Performance of residue number system based DS-CDMA over multipath fading channels using orthogonal sequences

This paper is concerned with a new direct-sequence spread-spectrum multiple-access communication system based on the so-called residue number system (RNS) or redundant residue number system (RRNS). The system operates in a multipath fading environment, and a RAKE receiver structure with maximum ratio combining is used for demodulation. Approximations to the error probabilities are given by using Gaussian statistics for the multiple-access and the multipath interference. Concatenated codes employing residue number system product codes (RNS-PC) as the inner codes and nonbinary Reed-Solomon (RS) codes as the outer codes are adopted to improve thc system performance. The performance of the system is determined for a range of different length RNS-PC schemes. The results show that, for a given outer RS code and a given number of moduli of the inner RNS-PC, the performance of the system ean be optimized by varying the relative number of information moduli and redundant moduli of the inner RNS-PC and the moduli's values. A direct-sequence code division multiple-access (DSCDMA) coinmunication system, in which a set of M-ary orthogonal pseudo-random noise (PN) sequences is assigned to each user has been proposed by Enge and Sanvate in [I], and the performance has been analyzed, when the channel impairments are considered as a combination of additive white Gaussian noise (AWGN) and 7 multiple-access interference. Under this communication model, k = log,M bits of information is transmitted in a symbol period. In [2], the above system has been discussed by Enge and Sarwate, when impulsive noise channels are considered. Instead of nonfading channels, Chase and Pahlavan have investigated the performance of the M-ary DS-CDMA system with diversity in the presence of multipath and multiple access interference in [3]. .In the above system, M = 2"N sequences are employed by each user to transmit k = log2M bits in a single symbol period, and the number of PN sequences of the system increases exponentially with increasing k. In other words, the number of the correlators or matched filters in the receiver of an M-ary DS-CDMA system increases exponentially with increasing k. when a correlation receiver or a matched filter receiver is con(') The work was supported by the Sino-British Fellowship Trust of Royal Society. The financial support of the European Community, Br~issels, Belgium and that of the Engineering and Physical Sciences Research Council, Swindon, UK is also gratefully acknowledged. sidered, respectively. This essentially limits the length k of every transmitted symbol, since the complexity of the receiver is proportional to the number of correlators or matched filters. For example, if we assume that there are 100 mobile users, who are transmitting information from a base station in a cell. and k = 8 bit symbols are transmitted from a set of M = 2R, then at least 51,200 (=2 x 100 x 28) PN sequences are utilized simultaneously in the cell for the forward channel (base-tomobile) and reverse channel (mobile-to-base). The base station has at least 25.600 correlators or matched filters and must synchronize with the 25,600 PN sequences of the mobiles in the cell. Each mobile has to acquire and track 256 PN sequences of the forward channel, and has at least 256 correlators or matched filters, in order to receive the X bit symbols sent by the base station. This complexity might be unacceptable concerning both the mobiles and the base stations. To combat this problem, various spread-spectrum schemes capable of sending multiple bits per symbol period have been proposed in [4, 51. In [4], Vandendorpe has proposed a communication system based on the combination of multitone modulation and DS spread-spectrum techniques, in which multiple information bits are transmitted in parallel and each bit modulates a carrier. Another multiple bit per symbol transmission method has been proposed in [5] using the combination of a set of random orthogonal sequences (or codes) according to combinational mathematics. In recent years, the so-called residue number system Vo1.9. No. h November December 1998 525 (RNS) arithmetic has attracted considerable attention for designing high-speed special-purpose digital hardware that is suitable for very large scale integration (VLSI). Digital systems that are structured around RNS arithmetic units may play an important role in ultra speed dedicated real-time systems that support pure parallel processing of integcr-valucd data [6 141. The RNS has two important properties for digital processing applications [6]: the ability to usc cany-free arithmetic and the lack of ordered significance among residue digits. The first property allows each digit of the representation to be processed separately from the others by a dedicatcd module. The second property implies that any erroneous digit can be discarded without affecting the result, provided that sufficient resolution remains in the reduced system in order to unambiguously represent the result. With these inherent properties of the RNS, residue arithmetic offers a variety of new approaches to the realization of digital signal processing algorithms [7 91, such as digital modulation and demodulation [lo], and the fault-tolerant design of arithmetic units [I I]. It also offers new approaches to the design of error-detection and error-correction codes [12 141. In this paper, a new communication system based on the combination of the so-called residue number system (RNS) or the so-called redundant residue number system (RRNS) [6] and spread-spectrum techniques is proposed. We will show that the complexity of the receiver can be reduced by decreasing the number of corrclators or matched filters. However, signal processing units for binary-to -residue and residue-to-binary conversion have to be designed for the transmitter and the receiver. respectively, as we will highlight during our forthcoming discussions with reference to Figs. 1 and 2. The performance of the proposed system will be analyzed, when random signature sequcnces are applied by using Gaussian approximations. Specifically, we are concerned with the bit error probabilities of the demodulated signals or the decoded signals at the output of the receiver. The channel itself is modeled as a multipath Rayleigh fading channel, and diversity reception based on maximum ratio combining (MRC) is adopted in the receiver, in order to improve the performance. The well-known concatenated code that employs an RNS product code [IS] (RNS-PC) as inner code and a nonbinary RS code as outer code is adopted for error correction and error detection. The inner code is used to detect andlor correct the residue errors, and the nonbinary RS code with errors-only decoding or errors-and-erasures decoding is used to correct the symbol errors or to fill the syrribol erasures. The remainder of the paper is outlined as follows. In section 2, we present the description of the system and the channel model. Section 3 describes the receiver structure. System analysis and derivation of the conditional symbol error probability are given in section 4. In section 5, the performance of the uncoded system is analyzed, while in section 6, the concatenated coded system is evaluated. Numerical results are presented in section 7. Finally, in section 8 we present our conclusions. A residue number system is defincd 11 2, 131 by the choice of v possible integers mi, (i = 1,2,. . . , v) referred to as moduli. If all the moduli are painvise relative primes, any integer X, describing a message in this paper, can be uniquely and unambiguously represented by the so-called residue sequence (r,, rz,.. ., r,.) in the range 0 5 X < M, where r, = X (mod mi) represents the residue of X upon division by mi, and M = n,:, rn, is the dynamic range. According to the so-called Chinese reminder theorem (CRT) [13 151. for any given 13-tuple (r , , r,,. . ., r,,), where 0 2 r, < m , i = 1, 2.. ... v there exists one and only one integer X such that 0 < X < M and r, = X (mod m,). It can be shown that the numerical value of M can be computed [13 151 by using 1' x = C ~ T M , (modM)