A new scalable decoder for linear space-time block codes with intersymbol interference

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1 A ew scalale der for liear space-time lock s with itersymol iterferece Marc Kuh, Armi Wittee Uiversity of Saarlad, Istitute of Digital Commuicatios, D 6604 Saarruecke, Germay marc.kuh@lnt.ui-saarlad.de, armi.wittee@lnt.ui-saarlad.de Astract - Space-Time Codes represet a key techology for future roadad wireless commuicatio systems. I this paper a class of space-time lock s accordig to [] is used which ca lead to itersymol iterferece () due to a optimized diversity performace. Therefore compesatio is a importat task of the decodig process of these s. Several kow compesatio methods ca e applied for example MMSE equalizatio or parallel or serial cacellatio. I this paper a ew scalale cacellatio method is preseted, that y usig a posteriori iformatio achieves almost the performace of the maximum likelihood der ut with a much lower complexity. The key idea is to use estimated a posteriori proailities to determie the order of the cacellatio process. I each iteratio the der joitly des a variale umer of symols, which meet a specified proaility threshold. By varyig the threshold, the der is scalale i such a way that the complexity of the der ad cosequetly the availale data flow rate ca e adapted i a wide rage to the requests of the trasmissio, to a give ode complexity or a required quality of service. I. INTRODUCTION The statistical chages of the commuicatio (fadig) ofte foud durig the trasmissio of digital iformatio ca affect the average reliaility of the iformatio trasmissio for a give sigal power. Space-Time s comat these fadig effects y utilizig the diversity of the commuicatio give for example y the use of a atea array at the trasmitter ad / or at the receiver [], []. igh rate space-time s icrease the data rate over Multiple Iput Multiple Output (MIMO) s without icreasig the adwidth y usig the spatial sus which are availale i rich diversity [], [3], [4]. I this paper a special class of space-time s accordig to [] is used. Oe of the tasks of the der for these s is the compesatio of, that ca result from a optimized diversity performace ad from iterferig spatial sus [3], [4]. Because otherwise the performace could e heavily affected y the. I this paper a ew iterferece cacellatio method is preseted usig the compesatio of the employed space-time as a example. O pricipal, this method ca e used for other applicatios too. The outlie of the paper is as follows: i Sectio II the cosidered class of space-time lock s is descried. The ew cacellatio method is discussed i Sectio III ad i Sectio IV simulatio results are preseted. II. SPACE-TIME CODE The liear space-time lock s accordig to [] are highly flexile ad adaptive. No a priori kowledge is required at the trasmitter. Fig. a) shows a system lock diagram of the used liear space-time lock. Such a cosists of two cocateated ut decoupled liear lock s, the ier ad the. The ier is used for a adaptatio to the applied umer of trasmit ad receive ateas. Efficiet matrices ca e used for trasmit (TX) diversity ad joit TX diversity ad spatial suig. The ier is optimized with respect to the variatio of the istataeous capacity coceived y the. The is optimized for diversity performace ad achieves a high diversity gai ad a excellet performace i a fadig eviromet eve at rate. Due to the cocateatio the diversity performace optimizatio ad coditioig (adaptatio to the umer of TX ad RX ateas, pure TX diversity, pure spatial sus, joit suig ad TX diversity) are decoupled.

