Multipath diversity of precoded OFDM with linear equalization

Transcription

1 Uvrsty of Wollogog Rsarch Ol Faculty of Iformatcs - aprs (Archv) Faculty of Egrg ad Iformato Sccs 8 ultpath dvrsty of prcodd OFD wth lar qualzato Xaojg uag Uvrsty of Wollogog, huag@uow.du.au ublcato Dtals X. uag, "ultpath dvrsty of prcodd OFD wth lar qualzato," IEEE Itratoal Cofrc o Commucatos, 8, pp Rsarch Ol s th op accss sttutoal rpostory for th Uvrsty of Wollogog. For furthr formato cotact th UOW Lbrary: rsarch-pubs@uow.du.au

2 ultpath dvrsty of prcodd OFD wth lar qualzato Abstract Ths papr sttls a cotrovrsy ovr th multpath dvrsty prformac of th prcodd orthogoal frqucy dvso multplg systm wth th lar qualzato. Through th asymptotcal aalyss of th bt rror rats wth trm systm paramtrs, a comprhsv udrstadg of th lar qualzato's bhavor th frqucy-slctv multpath fadg chals s gad. Compard wth th optmum mamum-lklhood dtcto, th dvrsty prformac of th lar qualzato ca b wll dscrbd by a asymptotcal sgal-to-os rato (SR) dgradato, whch rvals that th lar qualzato ca achv th mamum multpath dvrsty wth a opratoal SR rag but wll los dvrsty advatag as SR crass. Kywords ultpath, dvrsty, prcodd, OFD, lar, qualzato Dscpls hyscal Sccs ad athmatcs ublcato Dtals X. uag, "ultpath dvrsty of prcodd OFD wth lar qualzato," IEEE Itratoal Cofrc o Commucatos, 8, pp Ths cofrc papr s avalabl at Rsarch Ol:

3 Ths full tt papr was pr rvwd at th drcto of IEEE Commucatos Socty subjct mattr prts for publcato th ICC 8 procdgs. ultpath Dvrsty of rcodd OFD wth Lar Equalzato Xaojg uag School of Elctrcal, Computr ad Tlcommucatos Egrg Uvrsty of Wollogog Wollogog, Australa Abstract Ths papr sttls a cotrovrsy ovr th multpath dvrsty prformac of th prcodd orthogoal frqucy dvso multplg systm wth th lar qualzato. Through th asymptotcal aalyss of th bt rror rats wth trm systm paramtrs, a comprhsv udrstadg of th lar qualzato s bhavor th frqucy-slctv multpath fadg chals s gad. Compard wth th optmum mamum-lklhood dtcto, th dvrsty prformac of th lar qualzato ca b wll dscrbd by a asymptotcal sgal-to-os rato (SR) dgradato, whch rvals that th lar qualzato ca achv th mamum multpath dvrsty wth a opratoal SR rag but wll los dvrsty advatag as SR crass. Kywords orthogoal frqucy dvso multplg (OFD); multpath dvrsty; lar qulazato. I. ITRODUCTIO Orthogoal frqucy dvso multplg (OFD) provds a ffct mas to mtgat th trsymbol trfrc (ISI) causd by th chal multpath sprad ad joys th smpl frqucy doma chal qualzato va fast Fourr trasform (FFT). owvr, addto to th larg pak-to-avrag powr rato (AR) ad th cssty for complcatd frqucy sychrozato, covtoal OFD systms suffr from a major dsadvatag of poor dvrsty prformac frqucy-slctv fadg chals. Chal codg has b tradtoally usd to mprov th dvrsty across frqucy ad tm, ad rctly lar prcodg ad block spradg ar troducd for OFD systms to ga frqucy dvrsty [-4]. Though thr ar som varatos prformg prcodg, th prcodd OFD systms shar th sam prcpl of applyg a utary matr to a group of data symbols bfor subcarrr mappg. Sc th prcodd data symbol modulatd o a subcarrr s ow a lar combato of th orgal data symbols, f ay subcarrr prcs a dp fad aftr trasmttg ovr a frqucy-slctv multpath chal, th orgal data symbols ca b stll rcovrd from othr rcvd subcarrrs, so that th systm prformac s mprovd du to th crasd dvrsty ordr. Svral authors hav provd, usg a parws rror probablty (E) aalyss, that th prcodd OFD ca achv th mamum dvrsty advatag wth th mamuklhood (L) dtcto f th prcodg data