Full Exploitation of Diversity in Space-time MMSE Receivers 1

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1 Full Exploaon of Dvery n Space-me SE Recever Joep Vdal, argara Cabrera, Adran Aguín Sgnal heory and Communcaon Deparmen, Unvera Polècnca de Caalunya Barcelona, Span {pepe,marga}@gp.c.upc.e Abrac A unfed and general von of dfferen paceme proceor preened. any popular recever can be accommodaed, lke V-RAKE recever, weghed V-RAKE, or paal narrowband beamformng. By makng approprae aumpon on he pace/me characerc of he nerference poble o enhance he performance of he recever hrough paal/emporal pre-proceor. hee recever wll be eed n he FDD mode of URA [ESI-URA]. Inroducon he adven of he 3rd generaon of moble communcaon yem ha been accompaned by he recognon of he large ncreae n yem capacy ha can be obaned from he ue of adapve anenna array. Care ha been aken n he defnon of he andard o nclude capable for pace-me proceng of he gnal ncomng and radaed from he bae aon. Secon 2 preen he gnal model. In econ 3 he dfferen pace-me recever are preened n a unfed von. I wll be een ha he ue of mulple anenna and he emporal correlaon dvery of mulple uer allow addonal degree of freedom for cancellaon of mulple acce nerference. Only ngle uer approache wll be nroduced alhough ome dea are ealy exendable o he muluer cae. All hee recever need a relable emae of he correlaon propere of he nerference, o wo dfferen opon are preened n econ 4. Performance are compared n erm of probably of error n econ 5. Fnally concluon and reference are repecvely preened n econ 6 and 7. 2 Sgnal odel he ngle-uer gnal model aumed for he gnal receved a enor afer mached flerng and chp-me amplng can be wren n column vecor form a: where each erm defned a: d y y Q p L+Q p- = = = = y = d + w () [ d p d ] [ y y2 L y ] [ y () y () L y ( N ) ] (k) hp, Np (k) h, N Q 2 = he pace-me channel of he dered uer, d nclude he N raffc ymbol (d ) and he N p plo ymbol (d p ) for h uer n he phycal channel of he FDD mode of URA. w he vecor accounng for noe plu nerference. N he number of chp n a ngle lo, Q p he DPCC preadng facor and Q he DPDC (2) h work nvolved n he EC -IS , SAURN projec. I ha alo been parally uppored by CICy of Span (IC98-42, IC98-73, IC99-849) and CIRI of Caalunya (998SGR-8).

2 preadng facor. arx conan he convoluon of he mpule repone of he channel een by enor (compued a chp me) and he preadng code. he effec of he long cramblng code can be repreened by he me varaon of he preadng code from ymbol o ymbol, whch denoed wh he upercrp (k): h h, p, = h c = h c p In equaon (2), 5 ymbol have been ploed for he plo channel and for he raffc channel. I aumed ha he emporal lengh of he phycal channel L chp. 3 A famly of pace-me recever. Wh h model n mnd and modelng he nerference-plu-noe a paally and emporally correlaed Gauan noe, poble o formulae he lkelhood funcon whch, appropraely mnmzed, gve he deecor for he unknown ymbol: J = ( y d ) R w ( y d ) = y Rw y 2 Re y Rw d + d Rw { } d Some aumpon are poble o a o mplfy he recever: A.. he correlaon marx of he noe-plu-nerference can be decompoed a he Kronecker produc of a paal correlaon marx and a me correlaon marx: R w = R w, R w, (5) (3) (4) whch agree wh he mo recen channel model [Pederen], n whch he me and angular pread are hown o be ndependen phenomena. A.2. he phycal channel pread (L chp) much horer han he lengh of he preadng code (whch he cae when degnng a DS/CDA yem), o he marx Rw almo dagonal and he la erm n (4) can be negleced n mnmzaon. In fac, h one of he reaon why hgh b-rae uer canno be allocaed n rural or hlly envronmen, where delay pread are uually long compared o he lengh of he preadng code. herefore only he mddle erm reman n (4) and conue a uffcen ac of he problem. I maxmzaon lead o he well known Rake recever when boh pace and me correlaon marce are aumed whe: dˆ = arg max Re I I = y = y y B { } R w d = y ( Rw, R w, ) d = / 2 / 2 / 2 / 2 ( R w, R w, )( R w, R w, ) B d d = he nroducon of he correlaon marx of w mple a prewhenng of boh he gnal vecor y (whch noed wh y B ) and he dered uer channel marx (whch noed wh B ). h operaon can be done eparaely n /2 me and pace (noe ha R w, apple only on he / 2 paal componen of y and R w, apple on he emporal componen). Of coure h recever could be fully mplemened by ung ample emae of boh correlaon marce, bu uually he cae ha he complexy of he reulng rucure doe no jufy he mprovemen obaned wh mplfed veron. hee dfferen recever can be formulaed from equaon (6) by dong ceran approxmaon on he correlaon marce. 