Regularized Blind Detection for MIMO Communications

Transcription

1 Regularized Blind Deecion for MIMO Communicaions Yuejie Chi, Yiyue Wu and Rober Calderbank Deparmen of Elecrical Engineering Princeon Universiy Princeon, NJ 08544, USA ychi, yiyuewu, Absrac Muliple-Inpu Muliple-Oupu (MIMO) sysems improve he hroughpu and reliabiliy of wireless communicaions. Perfec Channel Sae Informaion (CSI) is needed a he receiver o perform coheren deecion and achieve he opimal gain of he sysem. In fas fading and low SNR regimes, i is hard or impossible o obain perfec CSI, which leads he receiver o operae wihou knowledge of he CSI and perform blind deecion. In realiy CSI may be available o he receiver bu his CSI may be insufficien o suppor coheren deecion. In his paper, we fill he gap beween coheren and blind deecion by considering a more realisic model where he receiver knows he saisics of he channel, ha is Channel Disribuion Informaion (CDI). We propose a new deecion algorihm, called Regularized Blind Deecion (RBD), where coheren and blind deecion can be viewed as special cases in our model. The algorihm esimaes CDI from any raining symbols ha are available and maximizes performance given he esimaed CDI. Simulaions demonsrae significan improvemen in performance over blind deecion. Our work can be viewed as a sysemaic exploraion of space beween coheren and blind deecion wih a srong Bayesian saisic flavor. I. INTRODUCTION Muliple-Inpu Muliple-Oupu (MIMO) sysems [], [2] enable higher rae ransmission on fading channels hrough higher specral efficiency while mainaining reliabiliy hrough spaial diversiy. The use of space-ime codes, inroduced by Tarokh e al. [3], [4], has furher improved he reliabiliy of communicaion over fading channel by correlaing signals across differen ransmi anennas. owever, from he perspecive of Shannon capaciy [5], [6], full realizaion of sysem poenial is heavily dependen on knowledge of Channel Sae Informaion (CSI) boh a he ransmier and he receiver. To perform coheren deecion, he channel is usually esimaed a he receiver hrough he ransmission of a sufficienly long raining/pilo sequence.this sysem overhead reduces he available daa rae. assibi e al. [7] explored he value of raining by analyzing channel capaciy when he channel saisics are known and raining is sufficien o obain an esimae of he CSI. In fas fading scenarios and low SNR regimes, i requires more resources and bandwidh o acquire perfec CSI, and i may be impracical o inroduce he necessary overhead. In his case, he ransmier will ransmi few raining/pilo sequences (o resolve phase ambiguiy), and he receiver will perform blind deecion wih around 3dB SNR degradaion in performance. One of he mos imporan approaches is join maximum likelihood daa deecion and channel esimaion, which has been exensively sudied in [8]-[4]. The sphere decoding algorihm [5] can be used o find he laice poin in he signal consellaion o minimize he arge norm; in paricular, i is shown in [6] ha for a wide range of SNR he complexiy of he sphere decoding algorihm is polynomial, making i feasible in many applicaions. In pracice, requiring a choice beween coheren and blind deecion is oo resricive. In sysems where coheren deecion is impracical, i.e. no enough resources can be allocaed o esimae exac CSI, i is usually possible, and much easier o obain parial knowledge of he channel informaion, for example, he Channel Disribuion Informaion (CDI). When he channel is assumed o be Gaussian, i can be described by is mean and covariance marix. We provide a graceful Bayesian approach o esimae CDI ha does no require he ransmission of a minimum number of raining symbols. Once perfec CDI is assumed known a he receiver, we propose a new deecion algorihm based on a Bayesian