On the Performance of V-BLAST with Zero-Forcing Successive Interference Cancellation Receiver

Transcription

1 On he erformance of V-BLAST wh Zero-Forcng Successve Inerference Cancellaon Recever Cong Shen, Yan Zhu, Shdong Zhou, Jnng Jang Sae Key Lab on crowave & Dgal Communcaons Dep. of Elecroncs Engneerng, Tsnghua Unversy, Beng, CHINA, 84 Absrac- The performance analyss of he Zero-Forcng Successve Inerference Cancellaon (ZF-SIC) deecor n Vercal Bell Labs Layered Space-Tme (V-BLAST) s generally consdered dffcul manly because of he nonlnear error propagaon effec. In hs paper we frs presen he nearly exac b error rae (BER) analyss of ZF-SIC V-BLAST wh Bnary hase-shf Keyng (BSK) modulaon n he rchly scaered Raylegh-fadng mulple-anenna channel. We derve he closed-form expressons of he error probably of each subsream under a gven number of ransm anennas. We furher gve a general recursve procedure o calculae he BER of each subsream wh arbrary number of ransm and receve anennas. The procedure we gve s easy o follow and s feasble n evaluang he BER performance of V-BLAST wh ZF-SIC recever. I can be furher ulzed o develop opmal ransmsson sraegy for he open-loop V-BLAST. Our analycal resuls mae excellen maches wh he one Carlo smulaon resuls. I INTRODUCTION Vercal Bell Labs Layered Space-Tme (V-BLAST) was proposed [5] o acheve he very hgh specral effcency promsed by he mulple-anenna sysem []-[3]. In he orgnal V-BLAST sysem [5], parallel daa sreams are smulaneously ransmed hrough mulple anennas n he same frequency band, and decoded a he recever wh he Zero-Forcng Successve Inerference Cancellaon (ZF-SIC) deecor, whch helps o acheve he hgh specral effcency wh reasonable decodng complexy. Due o hese advanages, ZF-SIC V-BLAST has ganed los of research aenon n he pas few years [4]-[7]. The performance analyss of V-BLAST usng ZF-SIC Ths wor s suppored by Chna Naonal 863 Scence Foundaon, No. AA3 and Chna Naonal Scence Foundaon, No. 94. deecor s n general consdered dffcul. The dffculy manly comes from he nonlnear nerference cancellaon operaon, whch generaes he so-called error propagaon effec n praccal sysems and s hard o quanfy. relmnary research n [6] has repored he asympoc analyss and numercal one Carlo smulaon resuls. Alhough he numercal approach s useful n performance evaluaon, he analycal approach provdes deep nsgh and comprehensve undersandng of he essenal and ey pons of V-BLAST. Furhermore, analycal resuls are useful n developng opmal ransmsson schemes such as power allocaon. [7] derves closed-form expressons for sgnals a each deecon sep and performs sascal analyss n a Raylegh-fadng channel based on he perfec nerference cancellaon assumpon. [8] breas hs assumpon and presens an analyss on he on error rae and symbol error rae. However he closed-form expresson of b error rae (BER) of each subsream s sll unnown. In hs paper we are movaed o presen a nearly exac error probably analyss for he praccal ZF-SIC V-BLAST wh Bnary hase-shf Keyng (BSK) modulaon n he Raylegh-fadng mulple-anenna channel, n whch we wll ae he effec of error propagaon no accoun. Our analyss wll gve closed-form expressons of BER of each subsream when he number of ransm anennas s fxed. For example we wll presen he probably of b error of each subsream n he case of =. We wll also propose a general procedure o calculae he BER of each subsream wh arbrary number of ransm and receve anennas. one Carlo smulaons are used o verfy our analycal resuls. Especally, hs analycal resul s useful o desgn he opmal ransmsson sraegy for open-loop V-BLAST, such as he opmal open-loop power allocaon. The res of hs paper s organzed as follows. In Secon II, we nroduce he sysem model. Bref nroducon of ZF-SIC V-BLAST s presened n Secon III. Secon IV descrbes he performance analyss and smulaon resuls. Fnally, IEEE Communcaons Socey Globecom /4/$. 4 IEEE