3. emporal correlaon marx.. emporally whe nerference. I aumed uually and a reaonable aumpon f he number of nerferen uer can be condered hgh..2. ph order arkov model for he emporal correlaon. h aumpon no uually ued bu work well for a low number of uer n a low noe cenaro. he mplcy of h model refleced n he rucure of he emporal marx for he fr order cae, whch ake he form: (6) N ρ L ρ N 2 2 ρ L ρ R = σ w, (7) O N N 2 ρ ρ L h marx, and hoe obaned for hgher order model, have n fac a cloed expreon for nvere [Kay] whch could be ued n (6) and preven from marx nveron. owever, much more nereng and praccal o recognze ha he / 2 emporal whenng role of R w, wll be done exacly by a ph order FIR fler. Of coure, hgh order of he model, mply long FIR fler whch

3 nroduce addonal nerchp nerference and, a a conequence, reduce he valdy of aumpon A.2. Care hould be aken o ue hor lengh compared o he delay pread of he channel mpule repone. 3.2 Spaal correlaon marx approxmaon S.. Spaally whe nerference. h aumpon realc only n he cae of a hgh number of nerferer or n a hghly angular dperve cenaro. hen, he recever become he well-known VRAKE [VanEen]. S.2. Reduced rank approxmaon: R w B R, S B, where B C / wh R< (8) he paal correlaon marx now reduced o a number of componen and, f emporal whene for he nerference aumed, he overall recever operaon can be wren a: I = R = σ y ( b I)( b I) d (9) he egenvecor aocaed o he beamformer gve a meaure of he relably of he nformaon conveyed by he branch of he combner. 4 Spaal fron-end wo approxmae paal recever wll be developed n he equel. No aumpon are made on emporal correlaon marx, hank o he paal-emporal uncouplng of he problem aed n equaon (5). 4. Noe-plu-nerference marx nveron (NII) recever h recever baed on he reduced rank approxmaon of he nvere of he paal correlaon marx gven by equaon (8). I llurave o nerpre equaon (9) a a coheren combnng (maxmum rao combnng) of he oupu of R beamformer (ee fgure, ncludng alo emporal prewhenng). he naure of each ealy een from a mple cae: aume he cae of P< pon nerferer. If vecor b are aken a he noe egenvecor of R w, each one ac a a paal nerference canceller. Seen n h way, dfferen nerference canceller can be emaed accordng o dfferen crera. h recever aume he knowledge of he marx R w,. Prevouly o emaon Le u fr redefne he gnal model of equaon () a: Y = unvec ( N L+ ) ( y) = D + W () where marx D a oeplz marx bul a chp me from he QPSK complex preaded and crambled ymbol. y (n) y 2 (n) y (n) a a... a Fgure. he recever n equaon (6) wh emporal prewhenng (gven by he FIR fler affecng equally o each branch) and paal prewhenng and reduced ank approxmaon (>R). he gan a he oupu of each beamformer are gven by he aocaed egenvalue n equaon (8). d ( L ) d( L 2) L d() d L d L d ( ) ( ) L () ( N L+ ) L D = C/ O d( N ) d( N 2) L d ( N L ) () where N and for he number of chp n he plo, L he lengh of he emaed phycal channel, and all erm d(n) belong o he e {--j, -+j, +j, -j}. conan he repone of he phycal propagaon channel a chp me for all enor (noe he dfference wh marx n equaon (2). ( L+ ) [ h h2 L h] C/ = [ h () h () L h ( L ) ] = (2) h /σ RAKE b b R /σ R Re{.} RAKE R +

4 Noe however, ha accordng o he gnal rucure of he FDD mode of US, we canno compleely deermne marx D beforehand nce conan he known chp of he plo channel bu alo he unknown chp of he raffc channel: D 2 = βd p + β D (3) where β he weghng facor aocaed o he plo (known) chp and β 2 he one aocaed o he raffc (unknown) chp. Fr of all, worh menonng ha he channel n equaon () may be emaed conenly by applyng a lea quare emaon. he channel marx modeled n(4): ˆ = β ( D p D p ) D p Y (2β ) ( N L + ) D p Y (4) n whch ncorrelaon beween known and unknown chp aumed. Under hee preme, he pace correlaon marx of nerference and noe can be compued ergodcally a: w, = Y Y µ D D 2 2 Y Y 2µ ( β + β2 )( N L + ) = ˆ y, µ R, (5) where n he la equaly we have aumed emporal ncorrelaon beween he n-phae and quadraure componen of he crambled chp, and aken no accoun he dfferen amplude of he plo and raffc channel. he erm µ ncluded wh he followng purpoe: one of he horcomng of (5) he marx ubracon, an operaon ha may lead o nonpovene of (5) due o emaon error. hen, he µ facor can be choen convenenly o a: z z = z z µ z z > w, y,, z (6) If h equaon ha o be pove defne for every poble vecor z, hen he value of µ ha o be maller han he mnmum value of he Raylegh quoen, ha, maller han he mnmum egenvalue of he marx pencl z µ < mn z y,, z z (7) 4.2 ached dered mpule repone (DIR) recever A dfferen way o buld a paal recever o oban a combner b ha maxmze he SINR a oupu. h correpond o he DIR approach developed n [Laguna]: mn b R y, b b D Db = (8) b I hown here ha he choce of he beamformer obaned a he egenvecor b, aocaed o he mnmum egenvalue n (9): R y, b = λ D Db (9) he gnal power a he oupu of he beamformer b, fxed o be one. λ ake he value of he nvere of he gnal-plu-noe-plu-nerference power. Noe however ha here are egenvecor gven by (9) and each yeld a dfferen gnal wh dfferen qualy. h dvery can be coherenly combned ung he rake n fgure, wh he egenvalue ued a relably facor n he branche of he rake. In he DIR approach each beamformer end o pon o all drecon from where gnal are ncomng. For fgure o be vald, he noe componen beween branche have o be uncorrelaed. I eay o how ha h he cae for he DIR recever and for he NII recever. heorem. he noe a he oupu of he beamformer obaned wh he DIR approach are uncorrelaed, unle he egenvalue aocaed are equal. Proof. Le u ake (9) and recogne ha he ame oluon for he egenvecor can be obaned by ubung R y, for R w,. Now le u exend he equaon wh all he egenvecor a: R w, B = D DBS = Rd, BS (2) By lef-mulplyng wh he conjugae ranpoe of B we oban on he lef hand de of he equaon, he correlaon marx of he noe a he oupu of he dfferen beamformer and nce he lef-hand de of he equaon an herman marx and he egenvalue are real we can wre: B R w, B = B Rd, BS = SB Rd, B

5 Wh no lo of generaly aume ha S ha a mulple egenvecor σ, o f he produc above commuave can be wren a: SB R d, σi B = C S D D C = E D D σi E S (2) By operang eay o ee ha D= and ha E ha o be dagonal. herefore we can conclude ha: σc B Rw, B = S where S dagonal. 5 Expermenal Performance Evaluaon. 5. Propagaon Channel odel. In order o evaluae he recever n a realc moble cenaro, we have carred ou mulaon baed on a Gauan aonary uncorrelaed hypohe for he channel, aumng ndependence beween angular and Doppler pread, a ha been experenced from meauremen aken n downown Sockholm n he,8 Gz band [Pederen]. here, emprcally hown ha azmuh pecrum follow a Laplacan law, along wh Gauan drbuon for he drecon of arrval (φ) around he mean angular poon of he uer. he angular pread (ha he andard devaon of he Gauan, σ φ ) aken 8º. he number of ray mpngng he array fed a a Poon random varable of mean 25. An exponenal law found n [Pederen] for he power delay pread, bu our mulaon wll be baed on he pederan and vehcular model for emporal preadng recommended n he SG2 documen for URA. he amplude aocaed wh each propagaon pah (α) a complex Gauan random varable whoe power decreae a he me delay and he angular drecon of arrval wh repec o he moble poon ncreae. A clacal Clarke bah-haped Doppler pecrum obaned by aumng mulple reflecon cloe around he moble. he carrer frequency 2, Gz. All enor have fla paal repone n a ecored area of 2º, and are lnearly and unformly paced a d/λ=,5. All plo hown n he mulaon below are repreenaon of he performance of he lnk level whch can be ued laer, hrough convenen mappng, o oban FER (frame eraure rao) when conderng channel codng or oher yem level feaure [ämälänen]. 5.2 Smulaon A e of mulaon ha been performed ung up o 9 uer n he FDD mode of URA of preadng facor 6. All uer are aumed o have conrolled ranmed power wh no error and wo propagaon channel are condered: he pederan model, wh moble movng around a 3 km/h and he Vehcular model, wh moble movng around 5 km/h. DIR ha been eed wh dfferen number of egenvecor. he probably of error ploed n fgure 2 for he pederan channel and n fgure 3 for he vehcular channel. In all cae, he performance of NII and DIR wa uperor o he convenonal VRAKE recever, o ubanal gan from he ue of paal beamformng acheved. BER Eb/No = 5 db 8enor VRAKE NII DIR- ev DIR-2 ev DIR-3 ev DIR-4 ev USERS Fgure 2. Probably of error for a dfferen number of acve uer, all ranmng conrolled power, for he dfferen recever and dfferen number of combner. Pederan channel, v=3km/h. 5.3 Reul Evaluaon I found ha mlar performance obaned wh DIR and NII recever for he pederan channel cae, howng no mprovemen when gong from 2 o hgher number of egenvecor, fgure 2. When he mulaed channel he vehcular model, DIR how he be performance when he number of ued egenvecor greaer o one. h verfed n fgure 4, where he cumulave funcon of he rao of he ncreang egenvalue o he maxmum egenvalue of he DIR recever are depced. I clear ha he econd egenvalue (ha, he SNIR aocaed o he oupu of he econd egenvecor) alway gnfcan, alhough decreae lghly a he number of acve uer ncreae.