framework for join daa deecion and channel esimaion, called Regularized Blind Deecion (RBD), and we describe wo RBD varians. Convenional coheren and blind deecion are special cases of choosing he regularizaion parameer. Maximum likelihood deecion can be realized using a modified version of sphere decoding, so he complexiy is he same as blind deecion. In he simulaion, our algorihm performs very close o coheren deecion in he low SNR regime, and sill much beer han blind deecion in he high SNR regime. Our work can be viewed as a sysemaic exploraion of space beween coheren and blind deecion wih a srong Bayesian saisic flavor. I is worh menioning ha Bayesian deecion is also considered in [7] for inerference cancellaion in MIMO sysems using Alamoui signaling. The paper is organized as follows. In Secion II we describe he MIMO model used in his paper and explain boh coheren and blind deecion. In Secion III we assume perfec CDI a he receiver and presen wo varians of he RBD algorihm wihin a Bayesian framework. In Secion IV we analyze how o exrac CDI from raining daa wihin our Bayesian framework and inerpre he classical use of raining daa as selecion of regularizaion parameers. Secion V provides numerical resuls. Finally, Secion VI draws he conclusion. A noe on noaion: We use capial boldface o denoe

2 marices and vecors, and use F for he Frobenious norm. For a marix A, A denoes is Penrose-Moore pseudo-inverse, A denoes is conjugae ranspose, and Tr(A) denoes is race. I denoes he ideniy marix. II. SYSTEM MODEL We consider a general MIMO wireless communicaion sysem wih N ransmi anennas and N r receive anennas in a block fading channel model, where he channel is consan over T consecuive blocks, afer which i changes o an independen consan for anoher T consecuive blocks. The received signals Y C Nr T over T consecuive blocks a he receiver is given by Y = X + N, () where C Nr N is he channel marix, N is he addiive noise marix wih i.i.d. complex Gaussian random variable enries, and X = X, X 2,, X T C N T is he ransmied symbol marix wih X i, i =,,T as one coding block whose enries are aken from a signal consellaion Λ such as QPSK and QAM. A. Coheren Deecion Coheren deecion handles he case when is perfecly known a he receiver. The coheren maximum likelihood (ML) decoding rule are given as Y X 2 F (2) where F is he Frobenius norm and K is he number of symbols in one coding block. Since he noise is independen from one coding block o anoher, he decoding scheme in equaion (2) can be decomposed ino single coding blocks as ˆX i = argmin X i Λ K Y i X i 2 F (3) where Y i = X i +N i wih N i as he corresponding noise for coding block i. B. Blind Deecion Blind deecion handles he case when is unknown a he receiver. In convenional blind deecion, he decoding rule is ˆX = argmin, C Nr N = argmin Y X 2 F min Y X 2 C Nr N F. (4) In equaion (4), he inner minimizaion is a leas square problem given X, so he close form for he esimae Ĥ is given by [4] Ĥ = YX [X X ]. (5) In he case of orhogonal codes where X i X i (5) can be reduced o = KI, equaion Ĥ = KT YX. (6) Subsiuing equaion (6) ino equaion (4), we have Y 2 F KT YX 2 F = argmax TrYX X Y. (7) Remark: Blind deecion is acually no oally blind. In order o solve he ambiguiy, he firs coding block is normally assumed known. In anoher word, blind deecion sill requires he use of limied piloing signals. In boh cases, sphere decoding [5] is a general echnique which can efficienly reduce he average compuaional complexiy of maximum likelihood decoding. Some oher opimal decoding algorihms (cf [8] ec) achieve low complexiy by aking advanage of coding srucures. Noe ha inroducing correlaion a he ransmier hrough space-ime codes leads o correlaion a he receiver which is a form of