2 Secon V conans our conclusons. II SYSTE ODEL We consder a sngle-user, pon o pon Raylegh-fadng communcaon channel wh ransm and N receve anennas. We assume ha he channel s fla fadng and quas-sac, namely, he channel s consan over a frame, and vares from one frame o anoher. The fadng coeffcen h s he complex pah gan from ransm anenna o receve anenna. We assume ha he coeffcens are ndepenly complex crcular symmerc Gaussan wh N un varance, and wre H = [ h ]. H s assumed o be nown o he recever o allow coheren deecon, bu no a he ransmer. We assume N so ha he channel marx H s full column ran wh probably one. The oal ransmsson power s assumed o be regardless of, and s equally allocaed on each ransm anenna. The nose s assumed o be addve whe Gaussan nose (AWGN) wh zero mean and double-sded power specral densy N. The SNR s hen gven by SNR = N. Specfcally, he followng dscree-me equvalen model s used: y = H s+ n () where s = [ s, s,, s ] s a vecor whose he -h componen represens he sgnal ransmed from he -h anenna and belongs o he uncoded normalzed (un average energy) BSK sgnal consellaon { +,. The receved sgnal and nose vecor are boh N vecors whch are denoed by y and n, respecvely. We also assume perfec synchronzaon and mng a he recever. The followng noaons wll be used hroughou hs paper: ' for ranspose, I n for he n n deny marx, E[] for expecaon, Cov( ) for covarance, rob{ for probably, bold lowercase leers for vecor, and bold uppercase leers for marx. Fnally, we shall also fnd convenen o paron he channel marx no s columns as H = [ h, h,, h ]. III ZF-SIC RECEIVER FOR V-BLAST The ZF deecor s a lnear nullng echnque and has he characersc ha s performance approaches ha of he mnmum mean-squared error (SE) deecor a hgh SNRs. SIC s a nonlnear echnque frs nroduced n he heory of muluser deecon (UD). I explos he benef of mng synchronsm and uses symbol cancellaon o mprove he dversy order [9][] of he ye o be deeced symbols for superor performance. erformance analyss of SIC n UD has been repored n many leraures such as [][]. However hose mehods may no be applcable n he V-BLAST scenaro and a precse analyss of he ZF-SIC over he mulple-anenna channel s sll unavalable. The orgnal srucure of he ZF-SIC recever also comprses he opmal orderng procedure o furher enhance he performance. However he opmal orderng wll mae our analyss much more complcaed. In [7] s also shown ha he opmal orderng does no resul n ncreased dversy order, bu only n a fxed SNR gan. Thus hroughou hs paper we only consder he ZF-SIC recever n a fxed deecon order (e.g., deecng accordng o he order of ransm anenna ndexes) nsead of he opmal order. I should be noed ha he analyss below can be exed n a sraghforward way o combne opmal orderng, a he expense of geng more complcaed expressons. We rewre () as y = h s + n = The ZF-SIC recever sars he processng of s and proceeds forwards o s. For s, he nerferng sgnal s = h s. We choose a wegh vecor w accordng o he ZF creron o lef mulply he receved sgnal y and ge he decson sasc r = w y for s. The wegh vecor w acually performs he so-called nerference nullng procedure [5]. We slce r o oban ŝ, and hen he conrbuon of s o he receved sgnal s oally subraced under he assumpon ha s ˆ = s. The processng connues for ŝ by nullng ou nerference from subsreams 3 o. Smlarly, ZF-SIC proceeds unl s has been deeced. Afer subracng he conrbuon of s,, s, we can wre he updaed receved sgnal as ( ) y = y h ˆ s = = h ( ˆ s + n+ h s s) = + = arge sgnal wh nerference equvalen nose I s easy o see from () ha he updaed receved sgnal ( ) y s composed of hree pars: he ye o be deeced symbols, he nose vecor and he poenal error propagaon () IEEE Communcaons Socey Globecom /4/$. 4 IEEE

