CAPACITY ANALYSIS OF ASYMPTOTICALLY LARGE MIMO CHANNELS. Georgy Levin

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1 CAPACITY ANALYSIS OF ASYMPTOTICALLY LAGE MIMO CANNELS by Geogy Levi The hesis submied o he Faculy of Gaduae ad Posdocoal Sudies i paial fulfillme of he equiemes fo he degee of DOCTO OF PILOSOPY i Elecical ad Compue Egieeig Oawa-Caleo Isiue fo Elecical ad Compue Egieeig, School of Ifomaio Techology ad Egieeig, Uivesiy of Oawa, Oawa, Oaio, Caada Copyigh Geogy Levi, Oawa, Caada, 8

2 Capaciy Aalysis of Asympoically Lage MIMO Chaels I he mouais he shoes oue is fom pea o pea, bu fo ha you mus have log legs. Fiedich Niezsche, Thus spoe Zaahusa. ii

3 Capaciy Aalysis of Asympoically Lage MIMO Chaels ABSTACT Muliple-Ipu-Muliple-Oupu (MIMO) wieless commuicaio sysems have aaced a eomous iees i boh academy ad idusy due o he poeial o povide emaable specal efficiecy by aig advaage of mulipah fadig isead of combaig i. Aalysis of classic chael models ude basic assumpios has demosaed possibiliy o scale up liealy he ifomaio ae by deployig muliple asmiig ad eceivig aeas wihi he same fequecy badwidh. ece wos show ha he achievable pefomace of pacical MIMO sysems depeds heavily o he udelyig fadig disibuio ad sysem cofiguaio. The specal efficiecy may seveely degade due o he high coelaio bewee mulipah compoes, chael a deficiecy o powe imbalace. The objecive of his hesis is o sudy he capaciy of coelaed, a-deficie ad full-a MIMO chaels. While i may cases he exac capaciy expessios ae complicaed ad do o allow fo sigifica isigh, a asympoic appoximaio fo a lage umbe of aeas is used o obai simple ad well acable esuls fo a boad class of MIMO chaels. Saig fom he sigle eyhole chael as a basic ad simple model of a a-oe MIMO chael, he aalysis is exeded o a family of highe-a chaels (icludig also he caoical full-a ayleigh-fadig oe) via a asiio model which icludes a umbe of saisically idepede eyholes (muli-eyhole chaels). I is show ha ude ceai mild codiios o coelaio, he ouage capaciy disibuio of he sigle eyhole ad muli-eyhole chaels is asympoically Gaussia. The geeal codiios ad popagaio-based implicaios of he covegece ae sudied. I some cases, he asympoic ouage capaciy disibuio follows closely he exac oe fo a easoably small umbe of asmi ad eceive aeas. A umbe of applicaios of he asympoic heoy ae discussed. (i) A ew scala measue of coelaio ad powe imbalace is ioduced o quaify he impac of coelaio o he capaciy ad evaluae effecive degees of feedom i MIMO chaels. The measue is simple, well acable ad fullodeig (ay wo chaels ca be compaed). (ii) Fiie SN size-asympoic divesiy-muliplexig ade-off (DMT) is aalyzed fo he muli-eyhole chaels. Ulie he SN-asympoic DMT of Zheg ad Tse, he size-asympoic oe accuaely epeses he effec of coelaio ad powe imbalace o iii

4 Capaciy Aalysis of Asympoically Lage MIMO Chaels he capaciy. (iii) Telaa s cojecue is pove fo muli-eyhole chaels wih a lage umbe of aeas. (iv) The bes mulipah agula desiy, which maximizes he asympoic capaciy of a boad class of MIMO chaels wih liea uifom aea aays, is deived. The desiy is o-uifom, which implies ha he popula Clae s (Jaes) model does o epese he bes case sceaio. Usig he igoous mehods of hypohesis esig, i is demosaed ha he ouage capaciy disibuio of some measued 5.Gz idoo MIMO chaels is saisically Gaussia wih a easoable sigificace level aleady fo wo aeas a each ed. The lae ca seve as a empiical validaio of he obaied heoeical esuls ad implies ha he asympoic aalysis wih espec o he umbe of aeas o oly offes a sigifica isigh ad simplificaio, bu also ca be applied o ealisic sysems of a modeae size. iv

5 Capaciy Aalysis of Asympoically Lage MIMO Chaels ACKNOWLEDGEMENTS I wish o expess my deepes gaiude fo he guidace, suppo ad cosucive ciicism o my hesis supeviso D. Segey Loya, whose elega way of appoachig poblems has cosideably iflueced my hiig. I am idebed o D. Pee Galo fom he Uivesiy of Oawa fo may helpful suggesios, D. Vidmaas Beus fom he Vilius Uivesiy fo valuable commes o Ceal Limi Theoems, D. Es Boe ad his eam a he Viea Uivesiy of Techology fo idly allowig me o use hei measuemes of MIMO chaels. I ha my paes, paes i law, my sise ad fieds fo hei suppo ad ecouageme duig he couse of his hesis. Fially, I ha my adoable childe Polia ad Michael ad my dea wife Taia, who sood by me duig fusaig ad had imes, ad wihou whom his wo would o have bee compleed. v

6 Capaciy Aalysis of Asympoically Lage MIMO Chaels CONTENTS Absac... iii Acowledgemes...v Coes... vi Lis of Acoyms... ix Lis of Symbols... xi Chape I: Ioducio..... Muli-Aea Wieless Commuicaios..... Moivaio ad Pio Wo Thesis Ogaizaio ad Coibuio Publicaios...3 Chape II: Lieaue eview MIMO Chael Capaciy Coelaio Sucue of MIMO Chaels Capaciy of ayleigh-fadig MIMO Chaels Ucoelaed ayleigh-fadig Chaels Semicoelaed ayleigh-fadig Chaels Double Coelaed ayleigh-fadig Chaels Asympoic Capaciy Disibuio of MIMO Chaels Ouage Capaciy Disibuio of Ucoelaed ayleigh-fadig MIMO Chaels Ouage Capaciy Disibuio of Coelaed ayleigh-fadig MIMO Chaels Ouage Capaciy Disibuio of Geeic MIMO Chaels Keyhole MIMO Chaels...3 Chape III: Capaciy of Keyhole MIMO Chaels Exac Ouage Capaciy Disibuio...36 vi

7 Capaciy Aalysis of Asympoically Lage MIMO Chaels 3.. Asympoic Ouage Capaciy Disibuios Bouds o Mea Capaciy Summay...53 Chape IV: Capaciy of Muli-Keyhole Chaels Mahemaical Model Full-a Muli-Keyhole Chaels a-deficie Muli-Keyhole Chaels Summay...64 Chape V: Measue of Coelaio ad Powe Imbalace Basic Popeies Impac o Asympoic Ouage Capaciy Disibuio Numbe of Effecive Degees of Feedom via he Measue of Coelaio ad Powe Imbalace Summay...73 Chape VI: Asympoic Nomaliy of ayleigh Chael Capaciy Geealized Lyapouov-Type Covegece Codiio Accuacy of Gaussia Appoximaio Covegece Codiio fo Chaels wih Toepliz Coelaio Sucue Covegece fo Some Popula Coelaio Models Summay: O Pacical Uiliy of Gaussia Appoximaio...8 Chape VII: Wha is he Bes Agula Desiy of Mulipah i MIMO Chaels? Asympoic Capaciy of MIMO Chaels Bes Agula Desiy i -D Space Bes Agula Desiy i 3-D Space Summay...94 Chape VIII: Applicaios of Asympoic Aalysis Fiie SN Divesiy-Muliplexig Tadeoff...95 vii

