{"title": "Analysis of Bit Error Probability of Direct-Sequence CDMA Multiuser Demodulators", "book": "Advances in Neural Information Processing Systems", "page_first": 315, "page_last": 321, "abstract": null, "full_text": "Analysis of Bit Error Probability of \nDirect-Sequence CDMA Multiuser \n\nDemodulators \n\nDepartment of Electronics and Information Engineering \n\nToshiyuki Tanaka \n\nTokyo Metropolitan University \nHachioji, Tokyo 192-0397, Japan \n\ntanaka@eeLmetro-u.ac.jp \n\nAbstract \n\nWe analyze the bit error probability of multiuser demodulators for direct(cid:173)\nsequence binary phase-shift-keying (DSIBPSK) CDMA channel with ad(cid:173)\nditive gaussian noise. The problem of multiuser demodulation is cast \ninto the finite-temperature decoding problem, and replica analysis is ap(cid:173)\nplied to evaluate the performance of the resulting MPM (Marginal Pos(cid:173)\nterior Mode) demodulators, which include the optimal demodulator and \nthe MAP demodulator as special cases. An approximate implementa(cid:173)\ntion of demodulators is proposed using analog-valued Hopfield model \nas a naive mean-field approximation to the MPM demodulators, and its \nperformance is also evaluated by the replica analysis. Results of the per(cid:173)\nformance evaluation shows effectiveness of the optimal demodulator and \nthe mean-field demodulator compared with the conventional one, espe(cid:173)\ncially in the cases of small information bit rate and low noise level. \n\n1 Introduction \n\nThe CDMA (Code-Division-Multiple-Access) technique [1] is important as a fundamental \ntechnology of digital communications systems, such as cellular phones. The important ap(cid:173)\nplications include realization of spread-spectrum multipoint-to-point communications sys(cid:173)\ntems, in which multiple users share the same communication channel. In the multipoint-to(cid:173)\npoint system, each user modulates his/her own information bit sequence using a spreading \ncode sequence before transmitting it, and the receiver uses the same spreading code se(cid:173)\nquence for demodulation to obtain the original information bit sequence. Different users \nuse different spreading code sequences so that the demodulation procedure randomizes \nand thus suppresses multiple access interference effects of transmitted signal sequences \nsent from different users. \n\nThe direct-sequence binary phase-shift-keying (DSIBPSK) [1] is the basic method among \nvarious methods realizing CDMA, and a lot of studies have been done on it. Use of \nHopfield-type recurrent neural network has been proposed as an implementation of a mul(cid:173)\ntiuser demodulator [2]. In this paper, we analyze the bit error probability of the neural \nmultiuser demodulator applied to demodulation of DS/BPSK CDMA channel. \n\n\fSpreading Code Sequences \n{ '7~ } { '7~} \u2022\u2022\u2022 {'7~} \n\nGaussian Noise \n{Vi} \n\n~1 \n\n~2------~X~-----+--~ \n\nReceived Signal \n{i} \n\n~N------~ \n\nInformation Bits \n\nFigure 1: DSIBPSK CDMA model \n\n2 DSIBPSK CDMA system \n\nWe assume that a single Gaussian channel is shared by N users, each of which wishes \nto transmit his/her own information bit sequence. We also take a simplifying assumption, \nthat all the users are completely synchronized with each other, with respect not only to the \nchip timing but also to the information bit timing. We focus on any of the time intervals \ncorresponding to the duration of one information bit. Let ~i E {-I, I} be the information \nbit to be transmitted by user i (i = 1, ... , N) during the time interval, and P be the number \nof the spreading code chips (clocks) per information bit. For simplicity, the spreading code \nsequences for the users are assumed to be random bit sequences {'7:; t = 1, ... , P}, where \n'7:'s are independent and identically distributed (i.i.d.) binary random variables following \nProb['7f = \u00b11] = 1/2. \nUser i modulates the information bit ~i by the spreading code sequence and transmits the \nmodulated sequence {~i '7f; t = 1, ... , P} (with carrier modulation, in actual situations). \nAssuming that power control [3] is done perfectly so that every transmitted sequences ar(cid:173)\nrive at the receiver with the same intensity, the received signal sequence (after baseband \ndemodulation) is {yl; t = 1, ... , P}, with \n\nN \n\nl = L'7:~i + Vi, \n\ni=1 \n\n(1) \n\nwhere Vi ~ N(O, a}) is i.i.d. gaussian noise. This system is illustrated in Fig. l. \n\nAt the receiver side, one has to estimate the information bits {~i} based on the knowledge of \nthe received signal {i} and the spreading code sequences {'7f} for the users. The demodu(cid:173)\nlator refers to the system which does this task. Accuracy of the estimation depends on what \ndemodulator one uses. Some demodulators are introduced in Sect. 3, and analytical results \nfor their performance is derived in Sect. 4. \n\n\f3 Demodulators \n\n3.1 Conventional demodulator \n\nThe conventional demodulator (CD) [1-3] estimates the information bit ~i using the spread(cid:173)\ning code sequence {11:; t = 1, . .. , P} for the user i , by \n\nWe can rewrite hi as \n\n1 P \n\nhi == N I>l;i. \n\n1= 1 \n\n(2) \n\n(3) \n\nThe second and third terms of the right-hand side represent the effects of multiple ac(cid:173)\ncess interference and noise, respectively. CD would give the correct information bit in the \nsingle-user (N = 1), and no noise (V i == 0) case, but estimation may contain some errors \nin the multiple-user andlor noisy cases. \n\n3.2 MAP demodulator \n\nThe accuracy of the estimation would be significantly improved if the demodulator knows \nthe spreading code sequences for all N users and makes full use of them by simultane(cid:173)\nously estimating the information bits for aLI the users (the multiuser demodulator). This \nis the case, for example, for a base station receiving signals from many users. A common \napproach to the multiuser demodulation is to use the MAP decoding, which estimates the \ninformation bits lSi = ~;} by maximizing the posterior probability p({~;}I{y l}). We call \nthis kind of multiuser demodulator the MAP demodulator 1 . \n\nWhen we assume uniform prior for the information bits, the posterior probability is explic(cid:173)\nitly given by \n\n(4) \n\n(5) \n\nwhere \n\np(sl{i }) = Z - I exp(-flsH(s\u00bb), \n\nfl. == N fa}, s == (Si), h == (hi), and W == (wij) is the sample covariance of the spreading \ncode sequences, \n\nP \n\nWij = ~ I>:11j. \n\n1= 1 \n\n(6) \n\nThe problem of MAP demodulation thus reduces to the following minimization problem: \n\nA (MAP) \n~ \n\n= arg min H(s). \n\nsE{- I,I} N \n\n(7) \n\n3.3 MPM demodulator \n\nAlthough the MAP demodulator is sometimes referred to as \"optimal,\" actually it is not so \nin terms of the common measure of performance, i.e., the bit error probability Ph, which is \n\nIThe MAP demodulator refers to the same one as what is frequently called the \"maximum(cid:173)\n\nlikelihood (ML) demodulator\" in the literature. \n\n\frelated to the overlap M == (1/ N) L~l ~i~i between the original information bits {~i} and \ntheir estimates {~d as \n\n(8) \nThe 'MPM (Marginal Posterior Mode [4]) demodulator,' with the inverse temperature /3, is \ndefined as follows: \n\nI-M \nPb=-2-' \n\n~i(MPM) = sgn(('\\\u00b7i},B), \n\n(9) \n\nwhere ('},B refers to the average with respect to the distribution \n\nP,B(s) = Z(/3)-1 exp( -/3H(s\u00bb) . \n\n(10) \nThen, we can show that the MPM demodulator with /3 = /3s is the optimal one minimizing \nthe bit error probability Pb. It is a direct consequence of general argument on optimal \ndecoders [5]. Note that the MAP demodulator corresponds to the MPM demodulator in the \n/3 --* +00 limit (the zero-temperature demodulator). \n\n4 Analysis \n\n4.1 Conventional demodulator \n\nIn the cases where we can assume that Nand P are both large while a == P / N = 0(1), \nevaluation of the overlap M, and therefore the bit error probability Pb, for those demodu(cid:173)\nlators are possible. For CD, simple application of the central limit theorem yields \n\nM = erf ( a) \n\n, \n\n2(1+1//3,,) \n2 r \n\n2 \n\nerf(x) == .rn 10 e-I dt \n\nwhere \n\nis the error function. \n\n4.2 MPM demodulator \n\n(11) \n\n(2) \n\n(3) \n\n(4) \n\nFor the MPM demodulator with inverse temperature /3, we have used the replica analysis \nto evaluate the bit error probability Pb. Assuming that Nand P are both large while a == \nP / N = 0(1), and that the macroscopic properties of the demodulator are self-averaging \nwith respect to the randomness of the information bits, of the spreading codes, and of the \nnoise, we evaluate the quenched average of the free energy ((log Z)} in the thermodynamic \nlimit N --* 00, where ((.