3 sgnal. The las wo pars mae up of he equvalen nose. Here we frs consder he deal case where no error propagaon s presen, and gve he performance analyss resul of hs deal scenaro. The resul here wll serve as a prelmnary for our fuure analyss n he nex secon where error propagaon s consdered. Assumng perfec feedbac, he ZF-SIC creaes ndepen one-dmensonal sub-channels. The -h sub-channel has he dversy order of D = N - +. [3] calculaes he exac probably of b error on he -h subsream wh BSK modulaon as e D D D + = ( µ ) ( µ ) + = (3) where Q( ) denoes he Q-funcon [3], ρ = N, µ ρ ρ ρ = N. = ( + ), and ( ) IV ERFORANCE ANALYSIS In hs secon we wll presen he maor conrbuon of our wor by gvng he nearly exac BER analyss of he ZF-SIC deecor wh error propagaon. Equaon (3) reveals he error probably of a subsream s oally deermned by he dversy order D and he subsream SNR ρ. Ths resul wll be used exensvely n he followng analyss. So for he sae of smplcy we defne a funcon A. Calculaon of rob{ s ˆ s A Here we are neresed n calculang he condonal probably of error gven ha sˆ sˆ have wrong decsons and rgh decsons. Consderng ha we do no now exacly whch decsons ou of sˆ sˆ are wrong, we defne a map funcon g (). I could be any necon from {,,, o {,,,. Thus we can wre he equvalen nose as ˆ A = + g ( ) sg ( ) sg ( ) = n n h ( ) (7) The codeboo of each ransm anenna s { +,. So s ˆ g ( ) sg ( ) can only be chosen from { +,. I s easy o see ha n A s no Gaussan dsrbued snce he even A wll brng resrcons o n and h g ( ). However we assume here n A s whe Gaussan dsrbued o connue our analyss. Ths assumpon s que reasonable consoldaed by our fuure smulaon resuls. We calculae he mean and covarance marx of n as A n A =, N N E( ) 4 Cov( A ) n = N + I As n A s sll whe Gaussan dsrbued, we can rob s sˆ A as drecly apply (4) o calculae { D D D + e( D, ρ ) = ( µ ) ( µ ) + = (4) { ˆ rob s s A = e N +, N + 4 (8) o represen (3). We are movaed o calculae he probably of b error of he -h subsream e, =,,. ( e can be drecly obaned from (3).) Frs we can wre = { ˆ = rob s s e { s ˆ s A rob{ A = rob where we defne he even { There are exacly deecon errors n ˆ ˆ. A = s s (6) I s clear ha f we can ge he wo correspondng pars n (5), we are able o calculae. e (5) B. Calculaon of rob{ A I s sraghforward o ge he followng resuls for =,,. When =, we have { A { ˆ = s = s A = rob{ sˆ = s rob{ A A rob rob, = e N +, rob A N When =, we have { (9) IEEE Communcaons Socey Globecom /4/$. 4 IEEE

4 { A { ˆ = s s A = rob{ s ˆ s A rob{ A rob rob, = e N +, rob A N + 4 ( ) { () When =,,, we have () a he boom of hs page. =, whch can be drecly calculaed from (3). Thus (9)-() are all calculable by recurson from =. We can ge { rob A = and noce ha rob{ A e C. Calculaon of e Snce rob{ s ˆ s A and rob{ A are boh calculaed, e s drecly acheved usng (5) for =,,. Obvously, f s no pror gven, we are unable o gve he explc expressons of e because he number of recurson for calculang (9)-() canno be deermned. On he conrary, once s fxed, we are able o ge he closed-form expressons of rob{ A and hence ge e. For example, we gve he closed-form expressons of he BER of each subsream when =, N as follows. e N N N + = ( µ ) ( µ ) + = N N N + = ( e ) ( µ ) ( µ ) + = e N N N + + e ( µ ) ( µ ) + = where µ = ρ ( + ρ ) and ( N 4 ( ) ) ρ = +, =,. Obvously, he closed-form expressons become much more complcaed as ncreases. So we are movaed o gve a general procedure o calculae he nearly exac BER of each subsream for arbrary number of ransm and receve anennas. The whole algorhm comes from (4)-(5), (8)-() and can be descrbed compacly hrough he recursve procedure as follows. Inalzaon: ρ () = N e Recurson:, =,, ( ) = e( N +, ρ()) { rob A = for =: f > { A = ( N + ρ ) { A K { A = ( N + ρ ) { A rob e, () rob rob e, ( ) rob for =:- for =:- { A { ˆ = s s A { A + rob{ s ˆ s A rob{ A rob rob rob { s ˆ s A = ( N + ρ + ) rob e, ( ) { ˆ rob{ = rob s s A A e = Funcon e( D, ρ ) s defned n (4). { A { ˆ, { ˆ = s s A + s = s, A rob rob rob { s ˆ { { ˆ s { A A s s A A { s ˆ { { ˆ s { A A + s s A A = rob rob + rob = rob = rob rob rob rob () IEEE Communcaons Socey Globecom /4/$. 4 IEEE

