by log b, the natural logarithm by ln. The Kronecker product of two matrices is denoted by Ω.

Transcription

1 00 Confrnc on Informaton Scncs and Systms, Th Johns Hopkns Unvrsty, March, 00 Cohrnt Multusr Spac-Tm Communcatons: Optmum Rcvrs and Sgnal Dsgn Matthas Brhlr and Mahsh K. Varanas -mal: fbrhlr, Unvrsty of Colorado, Bouldr, CO 8009 Abstract Th jontly optmum rcvr s obtand for multusr communcatons n a frquncy non-slctv Raylgh fadng channl wth N T transmt antnnas pr usr and N R rcv antnnas. Basd on a gnral analyss of quadratc rcvrs n zro-man complx Gaussan vctors, asymptotcally tght xprssons (for hgh SNR) for th par-ws rror probablts ar drvd. Consquntly, t s shown that N T -dmnsonal sngl-usr sgnalng sufcs to provd full dvrsty ordr N N T N R for all th usrs. In othr words, th prsnc of othr usrs dos not ncras th mnmum dmnson rqurd byond what s ndd for th sngl-usr spac-tm channl. For th spcal cas of low-rank CDMA sgnalng wth N T and provdd th sgnaturs of any two usrs ar lnarly ndpndnt, t s shown that th rror probablty of a K-usr systm asymptotcally approachs sngl-usr lk prformanc for vry usr. Rmarkably thrfor, an ncras n th numbr of usrs, and hnc an ncras n th aggrgat spctral fcncy, dos not rqur th usrs to transmt wth mor powr to achv sngl-usr lk prformanc asymptotcally. A sgnal dsgn algorthm s proposd to llustrat ths pont and xampls ar gvn. Ths rsults ar thn gnralzd to th multpl transmt antnna cas. A nw (N T +)-dmnsonal sgnalng stratgy s proposd for th multusr channl that lvrags xstng sngl-usr spac-tm sgnal dsgns whl nsurng a full dvrsty ordr and sngl-usr lk prformanc asymptotcally for vry usr. I. INTRODUCTION Multpl antnna communcaton has rcvd consdrabl attnton n rcnt yars du n larg part to th nformaton thortc work n [], [], whch showd that th us of multpl transmt and rcv antnnas could achv consdrabl gans on th Raylgh fadng channl whn th rcvr has prfct sd nformaton about th channl stat. Motvatd by ths promss, svral rsarchrs hav rcntly proposd mult-antnna codng and modulaton schms for cohrnt sngl-usr channls (cf. [] []) to show that dvrsty communcaton systms, whn dsgnd ntllgntly, can yld sgncant mprovmnts ovr sngl antnna channls. Th nformaton thory of th sngl-usr spac-tm channl asly xtnds to th multusr mult-antnna channl [] and t can b nfrrd that th gans n th capacty rgon ar vry bt as dramatc for th multusr channl as wll. Ths work was supportd n part by NSF Grants ANIR-958 and CCR and by ARO Grant DAAD In ths papr, w prsnt a thory of modulaton and dtcton for multusr spac-tm communcaton. In Scton II, w dscrb a vry gnral K usr, N T transmt, N R rcv antnna systm modl. Basd on our gnral rsults on th asymptotc analyss of quadratc rcvrs n complx zroman Gaussan vctors [8], w analyz, n Scton III, th optmum multusr rcvr and obtan asymptotcally tght xprssons for th par-ws rror probablts. In Scton IV w ntrprt ths probablts for th sngl transmt (IV-A) and multpl transmt antnnas (IV-B) cass n trms of th mnmum dmnson ndd to achv full ordr of dvrsty for all usrs. W also propos nw optmzd multusr sgnal dsgn stratgs that lvrag sngl-usr spac-tm dsgns n ordr to dlvr sngl-usr lk prformanc n th hgh SNR rgm. Notaton: Throughout th papr T dnots transpos and y complx conjugat transpos. Th mult-varat crcularly symmtrc, complx Gaussan dstrbuton wth manvctor m (and covaranc matrx K) s dnotd by CN(m) (CN(m; K)). E[ ] dnots th xpctd valu of th xprsson n brackts. For any matrx A w wrt ts dtrmnant as jaj and ts trac as tr (A). For any vctor a, w wrt ts ` norm as p a y a kak. Th logarthm to th bas b s dnotd by log b, th natural logarthm by ln. Th Kronckr product of two matrcs s dnotd by Ω. II. MULTI-ANTENNA, MULTIUSER DISCRETE TIME SYSTEM MODEL W dscrb a systm modl for K usrs communcatng smultanously n a common D dmnsonal sgnal spac. Each of th K usrs mploys N T transmt antnnas to snd nformaton symbol-synchronously to an N R rcv antnna array of th bas-staton. Snc thr ar N T transmt antnnas and D dmnsons, ach usr transmts on out of M possbl D N T complx-valud sgnal matrcs, drawn from th st S k fs k ; S k ;:::;S km g wth S km IC D NT. Th sgnal matrcs S km may b thought of as spac-tm (block-) codwords whr ach lmnt of th matrx s drawn from a nt, QAM-lk constllaton wth S k bng th k th usr s codbook, or thy may b thought of as supr-symbols of som arbtrary constllaton S k. Hnc, th rcvr s a dcodr or a dtctor n th two cass, rspctvly. W wll rfr to th matrcs S km as (supr-) symbols or sgnals or codwords as s approprat and us th gnral trm rcvr whn th trms W suggst th trm spac-dmnson communcaton rathr than spac-tm communcatons. Th lattr mpls a bass of tm-translats of a sngl wavform (so that D corrsponds to th lngth of th cohrnc ntrval n symbol duratons), whch s too rstrctv as pontd out n [8].