group sz s largr tha or qual to th chal dvrsty ordr [,]. Sc th L dtcto s hghly computatoally complcatd, spcally wh th data group sz s larg, a lar qualzato, such as th mmum ma squard rror (SE) qualzato ad th zro-forcg (ZF) qualzato, followd by a hard dcso s prabl practc. owvr, rgardg th prformac of th prcodd OFD wth lar qualzato, thr s a cotrovrsy foud th ltratur. Tpdlloglu clams [3], basd o th E aalyss, that a prcodd OFD systm ca always achv mamum multpath dvrsty through lar qualzato, whras ccloud cocluds [4], by asymptotcal rror prformac aalyss for oly ZF qualzato, that th lar qualzato dos ot ga ay dvrsty advatag for block sprad OFD (whch s also a prcodd OFD). I ths papr, a dffrt asymptotcal approach s tak to vstgat th dvrsty prformac of th lar qualzato for th prcodd OFD. W frst aalyz th bt rror rat (BER) of th lar qualzato udr two systm paramtrs,.., th data group sz ad th chal dvrsty ordr, assumg a quadratur phas shft kyg (QSK) subcarrr mappg. Th, w drv th asymptotcal prformac by puttg th two paramtrs to dffrt trm codtos. W also dtrm a prformac lowr boud of th L dtcto, so that th prformac of th lar qualzato ca b compard, ad th dvrsty advatag ca b assssd. Aftr dfg a asymptotcal SR dgradato, a comprhsv udrstadg of th lar qualzato s bhavor s gad, ad thus th cotrovrsy ovr th dvrsty prformac of th lar qualzato s sttld. Th rst of th papr s orgazd as follows. I Scto II, th prcodd OFD systm modls ar prstd. I Scto III, th BER of th lar qualzato s formulatd as a fucto of th data group sz ad th multpath dvrsty ordr. Scto IV s dvotd to th drvato of th closdform asymptotcal BER of th lar qualzato. Scto V provds th valuato rsults of th drvd asymptotcal BER ad dfs th asymptotcal SR dgradato. Fally, coclusos ar draw Scto VI. II. SYSTE AD CAEL ODELS Th prcodd OFD trasmttr basbad modl s show Fg. (a), whr a block of put data symbols ( ad ar chos as tgr powrs of for covc) s dotd as a squc [],,,,, or vctor. Ths rsarch s supportd by th Australa Rsarch Coucl Dscovry rojct D /8/$5. 8 IEEE 37

4 Ths full tt papr was pr rvwd at th drcto of IEEE Commucatos Socty subjct mattr prts for publcato th ICC 8 procdgs. Bfor prcodg, [] s frstly dvdd, through sral-toparalll covrso (S/), to groups of sz wth th th group,,,,, bg dotd as a vctor ( [ ], [ + ],, [ + ] ) T whr T [] dots th matr trasposto. S/ C Rmoval or Ovrlap-Add r[] U U U rcodg S/ Itrlavg FFT Y (a) IFFT y y[] /S C or Z r R ˆ ˆ [] (b) D-trlavg R R R Equalzato ad dtcto ˆ ˆ ˆ Fg.. rcodd OFD systm modls: (a) trasmttr ad (b) rcvr. Th prcodg procss s to apply a U, whch satsfs th proprty U U U /S utary matr U I, whr dots th matr trasposto ad compl-cojugato oprato ad I s th dtty matr of ordr, to ach vctor to produc a prcodd vctor U, ad thus ach lmt th prcodd vctor s a lar combato of th symbols vctor. To bttr plot frqucy dvrsty, th prcodd symbols ar prably mappd oto subcarrrs qually spacd across th trasmttd badwdth []. Ths mappg ca b mplmtd by prformg a block trlavg oprato amog th prcodd vctors ad th takg a IFFT of lgth o th trlavd vctor Y. Th rsultg vctor s dotd as y. Aftr paralll-to-sral covrso (/S), y s covrtd to a tm doma squc y [],,,,. To form a prcodd OFD symbol, thr a cyclc p (C) or a zropaddd (Z) suff of suffct lgth (logr tha th mamum chal multpath dlay L sampls) d to b addd to y [] to avod trfrc btw adjact prcodd OFD symbols ad tur th lar covoluto of th trasmttd sgal wth th chal mpuls rspos to a crcular o. Th prcodd OFD sgal s th trasmttd ovr a frqucy-slctv