Massive MIMO Sysems wih on-ideal Hadwae: negy fficiency, simaion, and Capaciy Limis mil Bjönson, Membe, I, Jakob Hoydis, Membe, I, Maios Kounouis, Membe, I, and Méouane Debbah, Senio Membe, I axiv:307.584v3 [cs.it] Sep 04 Absac The use of lage-scale anenna aays can bing subsanial impovemens in enegy and/o specal efficiency o wieless sysems due o he gealy impoved spaial esoluion and aay gain. Recen woks in he field of massive mulipleinpu muliple-oupu MIMO show ha he use channels decoelae when he numbe of anennas a he base saions BSs inceases, hus song signal gains ae achievable wih lile ine-use inefeence. Since hese esuls ely on asympoics, i is impoan o invesigae whehe he convenional sysem models ae easonable in his asympoic egime. This pape consides a new sysem model ha incopoaes geneal ansceive hadwae impaimens a boh he BSs equipped wih lage anenna aays and he single-anenna use equipmens Us. As opposed o he convenional case of ideal hadwae, we show ha hadwae impaimens ceae finie ceilings on he channel esimaion accuacy and on he downlink/uplink capaciy of each U. Supisingly, he capaciy is mainly limied by he hadwae a he U, while he impac of impaimens in he lage-scale aays vanishes asympoically and ine-use inefeence in paicula, pilo conaminaion becomes negligible. Fuhemoe, we pove ha he huge degees of feedom offeed by massive MIMO can be used o educe he ansmi powe and/o o oleae lage hadwae impaimens, which allows fo he use of inexpensive and enegy-efficien anenna elemens. Index Tems Capaciy bounds, channel esimaion, enegy efficiency, massive MIMO, pilo conaminaion, ime-division duplex, ansceive hadwae impaimens. I. ITRODUCTIO The specal efficiency of a wieless link is limied by he infomaion-heoeic capaciy [], which depends no only on Copyigh c 04 I. Pesonal use of his maeial is pemied. Howeve, pemission o use his maeial fo any ohe puposes mus be obained fom he I by sending a eques o pubs-pemissions@ieee.og. The Malab code ha epoduces all simulaion esuls is available online, see hps://gihub.com/emilbjonson/massive-mimo-hadwae-impaimens/. Bjönson was wih he Alcael-Lucen Chai on Flexible Radio, Supélec, Gif-su-Yvee, Fance, and wih he Depamen of Signal Pocessing, KTH Royal Insiue of Technology, Sockholm, Sweden. He is cuenly wih he Depamen of lecical ngineeing ISY, Linköping Univesiy, Sweden email: emil.bjonson@liu.se. J. Hoydis was wih Bell Laboaoies, Alcael-Lucen, Gemany. He is now wih Spaed SAS, Osay, Fance email: hoydis@ieee.og. M. Kounouis and M. Debbah ae SUPLC, Gif-su-Yvee, Fance email: maios.kounouis@supelec.f, meouane.debbah@supelec.f. This pape was pesened in pa a he Inenaional Confeence on Digial Signal Pocessing DSP, Sanoini, Geece, July 03 []. The wok of. Bjönson was funded by he Inenaional Posdoc Gan 0-8 fom The Swedish Reseach Council. This eseach has been suppoed by he RC Saing Gan 3053 MOR Advanced Mahemaical Tools fo Complex ewok ngineeing. Pas of his wok have been pefomed in he famewok of he FP7 pojec ICT-37669 MTIS. This wok was suppoed by he Fuue and meging Technologies FT pojec HIATUS wihin he Sevenh Famewok Pogamme fo Reseach of he uopean Commission unde FT-Open gan numbe 65578. he signal-o-noise aio SR bu also on spaial coelaion in he popagaion envionmen [3], [4], channel esimaion accuacy [5], ansceive hadwae impaimens [6], [7], and signal pocessing esouces [8], [9]. I is of pofound impoance o incease he specal efficiency of fuue newoks, o keep up wih he inceasing demand fo wieless sevices. Howeve, his is a challenging ask and usually comes a he pice of having sice hadwae and ovehead equiemens. A new newok achiecue has ecenly been poposed wih he emakable poenial of boh inceasing he specal efficiency and elaxing he afoemenioned implemenaion issues. I is known as massive MIMO, o lage-scale MIMO, and is based on having a vey lage numbe of anennas a each BS and exploiing channel ecipociy in ime-division duplex TDD mode [9] [3]. Some key feaues ae: popagaion losses ae miigaed by a lage aay gain due o coheen beamfoming/combining; inefeence-leakage due o channel esimaion eos vanish asympoically in he lage-dimensional veco space; 3 low-complexiy signal pocessing algoihms ae asympoically opimal; and 4 ineuse inefeence is easily miigaed by he high beamfoming esoluion. The amoun of eseach on massive MIMO inceases apidly, bu he impac of ansceive hadwae impaimens on hese sysems has eceived lile aenion so fa alhough lage aays migh only be aacive fo newok deploymen if each anenna elemen consiss of inexpensive hadwae. Cheap hadwae componens ae paiculaly pone o he impaimens ha exis in any ansceive e.g., amplifie nonlineaiies, I/Q-imbalance, phase noise, and quanizaion eos [4] [3]. The influence of hadwae impaimens is usually miigaed by compensaion algoihms [4], which can be implemened by analog and digial signal pocessing. These echniques canno emove he impaimens compleely, bu hee emain esidual impaimens since he ime-vaying hadwae chaaceisics canno be fully paameeized and esimaed, and because hee is andomness induced by diffeen ypes of noise. Tansceive impaimens ae known o fundamenally limi he capaciy in he high-powe egime [6], [4], while hee ae only a few publicaions ha analyze he behavio in he lage numbe of anenna egime. Lowe bounds on he achievable uplink sum ae in massive single-cell sysems wih phase noise fom fee-unning oscillaos wee deived in [5]. The impac of amplifie non-lineaiies in a ansmie can be educed by having a low peak-o-aveage powe aio PAPR. The excess degees of feedom offeed by massive MIMO wee used in [6] o opimize he downlink pecoding fo low
