Code Design for the Low SNR Noncoherent MIMO Block Rayleigh Fading Channel

Transcription

1 Cod Dsgn for th Low SNR Noncohrnt MIMO Block Raylgh Fadng Channl Shvratna Gr Srnvasan and Mahsh K. Varanas -mal: {srnvsg, Elctrcal & Computr Engnrng Dpartmnt Unvrsty of Colorado, Bouldr, CO Abstract Cod dsgn for th low SNR MIMO noncohrnt corrlatd Raylgh fadng channl s consdrd. Dsgn ruls whch xplot th corrlatons n th transmt antnnas n th MIMO cas, to provd gans ovr th corrspondng SIMO cas ar prsntd. Th Chrnoff bound on th avrag parws rror probablty APEP) s usd to study th ffct of th rcv corrlaton matrx on systm prformanc at dffrnt SNR rgms. Basd on a lowr bound on th APEP, whch s rlatd to th Bhattacharya coffcnt, a tchnqu s proposd to dsgn cods for us wth transmt bamformng, wth codwords havng unqual pror probablts. Th motvaton for such cods wth unqual prors arss from rcnt nformaton thortc rsults on th low SNR channl. Such constllatons ar shown to prform substantally bttr than constllatons dsgnd assumng qual prors, at low SNRs. I. INTRODUCTION AND SYSTEM MODEL Th problm of communcaton ovr a low SNR noncohrnt..d. Raylgh fadng channl undr an avrag powr constrant has bn rcntly studd from an nformaton thortc vwpont n [, ]. [3] studs th corrlatd Raylgh fadng modl and provds som charactrzatons of th capacty achvng dstrbuton. Rsults n [, 4] ndcat that at suffcntly low SNR, th mutual nformaton s mzd by kpng only on transmt antnna on. Howvr, n practc, ralstc MIMO channls xhbt corrlatons among th antnna path gans whch can b xplotd to provd gans. Ths s on ky da xplord n ths papr. Th codng-modulaton vwpont of low SNR noncohrnt Raylgh fadng channls s rlatvly nw. Rcntly, Borran, t al. [5] prsntd tchnqus to dsgn constllatons wth qual pror probablts for th MIMO Raylgh..d. fadng usng th Kullback-Lblr KL) dstanc as a dsgn crtron. Ths cods wr shown to prform bttr than som xstng noncohrnt cods at low SNR. Howvr, nformaton thortc rsults ovr th low SNR noncohrnt..d. Raylgh fadng channls undr an avrag powr constrant ndcat that th capacty achvng dstrbuton gts paky at suffcntly low SNR c.f. [, ]). Consquntly, w allow for codwords wth unqual prors n a cod and optmz ovr th pror probablts to achv prformanc gans. Ths s th scond ky da xplord n ths work. W also fnd that th Bhattacharya dstanc ylds bttr rsults for cod dsgn than dos th KL dstanc. Ths work was supportd n part by NSF grants CCF and CCF W consdr a communcaton systm wth N t transmt antnnas and N r rcv antnnas. In our channl modl, th channl matrx H IC Nt Nr s assumd to b constant for a duraton of T symbols aftr whch t changs to an ndpndnt valu. H has corrlatd crcularly symmtrc complx gaussan ntrs and s rprsntd by H Σ t ) / H W Σ r ) /, whr Σ t and Σ r ar th transmt and rcv array corrlaton matrcs rspctvly. H W has..d. crcularly symmtrc CN0, ) ntrs. Th channl matrx s assumd to b unknown to th transmttr and th rcvr, whl th long trm channl statstcs ar assumd to b known at th transmttr. Substtutng th gnvalu dcompostons for Σ t U t Λ t U t ) and Σ r U r Λ r U r ), whr th gnvalus ar arrangd n th dscndng ordr along th dagonal, w gt that H U t Λ t ) / U t ) H W U r Λ r ) / U r ) U t HU r ), whr H Λ t ) / U t ) H W U r Λ r ) /. Assumng that th transmttd symbol s X IC T Nt, th output of th channl can b wrttn as Ỹ X H + Ñ ) XU t HU r ) + Ñ ) Hr, th symbols { X} ar normalzd so that th transmt powr constrant T E[tr X X )] P holds. Aftr postmultplyng ) by U r, dnotng XU t by X, and dnotng ỸU r by Y, w gt Y XH + N, 3) whch dscrbs th suffcnt statstcs of th rcvd sgnal. Clarly, E[trXX )] E[tr X X )], and hnc th prcodd cod {X} satsfs th sam powr constrant as th orgnal cod { X}. W may hnc consdr 3) to b our ffctv channl modl, and us th notaton X to dnot a codword prcodd by th gnmatrx of Σ t. Rcognzng that U t ) H W U r has th sam dstrbuton as H W, t can b asly sn that wth h vch), Σ E[hh ] Λ r Λ t, and N has..d. crcularly symmtrc CN0, ) ntrs. Applyng a vc opraton to 3), w gt y I Nr X)h+n Xh+n, whr y vcy) and n vcn). II. THE MAP DECODER AND ITS WEP W nxt dscrb th mum a-postror MAP) dcodr for a cod {X } L, wth pror probablts {π } L. Snc X I Nr X I Nr X U t, t s undrstood that X, X and X rfr to th th codword. Th pdf of y condtond on codword X bng snt s gvn by

2 p y) π TNr I + X ΣX y Th MAP dcodng rul s gvn by I+X ΣX ) y. î arg j...l) π jp j y). 4) Lt ɛ b th vnt MAP dcodd codword codword transmttd and Pɛ) dnot th avrag codword rror probablty WEP). Th WEP for th cod s gvn by Pɛ) Σ L π Pɛ/) Σ L π p y), 5) whr Γ c s th complmnt of th dcodng rgon of th th codword and s gvn by Γ c {y : arg π jp j y)}. 6) j,...,l) It should b notd that th condtonal pdf of th obsrvaton p y) dpnds on th sgnal transmttd X, only through th trm X Λ t X. Thrfor, basd on 5), th WEP dpnds on th sgnals only through {X Λ t X }L and thr rspctv pror probablts {π } L. Th followng proposton hlps n rstrctng th doman ovr whch sgnals ar pckd, wthout changng th WEP. Proposton : Gvn any cod C {X } L whch has a WEP Pɛ), thr xsts a cod C {Φ V } L wth th sam WEP. Hr Φ IC T T s a untary matrx and V R T Nt s a dagonal matrx wth postv dagonal ntrs. Ths rsult may b vwd as a codng-modulaton analogu of th capacty achvng dstrbuton for th corrlatd Raylgh fadng channl gvn n [3] Proposton : Consdr a systm wth Σ t of rank M N t. Thn, any cod C {S } L should hav codwords wth only th frst M columns non-zro, for othrws thr xsts a cod C {X } L xhbtng th sam WEP as C, whl rqurng lssr transmttd powr than that of C. Du to Proposton, w s that th codwords {X } L should ssntally not hav mor than rankσ t ) non-zro columns. W say that th th column of th codword s transmttd ovr th th gnbam, for M. In th followng propostons, w assum that cods ar dfnd ths way, and that thr ar no two codwords such that X X j. Proposton 3: Lt C {x } L b any cod wth codwords x IC T, satsfyng th transmt powr constrant E[trxx )] P. Lt ths cod b usd on a SIMO channl wth rcv corrlaton matrx Σ r. Consdr a MIMO channl havng N t and N r transmt and rcv antnnas, and Σ t I and Σ r as transmt and rcv corrlatons, rspctvly. Thn thr xsts a MIMO cod C {X } L for ths channl, wth th sam WEP and has lowr avrag nrgy than C. Proof: Th rcvd sgnal gvn that th sgnal x s snt on th SIMO channl would smply b Y x h T W Λr ) / + N, 7) Γ c whr h W IC Nr. Dfn ach codword of