multpath fadg chal dscrbd by th dscrt chal mpuls rspos h [],,,, L, ad rcvd at th rcvr basbad. By rmovg th C or prformg a ovrlap-add oprato, -pot rcvd sampls r [],,,,, ar producd, whch ar rprstd as vctor r aftr S/, ad furthr trasformd to frqucy doma by FFT to yld a vctor R. Aftr dtrlavg, th dscrt-tm rcvd sgal ca b prssd th frqucy doma as R U + V,,,,, whr R ( [] [ ] [ ]) T R, R +,..., R + (3) s a vctor of lmts whch ar dcmatd from R by a dow-samplg factor startg from R [], th th lmt of R, dag( [], [ + ],..., [ ( ) + ] ) (4) s a dagoal matr wth dagoal lmts dcmatd from th chal dscrt frqucy rspos (th -pot dscrt Fourr trasform of h [] ) [] k, k,,,, by a dow-samplg factor startg from [], ad V s a zro-ma Gaussa os vctor wth covarac matr E{ V V } σ VI, whr E {} dots smbl avrag. It s also assumd that th data symbols ar mutually dpdt wth avrag sgal powr σ so that E{ } σ I. Fally, to rcovr ach trasmttd data vctor, qualzato ad dtcto must b prformd o ach rcvd sgal vctor R. Dotg th stmatd data vctors as ˆ, th output data squc ˆ [],,,,, s th T T T obtad from vctor ˆ ( ˆ ˆ ) T aftr /S. ˆ Rgardg th frqucy-slctv multpath fadg chal, w us a ormalzd tappd dlay l modl ad assum a full multpath dvrsty of ordr L. That s, all chal tap coffcts h [],,,, L, ar dpdt ad dtcally dstrbutd (..d.) compl Gaussa radom varabls wth zro-ma ad varac. L III. BERS OVER ULTIAT FADIG CAELS A. BER Lowr Boud of L Dtcto To st up a bchmark for prformac comparso, w frst gv a BER lowr boud usg th L dtcto. Assumg prfct kowldg of th chal at th rcvr, th L stmat of th th data vctor ca b obtad by mmzg th quatty ( R Uˆ )( R Uˆ ),,,,, (5) through haust sarch from all possbl dat vctors ˆ. Followg a wll stablshd procdur, a lowr boud of th avrag BER for th L dtcto ovr frqucy-slctv multpath fadg chals s gv by 38

5 Ths full tt papr was pr rvwd at th drcto of IEEE Commucatos Socty subjct mattr prts for publcato th ICC 8 procdgs. [ + ], L Eh Q l (6) l whr E h{} dots th smbl avrag ovr all chal coffcts h [],,,, L, th Q-fucto s dfd as Q( ) t σ dt, ad s th put SR. To σ π V dscrb th mpact of th data group sz ad th multpath lgth L o th dtcto prformac, w hav dotd th BER lowr boud (6) as a fucto of ad L. B. BER of Lar Equalzato Lar qualzato s prabl practc sc t ca smply us a o-tap qualzr for ach subcarrr th frqucy doma. Lt C [] k dot th o-tap qualzr coffct to b appld to R [] k o subcarrr k ad C dag( C[], C[ + ],..., C[ ( ) + ] ) (7) dot a dagoal matr wth dagoal lmts C [ l + ], l,,,. Th lar qualzato ad dtcto procss ca b dscrbd as follows. Frst, applyg C to R producs a qualzd vctor C R. Scod, usg U to rmov prcodg ylds th dcso varabl vctor d U CR. Fally, a stmat of th trasmttd data vctor s obtad aftr hard dcso o d. Wh th SE crtro s usd,.., dsgg C so that th ma squard rror (SE) btw d ad ε E{ ( d )( d )} (8) s mmzd, th dagoal lmt C s foud to b [ ] [ l + ] C l + (9) [ l + ] + ad th avrag BER ca b valuatd as whr mms, L E Q h l [ l + ] + s th ormalzd mmum SE [5]. As s from (9), th SE qualzato rqurs th kowldg of th put SR. If ths kowldg s ot avalabl, th ZF qualzato ca b prformd by assumg o os s prst at th rcvr ad slctg th qualzr coffcts C[ l + ] to forc th ISI to zro. [ l + ] owvr, ths ZF qualzato hacs os powr wh th put SR s low ad causs dvdg-by-zro problm at ull subcarrrs. I practc, w ca dsg th lar qualzr coffcts accordg to a prdtrmd or stmatd rc SR as l + [ + ] C l [ ] [ + ] + l. Appartly, wh ths lar qualzato bcoms th SE qualzato, ad as t bcoms th ZF qualzato. Usg th