Downlink: Pilos & Daa Uplink: Pilos & Daa sochasic vaiable x is denoed x x, q, whee x is he mean and q is he vaiance. A ciculaly symmeic complex Gaussian sochasic veco x is denoed x C x, Q, whee x is he mean and Q is he covaiance maix. The empy se is denoed by. The big O noaion fx = Ogx means ha is bounded as x. fx gx Base saion Use equipmen Fig. : Illusaion of he ecipocal channel beween a BS equipped wih a lage anenna aay and a single-anenna U. PAPR, while [7] consideed a consan-envelope pecoding scheme designed fo vey low PAPR. This pape analyzes he aggegae impac of diffeen hadwae impaimens on sysems wih lage anenna aays, in conas o he ideal hadwae consideed in [0] [3] and he single ype of impaimens consideed in [5] [7]. We assume ha appopiae compensaion algoihms have been applied and focus on he esidual hadwae impaimens. Moivaed by he analyic analysis and expeimenal esuls in [4] [8], he esidual hadwae impaimens a he ansmie and eceive ae modeled as addiive disoion noises wih ceain impoan popeies. The sysem model wih hadwae impaimens is defined and moivaed in Secion II. Secion III deives a new pilo-based channel esimao and shows ha he esimaion accuacy is limied by he levels of impaimens. The focus of Secion IV is on a single link in he sysem whee we deive lowe and uppe bounds on he downlink and uplink capaciies. Ou esuls eveal he exisence of finie capaciy ceilings due o hadwae impaimens. Despie hese discouaging esuls, Secion V shows ha a high enegy efficiency and esilience owads hadwae impaimens a he BS can be achieved. Secion VI pus hese esuls in a mulicell conex and shows ha ine-use inefeence including pilo conaminaion basically downs in he disoion noise fom hadwae impaimens. Secion VII descibes he impac of vaious efinemens of he sysem model, while Secion VIII summaizes he conibuions and insighs of he pape. To encouage epoducibiliy and exensions o his pape, all he simulaion esuls can be geneaed by he Malab code ha is available a hps://gihub.com/emilbjonson/massive- MIMO-hadwae-impaimens/ oaion: Boldface lowe case is used fo column vecos, x, and uppe case fo maices, X. Le X T, X, and X H denoe he anspose, conjugae, and conjugae anspose of X, especively. X X means ha X X is posiive semi-definie. A diagonal maix wih a,..., a on he main diagonal is denoed diaga,..., a and I denoes an ideniy maix of appopiae dimensions. The Fobenius and specal noms of a maix X ae denoed by X F and X, especively, while x k denoes he L k nom of a veco x. A sochasic vaiable x and is ealizaion is denoed in he same way, fo beviy. The expecaion opeao wih espec o a sochasic vaiable x is denoed x, while x y is he condiional expecaion when y is given. A Gaussian II. CHAL AD SYSTM MODL Fo analyical claiy, he majo pa of his pape analyzes he fundamenal specal and enegy efficiency limis of a single link, which opeaes unde abiay inefeence condiions. The link is esablished beween an -anenna BS and a single-anenna U. A main chaaceisic in he analysis is ha he numbe of anennas can be vey lage. We conside a TDD poocol ha oggles beween uplink UL and downlink DL ansmission on he same fla-fading subcaie. This enables efficien channel esimaion even when is lage, because he esimaion accuacy and ovehead in he UL is independen of [9]. The acquied insananeous channel sae infomaion CSI is uilized fo UL daa deecion as well as DL daa ansmission, by exploiing channel ecipociy; see Fig.. In Secion VI, we pu ou esuls in a mulicell conex wih many uses, ine-cell inefeence, and pilo conaminaion. We assume a block fading sucue whee each channel is saic fo a coheence peiod of T cohe channel uses. The channel ealizaions ae geneaed andomly and ae independen beween blocks. Fo simpliciy, T cohe is he same fo he useful channel and any inefeing channels, and he coheence peiods ae synchonized. We conside he convenional TDD poocol in Fig., which can be found in many pevious woks; see fo example [8] and [9]. ach block begins wih UL pilo/conol signaling fo T UL pilo channel uses, followed by UL daa ansmission fo Tdaa UL channel uses. ex, he sysem oggles o he DL. This pa begins wih Tpilo DL channel uses of DL pilo/conol signaling. These pilos ae ypically used by he Us o esimae hei effecive channel wih pecoding and he cuen inefeence condiions, which enables coheen DL ecepion. oe ha hese quaniies ae scalas iespecive of, hus he DL pilo signaling need no scale wih. The coheence peiod ends wih DL daa ansmission fo Tdaa DL channel uses. The fou paamees saisfy Tpilo UL UL DL DL +Tdaa +Tpilo +Tdaa = T cohe. The analysis of his pape is valid fo abiay fixed values of hose paamees, bu we noe ha hese can also be opimized dynamically based on T cohe, use load, use condiions, aio of UL/DL affic, ec. The sochasic block-fading channel beween he BS and he U is denoed as h C. I is modeled as an egodic pocess wih a fixed independen ealizaion h C 0, R in each coheence peiod. This is known as Rayleigh block fading and R = hh H C is he posiive semi-definie covaiance maix. The saisical disibuion is assumed o The physical channels ae always ecipocal, bu diffeen ansceive chains ae ypically used in he UL and DL. Caeful calibaion is heefoe necessay o uilize he ecipociy fo ansmission; see Secion VII-.
3 Uplink Pilo & Conol Signals T UL pilo Uplink Daa Tansmission T UL daa Downlink Pilo & Conol Signals T DL pilo Downlink Daa Tansmission T DL daa Coheence Peiod T cohe Fig. : Cyclic opeaion of a block-fading TDD sysem, whee he coheence peiod T cohe is divided ino phases fo UL/DL pilo and daa ansmission. be known a he BS. In he asympoic analysis, we make he following echnical assumpions: The specal nom of R is unifomly bounded, iespecive of he numbe of anennas i.e., R = O; The ace of R scales linealy wih i.e., 0 < lim inf R lim sup R < and R has sicly posiive diagonal elemens. The fis assumpion is a necessay physical popey ha oiginaes fom he law of enegy consevaion. I is also a common enable fo asympoic analysis cf. []. The second assumpion is a ypical consequence of inceasing he aay size wih and heeby impoving he spaial esoluion and apeue [9]. These assumpions imply 0 < lim inf ankr, which means ha R can be ank deficien bu he ank inceases wih such ha c ankr fo some c > 0. We sess ha R is geneally no a scaled ideniy maix, bu descibes he spaial popagaion envionmen and aay geomey. I migh be ank-deficien e.g., have a lage condiional numbe fo lage aays due o insufficien ichness of he scaeing [3], [4]. A. Tansceive Hadwae Impaimens The majoiy of papes on massive MIMO sysems consides channels wih ideal ansceive hadwae. Howeve, pacical ansceives suffe fom hadwae impaimens ha ceae a mismach beween he inended ansmi signal and wha is acually geneaed and emied; and diso he eceived signal in he ecepion pocessing. In his pape, we analyze how hese impaimens impac he pefomance and key asympoic popeies of massive MIMO sysems. Physical ansceive implemenaions consis of many diffeen hadwae componens e.g., amplifies, convees, mixes, files, and oscillaos [30] and each one disos he signals in is own way. The hadwae impefecions ae unavoidable, bu he seveiy of he impaimens depends on engineeing decisions lage disoions can be delibeaely inoduced o decease he hadwae cos and/o he powe consumpion [7]. The non-ideal behavio of each componen can be modeled in deail fo he pupose of designing compensaion algoihms, bu even