Channel estimation and user-wise symbol detection for space-time ...

E[¥[Rf(RRR¥>>@>AE[f 2fj55 >(XR2 ff/55 >E@(^¨§R9¨R¨§9'9Rconsist of the received signal from time slots to © ,at jk¤§¦ the receiver antenna.response of the £ user andgQ¥]¨§'¥"¨$% denotes the channel ¥"¨¡ SUSDSis the code vector of £¥¤§¦ the user, ""¨,-d. withX [and_ fQf ¨'h _ f ¨% _SDSDS _ f ¨©i! (4)(5)0 %1 ;is the additive noise vector at j)¤§¦ the receiver! denotes transpose operator.antenna.¨'*+For single user, the Alamouti scheme, (two transmit antennas),STBC is given bywhere © ¨is prior on . We assume that prioron is zero mean Gaussian (which can be correlatedor uncorrelated). We will use MAP method forRchannel estimation for correlated and for Runcorrelatedcase. MAP method is superior than maximum likelihood(ML) because we take into account the prioriknowledge about the channel matrix. The probability ofchannel matrix when there are correlations both transmittingand at the receiver end is given © ¨R¨cR [by, . Where i0 ¥ ¨are correlation< ¨matrices and is a constant. Taking logarithm of Eq.10,plugging the corresponding expressions and differentiatingwith respect to givesR < f ¨§ f ¨$%( < ¨' ¨$%! ¨ ! ¨'h)( Y§*) > * (12) 2¦ §#"§$§#""&%'("(% _ f ¨§ ^f ¨$% ( _(6)Where % represents Kronecker product and vec(.) isvectorization operator. In case of uncorrelated antennaswe haveR ¦ § "¨ ! Z¨ ¨§¨ @ ¨It can be further written as ) > (13)f ¨§

R>>^>XX R>R>>> >R>>@1[>R>R>RR> >>[>R>R>>auxiliary¤£¦¥aQ 9 9 § §0¢¡ ,function , which¨-,e0¢¡ , + ¨-,/.is the expected value of the complete data likelihood,given the observed ¡ , data D and the parameter computedat the previous iteration. Intuitively, computing correspondsto filling the missing data using the knowledgeof the observed data and previous parameters. The auxiliaryfunction is deterministic and can be maximized.An EM algorithm iterates the following two steps, fork=1,2,...., until a local or global maximum of the likelihoodis found.Expectation: Computeof the rest of the users. Without loss of the generality, wedetect the symbols for user 1 first, and for user in thelast, i.e., in the ascending order of the users. Given theabove model we are now ready to define complete dataset. We choose complete data set /< 0 1 as . Thederivation of the algorithm is as follows: The pdf of thecomplete data set can be written as,where ¨

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.,and 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 form we can write the above equation asZ0 (39)j 3©3 6Uj 63 I0§,0l_ ¦`¨§0l "!^¨ R.

0.05channel estimation 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: Channel estimate error vs ¡¡ N¨6. Conclusions and simulationsIn this work, we proposed channel estimation andsymbol detection for the Space-time block coded multiusersystem. The channel is estimated by MAP procedureusing short training sequence. After estimatingthe channel the symbol detection is performed user-wiseconsidering other users as interferers. Discrete prioris assumed on the interfering users’ bits. In this case,the complexity of computing the posteriori probabilitiesgrows exponentially in the number of interfering userstimes the symbols per user. 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 users. We consider thecase of two transmit and four receive antennas. Figure 1shows the channel estimation error for the system. Figure2 shows the BER versus SNR for two users withperfect channel state information 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 for the same system parametersas in figure 2 but with estimated channel. Thisfigure shows that the performance degrades when thereis channel estimation error. It is also clear from this figurethat linear response theory outperforms naive meanfield theory.REFERENCES[1] M. Opper and D. Saad, eds,”Advanced mean fieldmethods: Theory and practice”. MIT press, 2001.[2] B. Lu and X. Wang, “Iterative receivers for multiuserSpace-time coding systems,” IEEE JSAC vol.18, No. 11, November 2000.[3] A. B. Dempster, N. B. Liard, and 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 and 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, and A. R. Calderbank,“Space-time block coding for wireless communications:performance results,” IEEE JSAC vol. 17, No.3, March 1999.[5] E.de. Carvalho and D.T.M. Slock, “Maximum likelihoodFIR multi-channel estimation with Gaussianprior for the symbols,” Proc. ICASSP, Munich, Germany,April 1997.[6] G. McLachlan and T. Krishnan, “The EM algorithmand extensions”, John Wiley ad sons Inc. 1997.[7] V. Tarokh, A. Naguib, N. Seshadri, and A. R. calberbank,“Combined array processing and space-timecoding”, IEEE Tran. Information Theory, vol. 45,No. 4, May 1999.[8] G. Parisi, ”Statistical field theory”, Addison Wesley,Redwood, CA, 1988.[9] A. Grant, ”Joint decoding and channel estimation forlinear MIMO channels”, Wireless comms. and networkingconf. , Chicago, Sep. 2000.[10] P. Bohlin, ”Iterative least square techniques withapplication to adaptive antennas and CDMAsystems”,Licentiate thesis, Chalmers Univ. of Tech.,Goteborg, Sweden, 2001. .[11] S. Talwar, M. Viberg, and 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.

10 010 −1Av. BER10 −210 −30 1 2 3 4 5 6 7 8SNR in dBFigure 3: Av. BER of K=2, N=2, M=4 vs ¡¢ W¨ .The solid line represents ML by enumeration when channelis exactly known. Dashed line is ML using linearresponse theory using estimated channel and star-solidline is ML using naive mean field theory using estimatedchannel.[12] J. A. Fessler, and A. O. Hero, ”Space-AlternatingGeneralized Expectation Maximization Algorithm”, IEEE Trans. on Signal Processing, vol.42(10). pp.2664-2677, Oct. 1994.[13] P. F. R. Sorensen, O. Winther, and L. K. Hansen,”Mean field approaches to independent componentanalysis”, Neural Computation, pp.889-918, 2002.[14] M. Opper, and O. Winther, ”Variational linear response”, In advances in Neural Information Processing(NIPS), 2003.