8 Capaciy Aalysis of Asympoically Lage MIMO Chaels 8.. Ouage Capaciy ad Bloc Eo ae Telaa s Cojecue fo Lage MIMO Chaels Moivaio fo Koece Coelaio Model Schedulig Gai ad Feedbac ae i Muliuse Eviome Summay...4 Chape IX: Saisical Aalysis of Measued MIMO Chaels Ioducio o Saisical Aalysis Saisical Aalysis of ayleigh-fadig Chaels Saisical Aalysis of he Measued Chael Summay...5 Coclusio ad Fuhe eseach Coclusio Fuhe eseach...7 Appedix A: Keyhole Chaels... Appedix B: Muli-Keyhole Chaels...9 Appedix C: Measue of Coelaio ad Powe Imbalace...37 Appedix D: Asympoic Nomaliy...38 Appedix E: Bes Agula Desiy...4 Bibliogaphy...44 viii

9 Capaciy Aalysis of Asympoically Lage MIMO Chaels LIST OF ACONYMS AOA: AWGN: BE: BLE: BS: CDF: CF: CSI: FMK: IND: LS: LOS: MC: MIMO: MISO: MC: MU: PDF: PSK: QAM: DMK: S: x: Agle of Aival Addiive Whie Gaussia Noise Bi Eo ae Bloc Eo ae Base Saio Cumulaive Disibuio Fucio Chaaceisic Fucio Chael Sae Ifomaio Full a Muli-Keyhole Idepede Lef ad Side Lie of Sigh Moe-Calo Muliple Ipu Muliple Oupu Muliple Ipu Sigle Oupu Maximum aio Combiig Mobile Use Pobabiliy Desiy Fucio Phase Shif Keyig Quadaue Ampliude Modulaio a Deficie Muli-Keyhole igh ad Side eceive ix

10 Capaciy Aalysis of Asympoically Lage MIMO Chaels SE: SIMO: SISO: SN: STBC: Tx: UIU: ULA: Symbol Eo ae Sigle Ipu Muliple Oupu Sigle Ipu Sigle Oupu Sigal o Noise aio Space Time Bloc Codig Tasmie Uiay Idepede Uiay Uifom Liea Aay x

11 Capaciy Aalysis of Asympoically Lage MIMO Chaels LIST OF SYMBOLS Lai: A : A : a : b : C : C : C : C ɶ : C : Diagoal maix of eyhole complex gais Coefficies of paial facio decomposiio Complex gai of h eyhole Numbe of bis pe feedbac chael use Covaiace maix Isaaeous capaciy Isaaeous capaciy pe x aea Nomalized isaaeous capaciy Mea (egodic) capaciy C max : Maximal capaciy pe x aea D d : d( ) : Disace bewee a pai of aeas Aea spacig i ULA (Chape VII) Divesiy gai vs. muliplexig gai d ( ) : Diffeeial divesiy gai vs. muliplexig gai γ FX ( x ) : Cumulaive desiy fucio of a adom vaiable X f X ( x ) : Pobabiliy desiy fucio of a adom vaiable X f ( ) Mulipah wave-umbe desiy f θ ( θ ) : Mulipah agula desiy f θφ ( θ, φ ) : Joi mulipah agula desiy f ψ ( ψ ) : Desiy of phase diffeece bewee wo adjace aeas g (eihe g o g ): g : : Zeo-mea complex veco wih idepede eies Schedulig gai MIMO chael maix xi

12 Capaciy Aalysis of Asympoically Lage MIMO Chaels w : ɶ : : : : : h (eihe h o h ): i.i.d. cicula symmeic Gaussia maix adom maix wih idepede eies Maix of complex chael gais fom Tx ed o eyholes Maix of complex chael gais fom eyholes o x ed Null hypohesis Aleaive hypohesis Veco of chael complex gais fom Tx (x) ed o a eyhole h (eihe h o h ): Veco of chael complex gais fom Tx (x) ed o h eyhole I : K : K : M : (eihe o ): Ideiy maix Coelaio maix wihou powe imbalace compoe Numbe of ewo uses Wave-veco Numbe of eyholes Numbe of aeas (a eihe Tx o x ed) eff : Numbe of efficie degees of feedom P : Powe imbalace maix P ou : Ouage pobabiliy P ( ) e M : P T : P : Q : (eihe o ): Symbol eo ae vs. modulaio level M Toal asmied powe Toal eceived powe Ipu covaiace maix Coelaio maix (a eihe Tx o x ed) (eihe o ): Coelaio maix (a eihe Tx o x ed) associaed wih h eyhole : Tasmissio ae z ( δ ), ( δ ) : Covegece ae give δ z, : : Covegece ae Veco epeseig he measue of coelaio ad powe imbalace xii

13 Capaciy Aalysis of Asympoically Lage MIMO Chaels ( x ) : (eihe o ): Coelaio fucio Coelaio paamee (a eihe Tx o x ed), also muliplexig gai (Chape VIII, Secio 8.) T : U (eihe U, U, u (eihe u, u ): U T o U ): Tes saisics Eigebasis maix (a eihe Tx o x ed) Eigevecos Z ( ) δ : Lyapouov aio Gee: α : β (eihe β o β ): Equivale scala eyhole chael powe gai, also miss pobabiliy Equivale scala chael powe gai (a eihe Tx o x ed), also false alam pobabiliy Γ (eihe γ : Γ o Γ ): MIMO chael coelaio maix Aveage SN (eihe pe x aea o oal a x ed) γ eff : Effecive aveage SN θ : λ (eihe λ (eihe λ o λ o λ ): λ ): Elevaio agle Veco of eigevalues of coelaio maix (a eihe Tx o x ed) h eigevalues of coelaio maix (a eihe Tx o x ed) λ ( u) : Specum of Toepliz maix µ : Asympoic mea of isaaeous capaciy ν : σ : Efficiecy of usig aeas, also gaulaiy Asympoic vaiace of isaaeous capaciy ϒ (, ) : Toal measue of coelaio ad powe imbalace a Tx ad x eds Φ X ( ω ) : Chaaceisic fucio of a adom vaiable X φ : Azimuh agle Ψ ( ) : Measue of coelaio ad powe imbalace a Tx ed xiii

14 Capaciy Aalysis of Asympoically Lage MIMO Chaels CAPTE I: INTODUCTION.. Muli-Aea Wieless Commuicaios The fuue geeaios of wieless commuicaio sysems ae desigaed o offe high daa-aes ude igh powe, specum ad complexiy limis. To mee hese objecives, fudameal chages i sysem cofiguaio ad sigal pocessig echiques ae equied o eable ew ad effecive ways of sigal asmissio ad ecepio. The Muliple-Ipu-Muliple-Oupu (MIMO) achiecue saisfies may of hese demads. Aalysis of classic fadig MIMO chaels ude he assumpio of idepede ad ich scaeig has demosaed ha wieless sysems ae able o achieve emaable specal efficiecy by deployig a umbe of aeas asmiig ad eceivig wihi he same fequecy badwidh [3], [5]. The echiques, which use muliple aeas fo spaial divesiy combiig, ae o ew ad have bee used fo may yeas o comba mulipah fadig [34]. The idea behid he mode MIMO sysems is o ae advaage of mulipah fadig isead of combaig i. I is possible, i his case, o scale up he specal efficiecy liealy wih he umbe of aeas, i coas o he classic combiig echiques, whee he ifomaio ae pe ui badwidh ca be iceased oly logaihmically [8]. The excepioal icease i specal efficiecy of MIMO sysems has igied eseach aciviy i may diffee diecios icludig capaciy aalysis, space-ime codig, modulaio echiques, modelig of popagaio eviomes suiable fo muli-aea sysems, ec. I is ow widely ecogized ha he pefomace of pacical MIMO sysems depeds heavily o he chael codiios such as he udelyig mulipah disibuio, spaial coelaio, powe imbalace ad aea cofiguaio. Due o he eomous iees i his aea, a umbe of joual special issues ae dedicaed solely o MIMO sysems [3], [95], [96], [3]. The MIMO achiecue has bee aleady icopoaed i a se of sadads such as IEEE 8. (Wieless Local Aea Newo), IEEE 8.5 (Wieless Pesoal Aea Newo) ad IEEE 8.6 (Wieless Meopolia Aea Newo). I addiio o hese o-goig sadadizaio effos, idusial suppo has bee gaed fo his eseach. Fo isace, a umbe of