afe compensaion hee emain esidual ansceive impaimens [5], [7]; fo example, due o insufficien mod- Alhough hese assumpions make sense fo pacically lage [4], we canno physically le since he popagaion envionmen is enclosed by a finie volume [9]. eveheless, ou simulaions eveal ha he asympoic analysis enabled by he echnical assumpions is accuae a quie small. eling accuacy, impefec esimaion of model paamees, and ime vaying chaaceisics induced by noise. Fom a sysem pefomance pespecive, i is he aggegae effec of all he esidual ansceive impaimens ha is impoan, no he individual behavio of each hadwae componen. Recenly, a new sysem model has been poposed in [4] [9] whee he aggegae esidual hadwae impaimens ae modeled by independen addiive disoion noises a he BS as well as a he U. We adop his model heein due is analyical acabiliy and he expeimenal veificaions in [5] [7]. The deails of he DL and UL sysem models ae given in he nex subsecions, and hese ae hen used in Secions III VI o analyze diffeen aspecs of massive MIMO sysems. Possible model efinemens ae hen povided in Secion VII, along wih discussions on how hese migh impac he main esuls of his pape. B. Downlink Sysem Model The downlink channel is used fo daa ansmission and pilo-based channel esimaion; see Fig.. The eceived DL signal y C in a fla-fading muliple-inpu single-oupu MISO channel is convenionally modeled as y = h T s + n whee s C is eihe a deeminisic pilo signal duing channel esimaion o a sochasic zeo-mean daa signal; in any case, he covaiance maix is denoed W = ss H and he aveage powe is p BS = W. W is a design paamee ha migh be a funcion of he channel ealizaion h and he ealizaions of any ohe channel in he sysem e.g., due o pecoding; we le H denoe he se of channel ealizaions fo all useful and inefeing channels i.e., h H. Hence, W is consan wihin each coheence peiod bu changes beween coheence peiods since H changes. The addiive em n = n noise + n inef is an egodic sochasic pocess ha consiss of independen eceive noise n noise C 0, σu and inefeence n inef fom simulaneous ansmissions e.g., o ohe Us. The inefeence has zeo mean and is independen of he daa signal, bu migh depend on any channel in he sysem e.g., such ha cay inefeence. Hence, he condiional inefeence vaiance is n inef H = IH U 0 in he coheence peiod whee he channel ealizaions ae H. The long-em inefeence vaiance is denoed IH U. I is only fo beviy ha we use a common noaion n fo inefeence and eceive noise i does no mean ha he inefeence mus be eaed as noise a he U. A deailed inefeence model is povided in Secion VI. To model sysems wih non-ideal hadwae moe accuaely, we conside he new sysem model fom [4] [9] whee he eceived signal a he U is y = h T s + η BS + η U + n. The diffeence fom he convenional model in is he addiive disoion noise ems η BS C and η U C, which ae egodic sochasic pocesses ha descibe he esidual ansceive impaimens of he ansmie hadwae a he BS and he eceive hadwae a U, especively. We assume
4 ha hese ae independen of he signal s, bu depend on he channel h and hus ae saionay only wihin each coheence peiod. 3 In paicula, we conside he condiional disibuions fo given chan- and η U C 0, Υ BS and η U C 0, υ U nel ealizaions H. The Gaussian disibuions of η BS η BS have been veified expeimenally see e.g., [7, Fig. 4.3] and can be moivaed analyically by he cenal limi heoem he disoion noises descibe he aggegae effec of many esidual hadwae impaimens. A key popey is ha he disoion noise caused a an anenna is popoional o he signal powe a his anenna see [5] [7] fo expeimenal veificaions, hus we have Υ BS υ U = κ BS diagw,..., W 3 = κ U h T Wh 4 whee W ii is he ih diagonal elemen of W and κ BS, κ U 0 ae he popoionaliy coefficiens. The inuiion is ha a fixed poion of he signal is uned ino disoion; fo example, due o quanizaion eos in auomaic-gain-conolled analog-odigial convesion ADC, ine-caie inefeence induced by phase noise, leakage fom he mio subcaie unde I/Q imbalance, and ampliude-ampliude nonlineaiies in he powe amplifie [4], [], [3]. The popoionaliy coefficiens ae eaed as consans in he analysis, bu can geneally incease wih he signal powe; see Secion VII-B fo deails. Remak Disoion oise and VM. Disoion noise is an aleaion of he useful signal, while he classical eceive noise models andom flucuaions in he eleconic cicuis a he eceive. A main diffeence is hus ha he disoion noise powe is non-saionay since i is popoional o he signal powe p BS and he cuen channel gain h. The popoionaliy coefficiens κ BS and κ U chaaceize he levels of impaimens and ae elaed o he eo veco magniude VM [5]; fo example, he VM a he BS is defined as VM BS = η BS H s H = Υ BS W = κ BS. 5 The VM is a common qualiy measue of ansceives and he 3GPP LT sandad specifies oal VM equiemens in he ange [0.08, 0.75], whee highe specal efficiencies modulaions ae suppoed if he VM is smalle [3, Sec. 4.3.4]. LT ansceives ypically suppo all he sandadized modulaions, hus he VM is below 0.08. Lage VMs ae, howeve, of inees in massive MIMO sysems since such elaxed hadwae consains enable he use of low-cos equipmen. Theefoe, he simulaions in his pape conside κ-paamees in he ange [0, 0.5 ], whee small values epesen accuae and expensive ansceive hadwae. The sysem model in capues he main chaaceisics of non-ideal hadwae, in he sense ha i allows us o idenify some fundamenal diffeences in he behavio of massive 3 These ae model assumpions ha oiginae fom he expeimenal woks of [5] [7]. An analyic moivaion of he assumpions which should no be misinepe as a poof can be obained fom he Bussgang heoem; see Secion VII. MIMO sysems as compaed o he case of ideal hadwae. Howeve, i canno capue all pacical chaaceisics of esidual ansceive hadwae impaimens. Possible efinemens, and hei especive implicaions on ou analyical esuls and obsevaions, ae oulined in Secion VII. C. Uplink Sysem Model The ecipocal UL channel is used fo pilo-based channel esimaion and daa ansmission; see Fig. and Secions III IV. Simila o, we conside a sysem model wih he eceived signal z C a he BS being z = hd + η U + η BS + ν 6 whee d C is eihe a deeminisic pilo signal used fo channel esimaion o a sochasic daa signal; in any case, he aveage powe is p U = d. The addiive em ν = ν noise + ν inef C is an egodic pocess ha consiss of independen eceive noise ν noise C 0, σbs I as well as poenial inefeence ν inef fom ohe simulaneous ansmissions. The inefeence is independen of d bu migh depend on he channel ealizaions in H. Moeove, he inefeence