R55R@3¨>>) >§>©\@¨^R^R^>¨R0>¤RR6X Q356>>6¨3£3§The MF approximation is obtained by taking the approximatingfamily of probability distribution by all productdistribution, i.e.,<strong>and</strong> similarly variational energy can be written as© 3+6 j 3 j 6 j 3 , 3 (36)¨ ¡We now choose a distribution which is close to thetrue distribution, i.e., ¨ < . 0 . The parameter ofthe distribution is chosen so as to minimize Kullback-Leibler (KL) distance, i.e.,6 6 ¨ 6 Z* (24)In order to evaluate j 3 we have to minimize the variationalfree energy, i.e.,Q X Q XQ X(37) *Differentiating this equation with respect j3 to givesnonlinear fixed point equations, i.e.,^£¢ ¨ R . < ¨where ¨ ¦§£¨ ¨ 0¥¤0l¨ < .(25)/%0D%1 .¨ h'¡SUSDS(38)In the matrix <strong>for</strong>m we can write the above equation asZ0 (39)j 3©3 6Uj 63 I0§,0l_ ¦`¨§0l "!^¨ R.
0.05channel <strong>estimation</strong> error10 0 SNR in dB0.0450.040.03510 −1MSE0.030.025Av. BER0.0210 −20.0150.010.00510 −30 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8SNR in dBFigure 1: <strong>Channel</strong> estimate error vs ¡¡ N¨6. Conclusions <strong>and</strong> simulationsIn this work, we proposed channel <strong>estimation</strong> <strong>and</strong><strong>symbol</strong> <strong>detection</strong> <strong>for</strong> the Space-<strong>time</strong> block coded multi<strong>user</strong>system. The channel is estimated by MAP procedureusing short training sequence. After estimatingthe channel the <strong>symbol</strong> <strong>detection</strong> is per<strong>for</strong>med <strong>user</strong>-<strong>wise</strong>considering other <strong>user</strong>s as interferers. Discrete prioris assumed on the interfering <strong>user</strong>s’ bits. In this case,the complexity of computing the posteriori probabilitiesgrows exponentially in the number of interfering <strong>user</strong>s<strong>time</strong>s the <strong>symbol</strong>s per <strong>user</strong>. We derived low complexitymethod to circumvent this problem. The exact posterioriprobabilities are replaced by the approximate separabledistributions. The distributions are calculated by MFT(variational approach). For simulations we consider ascenario in which there are two <strong>user</strong>s. We consider thecase of two transmit <strong>and</strong> four receive antennas. Figure 1shows the channel <strong>estimation</strong> error <strong>for</strong> the system. Figure2 shows the BER versus SNR <strong>for</strong> two <strong>user</strong>s withperfect channel state in<strong>for</strong>mation at receiver. It is clearfrom the figure that the behavior of our algorithm is veryclose to the exact ML curve. It is also clear from figure2 that we obtain better BER with the linear response theoryin comparison to the naive mean field theory. Figure3 shows the BER versus SNR <strong>for</strong> the same system parametersas in figure 2 but with estimated channel. Thisfigure shows that the per<strong>for</strong>mance degrades when thereis channel <strong>estimation</strong> error. It is also clear from this figurethat linear response theory outper<strong>for</strong>ms naive meanfield theory.REFERENCES[1] M. Opper <strong>and</strong> D. Saad, eds,”Advanced mean fieldmethods: Theory <strong>and</strong> practice”. MIT press, 2001.[2] B. Lu <strong>and</strong> X. Wang, “Iterative receivers <strong>for</strong> multi<strong>user</strong>Space-<strong>time</strong> coding systems,” IEEE JSAC vol.18, No. 11, November 2000.[3] A. B. Dempster, N. B. Liard, <strong>and</strong> D. B. Rubin, “Maximum likelihood from incomplete data via EM .Figure 2: Av. BER of K=2, N=2, M=4 vs ¡¢ W¨ .The solid line represents ML by enumeration when channelis exactly known. Square-solid line is ML using linearresponse theory using exact channel <strong>and</strong> star-solidline is ML using naive mean field theory using exactchannel.algorithm”, Journal of Royal Statistical society B,vol.39, no.1, pp. 1-38, 1977.[4] V. Tarokh, H. Jafarkhani, <strong>and</strong> A. R. Calderbank,“Space-<strong>time</strong> block coding <strong>for</strong> wireless communications:per<strong>for</strong>mance results,” IEEE JSAC vol. 17, No.3, March 1999.[5] E.de. Carvalho <strong>and</strong> D.T.M. Slock, “Maximum likelihoodFIR multi-channel <strong>estimation</strong> with Gaussianprior <strong>for</strong> the <strong>symbol</strong>s,” Proc. ICASSP, Munich, Germany,April 1997.[6] G. McLachlan <strong>and</strong> T. Krishnan, “The EM algorithm<strong>and</strong> extensions”, John Wiley ad sons Inc. 1997.[7] V. Tarokh, A. Naguib, N. Seshadri, <strong>and</strong> A. R. calberbank,“Combined array processing <strong>and</strong> <strong>space</strong>-<strong>time</strong>coding”, IEEE Tran. In<strong>for</strong>mation Theory, vol. 45,No. 4, May 1999.[8] G. Parisi, ”Statistical field theory”, Addison Wesley,Redwood, CA, 1988.[9] A. Grant, ”Joint decoding <strong>and</strong> channel <strong>estimation</strong> <strong>for</strong>linear MIMO channels”, Wireless comms. <strong>and</strong> networkingconf. , Chicago, Sep. 2000.[10] P. Bohlin, ”Iterative least square techniques withapplication to adaptive antennas <strong>and</strong> CDMAsystems”,Licentiate thesis, Chalmers Univ. of Tech.,Goteborg, Sweden, 2001. .[11] S. Talwar, M. Viberg, <strong>and</strong> A. Paulraj, ”Blind separationof synchronous Co-channel digital signals usingan antenna array - Part I: Algorithms”, IEEETrans. on Signal Processing, vol. 44(5), pp. 1184-1197, May, 1996.
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