C as X [x 0 T Nt ] for ach. Thn, clarly trx x ) trx X ) and hnc C satsfs th sam powr constrant as C. Th rcvd sgnal condtond on X bng snt on th MIMO channl s Y [ x 0 T Nt )] Λ t ) / H W Λ r ) / + N 8) λ t )/ x H ) W Λr ) / + N, 9) whr H ) W IC Nr s th frst row of H W and hnc has th sam statstcs as h T W. Snc Σt I and trσ t ) N t, λ t >. Hnc, th xprsson for th rcvd sgnal dnsty s th sam as that n 7), xcpt for th ncras n sgnal powr n 9). Wth a sutabl rnormalzaton of C, th condtonal p.d.f.s of th rcvd sgnal corrspondng to codwords of both cods may b mad qual, wth C havng lowr avrag nrgy than C. Ths would mak th WEPs of C and C th sam, wth C havng lowr avrag nrgy than C. Proposton 3 ndcats that at any SNR, any sngl antnna cod dos bttr whn usd wth sutabl prcodng n a MIMO corrlatd Raylgh fadng channl havng Σ t I, than n a SIMO channl. Notc that th codword actually [ dployd on ] th orgnal channl gvn n ) would b x 0 T Nt ) U t ). Th prcodng ssntally rsults n th sngl antnna codwords bng bamformd usng th gnvctor corrspondng to th largst gnvalu of Σ t. W rfr to such cods throughout ths papr as sngl gnbam cods. Ths rsult s of sgnfcanc snc t s known that for low SNR and..d. Raylgh fadng spatal corrlatons, sngl transmt antnna cods mz th mutual nformaton [, 4]. Howvr, whn Σ t I, Proposton 3 provs that t s bttr to us a prcodd cod and hnc all transmt antnnas n gnral at low SNR. Furthrmor, n th low SNR rgm and whn thr s flxblty n choosng Σ t va antnna spacngs for xampl), t s bttr to hav Σ t I. Furthr, f th codng stratgy s fxd to b a sngl gnbam cod, Proposton 3 ndcats that th bst possbl transmt corrlaton matrx would b fully corrlatd, so that rankσ t ) and λ t N t. III. APEP AND THE CHERNOFF BOUND Th xprssons for th WEP ar hard to valuat n closd form and hnc w rsort to workng wth th APEP to dvlop dsgn crtra. W frst rlat th WEP of a cod to th APEPs. Dnot th parws dcodng rgon for codword j gvn that s snt as Dj {y : π jp j y) > π p y)}. Ths maks th PEP PDj /) p Dj y). W rfr to P ) π PDj /) + π jpd j /j) as th avrag parws rror probablty APEP) btwn codwords and j. Th followng bounds may b drvd usng standard argumnts Pɛ) Σ Σ j> P ) LL ) P ) 0) Pɛ) P ) ) Snc th uppr bound s a constant tms th lowr bound, a rasonabl crtron for cod dsgn amng to mnmz th WEP would b mn C {,...,L} P ).

3 For th noncohrnt channl, PDj /) PDj /j) and hnc any dstanc masur obtand usng th PEPs would b asymmtrc n gnral. Notc that snc P ) P j,), th APEPs ar symmtrc, and gv rs to a mor ntutv symmtrc dstanc masur. Whn th dsgn goal s to mz th mnmum dstanc btwn ponts n a constllaton, t would hnc hlp to us dstanc masurs drvd from th APEP or bounds on t. Snc no closd form xprssons ar known for th APEPs btwn codwords for th gnral noncohrnt MIMO Raylgh fadng channl, w rsort to uppr and lowr bounds on P ) for analyss. W frst study th Chrnoff bound on th avrag PEP btwn two codwords X and X j. Th followng quaton, whch s vald for [0, ], s th Chrnoff bound on th APEP for unqual prors, and th stps ladng to t ar gvn n [6] III.C.8). P ) π Π N n Γ [ ] pj y) p y)dy ) p y) π I+ )XjΛ t X j +XΛt X )λr n I+λ r n XΛt X I+λ r n XjΛt X j ) 3) Ch, j) 4) 3) can