15 Capaciy Aalysis of Asympoically Lage MIMO Chaels idusial alliaces ad cosoiums such as WiMax [3], WiFi [], Blueooh [4] ad WINNE [4], which iclude wold leadig commuicaio compaies, pomoe he compaibiliy ad ieopeabiliy of boadbad wieless poducs based upo he MIMO achiecue. Alhough hee has bee much ece pogess i he aea, may poblems sill emai ope. I paicula, he achievable pefomace ad coespodig cofiguaio of MIMO sysems have o bee sufficiely ivesigaed i may ealisic popagaio eviomes. Udesadig hese issues is a ey o he successful implemeaio of he MIMO achiecue i he fuue... Moivaio ad Pio Wo Oe of he majo pefomace chaaceisics of a MIMO chael is is capaciy, which gives he ulimae uppe limi o he eo-fee ifomaio ae. Capaciy epeses a impoa ifomaioheoeic boud which esablishes a bechma fo pefomace of pacical sysems ad ofe povides aalyical ools o maximize he daa ae ude diffee chael codiios [8], [6]. Fo ime-vaia, fadig chaels hee ae muliple capaciy defiiios icludig mea (egodic), ouage ad delay-limied capaciy. A excelle uoial o he fadig chael capaciy wih sigle asmi (Tx) ad eceive (x) aeas is give i [9]. This hesis focuses o he mea ad ouage capaciies as he wo mos geeal chaaceisics of MIMO fadig chaels. The mea capaciy is defied as he maximum muual ifomaio bewee he chael ipus ad oupus aveaged ove all possible chael ealizaios. This measue uppebouds he eo-fee ifomaio ae suppoed by egodic chaels, i.e. he chaels ha vay adomly duig asmissio ime ad he vaiaios ae egodic [6]. The ouage capaciy, i u, is a moe eleva pefomace measue of o-egodic chaels, ad i gives he uppe limi o he eofee ifomaio ae wih a give pobabiliy of ouage [3], [5], [6]. The mea ad ouage capaciies ae usually cosideed fo hee boad cases: (i) whe he chael sae ifomaio (CSI) is ow a boh asmi ad eceive eds, fo example [3], [], [69], (ii) whe CSI is available a he x ed oly, fo example [5], [7], [3], (iii) whe CSI is o ow a boh Tx ad x eds [6]. Appaely, he chael capaciy i case (i) is he highes, howeve i may pacical siuaios i does o

16 Capaciy Aalysis of Asympoically Lage MIMO Chaels povide a adequae pefomace measue sice a sysem wih a ifomed asmie is o always feasible as i equies a addiioal feedbac chael, which is o always available, fo example due o he badwidh limiaios, o he available feedbac chael is o capable o ac chael vaiaios ad/o o updae he asmie fas eough [5], [6]. A hoough eview o capaciy achievig asmissio ad eceivig saegies i MIMO chaels whe he SCI is available a boh Tx ad x eds, x ed oly, o o available a all ca be foud i [9]. The mea ad ouage capaciies of vaious MIMO chaels have bee exesively sudied duig he las decade. May aalyical ad empiical esuls have bee obaied. The ayleigh, ice ad Naagami disibued MIMO chaels have bee well ivesigaed ad closed-fom expessios fo hei capaciy ae ow available. Fo example, he mea ad ouage capaciy of ayleigh-fadig chaels wee sudied i [4], []. Seveal esuls o he mea capaciy of ayleigh ad ice MIMO chaels i lowpowe (widebad) egime ca be foud i [67]. The exac expessios fo he mea capaciy ad he ouage capaciy disibuio of semi ad double coelaed ayleigh-fadig chaels ae deived i [3], [99]. The capaciy aalysis alog wih seveal capaciy bouds fo ice ad Naagami MIMO chaels ae give i [4], [35], [38]. Much eseach has bee doe o evaluae he impac of spaial coelaio o he capaciy of ayleigh ad ice MIMO chaels. The impac of coelaio o he mea ad ouage capaciy of ayleigh-fadig chaels has bee sudied i [], [], [33], [6]. Capaciy bouds o he mea capaciy of coelaed ice MIMO chaels have bee obaied i [7]. The impac of coelaio o he ouage capaciy of geealized full-a MIMO chaels wih uiay-idepede-uiay (UIU) sucue has bee aalyzed i [9]. A umbe of effecive chael coelaio models have bee ioduced. The Koece coelaio model, which sigificaly simplifies he aalysis ad modelig of MIMO chaels by spliig he effec of coelaio bewee he asmi ad eceive eds, is poposed ad expeimeally validaed i [37]. Due o is simpliciy ad acabiliy, he model has bee used i may heoeical aalyses [3], [99], [69]. I some cases, howeve, he Koece model udeesimaes he chael capaciy [8]. A moe accuae bu also moe complex coelaio model, which aes i accou he joi coelaio of boh eds has bee poposed i []. 3

17 Capaciy Aalysis of Asympoically Lage MIMO Chaels I geeal, he impac of coelaio o boh mea ad ouage capaciies cao be chaaceized i a simple way. While i some cases he coelaio is beeficiay [], [77], [8], i some ohes i may sigificaly educe he capaciy of ayleigh-fadig chaels [], [], [6]. SN ad he coelaio sucue of he MIMO chael ae he ey facos i deemiig he effec of coelaio i each paicula sceaio [], [77]. To evaluae he impac of coelaio ad o ae i accou he coelaio sucue of a MIMO chael, a umbe of coelaio measues have bee poposed. A simple ad well acable measue of coelaio has bee poposed i [33] as a value ecipocal o he measue of spaial divesiy available i ayleigh-fadig chaels. I emais uclea, howeve, whehe his measue eflecs he impac of coelaio o he capaciy of he coespodig chaels. Aohe measue of coelaio, which is based o he majoizaio heoy, is poposed fo he ayleigh chaels wih a sigle Tx o x aea i []. The advaage of his measue is ha i clealy chaaceizes he impac of coelaio o he mea ad ouage capaciies. oweve, oly a subse of all possible chael coelaio sucues is measuable i his case, i.e. he measue does o posses a full odeig popey []. While he exac capaciy expessios fo ayleigh ad ice MIMO chaels ae complicaed ad usually do o allow fo sigifica isigh, a umbe of heoeical aalyses popose simple asympoic appoximaios of he mea ad ouage capaciies fo a lage umbe of aeas. Fo example, he asympoic expessios fo he mea ad ouage capaciy disibuio of ucoelaed ayleigh-fadig chaels ae deived i [3]. The capaciy of coelaed aow ad wide-bad ayleigh chaels has bee cosideed i [75], [74], [69]. The asympoic capaciy appoximaios fo icea MIMO chaels have bee aalyzed i [35]. The asympoic ouage capaciy disibuios of geeic full-a ucoelaed ad uiay-idepede-uiay (UIU) MIMO chaels have bee obaied i [8], [9], [9]. The commo coclusio i all hese wos is ha he ouage capaciy disibuio of a boad class of MIMO chaels is asympoically Gaussia. This idicaes ha he omal disibuio has a high degee of uivesaliy fo he aalysis of MIMO capaciy i geeal. I some cases he discepacy bewee he exac capaciy disibuio ad is Gaussia appoximaio has bee foud pacically idisiguishable aleady fo a modeae umbe of aeas []. A excelle suvey of diffee appoaches i asympoic aalysis of MIMO capaciy ca be foud i [9]. 4