CDI. III. REGULARIZED BLIND DETECTION In his secion, we formulae he decoding problem in a Bayesian probabilisic model, where we assume perfec CDI (namely, he disribuion and corresponding parameers of he channel) is known a he receiver. In pracice, he esimaion of CSI always involves uncerainy of he rue value, so ha i is reasonable o assume ha he rue channel follows cerain disribuion wih he esimaed CSI as he mean. From he Bayesian saisical viewpoin, he coheren deecion algorihm in equaion (2) is equivalen o maximizing he probabiliy of receiving Y given channel and daa X wih known, i.e. Pr(Y X, ); (8) similarly, he blind deecion algorihm in equaion (4) is equivalen o maximizing he same probabiliy given channel and daa X wihou assuming any prior on, which is equivalen o max C Nr N Pr(Y X, ), (9) where Pr(Y X, ) = Pr(X +N) N(X,σ 2 ni) is he pdf of he noise marix N. When is assumed o have a prior disribuion, namely a Gaussian prior on, N(Θ,Σ ), where Θ and Σ are assumed known, we propose wo variaions of algorihms for daa deecion: A. Join ML esimaion of channel and daa Join maximum likelihood esimaion of channel and daa X wih Gaussian prior on is given by: max C Nr N Pr(Y X, )Pr(). (0)

3 Wih Pr(Y, X) N(X,σnI), 2 and N(Θ,Σ ), Pr(Y X, )Pr() exp 2σn 2 Y X 2 F exp 2 ( Θ) Σ ( Θ). () Using he Cholesky decomposiion Σ = C C, he minimizaion problem in equaion (0) is equivalen o he following regularized esimaion: min Y X 2 C Nr N F + λ C( Θ) 2 F (2) where C( Θ) 2 F is he regularizaion erm and λ = σ2 n is he regularizaion facor. This is he reason he new algorihm is named by Regularized Blind Deecion. Rewriing he erms inside he bracke in equaion (2) as [ Y 0 ] [ X λc ] 2 F (3) where Y = Y ΘX and = Θ. Then he channel marix ha minimizes equaion (2) is: Ĥ = Θ + (Y ΘX)X (X X + λσ ). (4) Subsiuing his back ino equaion (2), we have he deecion rule for daa X : Y ĤX 2 F + λ C(Ĥ Θ) 2 F (5) = argmin Tr(Y ΘX)(I X (X X + λσ ) X) (Y ΘX). (6) B. ML esimaion of daa Maximum likelihood esimaion of daa X can be wrien as: Pr(Y X, )Pr()d (7) = argmax E Pr(Y X, ) (8) = argmax Pr(Y X). (9) When N(Θ,Σ ), his is compuable since E Pr(Y X, ) exp exp 2σn 2 Y X 2 F 2 ( Θ) Σ ( Θ) d X X + λσ /2 exp 2λ Tr(YX + λσ Θ)(X X + λσ (YX + λσ Θ) ) (20) Taking he logarihm of he above formula, he ML esimae is equivalen o X Λ 2 log X X + λσ + KT 2λ Tr (YX + λσ Θ)(X X + λσ ) (YX + λσ Θ). (2) C. Special Case: Orhogonal Codes As a special case, we consider he orhogonal codes X X = KTI and he channel covariance marix Σ = σh 2I. Le µ = λ/σh 2 = σ2 n/σh 2 ; he firs variaion of he algorihm in equaion (6) can be simplified as Tr YX X Y + µ ( YX Θ + ΘX Y ) = argmax Tr YX X Y + 2µR [ YX Θ ]. (22) Le W = Θ (YY ) Y, we have Tr (X + µw)(y Y)(X + µw). (23) Equaion (23) can be furher rewrien as Tr (X + µw)(ρi Y Y)(X + µw) (24) where ρ is a real consan greaer han all he eigenvalues of Y Y. Noe ha Y Y is posiive semidefinie, herefore we can perform Cholesky decomposiion of (ρi Y Y) such ha B B = ρi Y Y. (25) Therefore, equaion (24) can be reformulaed as a sandard sphere decoding problem wih a shif µw. B(X + µw) 2 F. (26) Similarly, he second variaion of he algorihm in equaion (2) becomes Tr YX X Y + µ ( YX Θ + ΘX Y ) = argmin B(X + µw) 2 F. (27) which coincides wih he firs variaion of RBD. Noice ha he variance of he channel plays a role in choosing he shif in sphere decoding. IV. EXTRACTING CDI FROM LIMITED TRAINING We now suppose some raining bis are ransmied o obain parial informaion abou he channel and we rewrie he received signals Y over T consecuive blocks a he receiver as Y = [Y τ Y d ] = [X τ X d ] + N (28) where X τ C N Tτ is he raining symbol marix, and X d C N T d is he daa symbol marix; respecively Y τ C N Tτ and Y d C N T d are he received symbol marix.