b drvd from ) usng tchnqus gvn n [7], whr th Chrnoff bound for gnral and for th corrlatd Rcan fadng modl s drvd assumng quprobabl sgnalng. W study Ch, j) for gnral and our conclusons wll clarly hold for th tghtst Chrnoff bound as wll, whnvr th mnmzng 0, ). Morovr, ths rsults hold for th spcal cas of, for whch th Chrnoff bound s popularly known as th Bhattacharya bound. Th Bhattacharya bound wll b usd n cod dsgn n Scton IV. W frst study th proprts of Ch, j) at low SNR. Consdr th logarthm of Ch, j), and apply th followng approxmaton log I + A tra) tra ), whch s vald for any Hrmtan matrx A wth small gnvalus, w gt Ch low, j) π π j 8 )ΣN n {λr n } )trx Λ t X XjΛt X j ), 5) whr w hav dnotd th Chrnoff bound at low SNR by Ch low, j). Ths approxmaton has bn usd on th Chrnoff bound n [7]. W show that ntrstng nsghts can b drawn from th low SNR Chrnoff bound n th corrlatd Raylgh fadng cas. W charactrz th bhavor of Ch low, j) wth rspct to Λ r n th nxt proposton. W mphasz that whnvr w rfr to Ch low, j), w assum that th SNR s suffcntly small to justfy th approxmaton mad to arrv at 5). Lt th vctor of th non-zro dagonal lmnts of Λ r b λ r. For clarty of xposton n th nxt proposton, w dnot Ch low, j) by Ch low j λ r ). W nd th followng two dfntons from [8] for th propostons that follow. Dfnton : For x,y R n, x s sad to b majorzd by y, dnotd by x y, f Σ k x [] Σ k y [], k,..., n, and Σ n x [] Σ n y [] whr x [] and y [] dnot th th largst componnts of x and y rspctvly. Dfnton : A ral valud functon f dfnd on a st A R n s sad to b Schur-convx on A f for any x,y A, x y fx) fy) Smlarly, f s dfnd to b Schurconcav on A f for any x,y A, x y fx) fy) If x and y ar vctors of gnvalus of two corrlaton matrcs Σ and Σ, x y would man that Σ s mor corrlatd than Σ. Ths noton of majorzaton has bn usd n many paprs studyng th ffct of fadng corrlatons on th MIMO channl. Proposton 4: For any par of codwords, Ch low j λ r ) s a Schur-concav functon of λ r. Proof : Omttd du to spac constrants. W nxt smplfy 3) assumng that th SNR s larg. For notatonal smplcty, w dnot Ch, j) at hgh SNR by Ch hgh j λ r ). Lt th non-zro gnvalus of A X Λ t X b {γ q } Q q, thos of B )X jλ t X j b {β r} R r, and thos of A+B X Λ t X + )X jλ t X j ) b {θ p} P p. Thrfor, 3) may b wrttn as Ch, j) π Π N n Π P p +λr n θp) [Π Q q +λr n γq)] [Π R r +λr n βr)] ), whch for hgh SNR may b approxmatd as Ch hgh j λ r ) π ) Π ) N P Q/ R/ Π P pθ p n λr n Π Q q γ q) Π R r β, 6) r) whr w hav assumd that th non-zro gnvalus dfnd abov ar all asymptotcally larg. Th followng proposton gvs th dpndnc of Ch hgh j λ r ) on λ r : Proposton 5: For any par of codwords, Ch hgh λ r ) s a Schur-convx functon of λ r. Proof: Π n λ r n ) s a Schur-convx functon of λ r by 3.E.) [8], snc /λ r n s a log-convx functon of λ r n. Snc Q ranka / ), R rankb / ), and P rank[a / B / ]), clarly P Q and P R. Ths maks P Q/ R/ 0. Hnc Π n λ r n) P Q/ R/) s also a Schur-convx functon of λ r by 3.B.) [8]. Ths n turn maks Ch hgh j λ r ) a Schur-convx functon of λ r. For cohrnt cods, t s provd n [9] that a mor corrlatd Σ r rsults n a hghr PEP for all pars n a cod and at all SNRs. In contrast, for noncohrnt cods, Propostons 4 and 5 ndcat that at suffcntly low SNRs, a mor corrlatd rcv corrlaton matrx rsults n a lowr Chrnoff bound on th APEP for all pars, whl at hgh SNR th oppost s tru. Ths rsults hnc xtnd to th unon uppr bound on th WEP gvn n 0). An obvous mplcaton of Propostons 4 and 5 s that at suffcntly low SNR, a rank on Σ r s optmal, and at suffcntly hgh SNR, Σ r I s optmal. j

