A Compact Representation of Spatial Correlation in MIMO Radio Channels

Transcription

1 A Compac epesenaon of Spaal Coelaon n MIMO ado Channels A. van Zels Endhoven Unves of echnolog P.O. Box MB Endhoven he Nehelands e-mal: A.v.Zels@ue.nl and Agee Ssems P.O. Box A Neuwegen he Nehelands e-mal: Zels@agee.com Absac Spaal fadng coelaon s a cucal mpamen fo paccal Mulple-Inpu Mulple-Oupu (MIMO) weless communcaon ssems. ence n ssem smulaons spaal coelaon should be aen no accoun. he man dsadvanage howeve s ha n geneal s epesened b a lage numbe of paamees namel he vaous coelaon max enes. In hs pape we noduce a compac epesenaon of he spaal coelaon havng a mos wo coeffcens whch neveheless esuls n exacl he same capac and B Eo ae (BE) pefomance. Moeove hs exac mappng allows one o pefom MIMO ssem smulaons wh spaal coelaon whle s no equed o explcl specf he anenna aa desgn and popagaon envonmen o nclude coelaon. Kewods MIMO ssems; spaal fadng coelaon. I. INODUCION In weless communcaons MIMO echnques have ecenl emeged as a new paadgm o acheve ve hgh bandwdh effcences []. he concep s based on usng mulple ansm and mulple eceve anennas along wh pope MIMO encodng and deecon algohms ([ ]). he specal effcenc ha can be exploed depends songl on he muldmensonal sascal behavo of he MIMO fadng channel pal chaacezed b he spaal fadng coelaon. Spaal coelaon n he MIMO conex has aaced a lo of aenon n leaue [3-9]. o ou nowledge up o hs momen when MIMO ssem smulaons ae pefomed he spaal coelaon s ncluded explcl b means of measued coelaon maces o based on a acng. hs appoach s cumbesome and has as majo dsadvanage ha he MIMO channel sascs ae epesened b a lage numbe of paamees namel he vaous coelaon max enes and heefoe s had o cove a wde ange of bes-case o woscase scenaos. In hs pape we noduce a compac mappng of he spaal coelaon o a mos wo coeffcens whch neveheless esuls n exacl he same capac and BE pefomance. hs smplfed coelaon model can be used n naowband smulaons decl o n wdeband conexs epesenng he MIMO fadng sub-pocesses pe dela ap. II. MIMO SINAL MODEL Consde a MIMO ssem wh N ansm (X) and N eceve (X) anennas. he ansme ems an N complex sgnal veco s. he eceve ecods an N complex veco x x s + n () whee s an N N complex popagaon max. Elemen (qp) of conans he fla-fadng channel coeffcen fom X anenna p o X anenna q wh vaance σ c. he veco n epesens zeo mean complex Addve Whe aussan Nose (AWN) wh covaance max E[nn ] σ n I whee (.) denoes he conjugae anspose of he coespondng veco o max and I epesens he den max (hee N N ). he oal ansm powe s E[s s] N σ s and se o P. he SN pe X anenna equals ρ N σ s /σ n. III. SPAIAL COELAION MODEL Fs we wll descbe he geneal spaal coelaon defnons and hen we wll deve a compac mappng of he spaal coelaon havng a mos wo coeffcens. In [3] he spaal fadng coelaon fo a naowband flafadng MIMO channel s defned as ( ) vec( ) ] E[vec () whee vec() denoes he N N veco composed b sacng he columns of. In chl-scaeed envonmens he spaal coelaon beween he ansm anennas ( X ) can be assumed o be ndependen fom he coelaon beween he eceve anennas ( X ) [6] heefoe can be wen as [3-6] X X (3) wh denong he Konece poduc. Fuhemoe (.) sands fo he anspose of he coespondng max and X and X ae defned as q q ( h ) ] fo all q N E [ h... (4) X hs wo s sponsoed b Agee Ssems he Nehelands and he Duch coopeave eseach pojec B4 Boadbandado@and BS0063.