saisics can be diffeen in he pilo and daa ansmission phases; fo example, i is common o assume ha each cell uses ime-division muliple access TDMA fo pilo ansmission, since his can povide sufficien CSI accuacy o enable spaial-division muliple access SDMA fo daa ansmission [9] [3]. Theefoe, we assume ha ν inef has zeo mean and S = ν inef ν H inef is ha he covaiance maix duing pilo ansmission. We assume ha S has a unifomly bounded specal nom, S = O, fo he same physical easons as fo R. Fo daa ansmission, we define he condiional covaiance maix Q H = ν inef ν H inef H, in a coheence peiod wih channel ealizaions H, and he coesponding long-em covaiance maix Q H. The covaiance maices S, Q H C ae posiive semi-definie. The specal nom of Q H migh gow unboundedly wih due o pilo conaminaion in muli-cell scenaios [9] [3]; see Secion VI fo fuhe deails. Simila o he DL, he aggegae esidual ansceive impaimens in he hadwae used fo UL ansmission ae modeled by he independen disoion noises η U C and η BS C a he ansmie and eceive, especively. These egodic sochasic pocesses ae independen of d, bu depend on he channel ealizaions H. The condiional disibuion fo a given H ae η U C 0, υ U and η BS C 0, Υ BS. Simila o 3 and 4, he condiional covaiance maices ae modeled as υ U Υ BS = κ U p U 7 = κ BS p U diag h,..., h. 8 oe ha he hadwae qualiy is chaaceized by κ BS a he BS and by κ U, κ U a he U. We can have κ BS and κ U κ U, κ BS κ BS since diffeen ansceive chains ae used fo ansmission and ecepion a a device. Geneally speaking, we would like o achieve high pefomance using cheap hadwae. This is paiculaly eviden in massive MIMO sysems since he deploymen cos of lage
5 anenna aays migh scale linealy wih unless we can accep highe levels of impaimens, κ BS, κ BS, a he BSs han in convenional sysems. This aspec is analyzed in Secion V. III. UPLIK CHAL STIMATIO This secion consides esimaion of he cuen channel ealizaion h by compaing he eceived UL signal z in 6 wih he pedefined UL pilo signal d ecall: p U = d. The classic esuls on pilo-based channel esimaion conside Rayleigh fading channels ha ae obseved in independen complex Gaussian noise wih known saisics [3] [35]. Howeve, his is no he case heein because he disoion noises and η BS effecively depend on he unknown sochasic channel h. The dependence is eihe hough he muliplicaion η U hη U o he condiional vaiance of η BS in 8, which is essenially he same ype of elaion. Alhough he disoion noises ae Gaussian when condiioned on a channel ealizaion, he effecive disoion is he poduc of Gaussian vaiables and, hus, has a complex double Gaussian disibuion [36]. 4 Consequenly, an opimal channel esimao canno be deduced fom he sandad esuls povided in [3] [35]. We now deive he linea minimum mean squae eo LMMS esimao of h unde hadwae impaimens. Theoem. The LMMS esimao of h fom he obsevaion of z in 6 is ĥ = d R Z z 9 A whee R diag = diag,..., consiss of he diagonal elemens of R and he covaiance maix of z is denoed as Z = zz H = p U + κ U R + p U κ BS R diag + S + σbsi. 0 The oal MS is MS = ĥ h = C, whee he eo covaiance maix is C = ĥ hĥ hh = R p U R Z R. Poof: The LMMS esimao has he fom ĥ = Az whee A minimizes he MS. The MS definiion gives MS = R dar d RA H + A ZA H whee he expecaions ha involve η U, η BS in MS = ĥ h ae compued by fis having a fixed value of h and hen aveage ove h. The LMMS esimao in 9 is achieved by diffeeniaion of wih espec o A and equaing o zeo. This veco minimizes he MS since he Hessian is always posiive definie. The eo covaiance maix and he MS ae obained by plugging 9 ino he especive definiions. Based on Theoem, he channel can be decomposed as h = ĥ + ɛ whee ĥ is he LMMS esimae in 9 and 4 Fo example, he ih elemen of η BS can be expessed as x i = h i ξ i, which is he poduc of he ih channel coefficien h i C 0, ii and an independen vaiable ξ i C 0, κ BS pu. The join poduc disibuion is complex double Gaussian wih he PDF fx i = K πµ 0 x i i µi, whee µ i = ii κ BS p U is he vaiance and K 0 denoes he zeoh-ode modified Bessel funcion of he second kind [36]. ɛ C denoes he unknown esimaion eo. Conay o convenional esimaion wih independen Gaussian noise cf. [3, Chape 5.8], ĥ and ɛ ae neihe independen no joinly complex Gaussian, bu only uncoelaed and have zeo mean. The covaiance maices ae ĥĥh = R C and ɛɛ H = C whee C was given in. We emak ha hee migh exis non-linea esimaos ha achieve smalle MSs han he LMMS esimao in Theoem. This sands in conas o convenional channel esimaion wih independen Gaussian noise, whee he LMMS esimao is also he MMS esimao [34]. Howeve, he diffeence in MS pefomance should be small, since he dependen disoion noises ae elaively weak. Coollay. Conside he special case of R = λi and S = 0. The eo covaiance maix in becomes p U λ C = λ I. 3 lim C = λ p U p U λ + κ U + κ BS In he high UL powe egime, we have + σ BS + κ U + κ BS I. 4 This coollay bings impoan insighs on he aveage esimaion eo pe elemen in h. The eo vaiance is given by he faco in fon of he ideniy maix in 3. I is independen of he numbe of anennas, hus leing gow lage neihe inceases no deceases he esimaion eo pe elemen. 5 The esimaion eo is clealy a deceasing funcion of he pilo powe p U = d, bu conay o he ideal hadwae case he eo vaiance is no conveging o zeo as p U. As seen in 4, hee is a sicly posiive eo floo of λ due o he ansceive hadwae +κ U +κ BS impaimens. Thus, pefec esimaion accuacy canno be achieved in pacice, no even asympoically. The eo floo is chaaceized by he sum of he levels of impaimens κ U and κ BS in he ansmie and eceive hadwae, especively. In ems of esimaion accuacy, i is hus equally impoan o have high-qualiy hadwae a he BS and a he U. on-ideal hadwae exhibis an eo floo also when R is non-diagonal and when hee is inefeence such ha S 0; he geneal high-powe limi is easily compued fom. In fac, he esuls hold fo any zeo-mean channel and inefeence disibuions wih covaiance maices R and S, because he LMMS esimao and is MS ae compued using only he fis wo momens of he saisical disibuions [3], [34]. A. Impac of he Pilo Lengh The LMMS esimao in Theoem consides a scala pilo signal d, which is sufficien o excie all channel dimensions in he UL and is used in Secion IV-B o deive lowe bounds on he UL and DL capaciies. Wih ideal hadwae and a oal pilo enegy consain, a scala pilo signal is also sufficien o minimize he MS [34]. In conas, we have non-ideal 5 The MS pe elemen is finie, i.e. C <, bu he sum MS behaves as C when since he numbe of elemens gows.