18 Capaciy Aalysis of Asympoically Lage MIMO Chaels While he limiig disibuio of he ouage capaciy is ow i may cases, much less aeio has bee paid o he igoous aalyical evaluaio of he accuacy of he asympoic (Gaussia) appoximaio whe he umbe of aeas is fiie. Some iiial esuls which demosae he covegece ae of he ouage capaciy disibuio of geealized full-a ucoelaed MIMO chaels o he Gaussia oe ae peseed i [8]. The covegece ae of he coespodig mea capaciy o is asympoic value is bouded fom above i [9]. The covegece ae ad appopiae bouds fo ohe ypes of chaels, especially coelaed oes, ae sill o be foud. Aohe impoa diecio i he cue eseach is he effecs of aea desig ad mulipah agula desiy o he MIMO capaciy. The sudy of hese effecs may help o fid he opimal cofiguaio of aea aays i a paicula popagaio eviome, ways o educe spaial coelaio, ad moe geeally, i may eveal he elaioship bewee he ifomaio ad elecomageic heoies (fo moe deails abou his elaioship see [56]). Ceai pogess i his aea has bee made i [], [73], [94], [64], [7]. A asympoic appoach has bee used i [87], whee by leig he umbe of aeas a Tx ed o go o ifiiy, he capaciy sauaio effecs i cicula aea aays wee sudied fo ayleigh chaels coelaed a eihe Tx o x ed. oweve, oe of he poblems ha has o bee addessed ye, is fidig he bes mulipah agula desiy ha maximizes he capaciy of a MIMO chael wih a give aea cofiguaio. Much iees has bee ecely paid o he divesiy-muliplexig ade-off (DMT) i MIMO chaels [7], [7]. A simple, well acable expessio, which has povided a deep isigh io he ade-off bewee he specal efficiecy (spaial muliplexig gai) ad he eo ae (divesiy ode) has bee deived i [7] fo ayleigh-fadig chaels a high SN egime ( SN ). I is ow widely ecogized ha he SN-asympoic DMT oveesimaes he chael pefomace fo fiie SN, especially whe he divesiy gai is high (low eo ae), ad/o he umbe of aeas is lage [76], [58], [59]. A moe accuae expessio fo he DMT i ayleigh-fadig chaels wih fiie SN has bee obaied i [76]. This expessio is based o he lowe boud o he coespodig ouage capaciy disibuio. Ulie he SN-asympoic DMT of Zheg ad Tse [7], he ade-off i [76] is o a closed-fom ad has a high degee of complexiy wihou much isigh. Fo example, he impac of 5

19 Capaciy Aalysis of Asympoically Lage MIMO Chaels coelaio o he DMT is difficul fo immediae evaluaio. ecely, he DMT has bee geealized fo a boad class of MIMO chaels (o ecessaily ayleigh-fadig) usig asympoic appoximaio of ouage capaciy disibuio whe he umbe of aeas is lage [58], [59]. The size-asympoic DMT is compac, well acable ad appoximaes wih easoable accuacy he ue DMT fo low o modeae SN. I coas o Zheg ad Tse DMT, he accuacy of he size-asympoic oe iceases wih he umbe of aeas. Due o he simpliciy, he size-asympoic DMT allows fo sigifica isigh. I paicula, i clealy demosaes ha spaial coelaio deceases he divesiy gai of a chael a ay give asmissio ae. While he lae suppos he iuiio, his fac does o follow fom he DMT i [7] due o he SN-asympoic appoximaio. The heoeical esuls meioed above have bee validaed i a umbe of measueme campaigs such as, fo example, he measuemes of 5.Gz idoo ad oudoo MIMO chaels [79], [8], [7], he measuemes udeae i Mahaa a.6gz [6], expeimeal ivesigaio of MIMO chael popeies i idoo picocell eviomes [37], ad some ohes. oweve, mos of he empiical esuls o MIMO capaciy ad ohe chael paamees wee o a subjec o a igoous saisical aalysis. ahe visual compaiso of he measued daa plos o he coespodig heoeical models was doe wih o sicly defied cieia. Such a appoach ca eihe accou fo he saisical eo due o he limied amou of daa available (his is especially poouced fo measued chaels, whee he umbe of daa pois measued a sigle fequecy i a paicula eviome is ypically limied o a mos [79]), o fo he cofidece pobabiliy of he coclusios. As a esul, diffee coclusios wee epoed by diffee auhos. Fo example, he validiy of he Gaussia appoximaio of he ouage capaciy of MIMO chaels wih a fiie umbe of aeas has o bee a subjec o he igoous saisical aalysis. Ulie ohe chaels, he eyhole MIMO chael has o bee sudied i sufficie deph ye. This chael was heoeically pediced i [5], [4] as a mulipah eviome, whee ceai popagaio mechaisms educe he chael a. I ca be epeseed as a cocaeaio of wo mulipah subchaels sepaaed by a eyhole whose dimesios ae much smalle ha he wavelegh. The pesece of he eyhole degeeaes he chael, i.e. is a is oe egadless of he umbe of Tx ad x aeas 6

20 Capaciy Aalysis of Asympoically Lage MIMO Chaels [5]. Cosequely, he capaciy of such chaels deeioaes sigificaly compaed o he full a ayleigh chaels wih he same umbe of Tx ad x aeas. Thee is a sigifica iees i eyhole chaels i ece lieaue as hey may appea i some pacically impoa popagaio sceaios. [4] suggess a eyhole sceaio, whee he li bewee Tx ad x eds is due o he -D diffacio oly. A example of he eyhole ealizaio is give i [7]. I shows ha whe he scaeig aoud Tx ad x eds causes local fadig, he chael a may be low if he scaeig igs ae oo small compaig o he sepaaio bewee he asmie ad he eceive,. A umbe of expeimeal wos suppos he heoeical pedicios above. Fo isace, measuemes of he chael capaciy alog a hallway, epoed i [89] shows he decease i capaciy wih disace, which is explaied by he eyhole effec i he hallways. The expeimeal veificaio of he heoeical capaciy epoed i [5] eveals he eyhole effec i a coolled fee space eviome whe he sepaaio bewee he Tx ad x aeas is lage. Aohe covicig expeimeal evidece of a eyhole chael is peseed i [5], [4], whee i is show, i paicula, ha he eyhole model descibes well wieless chaels whe he wave popagaes via waveguides. The eyhole chael ca also be useful o model amplify-ad-fowad elay ewos [39], whee he eyhole epeses a elay ode ahe ha a popagaio effec. oweve, his appae similaiy bewee he ewo sucue ad he eyhole popagaio eviome has o bee elaboaed i he lieaue ye. The sigificace of a eyhole MIMO chael is also due o is uique posiio as a chael wih oly oe o-zeo eigemode, which descibes he wos-case MIMO popagaio sceaio. ece, i addiio o he pacical impoace, he sudy of eyhole chaels is ecessay, as i eveals how well a sysem pefoms i chaels ohe ha he ayleigh oes ad how much he esuls esablished fo he classic ayleigh-fadig chaels apply elsewhee. Despie he iees i eyhole chaels, he lieaue dealig wih hei ifomaio-heoeic aalysis is ahe limied. Closed-fom expessios fo he mea capaciy of a spaially ucoelaed eyhole chael ae peseed i [97]. A igh lowe boud ad appoximaios of he mea capaciy of a spaially coelaed eyhole chael ae poposed i [] ad [6] especively. Pefomace aalysis of space-ime bloc codes (STBC) ove a ucoelaed eyhole chael is give i [98], [9], whee, i 7