2 dtctor or dcodr ar both applcabl. To succnctly wrt th dscrt-tm modl for ths systm w nd mor dntons. Lt H dnot th th hypothss wth»» M K. Wthout loss of gnralty lt dtrmn P unquly th K-tupl K ( ; ;:::; K ) accordng to ( k k )M k +. W lt hypothss H dnot that usr k transmts th sgnal S kk for ach k. DnthD KN T matrx of sgnals corrspondng to hypothss H as F [S ; S ;:::;S KK ]. Thus th dscrt-tm modl for th n th rcv antnna can b wrttn as y n F W hn + n ; () whr y n s th D dmnsonal vctor of obsrvatons, W dag fw ;w ;:::; w K gωi NT wth w k bng th k th usr s avrag nrgy, h n h T n ; ht n ;:::; T KnΛ ht s a KN T dmnsonal vctor of CN(0) dstrbutd fadng cofcnts wth h kn contanng th N T fadng cofcnts from th k th usr s transmt antnnas to rcv antnna n,and n s th D dmnsonal CN(0;ff I D ) dstrbutd addtv nos vctor. To obtan th sufcnt statstcs for all N R rcv antnnas, w smply stack th y n to obtan y y. y NR 5 INR Ω F W bh + ; () whr b h h T ;:::;h T N R Λ T contans th fadng cofcnts n rcvr-antnna ordr. For th analyss to com, t wll b mor convnnt to organz th fadng cofcnts usr-ws n th vctor h h T ;:::;h T N R ;:::;h T K ;:::;ht KN R Λ T, whch also rqurs th ntroducton of S km I NR Ω S km, F [S ; S ;:::;S KK ],andw W Ω I NR. Wth ths dntons th DN R sufcnt statstcs can b wrttn as y F W h + : () W dnot th corrlaton matrx of th fadng cofcnts as ± E hh yλ. ± kk dnots th k th dagonal N N block of ± and thus s th k th usr s fadng corrlaton matrx. Th sgnals and fadng procsss ar normalzd so that μfl k w k ff rprsnts th avrag rcvd sgnal-tonos rato (SNR) of th k th usr pr rcvr antnna and pr (supr-) symbol. Each usr s sgnals ar normalzd such that thr avrag nrgy ovr th transmt antnnas s unty,.., h E tr S y km S km N T 8 k; () whr th xpctd valu s takn ovr m. Th fadng cofcnts ar normalzd such that E h h y k Sy km S km h k N R 8 k; (5) whr hk Λ h T T k ;:::;ht kn R contans all of th k th usr s fadng cofcnts. For qu-probabl symbols ths condton can b wrttn as M m tr ±kk S y km S km MN R : () Not that wth ths normalzatons th avrag rcvd SNR s ndpndnt of th numbr of transmt antnnas. For xampl n..d. fadng th statd condtons lad to N T ± kk I N. III. OPTIMUM RECEIVER AND ANALYSIS In ths scton w spcfy th optmum rcvr n trms of a quadratc form n th obsrvatons and th fadng cofcnts. Ths allows us to mak us of th gnral rsults of [8] for th asymptotcally tght analyss of th par-ws rror probablts. A. Maxmum Lklhood Rcvr Th lklhood functon of th sufcnt statstcs y gvn th fadng cofcnts h and th tru hypothss H (.. F, s transmttd) s p(yjh;h ) ß DNR ff xp ff ky F W hk DNR () Dnng th nw (KN T +D)N R dmnsonal sufcnt statstc z ff h T y Λ T T and th matrx Q " W F # y F W W F y ; (8) F W 0 DNR th jontly optmum cohrnt rcvr Φ C can b xprssd as Φ C : ^ arg mn»»m zy Q C K z arg mn»»m K fc ; (9) whr f C s dnd mplctly. Not that th sufcnt statstcs z ar CN(0; K zzjh ) dstrbutd, whr K zzjh E zz yλ (0) " ff ± ff F W ± B. Bounds on Symbol and Bt Error Rat # ff ±W F y ff F W ±W F y + I : Lt E k (Φ C ) dnot th vnt that th rcvr Φ C dtcts usr k rronously. Thn Pr fe k (Φ C )jh g s th symbol rror probablty of th k th usr dtctd by rcvr Φ C condtond on th hypothss H. It s th probablty of th unon of th corrspondng (M )M K possbl vnts of th form Φ f C j <fc. Snc th probablty of th unon s usually not computabl, consdr th unon uppr bound Φ whch s th sum of th par-ws rror probablts Pr fj C <fc. A lowr bound n s obtand o by consdrng th par-ws probablty Pr fbj C <fc,whrh b j corrsponds to on of th M hypothss H j that rsult n an rror only for usr k whn compard to H. Th lowr boundo can b tghtnd by choosng H b j nf such that Pr bj C <fc s maxmzd. n o Pr f C <f C s only an rror probablty n a bnary hypothss tst. j Howvr, th trm par-ws rror probablty s customarly usd n th ltratur. :

