of achieving this goal. Different configurations of MC-CDMA systems are cited in the literature [2]. The MC-CDMA system, adopted in this manuscript, is the unique configuration that reduces the symbol rate essential in high data applications.1 At is sethe transmitting side, the bit stream with bit duration streams. The new bit duration rial-to-parallel converted into on each stream can suppress ISI. Each stream feeds parallel streams such that the same data stream exists on the branches (identical bit branches) to achieve frequency diversity and suppress ICI. On each of the latter streams, bits are spread . by a pseudonoise (PN) code with a suitable chip rate These streams then modulate orthogonal carriers with a successively overlapping bandwidth. binary phase-shift keying modulation is utilized to simplify the presentation. There have been a number of papers on the calculation of error probabilities for CDMA systems in the open literature. Most of these papers [3] use Gaussian approximation (GA) in

D

Paper approved by J. Wang, the Editor for Wireless Spread Spectrum of the IEEE Communications Society. Manuscript received February 12, 2000; revised July 19, 2000 and March 1, 2001. This paper was presented in part at the IEEE International Symposium on Wireless Communications (ISWC), Montreal, PQ, Canada, May 1998. B. Smida is with Microcell Connexions, Inc., Montreal, QC H5A 1K3, Canada. C. L. Despins is with Télébec Itée, Anjou, QC H1M 3N8 Canada. G. Y. Delisle is with the Department of Electrical and Computer Engineering, Laval University, Quebec, QC G1K 7P4, Canada. Publisher Item Identifier S 0090-6778(01)06923-9. 1It

is similar to the II-C.1 MC-DS-CDMA scheme presented by [2].

the probability of error calculations, claiming that the exact calculation is computationally difficult. The major difficulties in precisely evaluating MC-CDMA systems is the determination of the probability density function (pdf) of the interference. The exact calculation of the pdf of the interference is very difficult since the interference depends on the cross correlation of the spreading codes and on the random processes such as delay, phase shift, and information bit. Therefore [4] evaluated the upper and lowers bounds of the pdf. These bounds find through the density function of the random variable that models , is the the multiuser interference (MUI) [5] involve number of user, convolutions of the density function of the MUI. The complexity of this method would increase if applied to the MC-CDMA system. In this letter, we propose a method which does not resort to any assumptions with respect to the statistical or spectral nature of the interference. The bit-error rate (BER) is determined via the characteristic function (CF) of the interference and without explicit knowledge of the pdf. The CF method is more accurate and generalized. Moreover the complexity of the CF method is shown to be proportional to the number of user and the number of car. The CF method was first proposed by Helstrom [8] to rier compute the probability of error. This technique was later applied to the performance evaluation of DS-CDMA [6]. In this letter, this method is extended to MC-CDMA systems. II. THE SYSTEM AND CHANNEL MODEL The system model is described in detail in [3]. Here we repeat the basic elements of this model so that the concepts and notation which are necessary for the rest of the letter are introduced. The th transmitted signal for the binary MC-CDMA system is (1) mod such that is the bit stream where on the identical bit branches fed from the th branch of the serial . is the th carrier frequency, to parallel converter is a random phase for each carrier, and is the PN code are related by of user . The orthogonal frequencies (2) is the absolute carrier number in the system. Note where that if the relative carrier number within group is , where , then the absolute carrier number in the system . The receiver of user employs MF deis tectors, each tuned and synchronized to one of the carriers. We also consider the case when the detector of each carrier employs

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a RAKE receiver. For the system with asynchronous simultaneous transmitted signals, the received signal is given by (3) In (3), is the time delay for the communication link between is a the th transmitter and the first receiver. The process white Gaussian noise process with two-sided spectral density . Finally, is the fading channel output to input . of user The Rayleigh selective fading channel for carrier is described by the following impulse response: e . III. INTERFERENCE ANALYSIS A perfect power control is assumed. The receiver of user one consists of groups, one per bit of the parallel transmitted bits. . For a bit transEach group consists of paths, , the decision statistics of mitted on group , MFs are added (post-detection, equal-gain combining) the to form the final decision statistic (4) where represents the relative carrier number within group , is comand is the corresponding absolute carrier number. . is a posed of six terms: . Gaussian random variable with zero mean and variance is the desired output given by setting , , and . . is the self, same carpaths , on the rier interference due to the remaining from the same user . is self, other same carrier paths , carrier interference due to the remaining from the same user . is from the other carriers other users, same carrier interference due to the paths from from the others users . is other the same carrier users, other carrier interference due to the paths from the other from the others users . carriers

