aisics-based Anenna elecion for Muli-Access MIMO sysems Pallav udarshan, Huaiyu Dai, Brian L. Hughes Elecrical and Compuer Engineering Deparmen, Norh Carolina ae Universiy, Raleigh, NC, UA. psudars, huaiyu dai, blhughes}@ncsu.edu Absrac Receive anenna selecion based on aisical Channel Knowledge CK) is considered for a muliuser uplink sysem. For spaial muliplexing ransmission, we derive he opimal CK-based anenna selecion crieria ha maximizes he ergodic capaciy in he high signal o noise raio regime. Two differen receiver archiecures are considered. The firs receiver uses a MME fron-end o suppress he inerference, followed by ML decoding o opimally decode he desired user s daa sreams, and he second receiver performs a join ML deecion of all he users daa sreams. In a sufficienly correlaed channel, CK-based selecion gives gains close o exac channel knowledge-based selecion. We compare he performance of anenna selecion wih ha of full complexiy FC) receiver, ha uses all he available anennas for demodulaion. We show ha, unlike in he singleuser sysem, in a muli-access sysem, he capaciy achieved by a MME receiver wih anenna selecion can be higher han ha of a MME FC receiver. I. INTRODUCTION I is well esablished ha spaial muliplexing subsanially improves he sysem capaciy of a communicaion sysem, as compared o a single anenna sysem []. However, his comes a he expense of increased hardware and signal processing complexiy. Anenna selecion, which reduces he hardware coss of a muliple inpu muliple oupu MIMO) sysem, has gained lo of ineres recenly. Anenna selecion adapively selecs a subse of available anennas a he receiver ransmier) for down-conversion up-conversion). This reduces he number of RF chains required by a MIMO sysem, while mainaining mos of he advanages of using muliple anennas. The anenna selecion algorihms proposed in he lieraure opimize differen performance merics, like, informaion heoreic capaciy [6], [7], [8], [7], probabiliy of error [5], [], [4], ec. The work on anenna selecion can be furher classified depending upon wheher i uses he exac channel knowledge ECK) [6], [7], [8], [4], or saisical channel knowledge CK) [], [5], [6], [7]. In [6], [7], Gorokhov, e al. quanify he loss in he capaciy due o anenna selecion a he receiver when compared wih a full complexiy sysem ha uses all he available anennas). They also propose fas near-opimal pracical algorihms o compue he bes anenna subses. Based on he ECK, [8] derives he crieria for selecing he opimal subse of ransmi anennas o maximize he capaciy. A low complexiy, ECK-based anenna selecion algorihm ha minimizes he error probabiliy for a spaial muliplexing sysem wih linear zero-forcing ZF) receiver is considered in [4]. For a correlaed channel wih anenna selecion based on CK, he selecion crieria ha minimizes he probabiliy of error for differen receiver archiecures for a spaial muliplexing sysem is derived in [], [5], [6], [7]. In paricular, [], [5] derives he selecion crieria, based on he minimum probabiliy of error, for a maximum likelihood ML) receiver and a linear ZF receiver. While mos of he previous work on anenna selecion has focussed on deriving opimal selecion crieria, and on low complexiy fas algorihms for a single-user sysem, lile aenion has been given o anenna selecion for muliuser sysems. In [3], [4] he opimal signalling sraegy wih anenna selecion in presence of inerference is obained. For a muliuser sysem, he opimal selecion crieria ha maximizes he capaciy has no been looked a, o he bes of our knowledge. Careux, e al. show in [8] ha in an inerference-limied environmen, he capaciy of a MIMO sysem is no significanly more han ha achieved by a receiver employing smar anennas. However, a join archiecure, ha cancels inerference and deecs he desired daa sream, can significanly improve he performance of an inerference-limied MIMO sysem [9]. In his work, we derive he opimal receiver anenna selecion crieria for a join design in muliuser uplink environmen. We consider wo receiver archiecures: ) The receiver basesaion) employs MME processing o suppresses he inerference, followed by opimal ML decoding o esimae he desired user s daa sream. We refer o his archiecure as MME-ML receiver. ) The receiver performs a join ML esimaion o decode all he users daasreams. We refer o his as ML receiver. For hese wo sysems, we derive he crieria, based on CK, for choosing he selecion marix, such ha he pos-deecion expeced muual informaion is maximized. The res of he paper is organized as follows. ecion II inroduces he sysem and he channel model. In ecion III and IV, we derive he opimal selecion crieria for MME- ML receiver and ML receiver, respecively. We compare he performance of anenna selecion and full complexiy receiver in ecion V. imulaion resuls are presened in ecion VI and he conclusions follow in ecion VII.