2 X [ h ph p ] fo all p E... N (5) whee h q s he q-h ow of and h p s he p-h column of. o geneae ndependen naowband fla-fadng MIMO channel ealzaons wh spaal coelaon he followng expesson can be used ([4]) unvec g (6) whee g s an N N sochasc veco wh..d. zeo-mean un vaance complex aussan elemens and unvec(.) s he evese of he vec(.) opeaon. B usng some specal popees of and a Konece poduc den we can we (6) n a moe commonl used fom. Noe ha s eman and nonnegave defne. ence we ma we U U U U U fom whch we oban he "squae-oo" of. Such decomposons also hold fo X and X so fom (3) follows ha X X X X X X X U X X X (7) (8) whee unvec(g) s a sochasc N N max wh..d. complex aussan zeo-mean un vaance elemens. hs esul s equvalen o he coelaon model noduced n [7]. When ssem smulaons need o be caed ou one wa o poceed s o explcl sae specfc coelaon maces X and X coveng vaous popagaon scenaos. o oban hese specfc coelaon maces ehe a acng o coelaon measuemens have o be pefomed epesenng dffeen scenaos. hs appoach s cumbesome and has as majo dsadvanage ha he MIMO fadng coelaon sascs ae epesened b a lage numbe of paamees namel he vaous coelaon max enes and heefoe s had o cove a wde ange of bes-case o wos-case scenaos. hs leads o he queson how o educe hs amoun of paamees. We sa wh he obsevaon ha he capac and BE pefomance ae fequenl used measues o evaluae MIMO ssems. In he nex secons we noduce a compac epesenaon of he spaal coelaon ha neveheless esuls n an equvalen capac and BE pefomance. IV. MAPPIN OF E SPAIAL COELAION WI ESPEC O CAPACIY he capac of an N N naowband MIMO channel s gven b ([]) ρ C log de I N + bs/s/z. (0) N When spaal coelaon s pesen he capac equals C log de I N ρ + N X X X X. () Wh he equal de(i + AB) de(i + BA) hs can be ewen o whee denoes he pon-wse conjugaon. Fuhemoe he pope s used ha fo an max A B C and D wh pope dmensons (AB) (CD) (A C)(B D). Based on he Konece poduc den ha fo an (complex) M N max A N P max B and P Q max C vec(abc) (C A)vec(B) (6) can be ewen as ρ C log de I N + X X N Fo hgh SNs we ge bs/s/z. () unvec unvec X g X X X vec ( ) (9) C log log ρ de N ρ de I N N X de X ( ) de( ) de( ) de( ) X X (3) snce he deemnan of a poduc s he poduc of he deemnans. So appaenl he capac dsbuons of wo dffeen suaons wll be he same when he deemnan of

3 he X 's and X 's ae equal. O mus be possble o noduce a model ha s n a capac sense a mappng of measued coelaon maces. o ha end we eque ( ) de( ) de (4) X mod Xmeas ( ) de( ) de. (5) X mod Xmeas Noe ha n case he coelaon maces of a possble model would be se equal on boh sdes of he communcaon ln.e. Xmod Xmod mod we would ge he ceon ( ) de( ) de( ) de. (6) mod Xmeas Xmeas Now he queson s f hee exss a unque soluon fo he equemens (4) and (5). o answe ha queson noe ha le boh X and X ae nonnegave defne. Accodng o adamad's nequal fo an N N nonnegave defne max A ([0]) N a ( ) de A (7) whee a epesens he -h dagonal elemen of A. Fo he coelaon maces X and X hs means ha de( X ) and de( X ). Fuhemoe snce he deemnan of a max s he poduc of he egenvalues and snce he egenvalues of a nonnegave defne max ae eal and nonnegave hs elds de( X ) 0 and de( X ) 0. So he deemnan of he (measued) coelaon maces wll alwas be eal lage han o equal o zeo and less han o equal o one. Nex s shown ha a unque mach can be found wh espec o capac usng he followng smple and genec defnons fo he ansme and eceve coelaon : X N X X X X Xmod X X X (8) X N X X X X N X X X X Xmod X X X (9) X N X X X Noe ha a smla model has been noduced n [9] wh he dffeence ha n [9] he coelaon s defned as E[ ]. whee X and X epesen (eal-valued) coelaon coeffcens. he mos poweful pope of hs model s ha when angng he coeffcens beween 0.0 and.0 we can n a conolled wa go fom full uncoelaed scenaos (all offdagonal elemens of boh maces equal o 0.0) o full coelaed scenaos (all enes equal o.0). Anohe useful pope s he smple fom of he deemnans of hese maces. he deemnan of e.g. Xmod can be shown o be N ( ) ( ) de. (0) X mod Fnall can be shown ha he deemnan fo he modeled maces s monooncall deceasng n he ange of nees e.g. Xmod as funcon of X s monooncall deceasng fo 0 X (see Fgue fo N s 3 and 4). Based on hese obsevaons can be concluded ha hee wll alwas be a unque mappng ha sasfes he cea (4) and (5). D eemnan of Xm od N 4 N C oelaon coeffcen X X N Fgue : he deemnan of he coelaon model max Xmod vesus he coelaon coeffcen X fo a vaous numbe of X anennas. Snce a mahemacal ln s found o mach he MIMO capac of measued coelaon maces wh ha of he model we can suffce wh one example. he esul s pesened fo complex coelaon maces measued n a pcocell envonmen ([4]) and gven b () and (). Fo hese measued maces can be shown ha de( Xmeas ) 0.37 and de( Xmeas ) especvel. Fom he cea (4) (5) 0 X and 0 X we oban X 0.67 and X Wh hese esuls he capac of he measued coelaon maces can be compaed wh ha of he model. Noe ha fo eve ealzaon of () poduces a dffeen nsananeous capac value. he aveage of hese capac values.e. he egodc capac as funcon of he aveage SN pe eceve anenna s shown n Fgue fo he measued and modeled coelaon maces. Fom hese cuves we ndeed see ha he mach s pefec even fo low SN values.