6 hadwae and pe-symbol enegy consains in his pape. In his case we can impove he MS by inceasing he pilo lengh. Suppose we use a pilo signal d C B ha spans B Tpilo UL channel uses and whee each elemen of d has squaed nom p U. A simple esimaion appoach would be o compue B sepaae LMMS esimaes, ĥi = h ɛ i fo i =,..., B, using Theoem. By aveaging, we obain ĥ = B B ĥ i = h B i= B ɛ i. 5 i= If he disoion noises ae empoally uncoelaed and idenically disibued, he MS of he esimae ĥ is B H B ɛ i ɛ j B B = C B. 6 i= j= Hence, he MS goes o zeo as /B when we incease he pilo lengh B, alhough he MS pe pilo channel use is limied by he non-zeo eo floo demonsaed in Coollay. This is ineesing because one pilo signal wih enegy Bp U exhibis a noise floo, while B pilo signals wih enegy p U pe signal does no. 6 This sands in conas o he case of ideal hadwae, whee he MS is exacly he same in boh cases [34]. The eason is ha we can aveage ou he disoion noise simila o he law of lage numbes when we have B independen ealizaions. Despie he aveaging effec, we sess ha B T cohe and hus hee is always an esimaion eo floo fo nonideal hadwae we can, a mos, educe he floo by a faco /T cohe by inceasing he pilo lengh. Moeove, he deivaion above is based on having empoally uncoelaed disoions, bu he disoions migh be empoally coelaed in pacice especially if he same pilo signal d is ansmied muliple imes hough he same hadwae. In hese cases, he benefi of inceasing B is smalle and ĥ should be eplaced by an esimao ha explois he empoal coelaion by esimaing h joinly fom all he B obsevaions. Finally, we noe ha i is of gea inees o find he B ha maximizes some measue of sysem-wide pefomance, bu his is ouside he scope of ou cuen pape. We efe o [34], [35], [37], [38] fo some elevan woks in he case of ideal hadwae. B. umeical Illusaions This secion exemplifies he impac of ansceive hadwae impaimens on he channel esimaion accuacy. In Fig. 3, we conside = 50 anennas a he BS and no inefeence i.e., S = 0. The channel covaiance maix R is geneaed by he exponenial coelaion model fom [39], which means ha he i, jh elemen of R is δ j i, i j, [R] i,j = δ i j 7, i > j, 6 Since we have pe-symbol enegy consains, wha we eally compae is one sysem ha has an aveage symbol enegy of Bp U and one wih p U. whee δ is an abiay scaling faco. This model basically descibes a unifom linea aay ULA whee he coelaion faco beween adjacen anennas is given by fo 0 and he phase descibes he angle of aival/depaue as seen fom he aay. The coelaion faco deemines he eigenvalue spead in R, while deemines he coesponding eigenvecos. Since we simulae channel esimaion wihou inefeence, he angle has no impac on he MS and we can le be eal-valued wihou loss of genealiy. We conside a coelaion coefficien of = 0.7, which is a modes coelaion in he sense of behaving similaly o an aay wih half-wavelengh anenna spacings and a lage angula spead of 45 degees cf. [40, Fig. ] which shows how pacical angula speads map non-linealy o. Fig. 3 shows he elaive esimaion eo pe channel elemen, MS el = MS R, as a funcion of he aveage SR in he UL, defined as SR UL U R = p. 8 σ BS Based on he ypical VM anges descibed in Remak, we conside fou hadwae seups wih diffeen levels of impaimens: κ U = κ BS 0, 0.05, 0., 0.5. We compae he LMMS esimao in Theoem wih he convenional impaimen-ignoing MMS esimao fom [3] [34]. 7 Fig. 3 confims ha hee ae non-zeo eo floos a high SRs, as poved by Coollay and he subsequen discussion. The eo floo inceases wih he levels of impaimens. The esimaion eo is vey close o he floo when he uplink SR eaches 0 30 db, hus fuhe incease in SR only bings mino impovemen. This ells us ha we need an uplink SR of a leas 0 db o fully uilize massive MIMO, because coheen ansmission/ecepion equies accuae CSI. Lowe SRs can be compensaed by adding exa anennas see Fig. 6 in Secion IV, bu he pacical pefomance no as lage. Moeove, Fig. 3 shows ha he convenional impaimenignoing esimao is only slighly wose han he poposed LMMS esimao. This indicaes ha alhough hadwae impaimens gealy affec he esimaion pefomance, i only bings mino changes o he sucue of he opimal esimao. The influence of he esimaion eo floos depend on he anicipaed specal efficiency, he uplink SR, and he numbe of anennas. To gain some insigh, suppose we have ideal hadwae and ha he facion of channel uses allocaed fo UL daa ansmission is Tdaa UL /T cohe = 0.45. The uplink specal efficiency can hen be appoximaed as 0.45 log + MS el MS el + SR UL 9 by using [4, Lemma ]. When he numbe of anennas is lage, such ha SR UL, his appoximaion gives a specal efficiency of.5 [bi/channel use] fo MS el =0 and 4.5 [bi/channel use] fo MS el = 0 3. The impac of he esimaion eos on sysems wih non-ideal hadwae is 7 oe ha he MS of any linea esimao Ãz can be compued by plugging he maix à ino he geneal MS expession in. The diffeence in MS is easily quanified by compaing wih C using.
7 Relaive simaion o pe Anenna 0 0 0 0 0 3 0 4 Convenional Impaimen Ignoing LMMS simao o Floos κ U = κ BS = 0.5 κ U = κ BS = 0 κ U = κ BS = 0. κ U = κ BS = 0.05 0 5 0 5 0 5 30 35 40 Aveage SR [db] Fig. 3: simaion eo pe anenna elemen fo he LMMS esimao in Theoem and he convenional impaimenignoing MMS esimao. Tansceive hadwae impaimens ceae non-zeo eo floos. Relaive simaion o pe Anenna 0 0 0 0 0 3 5 db 30 db Fully Coelaed Disoion oise Uncoelaed Disoion oise Ideal Hadwae 0 4 3 4 5 6 7 8 9 0 Pilo Lengh B Fig. 4: simaion eo pe anenna elemen fo he LMMS esimao in Theoem as a funcion of he pilo lengh B. The levels of impaimens ae κ U = κ BS = 0.05 and diffeen empoal coelaions ae consideed. consideed in Secion IV, whee we deive lowe and uppe capaciy bounds and analyze hese fo diffeen SRs and numbe of anennas. ex, we illusae he possible impovemen in esimaion accuacy by inceasing he pilo lengh o compise B channel uses. As discussed in Secion III-A, i is no clea whehe he disoion noise is empoally uncoelaed o coelaed in pacice. Theefoe, we fix he levels of impaimens a κ U = κ BS = 0.05 and conside he wo exemes: empoally uncoelaed and fully coelaed disoion noises. The lae means ha he disoion noise ealizaions ae he same fo all B channel uses, since he same pilo signal is always disoed in he same way. The channel and inefeence saisics ae as in he pevious figue i.e., = 50, S = 0, and R is given by he exponenial coelaion model wih = 0.7. The elaive esimaion eo pe anenna elemen is shown in Fig. 4 as a funcion of he pilo lengh. We also show he pefomance wih ideal hadwae as a efeence. A a low SR of 5 db, hadwae impaimens have lile impac and hee is a small bu clea gain fom inceasing he pilo lengh because he oal pilo enegy inceases as Bp U. A a high SR of 30 db, he empoal coelaion has a lage impac. Only small impovemens ae possible in he fully coelaed case, since only he eceive noise can be miigaed by inceasing B. In he uncoelaed case he disoion noise can be also miigaed by inceasing B. This gives a logaihmic slope simila o he case of ideal hadwae. We sess ha he acual pefomance lies somewhee in beween he wo exemes. ex, we conside diffeen channel covaiance models: Uncoelaed anennas R = I. quivalen o he exponenial coelaion model in 7 wih = 0. xponenial coelaion model wih = 0.7. 