ITERATIVE RECEIVER FOR SPACE-TIME CODED FLAT FADING CHANNELS. Noura Sellami yz, Mohamed Siala y, David Declercq z, Inbar Fijalkow z

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1 ITERATIVE RECEIVER FOR SPACE-TIME CODED FLAT FADING CHANNELS Noura Sellami yz, Mohamed Siala y, David Declercq z, Ibar Fijalkow z y Frace Telecom R&D, rue du Gééral Leclerc Issy les Moulieaux, Frace z ETIS/ENSEA-UCP, 6 av. du Poceau Cergy Cedex Frace ABSTRACT I his paper, we prese a ieraive receiver for rasmissio over coded muliple-ipu muliple-oupu (MIMO) Rayleigh fla fadig chaels. We compare he performace of his receiver for parallel urbo-codes ad covoluioal codes. We show ha he use of parallel urbo-codes over ime varyig chaels yields a sigifica improveme compared o covoluioal codes.we also discuss he choice of he umber of ieraios of he urbo decoder for each ieraio of he receiver. Simulaio resuls show ha he performace of our receiver is sigificaly beer ha ha of he o ieraive receiver ad approaches he performace of he correspodig sysem where all ierferig rasmiers have bee suppressed. 1. INTRODUCTION The growig demad for ew services a high daa raes idicaes he eed for ew echiques o icrease chael capaciy. Foschii ad Gas [1][2] have demosraed he eormous capaciy poeial gai of wireless commuicaio sysems wih aea arrays i boh rasmier ad receiver. They have show ha he capaciy of fla fadig mulipleipu muliple-oupu (MIMO) chaels icreases liearly wih he umber N of rasmi aeas as log as he umber of receive aeas M is greaer ha or equal o N. I his paper, we cosider a MIMO sysem rasmiig parallel daa coded sreams simulaeously. The srucure of his rasmier was iiially proposed by Li e al. [8] ad referred o as verical codig for BLAST. A sigle code is used o ecode all he sigals o be rasmied. Our aim is o develop a efficie deecio ad decodig echique for his sysem. Recely, he ieraive processig priciple used i urbo codig [5] has bee successfully applied o a wide variey of joi deecio ad decodig problems [6][9]. The basic idea behid ieraive processig is o exchage exrisic iformaio amog he receiver modules i order o achieve successively refied performace. I his paper, we propose a ieraive receiver for MIMO Rayleigh fla fadig chaels. Durig each ieraio, he likelihood geeraor provides he sof-ipu/sof-oupu decoder wih codiioal probabiliies ad he decoder supplies a priori iformaio o he likelihood geeraor. We compare he performace of his receiver for parallel urbo-codes ad covoluioal codes over ime-varyig chaels. For comparable complexiies, he use of urbo-codes yields a sigifica improveme. We also give a clever combiaio of urbo-code decoder ieraios ad receiver ieraios i order o achieve a good complexiy/performace rade-off. I is see ha he performace of he ieraive receiver approaches he opimal performace achieved whe all spaial ierferece has bee perfecly suppressed. The paper is orgaized as follows. I secio 2, he sysem model is described. Secio 3 preses he ieraive receiver. I secio 4, a clever combiaio of urbo decoder ieraios ad receiver ieraios is give. Simulaio resuls are preseed i secio SYSTEM MODEL Geeral framework: We cosider a discree fla fadig MIMO chael wih N rasmi aeas ad M receive aeas. As show i Fig.1, he ipu iformaio bi sequece is firs ecoded wih a rae R = 1=N covoluioal ecoder. The oupu of he ecoder is demuliplexed io N sreams ha are ierleaved by differe ierleavers i, modulaed ad rasmied simulaeously by he N rasmiig aeas. We assume i he sequel of he paper ha he modulaio used is BPSK. b Ecoder DeMux Modulaor Modulaor Fig. 1. Trasmier srucure d d Trasmiig Aea Array The received sigal a aea j a ime is a liear combiaio of he N rasmied sigals perurbed wih oise r (j) = N i=1 h i;j ()d (i) + (j) where (j) are idepede samples of a zero mea whie Gaussia oise wih variace 2 = N0 ad h 2 i;j() is he ap gai from rasmi aea i o receive aea j a ime. Le d = (d ; :::; d ) T 2 f+1; 1g N be he N 1 vecor of coded symbols rasmied from he N aeas a ime ; r = (r ; :::; r (M) ) T be he M 1 received vecor ad = ( ; :::; (M) ) T be he M 1 oise vecor a he receive aeas. H is he chael marix which collecs he gais h i;j (). The discree ipu-oupu relaio of he chael ca be wrie i he followig vecor oaio