21 Capaciy Aalysis of Asympoically Lage MIMO Chaels paicula, he mome geeaig fucio of he isaaeous SN a he decode oupu is deived, ad SE fo vaious codes is evaluaed. Tigh lowe ad uppe Bofeoi-ype bouds o his SE ad he coespodig BE ae obaied i [5]. The divesiy ode of ucoelaed eyhole chaels has bee ivesigaed i [9]. May heoeical ad pacical quesios peaiig o eyhole chaels sill emai ope. Fo example, eyholes chaels wih subchaels ohe ha classic Naagami-m ad ayleigh-fadig have o bee cosideed. Ouage capaciy disibuio, impac of coelaio, ad divesiy-muliplexig adeoff i eyhole chaels have o bee ivesigaed. Addiioal sudy is equied o eveal he opimal asmissio ad eceivig saegies i eyhole chaels, ad whehe he coespodig saegies developed fo he caoic ayleigh-fadig chaels ae obus i he eyhole eviome. Thoough compaiso aalysis bewee he capaciy ad BE i eyhole ad ayleigh-fadig chaels has o bee coduced. Eve hough he exisece of a asiio model bewee he a-oe ad full-a MIMO chaels has bee suggesed i [7], hee is o a uifyig heoy ha esablishes a heoeical li bewee he a-oe eyhole ad classic full-a ayleigh-fadig MIMO chaels..3. Thesis Ogaizaio ad Coibuio Below we summaize he oigial coibuio coaied i Chapes III-IX. Chape III: Capaciy of Keyhole MIMO Chaels Closed-fom expessio fo he ouage capaciy disibuios of coelaed eyhole MIMO chaels is deived. A paicula bu commo case whee he coelaio maices a he Tx ad x eds ae osigula ad have disic eigevalues is cosideed. I is show ha he eyhole chael disibuio is diffee fom ha of adiioal divesiy chaels, which also have a oe. oweve, whe he umbe of eihe Tx o x aeas is lage, he eyhole chael capaciy achieves asympoically ha of he ayleigh divesiy chael wih a sigle Tx o x aea especively. The capaciy disibuio of he eyhole chael is uppe-bouded by hose of he equivale ayleigh divesiy chael. Fo a small umbe of aeas, spaial coelaio esuls i he loss of SN ad cosequely i smalle ouage 8

22 Capaciy Aalysis of Asympoically Lage MIMO Chaels capaciy. Fo a lage umbe of aeas, a compac ad well acable asympoic appoximaio of he ouage capaciy disibuio of eyhole MIMO chaels is deived. I is show ha ude ceai mild codiios o coelaio ad despie he degeeae aue of he eyhole chaels, he capaciy is asympoically Gaussia; he mea is affeced by aveage SN ad is idepede of coelaio, while he coelaio has a domia effec o he vaiace. igh coelaio iceases he vaiace which esuls i smalle capaciy a low ouage pobabiliies. Simulaios show ha he asympoic ouage capaciy disibuio follows closely he exac oe fo a easoably small umbe of Tx ad x aeas. Chape IV: Capaciy of Muli-Keyhole Chaels A mahemaical model of a muli-eyhole chael, which icludes a umbe of saisically idepede eyholes, is poposed o geealize ad expad he applicaio age of he sigle eyhole chael. The muli-eyhole chaels ae classified ad sudied i deail. I paicula, i is show ha he poposed model is complemeay o ha i [7], as i descibes a spase double-scaeig eviome, whee he scaees (eyholes) ae locaed fa apa fom each ohe. The diffee saisical behavio of a-deficie ad full-a muli-eyhole chaels is sessed ou. The muli-eyhole chael is show o be a asiio model ha lis he a-oe eyhole ad full-a ayleigh-fadig chaels. Whe a umbe of eihe Tx o x aeas ae lage, hee is a equivale ayleigh fadig chael, such ha he ouage capaciy of boh he muli-eyhole chael ad he ayleigh oe ae asympoically equal. Whe a umbe of boh Tx ad x aeas is lage, he ouage capaciy disibuio of boh a-deficie ad full-a muli-eyhole chaels is asympoically Gaussia. This fac implies ha Gaussia disibuio of ouage capaciy is a commo asympoic popey of a boad class of MIMO chaels. Chape V: Measue of Coelaio ad Powe Imbalace A ew scala measue of chael coelaio ad powe imbalace is ioduced based o he asympoic aalysis of muli-eyhole chael capaciy. The measue accous fo he oal coelaio ad 9

23 Capaciy Aalysis of Asympoically Lage MIMO Chaels powe imbalace bewee muliple aeas ad simulaeously affecs he asympoic ouage capaciy disibuio of he MIMO chael; he highe he measue, he lowe he capaciy a low ouage pobabiliies. I is show ha he poposed measue is compaible wih ohe measues of coelaio poposed fo ayleigh-fadig chaels [], [33], ad heefoe, descibes he impac of coelaio ad powe imbalace o he capaciy of a boad class of MIMO chaels. The advaages of he poposed measue ae simpliciy (o eigevalue decomposiio is equied), full odeig popey (ay wo chaels ca be compaed wihou excepios) ad acabiliy (i sepaaes he effec of coelaio ad powe imbalace). Aalysis of his measue idicaes ha he effecs of chael coelaio ad powe imbalace ae idepede, ad hei oal egaive impac o he ouage capaciy is chaaceized by he sum of he wo coespodig measues. I his sese, he impac of he powe imbalace ca be as bad as ha of he coelaio. Simulaios show ha he poposed measue of coelaio ad powe imbalace povides a adequae chaaceizaio of he impac of coelaio ad powe imbalace o he capaciy of he a-deficie ad full a muli-eyhole chaels wih a modeae umbe of aeas. Chape VI: Asympoic Nomaliy of ayleigh Chael Capaciy Geeal Lyapouov-ype codiio fo he asympoic omaliy of ayleigh fadig chael capaciy is discussed i deail, ad some physical implicaios of his codiio ae highlighed. I paicula, he covegece ae o he Gaussia disibuio is evaluaed. I may cases his ae is bouded fom below by /, i.e. he covegece is o slowe ha /, whee is he umbe of asmi aeas. The peseed aalysis povides heoeical ools o evaluae he accuacy of he Gaussia appoximaio whe he umbe of aeas is fiie. I is show ha fo he chaels wih Toepliz coelaio sucue, he Lyapouov-ype codiio is always saisfied, if he coelaio decays fase ha / D, whee D is he disace bewee he aeas. A umbe of popula coelaio models is cosideed o veify his esul.

24 Capaciy Aalysis of Asympoically Lage MIMO Chaels Chape VII: Wha is he Bes Agula Desiy i MIMO Chaels? The famewo poposed i [9] is geealized ad i is show ha whe he umbe of aeas is lage, he asympoic ouage capaciy of a boad class of MIMO chaels (o ecessaily ayleighfadig) wih a abiay coelaio sucue (o ecessaily UIU [9]) does o deped o a paicula chael disibuio, bu oly o he coelaio bewee aeas. Special cases iclude classic i.i.d. ayleigh-fadig chael [3], [5], ayleigh-fadig chael wih sepaable (Koece) coelaio sucue [37], ad i.i.d. zeo-mea (o ecessaily ayleigh-fadig) chael wih fiie fouh-ode saisics cosideed i [[8], Theoem.76]. Usig Szego Theoem [3], he mulipah agula desiy ha elimiaes he coelaio bewee aeas ad hus maximizes he asympoic capaciy of his class of MIMO chaels is deived, whe he eceive uifom liea aay (ULA) of isoopic aeas ad he mulipah ae locaed o a plae (- D). The capaciy-maximizig desiy is o-uifom. Sice he asympoic capaciy appoximaes easoably well he exac oe whe he umbe of aeas is modeae, i is cocluded ha he popula Clae s (Jaes) model [34] does o epese he bes case popagaio sceaio. Fo he opimal mulipah agula desiy, a simple expessio ha lis he measue of coelaio ad powe imbalace (ioduced i Chape V) o he disace bewee aeas, is obaied. The expessio explais he oscillaoy behavio of he capaciy as a fucio of aea spacig. The sudy is exeded o he mulipah disibued i he 3-D space (volume). I is show ha he capaciy-maximizig agula desiy i his case is also o-uifom. The lae povides guidelies fo a opimal locaio of a ULA aea i a 3-D mulipah eviome Chape VIII: Applicaios of Asympoic Aalysis of Ouage Capaciy Disibuio Secio 8.: Fiie SN size-asympoic divesiy muliplexig ade-off (DMT) is obaied fo he muli-eyhole chaels. I is show ha he DMT adequaely chaaceizes he impac of coelaio o he capaciy. Secio 8.: A simple ye easoably-accuae esimae of symbol eo ae (SE) i a fadig eyhole chael fo a vaiey of modulaio fomas is obaied. The esimae becomes especially