3 Th k th usr s symbol rror rat (SER) Pr fe k (Φ C )g for qu-probabl symbols s thn boundd as Pr fe k (Φ C )g M K M K» M K M K Pr fe k (Φ C )g M K M K Pr fe k (Φ C )jh g () 8jΛ (k) Pr Φ Pr fj C <f C ; () n f C bj <fc o ; () whr Λ (k) s th st of th (M )M K ndcs of hypothss n whch th k th usr s symbol dffrs from ts symbol corrspondng to th tru hypothss H. To obtan bounds on th avrag bt rror rat (BER) P b k of th k th usr, w ntroduc th vnt H! H j that hypothss H s dtctd as H j (n th prsnc of all othr hypothss). Snc th vnts H! H j ar mutually xclusv, th avrag bt rror rat can b wrttn as K M Pk b M K 8jΛ (k) b j(k) log M Pr fh! H j g ; () whr b j (k) s th numbr of rronously dtctd bts of usr k, whn hypothss H s dtctd as H j. An uppr bound on Pk b s obtand by uppr-boundng th probablts Pr fh! H j g by Pr Φ f C j <fc. A lowr bound on P b k s obtand by lowr boundng b j (k) by on and usng th fact that th nnr sum of probablts s qual to Prn fe k (Φ C )jh o g, whch n turn can b lowr boundd by Pr fbj C <fc,as n (). C. Par-ws Error Probablts Φ Th par-ws rror probablts Pr fj C <fc ar crucal for th bounds on th symbol as wll as th bt rror rat. Thy can b obtand va th calculaton of rsdus (cf. [8] []). Howvr, th rsdus dpnd on th gnvalus of C C j K zzjh (Q j Q ) and do not n gnral gv any nsght nto th dpndncs on th systm paramtrs of ntrst, such as th sgnal and fadng corrlatons. A rmdy for ths s offrd by th asymptotc (hgh SNR) analyss of th parws rror probablts n [8] whr w xamnd th asymptotc analyss of quadratc rcvrs n Raylgh fadng channls and found formulas for th asymptotc rror-probablts that rqur only th valuaton of th asymptotc gnvalus of C C j. Th structur of ths asymptotc gnvalus follows th structur obsrvd n [8]: half of th non-zro gnvalus ar postv and lnar n ff, and th othr half convrg to mnus unty. W stat nxt th par-ws rror probablts for nt SNR n th followng proposton, whch can b asly obtand from, for xampl, [8]. Φ Proposton (Exprsson for Pr fj C <fc ) Lt f l g L b th dstnct non-zro gnvalus of l CC j K zzjh (Q j Q ) wth multplcts fμ l g L,andltf l lg Ln l b ngatv and f l g L ll n+ postv, rspctvly. Thn Pr Φ f C j <fc L n k 0 Rs@ Q μl ; s k A : L s l μ l l s + k l Th rsdu of a functon f(s) n a pol a of multplcty m s dnd to b Rs (f(s); a) (m )! lm s!a d m ds m [(s a)m f(s)] : For ratonal functons th lmt s trval, bcaus th pols cancl wth th (s a) m trms. Not that th calculaton of th rsdus s numrcally unstabl for hgh-multplcts of gnvalus, so that for ths cass on must us, for xampl, a saddl pont ntgraton tchnqu []. As dscussd n dtal n [8], w must nd th asymptotc gnvalus of C C j to obtan th asymptotc xprsson for Pr Φ f C j <fc. To ths nd, w ntroduc som assumptons and notaton. W assum that th usrs ar ordrd such that usrs ; ;:::;suffr from an rror, f th rcvr would rronously dcd for hypothss H j whn hypothss H s transmttd. To avod a complcaton n notaton, w do not dnot ths usr-ordrng wth any spcal symbols, but assum t mplctly. Anothr notatonal convnnc s to splt up th transmttd sgnal nto two parts, th rst contanng th sgnals of th usrs that suffr from an rror rlatv to H j,and th scond part contanng th μ K sgnals corrspondng to th corrctly dtctd usrs,.., F [F F c ] ; F j F j F cλ ; (5) whr c sgns