qualzr coffcts dfd, th avrag BER for th lar qualzato ca b drvd as ( ), L Eh Q, L (3) whr, L [ ( )] [ ] ( [ ) ] + (4) s th output SR th dcso varabl ad. (5) l l + + ( [ ] ) IV. ASYTOTICAL BERS To show th rlatoshp btw th systm prformac ad th group sz as wll as th rlatoshp btw th systm prformac ad th chal dvrsty ordr, lt s work out two sts of asymptotcal BERs for th lar qualzato by asymptotcal aalyss. r, th word asymptotcal rs to th codtos wh som systm paramtrs tak o trm valus. Th frst st rprsts th BERs for dffrt block sz wh L. Th scod st dcats th BERs for a gv umbr of chal multpath L (rrd to as multpath dvrsty ordr) wth suffctly larg data block sz,..,. Wh both ad L approach fty, th asymptotcal BER wll dmostrat th bst dvrsty prformac that th lar qualzato could achv. A. Asymptotcal BER lowr bouds for L Dtcto For comparso purpos, w frst work out th asymptotcal BER lowr bouds for th L dtcto. To dtrm th asymptotcal BER lowr bouds udr codto L, w assum that th chal provds full multpath dvrsty,.., π j ( l + ) L. Sc [ l + ] h[], w hav { [ l + ] [ l ] } j E{ h[] h [ ]} E + π π ( l + ) j ( l + ) π j ( l l ), l l Ths mas that [ ]. (6), othrws l + bcoms a dpdt compl 39

6 Ths full tt papr was pr rvwd at th drcto of IEEE Commucatos Socty subjct mattr prts for publcato th ICC 8 procdgs. Gaussa varabl wth ut varac wh th chal has full dvrsty. Th, th smbl avrag (6) ca b prformd o [ l + ] drctly stad of h []. Dotg [ l + ] as l for covc, th asymptotcal BER lowr boud ca b valuatd as,, E Q l l E Q l (7) l whch s th BER wth dvrsty ordr (ot that th subscrpt l s gord sc th smbl avrags E Q l ar th sam for all ). Thus, t l s provd that gv a group sz th L dtcto ca oly achv dvrsty ordr v f th chal has ulmtd dvrsty. Th scod st of asymptotcal BER lowr bouds dcats th bst prformac for a gv multpath dvrsty ordr L but wth suffctly larg data block sz. I ths cas, w L hav [ ] π j l+ l + h[], ad thus l L L L [ l + ] h[] h[] h [ ] L [] l π ( + ) L π j l j ( l + ) [] h π ( ) π j j l ( ) l h, for L. (8) As, w hav (, L) (, ) L Eh Q h L L Eh Q h (9) whch s th BER wth dvrsty ordr L. Wh both ad h L l ad L, w hav l. Thor, from thr (7) or (9) th asymptotcal BER lowr boud of th L dtcto bcoms ( ) Q( ), whch s bst prformac a prcodd OFD systm could achv. W r to t as th mamum dvrsty prformac. It s th sam as th BER Gaussa chal wthout fadg. B. Asymptotcal BERs for Lar Equalzato Followg th sam procdur as that w usd for drvg (7), th frst st of th asymptotcal BERs (.., L ) ca b prssd as (, ) E { Q( (, ) )} whr (, ) has th sam prsso as (4) but th subscrpt s gord ad l l + ( ) l + ( l ). Th scod st of th asymptotcal BERs s rachd as udr a gv multpath dvrsty ordr L, ad ca b prssd as (, L) E h { Q( (, L) )} whr (, L) has th sam prsso as (4) but ( ) ( ) ( ) π π π ω ( ) j dω + (3) ω π d (4) ω ( ) j + jω whr ( ) s th Fourr trasform of h []. Wh both ad L, a closd-form prsso of th bst prformac that th lar qualzato could offr ca b drvd as (, ) Q ( (, ) ), whr (, ) has th sam prsso as (4) but ρ ( ) d ρ + ρ (5) ( ) ( ) ρ dρ (6) ( ρ + ) whch ca b drvd from ad rspctvly. For ampl, to drv (5), w otc that l s chsquar-dstrbutd wth two dgrs of frdom ad probablty dstrbuto fucto (pdf) ρ. As, th avrag ovr l ca b rplacd by smbl l l + ρ avrag d ρ + ρ. V. ASYTOTICAL BER AD SR DEGRADATIO Comparg th mamum dvrsty prformac of th L dtcto wth th asymptotcal BERs of th lar qualzato drvd abov, w ar ow abl to ga a full udrstadg of th dvrsty prformac of th lar qualzato. Fg. shows th (, ) curvs as fuctos of E b, by rplacg th put SR E b for QSK, whr E b Dfg a spcal fucto ( E ) ( ) altratvly valuatd by ( ) t t dt, (5) ad (6) ca b E for ad. 3