4 A. Tradiional Decoding Schemes wih Training. There are wo baseline approaches o using raining symbols ha do no make use of prior informaion abou channel saisics. The firs scheme is o obain he maximum likelihood deecion decoding rule using he formula below This is equivalen o min Y X 2 C Nr N F. (29) min Y d X d 2 C Nr N F + Y τ X τ 2 F (30) The second scheme is o firs use he raining symbols o ge an esimaed CSI, where hen use Ĥ o decode X d, i.e. Ĥ = argmin Y τ X τ 2 F C Nr N = Y τ Xτ (X τ Xτ ), (3) Y d ĤX d 2 F. (32) The leas square channel esimaion Ĥ from he raining daa lies in he span of Y τ, herefore i is necessary o ransmi enough raining symbols in order o ge a reasonable esimae, i.e. greaer han he number of ransmi anennas. B. Unified Approach I is possible o unify hese wo approaches by inroducing a penaly weigh λ o he second raining erm in equaion (30), given as min Y d X d 2 C Nr N F + λ Y τ X τ 2 F (33) hen he firs scheme is equivalen o choosing λ =, and he second scheme is equivalen o he limi soluion when λ goes o +. The general soluion o equaion (33) is essenially he same as he regularized blind deecion scheme discussed earlier under cerain parameerizaions. In fac, in he special case where he code is orhogonal and he channel variance marix Σ = σh 2 I is a scalar of ideniy marix, le Θ = Y τ X τ (X τ X τ ) = Y τ X τ /KT τ, i can be resaed as equaion (5). C. Esimaion of CDI hrough raining symbols. When aking advanage of he raining daa, he above decoding schemes essenially only obain a rough leas-square esimae of he CSI, and use i eiher wih full confidence (λ + ), or no confidence (λ = ) wihou disinguishing raining symbols from daa symbols. When here is limied raining available, he CSI can be very inaccurae. In fac we shall demonsrae ha i is more effecive o use low-rae raining daa over L blocks o esimae CDI and o hen use his esimae wihin our proposed RBD algorihms.., One way o esimae he mean Θ and variance Σ of he channel is via he following maximum likelihood esimaion, using available raining symbols from L blocks (Y τ,i, X τ,i ), i =,,L: Θ,Σ = argmax ˆΘ,ˆΣ = argmax ˆΘ,ˆΣ 2 2λ i= i= L E Pr(Y τ,i,x τ,i ) i= log X τ,i X τ,i + λˆσ + Tr (Y τ,i Xτ,i + λˆσ ˆΘ)(X τ,i Xτ,i + λˆσ ) (Y τ,i X τ,i + λˆσ ˆΘ) λ ˆΘ ˆΣ ˆΘ. (34) This can be furher simplified in he special case when X τ,i X τ,i = KT τi and Σ = σ 2 h I as (le ˆµ = σ2 n/ˆσ 2 h ): Θ,σh 2 = argmax L ˆΘ,ˆσ 2 log KT τ + ˆµ + h 2 2λ KT τ + ˆµ i= Y τ,i X τ,i + ˆµ ˆΘ 2 F ˆµL ˆΘ 2 F. (35) The choice of Θ does no depend on µ, and is given as Θ = KT τ L Y τ,i Xτ,i. (36) i= Plugging his back o equaion (35), and seing is derivaives wih respec o ˆµ o zero, we obain he maximum likelihood esimae of σ 2 h. V. NUMERICAL RESULTS In our simulaion, we consider a wireless communicaion sysem wih wo ransmi anennas and a single receive anenna which adops Alamoui signaling [4]. We consider he decoding of Alamoui signals during 3 consecuive coheren code blocks (namely, 6 consecuive ime slos). We assume ha he channel marix is complex Gaussian wih mean and covariance marix Σ = Θ = ( + i, + i) ( ). Fig. shows he comparison of differen deecion algorihms. For blind deecion, he phase ambiguiy is assumed o have been solved; for regularized blind deecion, CDI is assumed known a he receiver and for coheren deecion, CSI is assumed perfecly known a he receiver. The firs variaion of he proposed RBD algorihm is used. In he simulaion, our algorihm performs very close o coheren deecion in he low SNR regime, and sill much beer han blind deecion in he high SNR regime.