2 r = H d + (2) The problem we address is he o recover he iformaio bis b from he oisy observaio r : Simulaio framework: I our simulaios, we cocerae o a MIMO sysem wih 2 rasmi aeas ad 2 receive aeas. We use fla fadig chaels. To geerae he Rayleigh fadig, we cosider he Jakes model [3] wih as parameers he frequecy badwidh B d ad he symbol duraio T. The produc B d T characerizes he ime seleciviy of fla fadig chaels. Realisic values of B d T are used i our simulaios. For example, i a WCDMA FDD sysem ad for a speed of 65 km=h, B d T is aroud We assume ha he chael is perfecly kow. We plo he Bi Error Rae () wih respec o E b =N 0 per receive aea. The size of he o uiform ierleavers i is 4290: 3. ITERATIVE RECEIVER We prese a ieraive receiver based o successive esimaio of he coded bis d (i) ad decodig (Fig.2). The likelihood geeraor geeraes he codiioal likelihoods p(r jd (i) = d); d 2 f+1; 1g. These probabiliies are deierleaved ad muliplexed. The decoder uses his iformaio o calculae he a poseriori probabiliies (APP) for he coded bis p(d (i) = djr) usig he MAP algorihm [4]. Exrisic iformaio compued from a poseriori coded bis probabiliies is used as a priori iformaio for he likelihood geeraor o he ex ieraio. Afer some ieraios, hard decisios o he ipu iformaio bis are ake by he decoder. I he ex hree secios, we prese i deails each par of he receiver Compuaio of he codiioal likelihoods The likelihood geeraor uses he received sigals r j () o calculae he codiioal likelihoods p(r jd (i) p(r ; = d) = d(i) = d) p(d (i) = d) = d (d (i) =d) p(r jd ) NY j=1 j6=i p(d (j) ) (3) Accordig o (2), he codiioal probabiliy of r give d is he Gaussia probabiliy! 1 p(r jd ) = (2 2 ) M exp jr H d j 2 (4) O he firs ieraio, he ecoded bis are ukow so we use equal a priori probabiliies p(d (j) ) = 1=2: O he ex ieraios, p(d (j) ) are give by he exrisic values provided by he MAP decoder. r Geeraor p( r d ) p( r d ) ex ex -1-1 MU Fig. 2. Ieraive receiver 3.2. Decodig sep for covoluioal codes Decoder DeMux Decoded Bis (i) I he case of a covoluioal code, he a poseriori probabiliies of he coded bis are calculaed usig he MAP decodig algorihm proposed by Bahl e al. [4]. Le r be he whole received vecor, he ierleaved idex, S he rellis sae a ime ad B (i) he se of rasiios S 1 = m ;! S = m such ha d (i) is equal o d. The a poseriori probabiliies are expressed as where p(d (i) = djr) = (m ; ;m)2b (i) (m ; ; m) (5) (m ; ; m) = p (S 1 = m ; ; S = mjr) (6) Followig he oaios i [4] (m ; ; m) = 1 (m ; ) (m ; ; m) (m) (7) (m) ad (m) are recursively calculaed as (m) = (m) = m ; 1 (m ; ) (m ; ; m) (8) m ; +1 (m ; ) +1 (m; m ; ) (9) from he rasiio probabiliies (m ; ; m) ad +1 (m; m ; ). The decoder uses he iformaio provided by he likelihood geeraor o calculae he rasiio (m ; ; m) = p(s = mjs 1 = m ; ) NY j=1 ex p(r jd (j) = d(j) ) (10) where d (j) = d (j) are he values of he N coded bis correspodig o he rasiio S 1 = m ;! S = m whe i exiss. A he K h ieraio of he receiver, he exrisic value p (K) ex (d(i) ) is obaied by dividig he a poseriori probabiliy p(d (i) = djr) by he codiioal likelihood p(r jd (i) = d (i) ) provided by he likelihood geeraor. The exrisic iformaio p ex (d (i) ) is calculaed as he produc Q K k=1 p(k) ex (d(i) ) over previous ieraios ad provided o he likelihood geeraor as a priori iformaio.