4 Xmeas () Xmeas () Egodc capac (bs/s/z) C ap. of measued coelaon C ap. of modelled coelaon SN pe X anenna (db) Fgue : Egodc capac vesus SN pe X anenna fo measued and modelled spaal coelaon fo a 4 4 ssem. Obvousl he noduced spaal coelaon model ma no be an accuae model fo some eal-wold scenaos bu s a smple dual-coeffcen model ha allows one o sud he effec of coelaon on he MIMO capac n an explc wa. Moeove wh he cea (4) and (5) we found a smple mappng wh measued coelaon maces. In he nex secon a compac epesenaon of he spaal coelaon n BE pefomance evaluaons s obaned. V. MAPPIN OF E SPAIAL COELAION WI ESPEC O E BE PEFOMANCE In hs pape Maxmum Lelhood Deecon (MLD) [] s seleced as MIMO deecon scheme o fnd a compac epesenaon of he spaal coelaon n BE pefomance evaluaons. o ha end we wll use he Pawse Eo Pobabl (PEP) as a pefomance measue. Le s and s be wo possble spaal X vecos wh dmensons N and assume ha s s ansmed. hen wh (s' s' ) whee s' and s' ae he nomalzed vesons of s and s especvel such ha s' s /σ s and s' s /σ s and usng he same appoach as n [] he PEP of MLD can be shown o be [] P ( s s ) dei N s n σ + 4σ Q (3) whee Q s he covaance max of. Fo a hgh SN he PEP can be appoxmaed b P ( s s ) ρ de 4N Q. (4) ence n he asmpoc case he PEP (and hus he BE pefomance) depends nvesel on he deemnan of Q. Now he queson ases: wha s Q n scenaos wh spaal coelaon? o fnd he answe we sa b ewng : ( ' ) ) ( I ) vec( ) s. (5) Noe ha when s' and s' have a mean of zeo s also zeo mean. Now can be shown ha b aveagng ove he covaance max of equals Q E[ ] ) I N ) ) I ) N ) I N )( X X ) ) ) ) β. X X N ( I ) X N (6) Fom hs esul we can obseve ha n ode o have he same PEP fo he modeled and measued X he deemnans of boh maces mus be he same; de( Xmod ) de( Xmeas ). And b usng (9) we can deduce an X ha acheves an equvalen MLD pefomance compaed o he pefomance wh he measued spaal eceve coelaon Xmeas. egadng he spaal coelaon a he ansme s obvous ha β songl depends on (s s ). heefoe o fnd a ln beween Xmod and Xmeas one has o aveage ove all possble dffeence vecos (s s ) whch s equvalen o usng he oveall eo ae pefomance. An uppebound on he oveall eo ae pefomance can be found b aveagng ove

5 all PEP's b means of e.g. he unon bound. Snce he s 's ae aen fom a dscee se ha depends on he consellaon sze he eases and mos effecve wa o fnd a ln s hough numecal evaluaon. Because we found a (numecal) mahemacal mappng beween he MLD eo ae pefomance fo measued and modeled spaal coelaon maces one example showng he mach s suffcen. o ha end we wll agan use he measued spaal coelaon maces as gven b () and (). Cleal he machng ceon of he measued and modeled spaal coelaon a he eceve sde fo he MLD eo ae pefomance s equvalen o (5). So o ln (9) wh () X mus be se o Fuhemoe fom numecal evaluaon we found ha X mus be se o Fnall he mach s shown gaphcall n Fgue 3 n whch a pefec mach of he uppebounds can be obseved. he slgh msmach beween he smulaon cuves a hgh SN can be explaned manl b he lmed accuac. he cuves ae namel obaned b aveagng ove be paces. BE MLD BE of measued co. MLD BE of modelled co. MLD uppebound of meas. co. MLD uppebound of mod. co Aveage SN pe X anenna (db) Fgue 3: MLD BE pefomance and uppebound