25 Capaciy Aalysis of Asympoically Lage MIMO Chaels accuae whe he umbe of aeas is lage, ad/o he modulaio ode is high. Secio 8.3: Telaa s Cojecue [3] is pove fo he muli-eyhole chaels wih a lage umbe of aeas. Secio 8.4: A moivaio fo he Koece coelaio model [37] is povided by cosideig a ayleigh-fadig chael as a muli-eyhole oe wih a lage umbe of eyholes. I is demosaed ha he Koece sucue of he coelaio is jusified, whe, fo example, hee is a physical sepaaio (such a scee) of he coelaio-fomig mechaism io asmie (Tx) ad eceive (x) pas. Secio 8.5: Schedulig gai ad he equied feedbac ae i wieless ewos is evaluaed assumig ha he popagaio eviome is descibed by he muli-eyhole chael wih a sufficie umbe of aeas so ha he asympoic Gaussia appoximaio of chael ouage capaciy applies. I is show ha boh he schedulig gai ad he feedbac ae icease wih he measue of coelaio ad powe imbalace a Tx ed ad decease wih SN. Chape IX: Saisical Aalysis of Measued MIMO Chaels A igoous mahemaical famewo fo aalyzig he saisical chaaceisics of measued MIMO chaels i geeal ad hei ouage capaciy disibuio i paicula is poposed. The accuacy of a umbe of saisical ess ad hei suiabiliy fo he saisical aalysis is assessed. The ess ae fis applied o he coelaed ayleigh-fadig chaels obaied by he Moe-Calo simulaio, ad he o he measued 5.Gz idoo MIMO chaels [79], [8]. The igoous saisical aalysis shows ha he measued chaels is fequecy selecive ayleigh-fadig wih sigifica coelaio ( >.7 ) a he x ed. The ouage capaciy disibuio of some measued chael is saisically Gaussia wih a easoable sigificace level aleady fo wo aeas a each ed. Eve hough his secio does o aim o veify he validiy of he muli-eyhole chael model poposed i Chape IV, he fac ha he ouage capaciy disibuio is saisically Gaussia fo a easoably small umbe of aeas implies ha he asympoic aalysis wih espec o a umbe of aeas o oly offes a sigifica isigh, bu also ca be applied o ealisic sysems of a modeae size.

26 Capaciy Aalysis of Asympoically Lage MIMO Chaels.4. Publicaios Joual Papes: [] G. Levi, S. Loya, O he Ouage Capaciy Disibuio of Coelaed Keyhole MIMO Chaels, IEEE Tas. o Ifom. Theoy, vol.54, o.7, pp , July 8. [] G. Levi, S. Loya, Commes o Asympoic Eigevalue Disibuios ad Capaciy fo MIMO Chaels ude Coelaed Fadig, IEEE Tasacios o Wieless Commuicaios, vol.7, o., pp , Febuay 8. [3] S. Loya, G. Levi, O Physically-Based Nomalizaio of MIMO Chael Maices, submied o IEEE Tas. o Ifom. Theoy, 7, (3 double space pages, ude secod eview). [4] G. Levi, S. Loya, Fom Muli-Keyholes o Measue of Coelaio ad Powe Imbalace i MIMO Chaels, submied o IEEE Tas. o Ifom. Theoy, 7, (36 double space pages, ude fis eview). Cofeece Papes: [5] G. Levi, S. Loya, Wha is he Bes Agula Desiy of Mulipah i MIMO Chaels?, submied o IEEE Ieaioal Symposium o Ifomaio Theoy (ISIT8). [6] S. Loya, G. Levi, O Fiie-SN Divesiy-Muliplexig Tadeoff, i Poc. 7 IEEE Global Commuicaios Cofeece (Globecom7), Washigo, DC, Nov. 7. [7] G. Levi, S. Loya, O Asympoic Ouage Capaciy Disibuio of Coelaed MIMO Chaels, i Poc. he Ieaioal Symposium o Sigals, Sysems ad Elecoics 7 (ISSSE 7), Moeal, QC, July-Aug., 7. [8] S. Loya, G. Levi, Divesiy-Muliplexig Tadeoff via Asympoic Aalysis of Lage MIMO Sysems, i Poc. 7 IEEE Ieaioal Symposium o Ifomaio Theoy (ISIT7), Nice, Face, Jue 7. [9] G. Levi, S. Loya, Muli-Keyhole MIMO Chaels: Asympoic Aalysis of Ouage Capaciy, i Poc. IEEE 6 ISIT, 6 IEEE Ieaioal Symposium o Ifomaio Theoy, Seale, WA, July 6. [] G. Levi, S. Loya, Muli-Keyholes ad Measue of Coelaio i MIMO Chaels, i Poc. QBSC 6, 3 d Bieial Symposium o Commuicaios, Kigso, ON, May-Jue 6. [] G. Levi, S. Loya, O Coelaed Keyhole MIMO Chaels: SN ad Ouage Capaciy Disibuios', i Poc. IEEE CCECE 6, IEEE Caadia Cofeece o Elecical ad Compue Egieeig 6, Oawa, ON, May 6. [] G. Levi, S. Loya, O he Ouage Capaciy Disibuio of Coelaed Keyhole MIMO Chaels, i Poc. IEEE WCNC 6, IEEE Wieless Commuicaios ad Newoig Cofeece 6, Las Vegas, NV, Apil 6. 3

27 Capaciy Aalysis of Asympoically Lage MIMO Chaels [3] G. Levi, S. Loya, Capaciy Disibuio of a Coelaed Keyhole Chael, i Poc. CWIT 5, Caadia Woshop o Ifomaio Theoy, Moéal, QC, Jue 5. [4] G. Levi, S. Loya, Saisical Appoach o MIMO Capaciy Aalysis i a Fadig Chael, i Poc. IEEE VTC 4-Fall, IEEE Vehicula Techology Cofeece,, Los Ageles, CA, Sep. 4. [5] G. Levi, S. Loya, Saisical Aalysis of a Measued MIMO Chael, i Poc. IEEE CCECE 4, IEEE Caadia Cofeece o Elecical ad Compue Egieeig 4, Niagaa Falls, ON, May 4. 4