th common part n th two sgnals F and F j. Th matrcs F and F j ar D N T and F c s D μn T. Smlarly, w dn F, F j,andfc (whos szs ar multpld by N R whn compard to F, F j,andfc, rspctvly). Furthrmor, w dn ± and W as th N N upprlft block of ± and W, rspctvly (rcall N N T N R ). ± c and ± cc (W cc ) ar th corrspondng uppr- and lowr-rght blocks of ± (W). Wth ths and th dntons of A ± ± c F W ± + F c W cc± c 5 ; () B W F j F y ; () h C F jw F c W cc I DNR ; (8) Z 0 N μn 0 F j F W 0 0 DNR on nds aftr som tdous algbra that 5 ; (9) C C j ff ABC + Z: (0)

4 Snc th gnvalus of C C j ar th sam as th gnvalus of C b C j TC C jt for any nvrtbl matrx T, w ar fr to choos T whr T Λ F W ± 0 0 ± c ± I μn 0 F W F c W cc ± 0 0 ± c ± I μn 0 T Λ F c W cc ± +F c W F j F j F, w can calculat C b C j as bc C j h bc C j I DNR I DNR 5 5 ; () cc± c ± 0 ff ± W 0 0 μn 0 F jw ± ; (). Dnng F y j 5 ; 0 0 DNR h () whr bc C j ff ± W F y jf jw ±. Lt USV y b th conomy-sz sngular valu dcomposton (SVD) of F jw ±, whos rank s asly shown to b rn R,whrr s th rank of F j F. Hnc S s of sz rn R rn R. Thn t s not hard to show that th non-zro gnvalus of C b C j ar th non-zro gnvalus of M» ff S y S ff S y U y : () US 0 DNR Th gnvalus of M ar found by applyng a dtrmnantal qualty [, Scton 0.8.5],.., jm Ij ff S y S I (5) I US S y S ff I S y U y : For small ff th scond dtrmnant of th product can b approxmatd by I UU y, so that w nally arrv at th followng proposton, for whch w mad us of [, Thorm..0], whch stats that th non-zro gnvalus of Y ar qual to thos of Y. Proposton (Asymptotc Egnvalus of C C j ) Th asymptotc non-zro gnvalus of C C j K zzjh (Q j Q ) ar arbtrarly clos to th rn R non-zro gnvalus of ff W ± W F y jf j and mnus unty wth multplcty rn R. Wth ths gnvalus and th rsults of [8], on asly nds th asymptotc par-ws rror-probablty gvn n th nxt proposton. For as of notaton w ntroduc jj NZ as th product of th non-zro gnvalus of. Proposton (Asymptotc Par-Ws Error Probablty) For Φ cohrnt dtcton th par-ws rror probablty Pr fj C <fc of th optmum rcvr ΦC approachs arbtrarly closly ff rnr rn R rn R Pr a Φ f C j <fc W ± W F y jf j NZ as ff gos to zro. IV. INTERPRETATIONS AND SIGNAL DESIGNS Th asymptotc rsult of th prvous scton ncompasss many spcal cass of ntrst, som of whch w xplor n ths scton. For xampl, w spcalz to N T n Scton IV-A and gan som nsghts nto ths cas, whch hlp undrstand th multpl transmt antnna cas. In Scton IV- B w consdr multpl transmt antnnas but spcalz to K rst, bfor w dscuss th gnral K-usr, N T -antnna problm. W focus on gvng spcc rsults and ntrprtatons for th asymptotc par-ws rror probablts and assum that t s undrstood that th corrspondng optmum rcvr can b obtand by applyng th spccs to Φ C as dnd abov. A. On Transmt-Antnna pr Usr W dstngush btwn lnar vrsus gnral M-ary/blockcodd modulaton. In lnar modulaton, ach usr modulats ts sgnatur squnc by a symbol drawn from a xd alphabt, lk a QAM or PSK constllaton. In M-ary or blockcodd modulaton th k th usr s m th sgnal vctor may b a block-cod ovr a nt alphabt or a supr-symbol drawn from an arbtrary constllaton. A. Lnar Modulaton Multusr Dtcton and CDMA Sgnatur Squnc Dsgn If w spcalz F FB,whrB s a K K dagonal matrx contanng th usrs constllaton symbols, th systm modl of () corrsponds to a synchronous cod dvson multpl accss (CDMA) modl. Not that w do not mak any assumptons on th numbr of dmnsons D, so that ovrloadd systms wth D<Kar ncludd n th analyss. W dn F to b mad up from th columns (sgnaturs squncs) of th usrs that suffr from