3 One-ing model wih 0 degees angula spead [4]. 4 One-ing model wih 0 degees angula spead [4]. The exponenial coelaion model was defined in 7. The classic one-ing model assumes a ing of scaees aound he U, while hee is no scaeing close o he BS [4]. Fom he BS pespecive, he mulipah componens aive fom a main angle of aival hee: 30 degees and a small angula spead aound i hee: 0 o 0 degees. The BS is assumed o have a ULA wih half-wavelengh anenna spacings. An impoan popey of his model is ha R migh no have full ank as gows lage [43], [44], due o insufficien scaeing. The elaive esimaion eo pe channel elemen is shown in Fig. 5 fo hese fou channel covaiance models. We conside wo SRs 5 and 30 db, hadwae impaimens wih = 0.05, and show he esimaion eos as a funcion of he numbe of BS anennas. The main obsevaion fom Fig. 5 is ha he choice of covaiance model has a lage impac on he esimaion accuacy. I was poved in [34] ha spaially coelaed channels ae easie o esimae and his is consisen wih ou esuls; inceasing he coefficien in he exponenial coelaion model and deceasing he angula spead in he one-ing model lead o highe spaial coelaion and smalle eos in Fig. 5. Howeve, he eo floos due o hadwae impaimens make he diffeence beween he models educe wih he SR. Moeove, he esimaion eo pe anenna is viually independen of in he exponenial coelaion model, while inceasing impoves he eo pe anenna in he one-ing model. This is explained by he limied ichness of he popagaion envionmen in he one-ing model, which is a physical popey ha we can expec in pacice. κ U = κ BS Remak Acquiing Lage Covaiance Maices. The poposed channel esimao equies knowledge of he covaiance maices R and S. I becomes inceasingly difficul o acquie consisen esimaes of covaiance maices as hei dimensions gow lage [45]. Founaely, he channel saisics have a much lage coheence ime and coheence bandwidh han he channel ealizaion iself; hus, one can obain many moe obsevaions in he covaiance esimaion han in channel veco esimaion. Robus covaiance esimaos fo
8 Relaive simaion o pe Anenna 0 0 0 0 Case : Uncoelaed Case : xponenial Mod. = 0.7 Case 3: One-Ring, 0 degees Case 4: One-Ring, 0 degees 5 db 30 db 0 3 0 50 00 50 00 umbe of Base Saion Anennas Fig. 5: simaion eo pe anenna elemen fo he LMMS esimao in Theoem as a funcion of he numbe of BS anennas. Fou diffeen channel covaiance models ae consideed and κ U = κ BS = 0.05. lage maices wee ecenly consideed in [46]. The impac of impefec covaiance infomaion on he channel esimaion accuacy was analyzed in [47]. The auhos obseve ha he usual impovemen in MS fom having spaial coelaion vanishes if he covaiance infomaion canno be used, bu he MS degadaion is ohewise small if he esimaed covaiance maices ae obusified. Anohe poblem is ha he lage-dimensional maix invesion in 9 is vey compuaionally expensive, bu [47] poposed low-complexiy appoximaions based on polynomial expansions. Insead of acquiing he covaiance maix of a use diecly, he coveage aea can be divided ino locaion bins wih appoximaely he same channel saisics wihin each bin [48]. By acquiing and soing he covaiance maices fo each bin in advance, i is sufficien o esimae he locaion of a use and hen associae he use wih he coesponding bin. IV. DOWLIK AD UPLIK DATA TRASMISSIO This secion analyzes he egodic channel capaciies of he downlink in and he uplink in 6, unde he fixed TDD poocol depiced in Fig.. Moe pecisely, we deive uppe and lowe capaciy bounds ha eveal he fundamenal impac of non-ideal hadwae. These bounds ae based on having pefec CSI i.e., exac knowledge of h and impefec pilo-based CSI esimaion using he LMMS esimao in Theoem, especively. Since hese ae wo exemes, he capaciy bounds hold when using he channel esimaion echnique poposed in Secion III and fo any bee CSI acquisiion echnique ha can be deived in he fuue. We now define he DL and UL capaciies fo abiay CSI qualiy a he BS and U. We conside he egodic capaciy in bi/channel use of he memoyless DL sysem in. In each coheence peiod, he BS has some abiay impefec knowledge H BS of he cuen channel saes H and uses i o selec he condiional disibuion fs H BS of he daa signal s. The U has a sepaae abiay impefec knowledge H U of he cuen channel saes H and uses i o decode he daa. Based on he well-known capaciy expessions in [49], he egodic DL capaciy is C DL = T daa DL max Is; y H, H BS, H U T cohe fs H BS : s pbs 0 whee Is; y H, H BS, H U denoes he condiional muual infomaion beween he eceived signal y and daa signal s fo a given channel ealizaion H and given channel knowledge H BS and H U. The expecaion in 0 is aken ove he join disibuion of H, H BS, and H U. oe ha he faco Tdaa DL /T cohe is he fixed facion of channel uses allocaed fo DL daa ansmission. In addiion, he egodic capaciy in bi/channel use of he memoyless fading UL sysem in 6 is C UL = T UL daa T cohe max Id; z H, H BS, H U fd H U : d p U whee Id; z H, H BS, H U denoes he condiional muual infomaion beween he eceived signal z and daa signal d fo a given channel ealizaion H and given channel knowledge and H U. The condiional pobabiliy disibuion of he daa signal is denoed fd H U and he expecaion in is aken ove he join disibuion of H, H BS, H U. The facion of channel uses allocaed fo UL daa ansmission is H BS and covaiance maix Q H depend on he channel ealizaions H and change beween coheence peiods. We ae no limiing he analysis o any specific inefeence models bu ake cae of i in he capaciy bounds; he lowe bounds ea he inefeence as Gaussian noise, while he uppe bounds assume pefec inefeence suppession. Secion VI descibes he inefeence in muli-cell scenaios in deail. Secondly, we assume ha he disoion noises ae empoally independen, which is a good model when he daa signals ae also empoally independen. The nex subsecions sudy he capaciy behavio in he limi of infiniely many BS anennas, which bing insighs on how hadwae impaimens affec channels wih lage anenna aays. The DL and UL ae analyzed side-byside since he esuls follow fom simila deivaions. Tdaa UL /T cohe. Thee ae a few implici popeies in he capaciy definiions. Fisly, he inefeence vaiance I U H A. Uppe Bounds on Channel Capaciies Uppe bounds on he capaciies in 0 and can be obained by adding exa channel knowledge and emoving all inefeence i.e., IH U = 0 and Q H = 0. We assume ha he UL/DL pilo signals povide he BS and U wih pefec channel knowledge in each coheence peiod: H BS = H U = H. Since he eceive noise and disoion noises in and 6 ae ciculaly symmeic complex Gaussian disibued and independen of he useful signals unde pefec CSI, we deduce ha Gaussian signaling is opimal in he DL and UL [] and ha single-seam ansmission wih ankw = is sufficien o achieve opimaliy [6]; ha is, we can se s = ws