28 Capaciy Aalysis of Asympoically Lage MIMO Chaels CAPTE II: LITEATUE EVIEW This chape coducs a eview of he ece eseach o MIMO capaciy. A umbe of capaciy defiiios, which povide a adequae chael measue i diffee popagaio eviomes, ae cosideed. A lieaue suvey of exac (fiie umbe of aeas) ad asympoic (abiay lage umbe of aeas) capaciies of ucoelaed, coelaed ayleigh-fadig ad geeic MIMO chaels is peseed ude he assumpio ha he chael sae ifomaio (CSI) is available a he eceive bu o he asmi ed. A umbe of coelaio models, which have bee pove o chaaceize sufficiely well he sucue of MIMO chaels, ae eviewed. Fially, he mahemaical model of he eyhole chael is ioduced, ad he heoeical bacgoud esseial fo he aalysis i he followig chapes is give... MIMO Chael Capaciy Coside a MIMO chael wih Tx ad x aeas (see Fig..). Le be he chael asfe maix wih elemes, =... ; m =..., epeseig a complex chael gai fom he m h m asmi o he h eceive aea. Thee ae hee geeal ypes of [3]: (i) is a deemiisic maix. I his case, he capaciy pe ui badwidh of a fequecy fla MIMO chael wih addiive spaially whie Gaussia oise ad CSI available a he x ed oly is give i aual uis (as) by ( ) C = l de[ I + γ / ], (.) whee ( ) deoes he emiia aspose, de [ ] is he deemia, I is [ ] γ is he aveage SN pe x aea. (ii) is a egodic adom maix. Fom [3] whee E{} deoes expecaio. { l ( de[ / ] )} ideiy maix ad C = E I + γ, (.) 5

29 Capaciy Aalysis of Asympoically Lage MIMO Chaels (iii) is a o-egodic maix chose adomly a he begiig ad held cosa all he ime. I he classical Shao sese, he chael capaciy i his case is geeally zeo, sice hee is always ozeo pobabiliy (ouage pobabiliy) ha may have such values ha fo ay give γ o code would be able o povide abiay small pobabiliy of eo fo ay ae >. To defie chael capaciy fo a o-egodic, he cocep of ouage capaciy is used i [3], [5] as he maximal achievable ifomaio ae wih he give ouage pobabiliy F ( ), such ha: whee P{} deoes pobabiliy, ad C F ( ) P{ } C = C, (.3) ( ) C = l de[ I + γ / ] (.4) is he isaaeous capaciy, i.e. he capaciy of a give ealizaio of. I his defiiio C is a adom vaiable, ad F ( ) is he coespodig cumulaive desiy fucio (CDF). The mea of C C coicides wih (.), bu ulie he case whee is egodic, E{ C }, i his case, does o have opeaioal meaig, because sedig ifomaio wih he aes close o E{ C } is usually associaed wih high pobabiliy of ouage [3]. To disiguish bewee C i (.4) ad C i (.), he lae is emed he mea (egodic) capaciy of he chael ad will be fuhe deoed by C. Tx Ed MIMO Chael x Ed Fig.. MIMO Chael This hesis focuses o he capaciy defied fo adom, as may pacical wieless chaels ae adom ad ofe quasi-saic duig he asmissio ime (see, fo example, IEEE 8., IEEE 8.5 6

30 Capaciy Aalysis of Asympoically Lage MIMO Chaels ad IEEE 8.6). Fo he quasi-saic chaels, he ouage capaciy is a moe eleva pefomace measue fom a pacical pespecive (i.e. fo a give qualiy of sevice) as compaed o he mea capaciy. Fom (.) ad (.3), boh F ( ) ad C deped o he disibuio of. I paicula, he C coelaio bewee he elemes of has a cucial impac o he MIMO capaciy [], [9]. I he followig secio we ioduce a umbe of popula coelaio models ha ae used i Chapes III-IX o epese he coelaio sucue of... Coelaio Sucue of MIMO Chaels The followig defiiios epese commo coceps widely used i he lieaue ad ae ecessay fo he fuhe discussio. Defiiio.: A adom MIMO chael epeseed by maix is called ayleigh-fadig, if he elemes of ae joily disibued zeo mea cicula symmeic Gaussia. If he mea is o zeo, he chael is called icia. The icia ad ayleigh-fadig chaels ae appopiae models fo ich mulipah popagaio eviomes wih o wih o lie of sigh (LOS), especively. Defiiio.: If { a ( ) ( )} P = mi, =, (.5) he adom chael is called full-a, ohewise i is a deficie. Codiio (.5) is saisfied fo a boad class of chaels such as ucoelaed, coelaed ayleigh, ice ad geeic full-a MIMO chaels. Defiiio.3: Le { ( ) vec( ) } Γ = E vec, (.6) whee vec( ) epeses he opeao which ceaes a colum veco by sacig he elemes of columwise. If Γ = ci, whee I is ideiy maix, ad c is a omalizaio cosa, he chael is 7

31 Capaciy Aalysis of Asympoically Lage MIMO Chaels ucoelaed, ohewise i is said o be coelaed. Γ epeses he mos geeal ad full descipio of coelaio of chael. oweve, he aalysis of MIMO chaels wih abiay Γ is usually complicaed. ece, a umbe of moe aalyically fiedly coelaio sucues of Γ has bee poposed. A sepaable (Koece) model, which sigificaly simplifies he aalysis ad simulaio of coelaed chaels by allowig idepede modelig a he Tx ad x eds has bee ioduced fo ayleigh-fadig chaels ad expeimeally veified i [37]. Followig his model, Γ =, (.7) T whee supescip T deoes maix asposiio, ad is he Koece poduc [9] of he asmie ad eceive coelaio maices defied by { }; E{ } = E =, (.8) whee i is assumed, wihou loss of geealiy, ha he chael is omalized so ha ( ) = ( ) =, whee (.) sads fo ace. Eq. (.7) implies ha he coelaio bewee wo sigals colleced by a pai of x aeas is esiced o he same value iespecively of he asmiig aea he sigals oigi fom. The coelaio bewee a pai of Tx aeas has he same popey via dualiy of he poblem. Whe (.7) holds ue, ca be epeseed as [37] ( ), (.9) / / w whee meas ideically disibued, ad w is a adom maix of size composed of cicula symmeic Gaussia i.i.d. eies wih ui vaiace. Koece model has become a coesoe of a lage umbe of aalyses [3], [99], [69]. I Chape VIII we povide a moivaio fo he Koece model ad show explicily ha he Koece sucue of coelaio is due o sepaabiliy popey of he coelaio-fomig effecs io asmie ad eceive pas. The followig defiiio is possible due o he sepaabiliy popey of he Koece model. Defiiio.4 [3], [99]: A MIMO chael is called semicoelaed a eihe Tx o x ed, if eihe = I o = I. If boh I ad I, he chael is called double coelaed. 8

32 Capaciy Aalysis of Asympoically Lage MIMO Chaels The semicoelaed chaels descibe well pacical sysems whee he aeas ae sepaaed fa eough fom each ohe a oe ed (fo example a a base saio (BS)), so he coelaio bewee he aeas is egligible, bu he aea spacig a he ohe ed (fo example a a mobile ui (MU)) is small, which esuls i a high level of coelaio a ha ed. I addiio o he pacical value, he aalysis of semicoelaed chaels is ofe moe aalyically fiedly compaig o he double coelaed chaels. Fo his easo, he fis aalyical esuls o capaciy of coelaed MIMO chaels, which appeaed i he lieaue, wee obaied fo he semicoelaed oes [3], []. Thee ae cases, howeve, whe he coelaio sucue of a MIMO chael is o sepaable. The use of he Koece model may he udeesimae he chael capaciy [8]. A moe accuae coelaio model, which aes i accou he joi coelaio of boh Tx ad x eds has bee poposed i []: U ( Ω ) U, (.) w whee opeao deoes elemewise maix muliplicaio, especively, ad he elemes of Ω ae give as U ad U ae eigebases of ad m, = E, m, Ω { u u }, (.) whee u, ad u, m ae h ad m h eigevecos of U ad U especively. Ω is called a couplig maix sice is elemes specify he mea amou of eegy ha is coupled fom he mh U eigeveco of Tx ed o he h eigeveco of he x ed o vice vesa. The ecessay ad sufficie codiio fo his model o hold is ha he eigebasis a he x ed is idepede of he asmied sigal, ad he eigebasis a he Tx ed is idepede of he eceived sigal. Compaig o he Koece model, he oe i (.) pedics moe accuaely he chael capaciy, bu a he expese of a lage umbe of paamees o be evaluaed. As U ad U have o be foud fo boh models, he umbe of elemes i he couplig maix is, as compaed o + Tx ad x eigevalues i he Koece model. Fo MIMO chaels, whose coelaio sucue is sepaable bewee Tx ad x eds, he model i (.) educes o he Koece oe []. 9