3 3.3. Decodig sep for parallel urbo-codes I order o improve he performace of he receiver, oe may hik of replacig he covoluioal code by a parallel urbocode [5][7]. For he sake of simpliciy, Fig.3 shows he rasmier srucure whe N = 2: b Covoluioal ecoder Covoluioal ecoder (2) (0) Pucure =b Fig. 3. Trasmier srucure for a urbo-code We describe i he followig oe ieraio of he receiver we propose o use (Fig. 4). The urbo decoder akes he codiioal likelihoods provided by he likelihood geeraor. Afer a fixed umber of ieraios of he decoder is performed, Dec1 calculaes he APP for he reduda bis ad Dec2 calculaes he APP for he ucoded bis b ad for he reduda bis (2). The compuaio of hese a poseriori probabiliies is performed as i [5] ad he meric brach used is give by (10): I he ex receiver ieraio, he likelihood geeraor akes he exrisic values obaied from he a poseriori probabiliies of all he bis ( p ex (b ); p ex ( ) ad p ex ( (2) ) ) ad calculaes he codiioal likelihoods usig (3). differe ierleavers may improve he performace of he ieraive receiver over ime-varyig chaels. 4. IMPROVED RECEIVER WITH TURBO-CODE The likelihood values se o he decoder are o reliable a he firs ieraio of he receiver. Thus, if we perform may ieraios of he urbo decoder a he firs ieraio of he receiver, here are chaces ha he erroeous a priori iformaio provided o he likelihood geeraor would be ehaced. I ha case, he ex ieraios of he receiver are less likely o improve he codiioal likelihoods. Therefore, we cosider a varyig umber of ieraios of he urbo decoder i order o avoid his problem ad achieve a good complexiy/performace rade-off. Codiioal likelihoods provided o he decoder are more ad more reliable as receiver ieraios are performed. Thus, a good rade-off bewee complexiy ad performace is o icrease he umber of ieraios of he decoder wih he umber of ieraios of he receiver : for he h ieraio of he receiver; ieraios of he decoder are performed. 5. SIMULATION RESULTS We cosider he performace of he receiver for he covoluioal code wih 16 saes ad geeraor polyomials (23; 35) ad he parallel urbo-code(1; 5 ). We choose hese codes 7 sice heir decodig algorihms have comparable complexiies. ier1 ier2 ier3 p( r b) P( ( 1 ) r DEC1 1 2 p e(b) p (b) e p( r b) p( r DEC2 p e x ( ( 2 ) p e x (b) geeraor p e x ( geeraor Decoded bis Fig. 4. Turbo decoder As i [5], durig he ieraios of he decoder, Dec1 ad Dec2 exchage exrisic probabiliies o he sysemaic bis (p e (b ) ad he oher as a a priori iformaio, amely p(s = mjs 1 = m ; ) i (10): O he firs urbo ieraio, Dec1 does o have a priori iformaio ad assumes p(s = mjs 1 = m ; ) = 1=2: 3.4. Ierleaver desig To fully realize he poeial of he ieraive receiver, he sreams of coded symbols are fed o idepede radom ierleavers before hey are rasmied. Ierleavers help o break he correlaio of he ecoder oupu ad guaraee ha he a priori iformaio are almos idepede. The use of Fig. 5. Performace of he ieraive receiver : covoluioal code (23; 35), B d T = The opimal performace ha ca be achieved correspods o a sysem where all ierfereces from rasmi aeas k 6= k 0 are suppressed i he deecio of he sigal rasmied by aea k 0. This yields o a SIMO (sigle-ipu muliple-oupu) sysem. I he Figures, he label idicaes his opimal performace. Fig.5 shows performace for oe o hree ieraios of he receiver for B d T = 10 3 whe he covoluioal code is used. We oice ha mos of he improveme is achieved i he secod ieraio. Two ieraios are sufficie o approach closely he opimal performace.