5 Symbol Error Rae Blind Deecion Regularized Blind Deecion Coheren Deecion SNR(dB) Fig.. Comparison of differen deecion algorihms wih decoding block lengh as 3 coding blocks. VI. CONCLUSIONS We have inroduced and analyzed he performance of Regularized Blind Deecion, a new algorihm ha provides a sysemaic way of inerpolaing beween coheren and blind deecion. Significan performance benefis are possible in environmens where i may be impracical o obain full CSI. The algorihm requires knowledge of CDI a he receiver and we have described how CDI may be esimaed from any available raining symbols. Coheren and blind deecion are special cases wihin our Bayesian framework and i is CDI ha parameerizes he space in beween. Simulaions demonsrae ha our algorihm performs very close o coheren deecion in he low SNR regime, and sill much beer han blind deecion in he high SNR regime. [8]. Vikalo, B. assibi, and P. Soica, Efficien join maximum likelihood channel esimaion and signal deecion, IEEE Transacions on Wireless Communicaions, vol. 5, pp , [9] L. Tong, G. Xu, and T. Kailah, Blind idenificaion and equalizaion based on second-order saisics: a ime domain approach, IEEE Transacions on Informaion Theory, vol. 40, pp , 994. [0] N. Seshadri, Join daa and channel esimaion using fas blind rellis search, US Paen , 993. []. Bolcskei, J. R. W. eah, and A. J. Paulraj, Blind channel esimaion in spaial muliplexing sysems using nonredundan anenna precoding, Proc. of he 33rd Annual IEEE Asilomar Conf. on Signals, Sysems, and Compuers, vol. 2, pp , 999. [2] W.-K. Ma, B.-N. Vo, T. Davidson, and P. Ching, Blind ML deecion of orhogonal space-ime block codes: igh performance, efficien implemenaions, IEEE Transacions on Signal Processing, vol. 54, pp , [3] W. Xu, M. Sojnic, and B. assibi, On exac maximum-likelihood deecion for non-coheren mimo wireless sysems: A branch-esimaebound opimizaion framework, Proceedings of he Inernaional Symposium on Informaion Theory, pp , [4] E. G. Larsson, P. Soica, and J. Li, On maximum-likelihood deecion and decoding for space-ime coding sysems, IEEE Transacions on Signal Processing, vol. 50, pp , [5] M. Pohs, On he compuaion of laice vecors of minimal lengh, successive minima and reduced basis wih applicaions, ACM SIGSAM, vol. 5, pp , 98. [6] B. assibi and. Vikalo, On he sphere decoding algorihm: Par I, he expeced complexiy, IEEE Transacions on Signal Processing, vol. 53, pp , [7] S. Sirianunpiboon, A. R. Calderbank, and S. D. oward, Spacepolarizaion-ime codes for diversiy gains across line of sigh and rich scaering environmens, Proc. IEEE ITW, May [8] S. Sirianunpiboon, Y. Wu, A. R. Calderbank, and S. D. oward, Fas opimal decoding of muliplexed orhogonal designs by condiional opimizaion, o appear, IEEE Transacions on Informaion Theory, 200. ACKNOWLEDGEMENTS The auhors would like o hank Vanee Aggarval and Waheed Bajwa for helpful discussions. REFERENCES [] D. Tse and P. Viswanah, Fundamenals of wireless communicaion. New York, NY: Cambridge Universiy Press, [2] G. J. Foschini, Layered space-ime archiecure for wireless communicaion in a fading environmen when using muli-elemen anennas, Bell Labs Technical Journal, vol., pp. 4 59, 996. [3] V. Tarokh, N. Seshadri, and A. R. Calderbank, Space-ime codes for high daa rae wireless communicaion:performance crierion and code consrucion, IEEE Transacions on Informaion Theory, vol. 44, pp , 998. [4] S. M. Alamoui, A simple ransmi diversiy echnique for wireless communicaions, IEEE Journal on Seleced Areas in Communicaions, vol. 6, pp , 998. [5] E. Telaar, Capaciy of muliple-anenna gaussian channels, European Transacions on Communicaions, vol. 0, pp , 999. [6] A. Goldsmih, S. A. Jafar, N. Jindal, and S. Vishwanah, Capaciy limis of MIMO channels, IEEE Journal on Seleced Areas In Communicaions, vol. 2, pp , [7] B. assibi and B. ochwald, ow much raining is needed in mulipleanenna wireless links? IEEE Transacions on Informaion Theory, vol. 49, pp , 2003.