}) denotes averaging over the information bits and the noise. \nEvaluation of the overlap M (within replica-symmetric (RS) ansatz) requires solving \nsaddle-point problem for scalar variables {m, q, E, F}. The saddle-point equations are \n\nm = f Dz tanh(#z + E), \n\nq = f Dz tanh2(#z + E) \n\nE= _ _ a_/3 __ \n1 + /3(1 - q)' \n\nF-\n-\n\na/3 2 \n\n[l + /3(1 - q)]2 \n\n[ \n\n1 ] \n1 -2m -\n+ q + /3s \n\nwhere Dz == 0/ -J2ir)e- z2 / 2dz is the gaussian measure. The overlap M is then given by \n\nM = f Dzsgn(#z + E), \n\nfrom which Pb is evaluated via (8) . This is the first main result of this paper. \n\n\f4.3 MAP demodulator: Zero-temperature limit \n\nTaking the zero-temperature limit f3 --+ +00 of the result for the MPM demodulator yields \nthe result for the MAP demodulator. Assuming that q --+ I as f3 --+ +00, while f3 (1 - q) \nremains finite in this limit, the saddle-point equations reduce to \n\nM = m = erf(J 2(2 _ 2: + 1/f3s\u00bb) \n\n(15) \n\nIt is found numerically, however, that the assumption q --+ I is not valid for small a, so \nthat we have to solve the original saddle-point equations in such cases. \n\n4.4 Optimal demodulator: The case f3 = f3s \nLetting f3 = f3s in the result for the MPM demodulator gives the optimal demodulator \nminimizing the bit error probability. In this case, it can be shown that m = q and E = F \nhold for the solutions of the saddle-point equations (13). \n\n4.5 Demodulator using naive mean-field approximation \n\nSince solving the MAP or MPM demodulation problem is in general NP complete, we have \nto consider approximate implementations of those demodulators which are sub-optimal. A \nstraightforward choice is the mean-field approximation (MFA) demodulator, which uses \nthe analog-valued Hopfield model as the naive mean-field approximation to the finite(cid:173)\ntemperature demodulation problem2. The solution {mi} of the mean-field equations \n\nmi = tanh[f3(- LWijmj +hi )] \n\nj \n\n(16) \n\ngives an approximation to {(.\\'i) f! }, from which we have the mean-field approximation to \nthe MPM estimates, as \n\nA (MPA) \n~i \n\n= sgn(mi) . \n\n(17) \n\nThe macroscopic properties of the MFA demodulator can be derived by the replica analysis \nas well, along the line proposed by Bray et al. [6] We have derived the following saddle(cid:173)\npoint equations: \n\nm = f Dz fe z ), \n\nx = ~ f Dz zf(z), \n\nq = f Dz [f(z)]2 \n\naf3 \n\nE= - -\n1 + f3x' \n\nF-\n-\n\naf32 \n\n[ \n1 -2m -\n[l+f3X]2 \n+q+ f3s \n\nI ] \n' \n\nwhere fe z ) is the function defined by \n\nfe z ) = tanh [ flz - Ef(z ) + E]. \n\n(18) \n\n(19) \n\n(20) \n\nfez) is a single-valued function of z since E is positive. The overlap M is then calculated \nby \n\nM = f Dz sgn(t(z\u00bb). \n\nThis is the second main result of this paper. \n\n2The proposal by Kechriotis and Manolakos [2] is to use the Hopfield model for an approximation \nto the MAP demodulation. The proposal in this paper goes beyond theirs in that the analog-valued \nHopfield model is used to approximate not the MAP demodulator in the zero-temperature limit but \nthe MPM demodulators directly, including the optimal one. \n\n\f0.01 \n\n0.0001 \n\n0': \n\n10-6 \n\n...... \" ....... \n, \n\nOpt. - -\n10-8 MAP . _._----_._ .. \nMFA ------\nCD \n\n10-10 \n\n0.1 \n\na \n(a) f3s = 1 \n\n0.01 \n\n0 .0001 \n0': \n\n10-6 \n\n, \n... \n\n, \n\n.... \n\n>( , \n, \n, \n, \n, , \n, \n, , \n, , \n, \n, \n\\ \n, \n\\ \n, \n\nI \n\nI \n\n, \n, \n.'. \n\n'. \n\n, \n... \n\n, \n.'. \n, \n, \n., \n'. \n\nOpt. - -\n10-8 MAP --_._._----_ . \nMFA ------\nCD \n\n10-10 \n\n10 \n\n100 \n\n0.1 \n\n10 \n\n100 \n\na \n\n(b) f3s = 20 \n\nFigure 2: Bit error probability for various demodulators. \n\n4.6 AT instability \n\nThe AT instability [7] refers to the bifurcation of a saddle-point solution without replica \nsymmetry from the replica-symmetric one. In this paper we follow the usual convention \nand assume that the first such destabilization occurs in the so-called \"replicon mode [8] .