2 w a) α ier s ier ier (MIMO-) r + r ˆα der ier (MIMO-) + der cessig pro- processig ) α w x s r + r α ˆ de- + de- r r (.*) R D X R D X Fig.. Liear space-time lock This class of space-time s is very flexile, it is idepedet from the used modulatio alphaet ad ca easily e adapted to the requests of the trasmissio, for example to differet ode complexities, suet structures or trasmissio s. Furthermore the use of a large lock legth is possile []. These properties will e very importat for future heterogeeous roadad commuicatio systems. I the followig a asead represetatio accordig to Fig. ) is used ecause the ier is ot i the focus of this paper. The ier ad the MIMO are modeled as a Rayleigh fadig x uder the assumptio that fadig is idepedet for every trasmitted symol ( x is a sequece of statistically idepedet complex Gaussia radom variales with zero mea ad uit variace E[ x[ k ] ] = ). The is represeted y the matrix R which is orthoormal R R = I where I is the uit matrix. The vector α is the trasmitted symol vector, w cotais the samples of the oise. The matrix D x is a diagoal matrix of x. The received symol vector r ca e derived as follows: = " α + r R D D R R D w X X X r =Λ " α + () The fadig i D x itroduces ecause the orthoormality of R is destroyed. The icluded i the received sigal is liear ad represeted y the matrix Λ. I Fig. a example for the real part of such a iterferece matrix is show for lock legth 3. To compesate this a der usig a equalizatio method is eeded. III. INTERFERENCE COMPENSATION A maximum likelihood der is optimal for iterferece compesatio ut due to its high complexity it is i may cases ot suitale for the practical use. Therefore oly suoptimal methods are possile. These ca roughly e classified i liear methods (for example zero forcig or MMSE detectors) ad sutractive iterferece cacellatio methods, which agai ca e divided i serial (successive) ad parallel techiques. Sutractive iterferece compesatio methods are already preseted i may pulicatios, for example a serial method i [3] ad a parallel method i [5]. The serial method accordig to [3] first applies a liear equalizatio. The the ifluece (iterferece) of a estimated symol o the remaiig symols is compesated usig the kowledge at the receiver. The a liear equalizatio follows agai efore compesatig the ifluece of the ext estimated symol. O

3 4. Decidig the symols of the segmet of step 3 5. Compesatio of the iterferece of all symols of step 4 o the remaiig ot ded symols The step follows agai; oly ot ded symols will e take ito accout. These steps are repeated util all symols are ded. Fig.. Example of a matrix (real part) accout of sequece errors the order of this cacellatio process is importat. I [3] the order is determied y a priori symol error proailities. A. New scalale iterferece cacellatio method The iterferece compesatio method preseted i this paper is a comiatio of a parallel ad a serial method. After a liear equalizatio a segmet cosistig of received symols is parallel ded i each iteratio. This segmet cosists of a variale umer of symols, which have a a posteriori symol error proaility elow a certai threshold. Tha the of the symols of this segmet is compesated usig the kowledge of the. This is the key idea: to use estimated a posteriori proailities to determie the order of the cacellatio process. I the ext iteratio the a posteriori symol error proailities of the remaiig symols of the received lock oce more are estimated. Based o these values the ext segmet of symols is determied. The complexity of the der ca e adapted i a wide rage y choosig a particular threshold. Algorithm: Iterferece compesatio of a received lock of symols:. MMSE estimatio of all symols i cosideratio of the iterferece matrix. Determiig the a posteriori error proailities of all still ot ded symols o the asis of the MMSE estimatios 3. Determiig the segmet of the symols that will e joitly ded o the asis of the a posteriori error proailities I Fig. 3 this compesatio process is show for the cosidered space-time s. The received symol vector r is multiplied with the MMSE matrix G (). This matrix is calculated usig the kowledge of Λ ad σ ad employig the mea squared error (MMSE) criterio to miimize () mi{ e = G r α }. The elemets of vector () d are MMSE estimatios of all symols. For all these estimated values the a posteriori symol error proailities are calculated. The a posteriori error proaility of a received - PSK symol is P = ( + exp( d E )) () e, PSK σ where σ is the variace of the oise after MMSE equalizatio (assumed as AWGN), d is the asolute value of the estimated amplitude of the symol ad s = - s = E are the two possile sigal poits, E is the eergy per it [6]. I the case of the space-time s accordig to [] the a posteriori symol error proailities ca e approximated uder the assumptio of additive white Gaussia oise (AWGN), although the oise is colored. 4-QAM ca e viewed as two -PSK sigals I ad Q (i phase ad quadrature compoet) with statistically idepedet oise ad idepedet error proailities. The the a posteriori error proaility for a 4-QAM symol ca e approximated as: P = ( P P ) e,4 QAM ( I) ( Q) e, PSK e, PSK ( I) ( Q) ( I) ( Q) e, PSK e, PSK e, PSK e, PSK = P + P P P (3)