an rror. B, B j ar dagonal matrcs that contan th nformaton symbols of ths usrs. Corollary : (Asymptotc Par-Ws Error Probablty for Cohrnt CDMA Dtcton) Assumng ± I KNR and that any subst of D columns of F span th D-dmnsonal sgnal spac, w hav for» D that th par-ws rror probablty of th optmum dtctor Φ C approachs arbtrarly closly Pr a f C j <f C as ff gos to zro. For D» w hav Pr a fj C <fc ff NR N R N R W B j B y F y F B j N R B ff DNR DN R DN R F B j B W B j B y F y N R : In ths and som of th followng corollars, th assumpton of..d. fadng s only mad to avod havng to ntroduc mor notaton.

5 W not that for D and BPSK modulaton th scond bound spcalzs to th on gvn n []. Not that as th numbr of usrs ncrass, th aggrgat spctral fcncy K log M ncrass lnarly snc all th usrs mploy th sam sgnatur sgnal whch, wthout bng wastful of bandwdth, can b takn to b th mnmum bandwdth snc puls or a rasd cosn puls wth sufcnt roll-off to nsur robustnss to tmng jttr and usr quas-synchronsm. Th scond bound of Corollary mpls that thr s no loss of ordr of dvrsty compard to a sngl-usr channl for any of th usrs. Wthout any bandwdth xpanson compard to a sngl-usr channl, multpl usrs can b accommodatd wth no loss of dvrsty ordr. Thr would b, howvr, a loss of nrgy fcncy n that ach usr would hav to transmt at a somwhat hghr powr to achv th prformanc t would hav n th absnc of othr usrs, and ths loss would ncras wth th numbr of usrs. Consdr th cas D wth aggrgat spctral fcncy of K log M bps/hz whch also lnarly ncrass wth an ncras n th numbr of usrs albt at half th rat of th narrowband channl. In ths cas, t s asy to dsgn twodmnsonal sgnatur squncs ovr th ld of complx numbrs by smply nsurng that any two usrs ar assgnd lnarly ndpndnt sgnals so that not only do all th usrs achv full ordr of dvrsty but vn th abov-mntond loss of asymptotc ffctv nrgy rlatv to sngl-usr prformanc s lmnatd. Hnc, wth a bandwdth xpanson by a factor of two rlatv to a sngl-usr channl, an ncrasng numbr of usrs can b accommodatd and rcvd wth a rlablty that s asymptotcally quvalnt to sngl-usr lk prformanc for vry usr, n th sns that th uppr bound on th multusr BER convrgs to th sngl-usr uppr bound. To obtan such spradng sgnals for D > dmnsons w suggst a sgnal dsgn algorthm that mnmzs th maxmum of a pr-usr asymptotc prformanc crtron ovr all usrs. Ths crtron s drvd from th uppr bound on th k th usr s asymptotc bt rror rat, whch n turn rsults from () by uppr-boundng Pr fh! H j g by Pr a Φ f C j <fc for small ff,.., K M Pk b» M K D d 8jΛ (k) b j(k) log M Pra ff j <f g c d (k)ff dnr ; () whr c d (k) contans all th cofcnts wth dvrsty ordr d n th uppr bound on th k th usr s BER. Th sgnal dsgn algorthm must mnmz max»k»k c (k). Whl th trms c d (k) wth d > asymptotcally do not nflunc th BER, w conjctur that by mnmzng max»k»k c (k), th convrgnc of th uppr-bound to th lowr-bound s mprovd, so that th BER of a systm mployng optmzd sgnals s mprovd at nt SNRs (asymptotcally th BER dos not dpnd on th sgnals provdd that at last any two sgnatur squncs ar lnarly ndpndnt). Fgur shows th prformanc of multusr systms mployng optmzd sgnatur squncs n D dmnsons. Th qual-nrgy usrs (W I) transmt spradd BPSK symbols from on transmt antnna to on rcv antnna n..d. fadng (± I). Th dsgn algorthm yldd sgnal sts for whch th usrs prformancs ar dntcal for nt SNR, so that w plot th uppr bound and smulatd BER of on usr only. W s that asymptotcally sngl-usr prformanc s achvd. Howvr, for ncrasng numbr of usrs th asymptot s rachd for ncrasng