vesus aveage SN pe X anenna fo measued and modelled spaal coelaon fo a 4 4 ssem. VI. COELAION DELAY POFILE he spaal coelaon model descbed n hs pape s a naowband model. he maxmum of wo paamees howeve allow us o easl exend hs o a wdeband channel model. In geneal he coelaon changes ove he me neval of he wdeband channel mpulse esponse. hese vaaons can be capued n wha s efeed o as he Coelaon Dela Pofle (CDP). One can magne ha n eal-wold envonmens he fs channel aps ae manl deemned b a few song pahs e.g. he LOS pah and some domnan eflecons wheeas owads he las aps of he channel mpulse esponse he angula spead s moe omn-deconal wh man (equall song) conbung pahs. hs esuls n a hgh coelaon coeffcen a he begnnng of he CDP and low values a he end. VII. CONCLUSIONS We have noduced a smple epesenaon of spaal coelaon n MIMO ado channels. Fo he fequenl used evaluaon measues of a MIMO ssem namel capac and BE pefomance he amoun of paamees epesenng he spaal coelaon can be educed o a mos wo. Wh a pope choce of hese coeffcens he coelaon can be vaed conollabl fom he oall uncoelaed scenao o he full coelaed scenao. hs smplfed coelaon model allows one o pefom smulaons wh spaal coelaon whle s no equed o explcl specf he hadwae (e.g. anenna) seup and wave popagaon envonmen o nclude he spaal coelaon. Alogehe hs maes he model poweful e smple o use. ACKNOWLEDEMEN he auho would le o han J.S. ammeschmd fo he valuable dscussons. EFEENCES []. J. Foschn and M. J. ans On lms of weless communcaons n a fadng envonmen when usng mulple anennas Weless Pesonal Communcaons vol. 6 no. 3 Mach 998 pp [] A. van Zels "Space dvson mulplexng algohms" n Poc. of he 0h Medeanean Elecoechncal Conf. (MELECON) 000 vol. 3 Ma 000 pp. 8-. [3] Da-Shan Shu. J. Foschn M. J. ans and J. M. Kahn "Fadng coelaon and s effec on he capac of mulelemen anenna ssems" IEEE ansacons on Communcaons vol. 48 no. 3 Mach 000 pp [4] J. P. Kemoal L. Schumache K. I. Pedesen P. E. Mogensen and F. Fedesen "A sochasc MIMO ado channel model wh expemenal valdaon" IEEE Jounal on Seleced Aeas n Communcaons vol. 0 no. 6 Aug. 00 pp. -6. [5] L. Schumache e al "MIMO channel chaacesaon" Mea Delveable D IS /AAU-WP-D-V..doc Feb hp:// [6] Ka Yu M. Bengsson B. Oesen D. McNamaa P. Kalsson and M. Beach "Second ode sascs of NLOS ndoo MIMO channels based on 5. z measuemens" n Poc. of he IEEE lobal elecommuncaons Conf. (LOBECOM) 00 vol. pp [7] D. Chzh F. ashd-faoh J. Lng and A. Lozano "Effec of anenna sepaaon on he capac of BLAS n coelaed channels" IEEE Communcaons Lees vol. 4 no. Nov. 000 pp [8]. D. Dugn and. S. appapo "Effecs of mulpah angula spead on he spaal coss-coelaon of eceved volage envelopes" n Poc. of he 49h IEEE Vehcula echnolog Conf. (VC) 999 vol. pp [9] S. L. Loa "Channel capac of MIMO achecue usng he exponenal coelaon max" IEEE Communcaons Lees vol. 5 no. 9 Sep. 00 pp [0] on.a. and C.. Johnson Max Analss Cambdge Cambdge Unves Pess 985. [] V. aoh N. Seshad and A.. Caldeban "Space-me codes fo hgh daa ae weless communcaon: pefomance ceon and code consucon" IEEE ansacons on Infomaon heo vol. 44 no. 3 Mach 998 pp [] A. van Zels "MIMO OFDM fo weless LANs" Ph.D. dsseaon Depamen of Eleccal Engneeng Endhoven Unves of echnolog Apl 004.