4 IV. LOW SNR CODE DESIGN Whn th powr avalabl s suffcntly low, t sms ntutv that only th gnbam corrspondng to th largst gnvalu of Σ t ought to b usd, as th othr channls would provd thr smallr gans or would only attnuat th corrspondng gnbam. Ths approach sms vn mor approprat f n addton, Σ t s hghly corrlatd, rsultng n λ t bng much largr than th othr gnvalus. Proposton 3 ndcats that th..d. channl rsult on th numbr of antnnas to actually us, s fragl. Basd on th rasons gvn, wth transmt corrlatons and at suffcntly low SNR, th..d. rsult gnralzs naturally to ndcat th optmalty of MIMO sngl gnbam cods. Ths motvats ths scton on cod dsgn at low SNR. Th tghtst Chrnoff bound ovr howvr sms hard to calculat for gnral MIMO codwords, as a rsult of whch w us th spcal cas of / nstad. Th followng bounds ar known on th APEP for any channl modl [0]: mnπ, π j )ρ P ) π π j ρ, 7) whr ρ s rfrrd to as th Bhattacharya coffcnt. For th modl assumd n ths papr, ρ s gvn by ρ Π N I+ XjΛt X j +XΛt X )λn. 8) n I+X Λ t X λn / I+X jλ t X j λn / Th Bhattacharya uppr bound n 7) s got whn of th Chrnoff bound n 3) s assgnd th valu. In th cas of qual prors, mnmzng th largst P ) translats to th sam crtron for th uppr and lowr bounds, namly mnmzng th largst ρ among all pars, j. For unqual prors, th corrspondng dsgn crtron would b to mnmz th mum of thr th lowr or uppr bound n 7) ovr all pars, j. From smulatons, w fnd that th lowr bound s mor sutabl for our dsgn mthod and w us t n our llustratons hr. Lt a rat R bts/s/hz b fxd for th cod as a dsgn paramtr. Anothr dsgn paramtr that w wll fx s th cardnalty of th constllaton L. Snc th algorthm w wll dscrb allows for unqual prors, whch ar obtand as th output of an optmzaton, th choc of L has to ncssarly satsfy L RT. Snc w ar ntrstd n sngl gnbam cods, w consdr cods wth codwords dfnd by X [x 0 T Nt )]. Wth no chang n th WEP, th sgnal constllaton may b takn to b n th form ndcatd, wth x φ r,,..., L, whr {φ } L ar unt norm vctors, whl {r } L ar ral scalars by Proposton. Also dfn µ n λ t λr n, n,..., N. W may us ths n th lowr bound n 7) to gt aftr som smplfcaton, Bh LB, j) mn π,π j) ΠN n B n) ) 9), j) + Bn), j) whr, B n), j) +µ nrj +µ nr + +µ nr +µ nr clarly dpnds only on th magntuds of th sgnals r and r j, j and ) B n), j) µ n r r j 4 φ +µnr +µnr j φ ) s a scald functon of th angl btwn th two sgnals, th scalng bng a functon of th radus. W sk a cod whch has th mnmum P ) and usng th lowr bound nstad of P ) gvs us th followng dsgn crtron: mn K L, Σ K k π k l k r k P T, ΣK k l k L, 0<{π k } K k <, ΣK k l k π k log π k R, 0 r <...r K, Σ K k l k π k Bh LB, j) 0) Solvng ths problm sms hard n gnral, and w adopt a suboptmal procdur nstad. W prform ths optmzaton ovr th st of all constllatons such that th sgnal ponts l n concntrc sphrs C, C,..., C K, wth th probablts of sgnals lyng n th sam sphr bng qual. W assum that th sphrs hav rad r, r,..., r K, and th sphrs hav l, l,..., l K ponts wth probablts π, π,...