6 Noe ha he hrd evenvalue only gnfcan for a low number of uer, o can be dcarded n order o oban a good performance of he yem. approxmaon compued a hown n (9) he rank of he dered gnal correlaon marx can be approxmaed a hown n he followng: - Eb/No = 5 db 8enor rank( D D) rank( ) 2 PederanChannel Vehcular Channel (22) BER -2-3 In (22) ha been aumed good auocorrelaon propere for he ranmed crambled equence n marx D VRAKE NII DIR- ev DIR-2 ev DIR-3 ev DIR-4 ev USERS Fgure 3. Probably of error for a dfferen number of acve uer, all ranmng conrolled power, for he dfferen recever and dfferen number of combner. Vehcular channel, v=5km/h..8 λ uer Eb/No = 2 db λ 2 5 uer Eb/No = 2 db P / λmax < ab uer Eb/No = 2 db uer Eb/No = 2 db λ λmax db ( λ ) ploedv / ( ) Fgure 4. Cumulave funcon of he rao of he DIR egenvalue o he maxmum egenvalue. Vehcular channel. 8 Senor. o jufy he preceden reul, he delay pread of Vehcular and Pederan model have o be analyzed. For he Pederan cae (Fgure 2, Rank One approxmaon), here are only wo gnfcan ap, wh aenuaon below db, and he reulan delay pread approxmaely half a chp perod. For he Vehcular cae (Fgure 3,4, Rank wo approxmaon), here are hree gnfcan ap, wh aenuaon below db, and he reulan delay pread approxmaely en chp perod, producng a conderable level of ISI (Iner Symbol Inerference) when he Spreadng Facor 6. When a reduced rank 6 Concluon wo dfferen pace-me proceor have been preened whch boo he power of paal cancellng and coheren rake combnng. hey have been eed n a realc cenaro, ung US FDD mode and up-o-dae model for pao-emporal propagaon channel. Reul how a gnfcan mprovemen n he probably of error wh repec o convenonal approache, ha, only paal beamformng or only VRAKE combnng. Furher work nended o emporally rack he emaed channel and correlaon marce parameer o a o reduce b error when hgh peed cenaro are found. 7 Reference [ESI-URA] Submon of Propoed Rado ranmon echnologe: he ESI US erreral Rado Acce (URA) IU-R R Canddae Submon, ESI SG2. Dae of ubmon: 29//998. IU WWW hp:// [ämälänen] S. ämälänen, P. Slanna,. arman, A. Lappeelänen,. olma, O. Salonaho, A Novel Inerface beween Lnk and Syem Level Smulaon, Proc. of he ACS oble elecommuncaon Summ, Aalborg, Denmark, Ocober 997, pp [Kay] S. Kay, odern Specral Analy, Prence-all. [Laguna].A.Laguna, J. Vdal, A.I. Perez, Jon beamformng and Verb equalzer n wrele communcaon, Proc. 3 Alomar Conf. On Sgnal, Syem and Compuer, Nov [Pederen] K. Pederen, P. ogenen, B. Fleury, A Sochac odel of he emporal an Azmuhal Dperon een a he Bae Saon n Oudoor Propagaon Envronmen, ubmed o IEEE ran. on Vehcular echnology. [VanEen] W. Van Een, axmum Lkelhood Recever for ulple Channel ranmon Syem, IEEE ran. on Communcaon, Feb. 976, pp