9 fo s C 0, p BS and some uni-nom beamfoming veco w in he DL and d C 0, p U in he UL. This gives us he following iniial uppe bounds. Lemma. The downlink and uplink capaciies in 0 and, especively, ae bounded as C DL T daa DL T cohe log + h H κ BS D h + κ U hh H + σ U p BS I h C UL T daa UL 3 T cohe log + h H κ U hh H + κ BS D h + σ BS p U I h whee D h = diag h,..., h wih h = [h... h ] T. These uppe bounds ae achieved wih equaliy unde pefec CSI, using he beamfoming veco w DL uppe = κbs D h + σ U p I h BS κ BS D h + σ U p I 4 h BS in he downlink and by applying a eceive combining veco in he uplink. 8 w UL uppe = κbs D h + σ BS p I h U κ BS D h + σ BS p I h. 5 U Poof: The poof is given in Appendix C-A. oe ha he beamfoming veco in 4 and eceive combining veco in 5 only depend on he channel veco h, hadwae impaimens a he BS, and he eceive noise. Hadwae impaimens a he U have no impac on wuppe DL and wuppe UL since hei disoion noise essenially ac as an inefee wih he same channel as he daa signal; hus fileing canno educe i. The bounds in Lemma ae no amenable o simple analysis, bu he lemma enables us o deive fuhe bounds on he channel capaciies ha ae expessed in closed fom. Theoem. The downlink and uplink capaciies in 0 and, especively, ae bounded as C DL C DL uppe = T daa DL log T + cohe + C UL C UL uppe = T daa UL log T cohe G DL G DL G UL + κ U + κ U G UL 6 7 8 A eceive combining veco w is a linea file w H z ha ansfoms he sysem ino an effecive single-inpu single-oupu SISO sysem. whee,..., ae he diagonal elemens of R, σ G DL = κ BS σ U e U p BS κ BS ii σ p BS κ BS U, ii p BS κ BS ii G UL = i= i= κ BS and x = σ σ BS e BS p U κ BS ii p U κ BS ii σbs ii p U κ BS 8, e x d denoes he exponenial inegal. 9 Poof: The poof is given in Appendix C-B. These closed-fom uppe bounds povide impoan insighs on he achievable DL and UL pefomance unde ansceive hadwae impaimens. In paicula, he following wo coollaies povide some ulimae capaciy limis in he asympoic egimes of many BS anennas o lage ansmi powes. Coollay. The downlink uppe capaciy bound in 6 has he following asympoic popeies: lim C DL p BS uppe = T DL daa log T + cohe κ BS 30 + κ U uppe = T daa DL. 3 lim CDL T cohe log + κ U Poof: The diagonal elemens of R saisfy ii > 0 i, by definiion, hus G DL i= = as p BS κ BS κ BS fo fixed, giving 30. The posiive diagonal elemens also implies GDL G > 0 as, hus DL +κ GDL U GDL κ U 0 as which uns 6 ino 3. GDL This coollay shows ha he DL capaciy has finie ceilings when eihe he DL ansmi powe p BS o he numbe of BS anennas gow lage. The ceilings depend on he impaimen paamees κ BS and κ U, bu he U impaimens ae clealy imes moe influenial. oe ha even vey small hadwae impaimens will ulimaely limi he capaciy. In ohe wods, he eve-inceasing capaciy obseved in he high-sr and lage- egimes wih ideal ansceive hadwae cf. [9] [3] is no easily achieved in pacice. The nex coollay povides analogous esuls fo he UL. Coollay 3. The uplink uppe capaciy bound in 7 has he following asympoic popeies: lim C UL p U uppe = T UL daa log T + cohe κ BS + κ U 3 uppe = T daa UL. 33 lim CUL T cohe log + κ U Poof: This is poved analogously o Coollay. As seen fom Coollay 3, he UL capaciy also has finie ceilings when eihe he UL ansmi powe p U o he numbe of anennas gow lage. Analogous o he DL, he U impaimens ae imes moe influenial han he BS impaimens and hus dominae as. The uppe bounds in Coollaies and 3 show ha he DL and UL capaciies ae fundamenally limied by he ansceive
0 hadwae impaimens. To be ceain of he cause of hese limis, we also need lowe bounds on he channel capaciies. B. Lowe Bounds on Channel Capaciies We obain lowe capaciy bounds by making he poenially limiing assumpions of Gaussian codebooks, eaing inefeence as Gaussian noise, using linea single-seam beamfoming in he DL, using linea eceive combining in he UL, pilo-based channel esimaion as in Theoem, and he enopy-maximizing Gaussian disibuion on he CSI unceainy a he eceive of he DL and UL. 9 The esuling lowe bound is given in he following heoem. Theoem 3. Le HU and H BS denoe he CSI available in he decoding a he eceive in he downlink and uplink, especively. These ae degaded as compaed o H U and H BS o equal. The downlink and uplink capaciies in 0 and, especively, ae hen bounded as C DL C DL lowe = T daa DL log T + SIR DL lowev DL 34 cohe C UL C UL lowe = T daa UL log T + SIR UL lowev UL 35 cohe whee he beamfoming veco v DL = [v DL... vk DL ] T and he eceive combining veco v UL = [v UL... vk UL ] T ae funcions U of ĥ and have uni noms. The expecaions ae aken ove H and H BS, while he SIR expessions fo DL and UL ae given in 36 and 37, especively, a he op of he nex page. Poof: This heoem is obained by aking lowe bounds on he muual infomaion in he same way as was peviously poposed in [5] and [4]. This bounding echnique was applied o massive MIMO sysems wih ideal hadwae in [] [3] among ohes, by making he limiing assumpions lised in he beginning of his subsecion. The disoion noises fom non-ideal hadwae ac as addiional noise souces wih spaially coelaed covaiance maices, hus hese can easily be incopoaed ino he poofs used in pevious woks. This heoem is he key o he lowe capaciy bounding in his pape. The lowe bounds in 34 and 35 can be compued numeically fo any channel disibuion and any way of selecing he beamfoming veco in he DL and eceive combining veco in he UL fom he channel esimae ĥ, povided ha he condiional disibuion of h, ĥ given H can be chaaceized. 