(5)

where and are defined in [3]. , , and are inferred from (5) by setting , , and , respectively. The probability of error is determined using the CF method. The formulation of this method is described in detail in [8]. It aims to provide the probability of error from the knowledge of the (mgf) e of the decision moment-generation function variable . A. The MGF Function The mgf takes the following form

e , where and are, respectively, the mgfs of the desired . signal and noise components of the decision variable e

(6) e erfc

(7)

. After some manipulations, we Note that , see (8) and (9) get the mgf of , , and are inferred at the bottom of the page. , , from (8) by setting , respectively. and B. The Probability of Error depends on the environment and the carrier frequency used here in are obtained from the used. The values of wideband measurements carried out in SHF-band, high data rate, point-to-multipoint, fading channel application [7]. After relative to the data sequence , the probthe calculation of is given by . ability of error IV. NUMERICAL RESULTS AND DISCUSSION The results of the CF technique described herein closely match those of simulations, as shown in Fig. 1. The simulation results are Monte Carlo and are based on the calculation of the decision variable for each bit, and averaging the results over large number of bits (100 000 bits have been employed to achieve each value). After this validation, the results from the CF method are compared with those from the GA method for different types of signature sequences. All subsequent results are generated by numerical procedures and are sequence-depend: they vary with the

(8) (9)

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Fig. 1. BER of an MC-CDMA system over a Rayleigh fading channel.

Fig. 3. Accuracy of GA method with respect to different MC-CDMA configurations.

Fig. 2. Accuracy of GA method with respect to different types of signature sequences.

Fig. 4.

choice of the Walsh or Gold sequences. Fig. 2 shows the results , , and . The accuracy of the GA for is seen to improve with Gold sequences as the number of users decreases. This unexpected behavior is the consequence of orthogonal carriers related by (2). The transmitting over is more GA of the interferences due to the other carriers encountered inaccurate than the GA of the interferences in DS-CDMA system. The GA method considers the interferences between two sources, transmitting on two distinct orthogonal carriers to be white Gaussian noise. This approximation error increase with the number of users. On the other hand, the large difference in the cross-correlation function between two different sequences [the cross-correlation sidelobes of the Walsh sequences (18,26) are higher than those of (18,21)] causes discontinuity in the probability of error curves and strongly degrades not only the system performance but also the accuracy of the GA based method. Occasionally, adding a new user (with bad Walsh sequence cross correlation)

to the system generate a strong degradation of the probability of error of the desired user (we notice a hop in the curves). In other case, the degradation caused by adding new user is almost nonexistent. Fig. 3 considers two configurations of MC-CDMA system and . The figure shows that the accuracy of the GA methods depends on the number of carriers. GA becomes more accurate as the number of carriers decreases (and the system becomes closer to the DS-CDMA). The interference due to other distinct carriers (symbolized by ) is almost nonexistent even over a frequency-selective increases fading channel, and GA approximation error of . with Fig. 4 compares the relative accuracy of the GA-based method with different levels of background noise. The results dB are shown in the figure and it is seen, for not surprisingly, that presence of strong background white Gaussian noise improves the accuracy of the GA method.

BER of MC-CDMA system versus E =N .

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REFERENCES [1] R. A. Stirling-Gallacher and G. J. R. Povey, “Comparison of MC-CDMA with DS-CDMA using frequency domain and time domain RAKE receivers,” Wireless Pers. Commun., vol. 2, pp. 105–119, 1995. [2] R. Prasad and S. Hara, “An overview of multi-carrier CDMA,” in Proc. ISSSTA’96, Mainz, Germany, Sept. 1996, pp. 107–114. [3] E. A. Sourour and M. Nakagawa, “Performance of orthogonal multicarrier CDMA in multipath fading channels,” IEEE Trans. Commun., vol. 44, Mar. 1996. [4] A. W. Lam and F. M. Ozluturk, “Performance bounds for DS/SSMA communications with complex signature sequences,” IEEE Trans. Commun., vol. 40, pp. 1607–1614, Oct. 1992.

[5] J. S. Lehnert, “An efficient technique for evaluating direct-sequence spread-spectrum multiple-access communications,” IEEE Trans. Commun., vol. 37, pp. 851–858, Aug. 1989. [6] D. Liu, C. L. Despins, and W. A. Krzymien, “Efficient and accurate DS-SSMA deterministic signature sequence performance evaluation over wireless fading channels,” in Proc. IEEE ICC’97, Montreal, PQ, Canada, June 1997. [7] D. Lacroix, C. Despins, G. Delisle, and V. Spiegel, “Impulse response measurements in the UHF and SHF lands for outdoor microcellular quasistatic environments,” in Proc. IEEE GLOBECOM’97, Phoenix, AZ, Nov. 1997. [8] C. W. Helstrom, Elements of signal detection and estimation. Englewood Cliffs, NJ: Prentice-Hall, 1995.