SNR only. Fgur shows th prformanc of th sgnatur squncs for K 0 usrs of th prvous plot n comparson wth narrow-band communcatons (D ) and a sngl usr n on dmnson wth a spctral fcncy of 5bpsHz (thus th sngl usr mploys -QAM). As bfor w choos W I K and ± I K for on transmt and rcv antnna. Although an asymptotc prformanc crtron s optmzd n th dsgn procss for th sgnatur squncs, th advantag ovr narrow-band sgnalng at a BER of 0 s ovr 5 db for th K 0 narrowband systm and about db for th K 5 narrowband systm, whch has th sam spctral fcncy as th K 0, D CDMA systm. Whl for a BER 0 th gap to th sngl usr mployng -QAM s about db, th sngl usr s asymptotcally out-prformd by roughly db. Fgur shows bounds and smulatd BERs for a N R rcv antnna, K usr systm n whch ach usr mploys th most nrgy fcnt 8-QAM constllaton. Th dsgnd sgnals hav an dntcal absolut valu of th crosscorrlaton of 0:5 and th rsultng BER of ach usr s almost ndstngushabl from that of a sngl-usr channl. On th othr hand, at a BER of 0, th gap to narrow-band sgnalng s roughly 5 db. In all cass th bounds on th BERs ar not tght, du to th us of M 8-ary sgnalng. A. M-ary or Block Codd Modulaton In ths scton, w ntrprt th k th column of F as a suprsymbol of usr k whch can b thought of as blongng to som dns lattc (or mor gnrally to an arbtrary non-lattc constllaton), whos ndvdual scalar lmnts may b drawn from a rgular QAM-lk alphabt or may b arbtrary complx numbrs, not ncssarly rstrctd to b part of a nt alphabt. Whn th codword ntrprtaton s approprat, th rcvr may b thought of as a dcodr. W rwrt th gnral xprsson of th asymptotc parws rror probablty from Proposton. Corollary : (Asymptotc Par-Ws Error Probablty for Cohrnt Dcodng) Assumng ± I KNR and that F j F has rank r» mn(d; ), th par-ws rror probablty of th optmum dcodr Φ C approachs arbtrarly closly Φ Pr a fj C <fc as ff gos to zro. ff rnr rn R rn R W F j F y F j F N R NZ

6 Lt us rconcl ths rsult for th cttous cas whr all K usrs co-oprat so that w hav an quvalnt sngl-usr, K-transmt, N R -rcv antnna channl. In ths cas, th maxmum dvrsty ordr for a gvn K could b achvd f r K and th proposton corrsponds to th wll-known rank crtron [], for whch of cours w nd D K. Byth us of spac-tm cods such as th orthogonal dsgns of [5] or th algbrac cods of [5] whch satsfy ths rank crtron, on can achv full dvrsty ordr (namly KN R ). Howvr, th multusr rank crtron s vry dffrnt from th sngl-usr crtron bcaus whl n th sngl-usr channl wth K transmt antnnas, sgnals transmttd ovr th dffrnt transmt antnnas can b dpndnt (.., an suprnformaton symbol s ncodd nto a D N T matrx), th columns of ths matrx n th multusr channl ars from th ndpndnt transmsson of vctors of lngth D ach from th K dffrnt usrs. B. Multpl Transmt-Antnnas pr Usr Th classcal sngl-usr multpl transmt antnna spactm codng analyss also prots from our gnral analyss: n contrast to th arlr, Chrnoff bound basd approachs, our analyss provds asymptotcally tght xprssons for th parws rror probablty and consdrs possbly corrlatd fadng. Fnally, for th mult-transmt antnna, multusr spac tm channl, w propos a sgnal dsgn algorthm that nsurs sngl-usr lk prformanc asymptotcally. B. On Usr In ths cas on usr transmts a D N T sgnal matrx fs m g M m wth avrag nrgy w. ± smpls to th N N fadng corrlaton matrx assocatd wth all th antnnas. Rcallng that w dnd S m I NR ΩS m, Proposton asly smpls to th followng corollary. Corollary : (Asymptotc Par-Ws Error Probablty for Sngl Usr Rcpton) Assumng that (S j S ) has rank r» mn(d; N T ), th parws