π K. Whl ths assumpton nd not hold wth th optmal constllaton, t s rasonabl, as th capacty achvng dstrbuton for sngl gnbam constllatons s known [3] to b of th form [φv 0 T Nt )] for th corrlatd rcv antnna cas, whr φ s an sotropcally dstrbutd unt vctor, whl v s a scalar whos dstrbuton s not known n closd form. Ths suggsts th structur of th constllaton as assumd. Th othr smplfyng assumpton mad s that whnvr two concntrc sphrs C and C j hav non-zro numbr of codwords lyng on ach, thr s at last on codword n C that ls along th sam unt norm vctor as a codword n C j. Ths s a proprty that nd not b tru of th optmal cod. Howvr, ths assumpton smplfs th cod dsgn gratly. Notc that for any two ponts on th sam sphr wth radus r k, B n), j) and B n), j) µ n r4 k 4 + µ n rk φ j φ ) ) whr φ j φ ) s a masur of th angular dstanc btwn φ and φ j. For th par of codwords wth φ φ j btwn any two shlls at rad r and r j, B n), j) + µ n rj + µ n r and B n), j) µ n r + µ n r j Now dfnng th ntr-sphr dstanc as B ntr, j) mn π, π j ) ΠN + µ n rj n + µ n r + + µ n r + µ n rj, ) and th ntra-sphr dstanc as B ntra k) mn Π N n + µ n r4 k π k x,x j C k 4 + µ n rk φ j φ ) )) Π N n + µ n r4 k π k 4 + µ n rk mn φ j φ ) )). ) x,x j C k

5 Th smplfd -mn problm corrspondng to 0) may b xprssd as K L, Σ K k π k l k r k PT, ΣK k l k L, 0<{π k } K k <, ΣK k l k π k log π k R, 0 r <...r K, Σ K k l k π k mn whr S { {B ntr, j)} >j, {B ntra k)} K k}. S, 3) For a gvn {l k } K k such that ΣK k l k L, ths optmzaton may b dcoupld nto th problm of fndng th bst constllaton wthn ach sphr, and usng ths constllaton n 3), to gt {π k } K k and {r k} k k usng th fmn functon n MATLAB. Ths s bcaus mzng B ntra dos not dpnd on thr r k or π k as thy occur as constant multplcatv factors n th xprsson. Th problm of fndng th bst constllaton wthn ach sphr s studd for nstanc n [ 3], and so w may us ths constllatons and thr mnmum dstancs for ths stag of th optmzaton. Ths procdur may b adoptd for ach traton corrspondng to a st of {l k } K k wth th mnmum dstanc, {r k} K k and {π k } K k of ach rsultng constllaton stord. Thn th canddat that has th largst mnmum dstanc btwn pars s slctd as th bst constllaton. W plot th WEPs of constllatons dsgnd for a cohrnc tm of T 6, transmt powr P 0.5 n Fgur. For th mzaton of B ntra k), w us a sngl antnna untary dsgn of dmnson T 6 n [3]. For smplcty, w assum Σ r I whl N t 3, and th gnvalus of th transmt corrlaton matrx Σ t [ ; ; ] ar.09, , and For R 3, P 0.5, and an 8-pont constllaton havng qual prors, cod dsgn for both th MIMO and SIMO cass rsultd n th 8-pont sngl antnna untary dsgn wth all ponts at a radus of 3 as th consttunt gnbam. Wth unqual prors and an xpandd constllaton of 7 ponts wth th sam rat, th algorthm dscrbd ndcatd a constllaton that has a pont at th orgn wth probablty, and 6 ponts at a radus of 3 formng a sngl antnna untary dsgn ach wth probablty 3, for both th SIMO and MIMO cass. It s sn that th 7-pont cod dsgnd