4 Fig.6 ad Fig.7 show performace for oe o four ieraios of he receiver for B d T = 10 3 for he urbo-code respecively whe four ieraios of he decoder are performed a each receiver ieraio ad whe he reduced complexiy receiver is used. I ca be observed ha similar performaces are achieved afer four ieraios i he wo cases. Thus he combiaio proposed i secio 4 achieves a good complexiy/performace rade-off.. From Fig.5 ad Fig.6, we coclude ha for comparable complexiies he use of parallel urbo-codes yields a sigifica improveme compared o covoluioal codes. For a = 10 3 he urbo-code performs 0:75 db beer ha he covoluioal code. Neverheless, he opimal performace is approached more rapidly ad more closely whe he covoluioal code is used. affeced bu i sill approches he lower. I fac, whe B d T is low, he coefficies of he chael are almos cosa over a rasmied block so ime diversiy is decreased. ier1 ier2 ier3 ier4 ier1 ier2 ier3 ier4 Fig. 8. Performace of he ieraive receiver : urbo-code (1; 5 ), B 7 dt = 2:10 4 ; opimal combiaio of urbo decoder ieraios ad receiver ieraios Fig. 6. Performace of he ieraive receiver : urbo-code (1; 5 ), B 7 dt = 10 3, 4 ieraios of he decoder a each receiver ieraio. ier1 ier2 ier3 ier4 6. CONCLUSION I his paper, we have proposed a ieraive receiver for MIMO fla fadig chaels. Durig oe ieraio, he likelihood geeraor provides he sof-ipu/sof-oupu decoder wih codiioal probabiliies ad he decoder uses hese probabiliies o calculae a poseriori probabiliies for he coded bis. Exrisic values are obaied from hese APP. O he ex ieraio, he likelihood geeraor uses he exrisic iformaio as a priori iformaio. We have compared he performace of his receiver for covoluioal codes ad parallel urbo-codes. We also have proposed a improved combiaio of urbo decoder ieraios ad receiver ieraios ha reduces sigificaly he complexiy. Simulaio resuls have show ha he performace of he proposed ieraive receiver approaches he performace of a SIMO sysem. 7. REFERENCES Fig. 7. Performace of he ieraive receiver : urbo-code (1; 5 ), B 7 dt = 10 3, opimal combiaio of urbo decoder ieraios ad receiver ieraios. Fig.8 shows performace for oe o four ieraios of he receiver for B d T = 2:10 4 whe he improved combiaio of receiver ieraios ad urbo decoder ieraios is used. I is see ha he performace of he receiver is sigificaly [1] G.J.Foschii ad M.J.Gas, Capaciy whe usig diversiy a rasmi ad receive sies ad he marix chael is ukow a he rasmier, Wireless Persoal Commuicaios, Vol. 6, No. 3,1998, pp [2] G.J.Foshii, Layered space-ime archiecure for wireless commuicaios i a fadig evirome usig muli-eleme aeas, Bell Labs Sys. Tech. J:, vol.1, pp , Auum [3] W.C.Jakes, Microwave Mobile Commuicaios, New York: Wiley, [4] L.R.Bahl, J.Cocke, F.Jeliek, ad J.Raviv, Opimal decodig of liear codes for miimizig symbol error rae, IEEE Tras. Iform. Theory, Vol. IT-32, pp , March [5] C.Berrou, A.Glavieux ad P.Thimajshima, Near Shao limi error-correcig codig ad decodig: Turbo codes, IEEE I. Cof. Commuicaios, Geeva, Swizerlad, pp , May 1993.

5 [6] M.C.Reed, C.B.Schlegel, P.D.Alexader ad J.A.Asesorfer, Ieraive muliuser deecio for CDMA wih FEC: ear sigle user performace, IEEE Tras. Comm., Vol.46, No.12, pp , Dec [7] C.B.Schlegel, Trellis codig, Piscaaway, NJ: IEEE Press, [8] iaodog Li, Howard Huag, G.J.Foschii, ad Reialdo A. Valezuela, Effecs of ieraive deecio ad decodig o he performace of BLAST, GLOBECOM, Sa Fracisco, USA, November [9] A. Roumy, I. Fijalkow, D. Pirez, Turbo muliuser deecio for coded asychroous DS-CDMA over frequecy selecive chaels, IEEE Sixh Ieraioal Symposium o Spread Specrum Techiques & Applicaios (ISSSTA 00), pp , New Jersey, USA, Sepember 2000.