7 Ths full tt papr was pr rvwd at th drcto of IEEE Commucatos Socty subjct mattr prts for publcato th ICC 8 procdgs. s th sgal rgy pr bt ad s th os powr spctral dsty. Th rc ormalzd SR s st to ( E mms,,, 3, 4 db rspctvly. (, ) ad (, ) b ), whch s th prformac lowr boud of th lar qualzato, ar also dsplayd for comparso purpos. lowr tha th rc SR th dgradato s almost a costat, whch mpls th sam slop o th BER curv. Oc th SR gos byod th opratoal rag, th dgradato crass rapdly, whch dcats th loss of dvrsty. As th rc SR crass, th opratoal rag bcoms wdr but th SR dgradato crass too. 5 BER (E b / ) db db db 3 db 4 db SR dgradato (db) 5 (E b / ) db db db 3 db 4 db E / (db) b Fg.. Asymptotcal BERs of prcodd OFD wth lar qualzato udr dffrt rc ormalzd SRs (sold ls). Th asymptotcal BERs for L dtcto (dotd l) ad SE qualzato (dashd l) ar also dsplayd. As w kow, th dvrsty ordr dscrbs how fast th BER dcrass as th SR crass,.., t s rlatd to th slop of th BER curv. From Fg. w s that a BER curv of th lar qualzato for a gv rc SR has smlar slop to that of th mamum dvrsty BER curv for SRs blow th rc SR. As th SR crass byod th rc SR, th BER curv gradually gos flat. Ths obsrvato dcats that th lar qualzato achvs th sam dvrsty ordr as th L dtcto for SRs lowr tha th rc SR (though th BER tslf s wors tha that of th L dtcto) but th dvrsty s lost wh th SR s hghr tha th rc SR. W r to th SRs blow th rc SR as th opratoal SR rag. Furthrmor, w otc that th asymptotcal BERs (, ) ad (, ) ad (, ) ar valuatd by th sam Q-fucto but wth SR rspctvly. Thus th prformac dgradato ca b mor ffctly dscrbd by th SR dgradato. W call ths SR dgradato as th asymptotcal SR dgradato, whch s dfd as (, ) D log [ ] ( + [ ) ]( ) [ ] log. Fg. 3 shows th asymptotcal SR dgradato as a fucto of th ormalzd SR E b, E b wth,, 3, 4 db rspctvly. W s that wh th SR s E / (db) b Fg. 3. Asymptotcal SR dgradatos for lar qualzato udr dffrt rc ormalzd SRs (sold ls). Th asymptotcal SR dgradato for SE qualzato (dashd l) rprsts a lowr boud. VI. COCLUSIOS W hav show that a practcal lar qualzato achvs th mamum multpath dvrsty for th SRs wth a opratoal rag whch s dtrmd by th rc SR usd to dsg th lar qualzato coffcts. owvr, oc th SR gos across th opratoal thrshold (.., th rc SR), th dvrsty advatag s lost. If th SR s kow at th rcvr, th SE qualzato wll offr th lowr boud prformac. Th drvd closd-form asymptotcal BER ad SR dgradato prssos ot oly provd a ffct way to valuat th pottal prformac that th lar qualzato could achv but also srv as a usful tool for optmal rcvr dsg. REFERECES [] Z. Wag ad G. B. Gaaks, Larly prcodd or codd OFD agast wrlss chal fads?, rocdgs of Sgal rocssg Advacs Wrlss Commucatos Workshop, Taoyua, Tawa, arch 3,, pp [] Z. Lu, Y. X, ad G. B. Gaaks, Lar costllato prcodg for OFD wth mamum multpath dvrsty ad codg gas, IEEE Trasactos o Commucatos, Vol. 5, o. 3, arch 3, pp [3] C. Tpdlloglu, amum multpath dvrsty wth lar qualzato prcodd OFD systms, IEEE Trasactos o Iformato Thory, Vol. 5, o., auary 4, pp [4]. L. ccloud, Aalyss ad dsg of short block OFD spradg matrcs for us o multpath fadg chals, IEEE Trasactos o Commucatos, Vol. 53, o. 4, Aprl 5, pp [5] X. uag, Dvrsty prformac of prcodd OFD wth SE qualzato, prstd at th 7 Itratoal Symposum o Commucatos ad Iformato Tchologs (ISCIT7), Sydy, Australa, 6-9 Octobr, 7. 3