through th algorthm dscrbd, has a hghr probablty assgnd to th orgn and smallr probablts assgnd to ponts on a sphr. Th form of th cod rsmbls th capacty achvng nput dstrbuton at suffcntly low SNR. For th SIMO cas, th 7- pont constllaton wth unqual prors rqurd around 3 rcv antnnas fwr than th 8-pont constllaton wth qual prors for a WEP 0. Sgnfcant mprovmnts ar obsrvd whn th sam cods ar usd n th MIMO corrlatd channl wth prcodng. V. CONCLUSIONS W prsntd a mthod to dsgn sngl gnbam cods for th MIMO corrlatd Raylgh fadng channl, usng a lowr bound on th APEP rlatd to th Bhattacharya coffcnt. W showd how th Chrnoff bound on th APEP vard at dffrnt SNR rgms wth Σ r. W provd thortcally, and dmonstratd va smulatons that sgnfcant m- Probablty of word rror 0 0 Transmt powr 6 db, Rat 0.5 b/s/hz, T6 0 0 B q, Nt B unq, Nt B q, Nt3 B unq, Nt Nr Numbr of rcvd antnnas) Fg.. Prformanc of cods dsgnd usng th Bhattacharya lowr bound at P 0.5 provmnts n prformanc of SIMO cods ar possbl by usng thm wth a prcodng stratgy ovr MIMO channls wth a hghly corrlatd Σ t. Snc sngl gnbam cods ar optmal at suffcntly low SNR, th cods dsgnd hr ar mnntly sutd for us n ths SNR rgm. REFERENCES [] C. Rao and B. Hassb, Analyss of multpl-antnna wrlss lnks at low SNR, IEEE Trans. Inform. Thory, vol. 50, no. 9, pp. 3 30, Spt [] X. Wu and R. Srkant, MIMO channls n th low SNR rgm: Communcaton rat, rror xponnt and sgnal paknss, n Proc. IEEE Intl. Informaton Thory Workshop, San Antono, TX, 004. [3] S. A. Jafar and A. Goldsmth, Multpl-antnna capacty on corrlatd Raylgh fadng wth channl covaranc nformaton, to appar IEEE Trans. Wrlss Commun. [4] B. Hassb and T. L. Marztta, Multpl-antnnas and sotropcallyrandom untary nputs: Th rcvd sgnal dnsty n closd-form, IEEE Trans. Inform. Thory, vol. 48, no. 6, pp , Jun 00, Spcal Issu on Shannon Thory: Prspctv, Trnds, and Applcatons. [5] M. J. Borran, A. Sabharwal, and B. Aazhang, On dsgn crtra and constructon of noncohrnt spac-tm constllatons, IEEE Trans. Inform. Thory, vol. 49, no. 0, pp , Oct [6] H. V. Poor, Introducton to Sgnal Dtcton and Estmaton, Sprngr- Vrlag, Nw York, 994. [7] A. Dogandzc, Chrnoff bounds on parws rror probablts of spac-tm cods, IEEE Trans. Inform. Thory, vol. 49, no. 5, pp , May 003. [8] A. W. Marshall and I. Olkn, Inqualts: Thory of Majorzaton and Its Applcatons, Acadmc Prss, Nw York, 979. [9] J. Wang, M. K. Smon, M. P. Ftz, and K. Yao, On th prformanc of spac-tm cods ovr spatally corrlatd Raylgh fadng channls, IEEE Trans. Commun., vol. 5, no. 6, pp , Jun 004. [0] T. T. Kadota and L. A. Shpp, On th bst fnt st of lnar obsrvabls for dscrmnatng two Gaussan sgnals, IEEE Trans. Inform. Thory, vol. 3, no., pp , Apr [] V. Tarokh and I.-M. Km, Exstnc and constructon of noncohrnt untary spac tm cods, IEEE Trans. Inform. Thory, vol. 48, no., pp. 3 37, Dc. 00. [] M. L. McCloud and M. K. Varanas, Modulaton and codng for noncohrnt communcatons, Journ. VLSI Sgnal Procssng, vol. 30, no. 3, pp , Jan.-Fb.-Mar. 00. [3] B. M. Hochwald, T. L. Marztta, T. J. Rchardson, W. Swldns, and R. Urbank, Systmatc dsgn of untary spac tm constllatons, IEEE Trans. Inform. Thory, vol. 46, no. 6, pp , Spt. 000.