4 () α r d Λ ; σ Λ () a post G SER () p e k () P extract fid ethres, segm. segm. () d k () p e k slicer first ˆ α k ded segmet - ( ) G Λ :, k Fig. 3. Iterferece compesatio process Λ a post ; σ SER After calculatig these proailities a segmet () d [ k ] of estimated symols is parallel ded (simultaeously sliced). This segmet cosists of a variale umer of symols with a a posteriori symol error proaility elow a certai threshold () P ethres,. If all symols have a error proaility higher tha this threshold, the oly the symol with the lowest error proaility is ded. After that the of these ded symols cocerig the remaiig symols is caceled usig the kowledge of Λ. The ext iteratio starts with the calculatio of the MMSE matrix G () ad the estimatio of the remaiig symols ad their a posteriori proailities. This process stops whe all received symols are ded. Various refiemets of the iterferece compesatio process are coceivale e. g. a optimal choice of the der threshold value i each iteratio or successive forward-ackward iteratios, if i ay itermediate step the est estimated a posteriori symol error proaility is aove the threshold. Moreover the a posteriori symol error proailities are availale for further use at the receiver, e. g. as soft decisios for additioal FEC codig / decodig. IV. RESULTS I this sectio results are preseted ased o simulatios usig the accordig to [] i a Rayleigh fadig eviromet (Fig. )). Perfect kowledge at the receiver is assumed. The der (Maximum a posteriori DFE MAP-DFE) uses the ew cacellatio method. Fig. 4 ad Fig. 5 highlight the efficiecy of the ew der. They show the symol error rate (SER) of 4-QAM versus E / N 0 at the receiver for a Rayleigh fadig ad lock legth 0 ( r cosists of 0 elemets) ad lock legth 3 respectively. Besides they show the performace of 4-QAM for a AWGN as a referece. The der threshold is zero that meas oly the symol with the lowest a posteriori error proaility is ded i each iteratio. The ew der has a excellet performace, much etter tha a simple MMSE filter for compesatio ad clearly etter tha a der accordig to [3]. The differece etwee the ders icreases with growig lock legth as Fig. 5 shows compared to Fig. 4. Fig. 6 presets a compariso of the performace of the MAP-DFE ad a maximum likelihood sequece estimator (MLSE) for the cosidered spacetime s i a Rayleigh fadig eviromet; 4- QAM is used ad lock legth 6. The ew der achieves almost the performace of the maximum likelihood der ut with a much lower complexity eve with the der threshold P e,thres = 0. Fig. 7 shows the iterdepedece etwee the achieved SER ad a particular threshold for the der at E / N 0 = 0 db; 4-QAM is used ad the lock legth is 0 (as i Fig. 4). I additio it shows a iformatio aout the approximated complexity (cosidered are matrix iversios) i percet. For P e,thres = 0 the complexity is 00%, ad the simulated symol error rate is SER =.5 e(-4). For P e,thres = e(-0) the same SER is foud ut with a approximated complexity of 46%. V. CONCLUSIONS The ew der achieves almost the performace of the maximum likelihood der ut with a lower complexity. The complexity ad cosequetly the availale data flow rate - is scalale y varyig the der threshold (Fig. 7). So, the der ca e adapted to a give ode complexity or a required quality of service. The preseted iter-

5 Fig. 4. SER performace compariso of differet ders (lock legth 0) Fig. 6. Compariso of MLSE ad MAP-DFE for lock legth 6 complexity Fig. 5. SER performace compariso of differet ders (lock legth 3) Fig. 7. SER versus der threshold for E / N 0 = 0 db ferece compesatio method ca e used i a der for space-time s ut also for other applicatios, e. g. multi-user iterferece cacellatio for CDMA-systems. REFERENCES [] A. Wittee, Marc Kuh A ew cocateated liear high rate space-time lock, VTC 00 Sprig, i press. [] V. Tarokh,. Jafarkhai, A. R. Calderak, Space-time lock s from orthogoal Desigs, IEEE Trasactios o Iformatio Theory (vol 45, pp , July 999). [3] G. D. Golde, G. J. Foschii, R. A. Valeuela, P. W. Woliasky, Detectio algorithm ad iitial laoratory results usig V- BLAST space-time commuicatio architecture, Electroic Letters (vol. 7, Nov. 998). [4] G. J. Foschii, Layered space-time architecture for wireless commuicatio i a fadig eviromet whe usig multi-elemet ateas, Bell Las Techical Joural, autum 996. [5] A. Wittee, A ovel adwidth efficiet aalog codig/decodig scheme for data trasmissio over fadig s, Proceedigs ICC 994 [6] J. G. Proakis, Digital commuicatios, ISBN , McGraw-ill, Sigapore, 995.