rror probablty of th optmum rcvr Φ C approachs arbtrarly closly Pr a Φ f C j <fc as ff gos to zro. wff rnr rn R rn R ± (S j S ) y (S j S ) Not that n addton to rvalng th rank and dtrmnant crtron of [], [] for..d. fadng, ths formula s s also asymptotcally tght and consdrs th mor gnral cas of corrlatd fadng. As a consqunc of th asymptotc tghtnss, asymptotcally tght lowr bounds on symbol and bt rror rats can b obtand. For corrlatd fadng and full-dvrsty spac-tm cods, th fadng corrlaton dos not affct th dtrmnant crtron, bcaus ± (S j S ) y (S j S ) j±j(s j S ) y (S j S ) NR. Consquntly, full-dvrsty spac-tm cods that wr optmzd for..d. ar also asymptotcally optmal for corrlatd NZ fadng. Th analyss prsntd hr also mprovs on our work n [8] by provdng xact xprssons for th asymptotc parws rror probablts n cas S S j s low rank. B. Multpl Usrs Th obsrvatons w mad for th varous spcal cass allow us to nally draw som conclusons about Proposton for multusr communcaton whn ach usr mploys N T transmt antnnas. Most mportantly, f vry usr mploys a full-dvrsty spac-tm cod/constllaton (rqurng D N T )nthsamd dmnsonal sgnal spac, th multusr systm stll achvs asymptotcally a dvrsty ordr of N N R N T,.., no loss n dvrsty ordr occurs whn compard to th sngl-usr cas, wthout any bandwdth xpanson (ths was also ralzd ndpndntly n [5] by usng th wakr Chrnoff analyss that dos not yld asymptotcally tght bounds on par-ws rror rats). Howvr, a loss n nrgy-fcncy occurs whn mor usrs ar addd. Ths bhavor mrrors xactly th N T, D narrow-band cas dscussd abov. W saw that w could mprov on ths bhavor by xpandng th sgnal spac to D dmnsons to dsgn sgnals such that th optmum rcvr achvs snglusr lk prformanc asymptotcally. On way to gnralz ths da to th multpl transmt antnna cas s sgnal accordng to F F B 0 ::: 0. 0 B ::: ; () 0 ::: 0 B KK whr F s a D KD SU xd sgnatur matrx and B kk ar D SU N T sngl usr full-dvrsty cod matrcs. Dnot th D SU columns of F that corrspond to usr k as F k,.., F [F ; F ;:::;F K ]. Usng a rank nqualty ([, Scton 0..5]), on can show that f any compound matrx [F l ; F k ], l k, s of rank gratr than or qual D SU N T +, thn t s guarantd that th asymptotc probablty of an vnt f C j <f C nvolvng usrs has a dvrsty ordr of at last (N T +)N R, so that ths probablty (and all parws rror probablts nvolvng mor than two usrs) can b asymptotcally nglctd whn compard to th sngl-usr par-ws rror probablts whos dvrsty ordr s N T N R. Thus, vn n th mult-transmt antnna cas sngl-usr lk prformanc can b asymptotcally achvd. To nd such a sutabl sgnatur matrx F w nd of cours D D SU N T +. Furthrmor, th dsgn algorthm dscussd n Scton IV-A. can b adaptd to dsgn an optmzd sgnatur matrx F. V. CONCLUSIONS Th gnral analyss of [8] s appld to cohrnt multusr spac-tm rcpton. Consquntly, asymptotcally tght xprssons for th par-ws rror probablts ar obtand. Svral conclusons can b drawn from ths analyss: ffl In a on transmt antnna pr usr CDMA systm, all usrs can b dtctd wth asymptotc sngl-usr lk prformanc,