\" \nAs the stability condition of the RS saddle-point solution for the MPM demodulator, we \nobtain \n\na - E2 f D z sech4 (flz + E) = O. \n\nFor the MFA demodulator, we have \n\na - E2 D \n\nf z 1 + E(l -\n\n[ \n\n1 -\n\n2]2 \n\nfez) \n\nf(z)2) \n\n- 0 \n-\n. \n\n(21) \n\n(22) \n\nThe RS solution is stable as long as the left-hand side of (21) or (22) is positive. \n\n5 Performance evaluation \n\nThe saddle-point equations (13) and (18) can be solved numerically to evaluate the bit error \nprobability Pb of the MPM demodulator and its naive mean-field approximation, respec(cid:173)\ntively. We have investigated four demodulators: the optimal one (f3 = f3s), MAP, MFA \n(with f3 = f3s, i.e., the naive mean-field approximation to the optimal one), and CD. The \nresults are summarized in Fig. 2 (a) and (b) for two cases with f3s = 1 and 20, respectively. \nIncreasing a corresponds to relatively lowering the information bit rate, so that Pb should \nbecome small as a gets larger, which is in consistent with the general trend observed in \nFig. 2. The optimal demodulator shows consistently better performance than CD, as ex(cid:173)\npected. The MAP demodulator marks almost the same performance as the optimal one \n(indeed the result of the MAP demodulator is nearly the same as that of the optimal de(cid:173)\nmodulator in the case f3s = 1, so they are indistinguishable from each other in Fig. 2 (a\u00bb. \n\nWe also found that the performance of the optimal, MAP, and MFA demodulators is signif(cid:173)\nicantly improved in the large-a region when the variance a? of the noise is small relative \nto N, the number of the users. For example, in order to achieve practical level of bit error \nprobability, Pb '\" 10- 5 say, in the f3s = 1 case the optimal and MAP demodulators allow \ninformation bit rate 2 times faster than CD does. On the other hand, in the f3s = 20 case \nthey allow information bit rate as much as 20 times faster than CD, which demonstrates that \nsignificant process gain is achieved by the optimal and MAP demodulators in such cases. \n\n\fThe MFA demodulator with fl = fls showed the performance competitive with the optimal \none for the fls = 1 case. Although the MFA demodulator feU behind the optimal and MAP \ndemodulators in the performance for the fls = 20 case, it still had process gain which al(cid:173)\nlows about 10 times faster information bit rate than CD does. Moreover, we observed, using \n(22), that the RS saddle-point solution for the MFA demodulator with fl = fls was stable \nwith respect to replica symmetry breaking (RSB), and thus RS ansatz was indeed valid \nfor the MFA solution. It suggests that the free energy landscape is rather simple for these \ncases, making it easier for the MFA demodulator to find a good solution. This argument \nprovides an explanation as to why finite-temperature analog-valued Hopfield models, pro(cid:173)\nposed heuristically by Kechriotis and Manolakos [2], exhibited better performance in their \nnumerical experiments. We also found that the RS saddle-point solution for the optimal \ndemodulator was stable with respect to RSB over the whole range investigated, whereas \nthe solution for the MAP demodulator was found to be unstable. This observation suggests \nthe possibility to construct efficient near-optimal demodulators using advanced mean-field \napproximations, such as the TAP approach [9, 10]. \n\nAcknowledgments \n\nThis work is supported in part by Grant-in-Aid for Scientific Research from the Ministry \nof Education, Science, Sports and Culture, Japan. \n\nReferences \n[1] M. K. Simon, 1. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread Spectrum Commu(cid:173)\n\nnications Handbook, Revised Ed., McGraw-Hill, 1994. \n\n[2] G. I. Kechriotis and E. S. Manolakos, \"Hopfield neural network implementation of \nthe optimal CDMA multiuser detector,\" IEEE Trans. Neural Networks, vol. 7, no. 1, \npp. 131-141,Jan. 1996. \n\n[3] A. J. Viterbi, CDMA: Principles of Spread Spectrum Communication, Addison-Wesley, \n\nReading, Massachusetts, 1995. \n\n[4] G. Winkler, Image Analysis, Random Fields and Dynamic Monte Carlo Methods, \n\nSpringer-Verlag, Berlin, Heidelberg, 1995. \n\n[5] Y. 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Keams et al.(eds.), Advances \nin Neural Information Processing Systems, vol. 11, The MIT Press, pp. 246-252, 1999. \n\n\f", "award": [], "sourceid": 1927, "authors": [{"given_name": "Toshiyuki", "family_name": "Tanaka", "institution": null}]}