0 To bing explici insighs on he behavio when he numbe of anennas,, gows lage, we have he following esul fo he cases of appoximae maximum aio ansmission MRT in he DL and appoximae maximum aio combining MRC in he UL. Theoem 4. Assume ha no insananeous CSI is uilized fo decoding i.e., HBS = H U =. Fo v = ĥ he ems in ĥ 9 The linea pocessing assumpion is moivaed by is asympoic opimaliy as [9]. 0 Finding such a chaaceizaion is a challenging ask, excep fo he case H BS = H U = consideed in Theoem 4. 36 and 37 behave as h H v = ϕ R C + O 38 h H v = ϕ R C + O 39 h i v i = O 40 i= whee ϕ = + d η U R C A d + η U R + ΨA H 4 is a funcion of he sochasic vaiable η U while A = d R Z and Ψ = p U κ BS R diag +S+σBS I ae deeminisic maices. Poof: The poof is given in Appendix C-C. Simila asympoic behavios wee deived in [] [3] fo he case of ideal hadwae. In he geneal case wih hadwae impaimens, he expecaions of ϕ and ϕ mus be compued numeically, because he andomness of he scala disoion a he U emains even when gows lage. In he special case of κ U = 0 which implies η U = 0, 38 and 39 boh educe o R C + O. Fo κ U > 0, he ems in Theoem 4 ae easy o compue numeically. Based on his esul, we povide now an asympoic chaaceizaion of he downlink capaciy. noise η U Coollay 4. Conside he DL wih beamfoming veco v = ĥ and H U =. If I U ĥ H O n fo some n <, he lowe capaciy bound in 34 can be expessed as C DL T daa DL T cohe log + ϕ +O +κ U ϕ ϕ +O + n 4 and O n whee ϕ is given in 4. The ems O vanish when, while he ohe ems ae sicly posiive in he limi. Poof: The expession 4 is obained fom 34 by plugging in he expessions in Theoem 4 and muliplying each em by R C = = O. The p U R Z R I inefeence em becomes U H p BS R C =O. n Combining he uppe bound in Coollay wih he lowe bound in Coollay 4, we have a clea chaaceizaion of he DL capaciy behavio when. Boh bounds ae independen of κ BS in he limi, hus he ansmie hadwae of he BS plays lile ole in massive MIMO sysems. Conay o he uppe bound, he level of eceive hadwae impaimens We sess ha he assumpion in Theoem 4 ha decoding is pefomed wihou insananeous CSI is only made o enable closed-fom lowe bounds. The BS should ceainly exploi he channel esimae ĥ and he U migh eceive a downlink pilo signal ha enables esimaion of he effecive channel h H v DL. While his is elaively easy o handle wih ideal hadwae, whee he channel esimae and esimaion eo ae independen cf. [], he exension o non-ideal hadwae seems inacable due he saisical dependence beween he channel esimae and esimaion eo.
SIR DL SIR UL lowev DL = lowev UL = +κ U h H v DL H U +κ U h H v UL H BS h H v DL H U h H v DL H 36 U +κ BS h i vi DL H U + IU H H U p + σ BS U p BS i= h H v UL H BS h H v UL H BS +κ BS h i vi UL H BS + vul H Q H +σbs IvUL H BS p U i= 37 a he BS κ BS is pesen in he lowe bound 4, hough A and Ψ in ϕ. Howeve, he numeical esuls in Secion IV-C eveal ha he asympoic impac of BS impaimens is negligible also in he lowe bound. This can also be seen analyically in ceain cases; if κ U = 0 we ge ϕ = and heefoe lim CDL lowe = T DL daa T cohe log + κ U. 43 In his special case, he lowe bound acually appoaches he uppe bound in 3 asympoically, and any DL capaciy can be achieved by making κ U sufficienly small. The opposie is no ue; seing κ BS = 0 will no make he impac of U impaimens vanish. We heefoe conclude ha he DL capaciy limi is mainly deemined by he level of impaimens a he U, boh in he uplink esimaion κ U and he downlink ansmission κ U alhough he fome connecion was no visible in he uppe bound since i was based on pefec CSI. Fo he uplink, we have he following simila asympoic capaciy chaaceizaion. Coollay 5. Conside he UL wih eceive combining veco v = ĥ and H BS =. If Q H O n fo some n <, ĥ he lowe capaciy bound in 35 can be expessed as C UL T daa UL T cohe log + ϕ +O +κ U ϕ ϕ +O + n 44 whee ϕ is given in 4. The ems O and O n vanish when, while he ohe ems ae sicly posiive in he limi. Poof: The expession 44 is obained fom 35 by plugging in he expessions fom Theoem 4 and muliplying each em by R C = = O. The p U R Z R inefeence em becomes vh Q H v p U R C =O. n The uppe bound in Coollay 3 and he lowe bound in Coollay 5 povide a join chaaceizaion of he uplink capaciy when gows lage. The U impaimens manifes he behavio in boh bounds; he BS impaimens ae pesen in 4 since ϕ depends on A and Ψ, bu hei impac vanish when κ U 0. By making κ U appopiaely small, we can hus achieve any UL capaciy as gows lage. We heefoe conclude ha i is of main impoance o have high qualiy hadwae a he U, which is analog o ou obsevaions fo he DL. These obsevaions ae illusaed numeically in he nex subsecion and ae explained by he following emak. Remak 3 BS Impaimens Vanish Asympoically. The lowe and uppe bounds show ha i is he qualiy of he U s ansceive hadwae ha limis he DL and UL capaciies as. Thus, he deimenal effec of hadwae impaimens a he BS vanishes compleely, o almos compleely, when he numbe of BS anennas gows lage. This is, simply speaking, since he BS s disoion noises ae spead in abiay diecions in he -dimensional veco space while he inceased spaial esoluion of he aay enables vey exac ansmi beamfoming and eceive combining fo he useful signal. This is a vey pomising esul since lage aays ae moe pone o impaimens, due o implemenaion limiaions and he will o use anenna elemens of lowe qualiy o avoid having deploymen coss ha incease linealy wih. In conas, he U s disoion noises ae non-vanishing since hey behave as inefees wih he same effecive channels as he useful signals. Coollaies 4 and 5 assumed ha he ine-use inefeence saisfy IH U O n and Q H O n, especively, fo some n <. These condiions imply ha he inefeence ems only vanish asympoically if he scaling wih is slowe han linea. This is saisfied by egula inefeence which has consan vaiance i.e., n = 0, bu hee is a special ype of non-egula pilo conaminaed inefeence in muli-cell sysems ha scales linealy wih. This adds an addiional non-vanishing em o he denominaos of 4 and 44. We deail his scenaio in Secion VI. Finally, we sess ha he DL and UL capaciy bounds in Coollaies and 3, especively, have a vey simila sucue. The main diffeence is ha he UL is only affeced by UL hadwae impaimens i.e., κ U, κ BS, while he DL is affeced by boh DL and UL hadwae impaimens i.e., all κ-paamees due o he evese-link channel esimaion. C. umeical Illusaions ex, we illusae he lowe and uppe bounds on he capaciy ha wee deived ealie in his secion. We conside a scenaio wihou inefeence, Q H = S = 0 and IH U = 0, and define he aveage SRs as p U R and p BS R in σbs σu he UL and DL, especively. We conside diffeen fixed SR values, while we vay he numbe of anennas and he levels