7 f th common sgnal spac has at last dmnsonalty two. An algorthm to dsgn optmum spradng squncs s prsntd. ffl For M-ary/block codd modulaton wth on transmt antnna pr usr a sgnal/cod dsgn crtron s prsntd. ffl For th classcal sngl-usr N T transmt and N R rcv antnna spac-tm communcatons, w mprov on th prvous approachs by provdng asymptotcally tght bounds whl ncludng channls wth corrlatd fadng. ffl For th multusr spac-tm problm t s stablshd that vry usr achvs th total ordr of dvrsty N N T N R whn communcatng wth D-dmnsonal sngl-usr spactm cods n a common D-dmnsonal sgnal spac. To achv asymptotcally sngl-usr lk prformanc for multpl usrs, at last N T +dmnsons ar ncssary, as opposd to N T n th sngl-usr channl. A sgnal dsgn algorthm s gvn, that gnralzs th algorthm for N T. REFERENCES [] İ. E. Tlatar, Capacty of mult-antnna Gaussan channls, Europan Trans. on Tlcommun., vol. 0, no., pp , Nov. 999, Orgnally a Bll Laborators, Lucnt Tchnologs, Tchncal Rport, Octobr 995. [] G. J. Foschn and M. J. Gans, On lmts of wrlss communcatons n a fadng nvronmnt whn usng multpl antnnas, Wrlss Prsonal Communcatons, vol., no., pp. 5, Mar [] G. J. Foschn, Layrd spac-tm archtctur for wrlss communcaton n fadng nvronmnts whn usng multpl antnnas, Bll Labs Tch. J., vol., no., pp. 59, Autumn 99. [] J.-C. Guy, M. P. Ftz, M. R. Bll, and W.-Y. Kuo, Sgnal dsgn for transmttr dvrsty wrlss communcaton systms ovr Raylgh fadng channls, IEEE Trans. Commun., vol., no., pp. 5 5, Apr [5] V. Tarokh, H. Jafarkhan, and A. R. Caldrbank, Spac-tm block cods from orthogonal dsgns, IEEE Trans. Inform. Thory, vol. 5, no. 5, pp. 5, July 999. [] V. Tarokh, N. Sshadr, and A. R. Caldrbank, Spac-tm cods for hgh data rat wrlss communcatons: Prformanc crtron and cod constructon, IEEE Trans. Inform. Thory, vol., no., pp. 5, Mar [] A. R. J. Hammons and H. El Gamal, On th thory of spac-tm cods for PSK modulaton, IEEE Trans. Inform. Thory, vol., no., pp. 5 5, Mar [8] M. Brhlr and M. K. Varanas, Asymptotc rror probablty analyss of quadratc rcvrs n Raylgh fadng channls wth applcatons to a und analyss of cohrnt and noncohrnt spac-tm rcvrs, to appar IEEE Trans. Inform. Thory., 00. [9] J. K. Cavrs and P. Ho, Analyss of th rror prformanc of trllscodd modulatons n Raylgh fadng channls, IEEE Trans. Commun., vol. 0, no., pp. 8, Jan. 99. [0] E. Bglr, H. L. Own, and E. W. Zgura, Computng rror probablts ovr fadng channls: A und approach, Europan Trans. on Tlcommun., vol. 9, no., Fb [] M. J. Barrtt, Error probablty for optmal and suboptmal quadratc dtctors n rapd Raylgh fadng channls, IEEE J. Slct. Aras Commun., vol. 5, no., pp. 0 0, Fb. 98. [] C. W. Hlstrom, Elmnts of Sgnal Dtcton & Estmaton, Prntc- Hall, Englwood Clffs, NJ, 995. [] R. A. Horn and C. R. Johnson, Matrx Analyss, Cambrdg Unvrsty Prss, 99. [] E. A. Fan and M. K. Varanas, Dvrsty ordr gan for narrowband multusr communcatons wth pr-combnng group dtcton, IEEE Trans. Commun., vol. 8, no., pp. 5 5, Apr [5] M. O. Damn, Jont Codng/Dcodng n a Multpl Accss Systm, Applcaton to Mobl Communcatons, Ph.D. thss, Écol Natonal Supérur ds Télécommuncatons, Pars, Franc, 999. Som of th rfrncs ar avalabl for download from varanas BER K0, uppr bnd. K0, smul. BER K, uppr bnd. K smul. BER K, uppr bnd. K, smul. BER lowr bnd. on BER SNR pr bt n db Fg.. Whl an ncrasng numbr of usrs can b accommodatd n D dmnsons wth asymptotc sngl-usr prformanc, th asymptot s rachd for hghr SNR as K ncrass. BER uppr bnd., NB, K0 smul. BER, NB, K0 uppr bnd., NB, K5 smul. BER, NB, K5 uppr bnd. on BER, D smul. BER, D lowr bnd. on BER smul. BER, SU, QAM SNR pr bt n db Fg.. Wth only D dmnsons th 0 usr systm can asymptotcally achv sngl-usr prformanc and out-prform th K 0narrow-band systm by 5 db at a BER of 0. BER QAM, K usrs, N, N ant. T R uppr bnd. on BER, narrow band smulatd BER, narrow band uppr bnd. on BER, D smulatd BER, D smulatd BER, sngl usr lowr bnd. on BER SNR (w k /σ ) n db Fg.. For ncrasng constllaton sz, th multusr systm can stll asymptotcally achv sngl-usr lk prformanc.