5 We resor o one Carlo smulaons o verfy our analyss resul. We consder an uncoded V-BLAST sysem wh 4 ransm anennas and 4 receve anennas. BSK modulaon s adoped a he ransmer and ZF-SIC wh ndex-order deecon s performed a he recever. Uncoded BER n smulaon s obaned by averagng over large volumes of channel realzaons. Fg. gves he BER performance obaned from boh he smulaon and he analyss. I s clear ha he one Carlo smulaon maes a nearly perfec mach o our analyss resul, whch demonsraes he valdy of he calculang mehod we have proposed. To he bes of our nowledge, here s no such accurae calculaon repored. BER - sub-sream, smulaon - sub-sream, analyss sub-sream, smulaon sub-sream, analyss sub-sream 3, smulaon sub-sream 3, analyss sub-sream 4, smulaon sub-sream 4, analyss SNR (db) Fg.. Comparson beween smulaon resuls and our analyss. =N=4, BSK modulaon, uncoded ZF-SIC V-BLAST. V CONCLUSIONS We have solved he problem of calculang he nearly exac BER of ZF-SIC V-BLAST wh BSK modulaon n a rchly scaered Raylegh-fadng mulple-anenna channel. Gven he number of ransm anennas, we can gve he closed-form expressons of he average probably of b error of each subsream. We also proposed a general recursve procedure o calculae he BER of each subsream wh arbrary number of ransm and receve anennas. The procedure s easy o follow and s feasble n evaluang he BER performance of V-BLAST wh ZF-SIC recever. Smulaon resuls have demonsraed he valdy of our analyss. Our mehod can be exed o he hgh-order modulaon schemes by mang some approxmaons. The analycal resul can be furher ulzed o develop opmal ransmsson sraegy for he open-loop V-BLAST. We wll rea hese opcs n a fuure wor [4]. REFERENCES [] G. J. Foschn and. J. Gans, On he lms of wreless communcaons n a fadng envronmen when usng mulple anennas, Wreless ersonal Communcaons, vol. 6, no. 3, pp , 998. [] E. Telaar, Capacy of mul-anenna Gaussan channels, European Trans. on Telecommun., vol., no. 6, pp , Nov./Dec [3] T. L. arzea and B.. Hochwald, Capacy of a moble mulple-anenna communcaon ln n Raylegh fla fadng, IEEE Trans. Inform. Theory, vol. 45, no., pp , Jan [4] C. Shen, H. Zhuang, L. Da, S. Zhou, "Deecon algorhm mprovng V-BLAST performance over error propagaon", IEE Elecroncs Leers, Vol. 39, No. 3, pp. 7-8, Jun. 3. [5]. W. Wolnansy, G. J. Foschn, G. D. Golden and R. A. Valenzuela, V-BLAST: an archecure for realzng very hgh daa raes over he rch-scaerng wreless channel, n roc. ISSSE, pp. 95-3, 998. [6] G. J. Foschn, G.D. Golden, R.A. Valenzuela and.w. Wolnansy, Smplfed processng for hgh specral effcency wreless communcaon employng mul-elemen arrays, IEEE J. Selec. Areas Commun., vol. 7, no., pp , Nov [7] S. Loya, V-BLAST ouage probably: analycal analyss, n roc. IEEE VTC Fall Conf., vol. 4, pp. 4-8, Sep.. [8] N. rasad and. K. Varanas, "Analyss of Decson Feedbac Deecon for IO Raylegh Fadng Channels and Opmum Allocaon of Transmer owers and QA Consellaons", n roc. 39h Annual Alleron Conf. on Comm. Conrol, and Compu., Oc.,. [9] G. Care, G. Tarcco, J. Venura-Travese and E. Bgler, A muluser approach o narrowband cellular communcaon, IEEE Trans. Inform. Theory, vol. 43, no. 5, pp , Sep [] E. A. Fan and. K. Varanas, Dversy order gan for narrowband muluser communcaons wh pre-combnng group deecon, IEEE Trans. Commun., vol. 48, no. 4, pp , Apr.. []. Frenger,. Oren, T. Oosson, "B error rae calculaon for nonlnear nerference cancellaon", IEE Elecroncs Leers, vol. 33, no. 5, pp , Jul []. ael and J. Holzman, "Analyss of a smple successve nerference cancellaon scheme n a DS/CDA sysem", IEEE J. Selec. Areas Commun., vol., no. 5, pp , Jun [3] J. roas, Dgal Communcaons, 4h ed. New Yor: cgraw-hll. [4] C. Shen, S. Zhou, Y. Zhu, BER analyss of V-BLAST wh Zero-Forcng Successve Inerference Cancellaon Recever and he opmal power allocaon sraegy, n preparaon. IEEE Communcaons Socey Globecom /4/$. 4 IEEE