33 Capaciy Aalysis of Asympoically Lage MIMO Chaels While he coelaio models above ae specifically ailoed fo ayleigh-fadig MIMO chaels, [9] poposes a geeic coelaio sucue give by U U ɶ, (.) T whee U T ad U ae deemiisic uiay maices, ad ɶ is a adom maix wih idepede zeomea abiay bu o ecessaily ideically disibued elemes. Due o he uiay-idepede-uiay sucue i (.), his model is efeed as UIU ad ecompasses a boad class of zeo mea MIMO chaels as show below. (i) If ɶ =, i.e. i has i.i.d. cicula symmeic Gaussia eies, epese a caoical w ucoelaed ayleigh-fadig chael discussed, fo example, i [3], [5]. (ii) If he eies of ɶ ae cicula symmeic Gaussia wih he sepaable coelaio sucue, he UIU model eves o he Koece oe, ad ca be equivalely epeseed by (.9), whee UT = U ad U = U [9]. (iii) If U = I ad U = T I, he model i (.) educes o a idepede (IND) o ecessaily ideically disibued MIMO chael. This chael ca descibe, fo example, he use of polaizaio divesiy, whee despie small aea spacig, he level of coelaio is low [9]. (iv) If ɶ is IND cicula symmeic Gaussia, ad U T ad U ae Fouie maices, i.e. he elemes of eihe U T o U ae U, m j m / = e π,, m =..., whee is eihe o, he UIU model edes he viual chael epeseaio ioduced i [93] fo uifom liea aea aays (ULA), whee he colums of specific spaial diecios. (v) If U T ad U T ad U ae iepeed as seeig vecos asmiig ad eceivig eegy a U ae abiay uiay maices, while ɶ is IND cicula symmeic Gaussia, he model i (.) is equivale o ha i (.). Thee ae also chaels ha cao be epeseed by he UIU model: (i) The chaels wih diagoal coelaio discussed i [77]. These chaels ae cosideed explicily fo = = ad have a ceai fixed coelaio pae, which ca o be descibed by he UIU model.

34 Capaciy Aalysis of Asympoically Lage MIMO Chaels (ii) The eyhole chaels [4], [5], whee is he oue poduc of wo adom vecos. Thee he eies of ca be ucoelaed bu o idepede. While he eyhole chaels fall ouside he UIU model, hey ae oe of he mai subjecs of his hesis. Secio.5 i he cue chape povides a mahemaical model ad physical moivaio of a eyhole chael. The ouage capaciy disibuio ad effec of coelaio i eyhole chaels ae sudied i Chapes III ad V. To evaluae he effec of coelaio i a explici fom, he followig popula paameic coelaio models fo ad ae used houghou his hesis. Uifom Coelaio Model: This model epeses a simple case whe he coelaio bewee ay pai of aeas a Tx o x ed is equal ad eal, i.e. he elemes of, eihe o, ae give by [55] ; m = m =, -/( -) ; m, (.3) whee is a coelaio coefficie bewee wo aeas, is eihe o. The uifom model is somewha aificial sice i pesumes he same coelaio bewee ay pai of aeas egadless of ay specifics, while i pacice he coelaio deceases wih aea spacig. I his sese, he uifom model epeses he wos case coelaio sucue ad, i some cases, povides some isigh io he opeaio of MIMO achiecue [55]. Expoeial Coelaio Model: I his model he elemes of ae epeseed hough a sigle complex coelaio paamee, which is he coelaio bewee adjace aeas [86]: m ; m m =, <, (.4) m ; m < I his model is eal ad esiced fom below o esue ha is a coelaio maix, i.e. posiive semi-defiie [55].

35 Capaciy Aalysis of Asympoically Lage MIMO Chaels whee is he complex cojugae of. This model allows fo sigifica isigh ad has bee successfully used fo may commuicaios poblems. Despie is simpliciy, i is a physically-easoable model i he sese ha he coelaio deceases as disace m bewee aeas iceases. Quadaic Expoeial (QE) Model: This is a physically-moivaed sigle-paamee coelaio maix model, whee he elemes of ae give by [], [7]: ( m ) ; m m =, < (.5) ( m) ; m < QE model is icopoaed i he IEEE 8. Wieless LANs sadad [], ad epeses he sceaio wih a Gaussia pofile of mulipah agle-of-aival [], [7]. Compaig o he expoeial coelaio model, hee he coelaio bewee diffee aeas decays sigificaly fase wih disace m. Ti-diagoal Model: Whe he coelaio is sigifica oly amog adjace aeas, he elemes of ca be modeled as [86],, = m, = m π, m =, < cos, m (.6) = + +, ohewise This model offes a sigifica coveiece fo mahemaical aalysis i paicula because i has a simple eigevalue decomposiio [86]. oweve, followig (.6), he model is esiced by he ceai values of. Whe his esicio is o saisfied, becomes o posiive semi-defiie ad, heefoe cao epese a coelaio maix. Noe ha fo a lage umbe of aeas >>, such ha cos[ π /( + )], he valid values of < /, i.e. i (.6) is defied fo low coelaios oly. Ofe, whe he umbe of aeas >, a meaigful scala measue is equied o evaluae he amou of coelaio epeseed by a coelaio maix. A simple ad well acable measue of

36 Capaciy Aalysis of Asympoically Lage MIMO Chaels coelaio has bee poposed i [33]: / { } Ω ( ) =, (.7) whee is he L om of [9]. Ω( ) epeses a value ecipocal o he measue of spaial divesiy available i ayleigh-fadig chaels. Fo example, if is give by he uifom coelaio model (.3), Ω ( ) = [33]. oweve, his measue o oly epeses he coelaio coefficie bewee adjace aeas, bu also aes io accou he sucue of, e.g. he ae of coelaio decay wih disace. I emais uclea, howeve, whehe Ω( ) epeses adequaely he impac of coelaio o he chael capaciy. Aohe measue of coelaio is poposed fo he ayleigh chaels wih a sigle Tx o x aea i []. This measue is based o he majoizaio heoy [68], whee a coelaio maix is said o majoize (moe coelaed ha) (deoed as ), if m =..., whee () λ ad fo all m λ () m λ () = = () λ ae he eigevalues of ad especively soed i a descedig ode []. I is show, i paicula, ha i muliple ipu sigle oupu (MISO) ayleigh-fadig chaels wih pefec CSI available a he eceive oly, a icease i coelaio (highe measue of coelaio) esuls i lowe ouage capaciy, if he aveage SN γ e, whee he daa ae is give i as. I opposie, whe γ ( e )/, he impac of coelaio o he ouage capaciy is beeficial. Appaely, he advaage of he coelaio measue i [] is ha i clealy chaaceizes he impac of coelaio o he chael capaciy. oweve, oly a subse of all possible chael coelaio sucues is measuable i his case, i.e. he majoizaio-heoy-based measue is o full odeig []. I Chape V, we ioduce a ew full-odeig scala measue of coelaio ha applies fo MIMO chaels of abiay size ad clealy chaaceizes he impac of coelaio o he capaciy..3. Capaciy of ayleigh-fadig MIMO Chaels This secio summaizes he mai esuls available i he lieaue fo ucoelaed, semicoelaed ad double-coelaed ayleigh-fadig chaels. The pupose of his summay is o daw lae a 3