II. YTEM AND CHANNEL MODEL Consider a muli-access MIMO sysem, in which each of he K users has anennas, and ransmis o a common basesaion having N r receive anennas. The received vecor, y, can be wrien as K ρ k y = H k x k + n, ) k= where ρ k is he signal o noise raio NR) of user k, H k is he N r channel marix of he fading pah gains from user k o he receiver, x k is he daa sream ransmied by user k and n is noise vecor, wih i. i. d. complex Gaussian enries N c 0, ). The receiver has perfec channel sae informaion CI), while he ransmier has no CI. We use he widely adoped Kroncker correlaion model for he channel, which assumes ha he correlaion a he receiver and he ransmier is independen of each oher. We assume ha he columns of H are uncorrelaed, i.e., here is no ransmier-side correlaion. This is a reasonable assumpion because he mobile saion is ypically locaed in a rich scaering environmen. Then, according o his model, he channel, H k can be wrien as H k = R k H w,k ) where he enries of H w,k are i. i. d. complex Gaussian N c 0, ), and R k is he receiver correlaion marix for he k h user. Assume ha he receiver has only L N r demodulaor chains, so ha he bes L ou of N r available anennas need o seleced for down-conversion and subsequen baseband processing. Le be he selecion marix ha selecs L rows from H. The reduced channel marix H k can be wrien as H k = R L,k H L,k where R L,k is a L L sub-marix of R k formed by removing N r L rows and he corresponding columns from R k, and H L,k is L marix wih i. i. d. complex Gaussian enries. Noe ha since we are considering receiver selecion for he uplink sysem, he same selecion marix is chosen for all he users. III. ANTENNA ELECTION FOR MME-ML RECEIVER We firs consider a receiver archiecure where an MME fron end is used o suppress he inerference from oher users, followed by ML deecion of desired user s daa sream. The ergodic capaciy of a full complexiy receiver ha uses all he available anennas) is given by C MME-FC = E I L + ρ H H ) K ρ i I + H i H i, 3) i= where user is he user of ineres. Here E.} denoes expecaion, and.) denoes he Hermiian of a marix. When he number of demod chains is less han he number of receive anennas, he ergodic capaciy afer selecion can be wrien as C MME-sel = max E I L + ρ H H ) K ρ i I + H i H i. 4) i= Wihou loss of generaliy, assume K =. Then he maximizaion problem in 4) becomes C MME-sel =max E I + ρ } H L, N R L, Q R L, H L, where Q is defined as Q = I L + ρ R L, N H L,H L, R L,. Our goal is o find he opimal channel saisics-based selecion marix,, such ha 4) is maximized, or equivalenly, find he sub-marices R L, and R L, such ha 5) is maximized. In general, i is difficul o solve he above opimizaion problem involving ergodic capaciy. Insead, we maximize he following lower bound, which can be derived following he seps in [0]. This bound is shown o be very igh in he high NR regime [0]. Lemma : Le H = R H w be a given such ha R is a full rank marix, and N r. Then, he ergodic capaciy of his channel can be lower bounded as C = E N r I N r + ρ } HH N + ρ exp 5) E HH } )). N r We make he following assumpions o derive he opimal selecion crieria. Assumpions : Assume ha R L, is a full rank marix and L, and ha user is operaing in he high NR regime. Under assumpions, we can apply Lemma on 5) o ge C MME-sel L + ρ exp L R L, H L, H E L, 6) Q = L + ρ R L, L β exp L E I L + ρ })) R L, H L, H L,, 7) where β is defined as β = exp E }) H w H w L. The expression in 7) can be furher lower bounded
by using Jensen s inequaliy on he concave funcion I L + ρ R L, H L, H L,. Then, we ge C MME-sel L + ρ R L, L β N exp )) L I L + ρ R L, 8) ) = L + ρ R L, L β. 9) I L + ρ R L, L Eqn. 9) is maximized when α = R L, I L +ρ R L, is maximized. Thus he following proposiion holds. Proposiion : The ergodic capaciy of a MME-ML receiver, given in 5), is maximized when R L, α = I L + ρ R L, is maximized. Here R L, and R L, are he receive correlaion marices of user and user, corresponding o he rows seleced from H. IV. ANTENNA ELECTION FOR ML RECEIVER For a full complexiy ML receiver, he ergodic capaciy is given by C ML-FC = E I L + ρ H H N + ρ } H H. 0) When he receiver selecs L ou of N r anennas, he ergodic capaciy can be wrien as C ML-sel = max E I L + ρ H H + ρ } H H. ) Nex, we sae a Lemma ha will help us simplify ). Lemma : Consider channel marices H and H, such ha H = R H L, and H = R H L,. Then H H + H H can be wrien as H H + H H = R eq H w H wr eq, where R eq = R + R and H w has i. i. d. complex Gaussian enries. Proof: Define HH = H H + H H. ince H and H have no ransmi correlaion, H can be wrien as H = R eq H w. Then he receive correlaion of H is HH } R eq = E } = E R H L, H L, R + R H L, H L, R = R + R } ) where ) follows by observing ha E H L, H L, = I. To derive he opimal selecion crieria, we make he following assumpions: ) Assumpions : Assume ha ρ R L, + ρ R L, is a full-rank marix and L, and ha he sysem is operaing in he high NR regime. Under hese assumpions, ) becomes C ML-sel = max E I ρ L + R + ρ R H w H ρ w R + ρ ) } R L + ρ R L, + ρ L R L, exp ) 3) L E H w H })) w, 4) where 3) follows from Lemma and 4) follows by applying Lemma on 3). Eqn. 4) is maximized when γ = ρ R L, + ρ R L, is maximized. Thus he following proposiion holds. Proposiion : The ergodic capaciy of a ML receiver, given in 3), is maximized when γ = ρ R L, + ρ R L, is maximized. Here R L, and R L, are he receive correlaion marices of user and user, corresponding o he rows seleced from H. V. RELATION BETWEEN C EL AND C FC In his secion, we show he impac of selecion on he ergodic capaciy of muli-access sysems using ML receiver and MME-ML receiver. Le e i AA ) denoe he i h larges eigenvalue of A. Then he capaciy of full complexiy ML receiver can be wrien as N r C ML-FC = i= [ + e i ρ H H + ρ H H )]. 5) Using he propery ha i h larges eigenvalue of a marix is greaer han he i h larges eigenvalue of is sub-marix [], we ge he following lower bound on C ML-FC : C ML-FC L i= [ + e i ρ H H + ρ H H ) )]. 6) Noing ha he righ-hand side of he inequaliy in 6) is C ML-sel, we conclude ha C ML-FC serves as an upper bound for C ML-sel, i.e. selecion necessarily reduces he capaciy of muli-user ML receiver. For a MME-ML receiver, such an inference canno be made. The full complexiy MME receiver employs a linear fron-end o miigae inerference. C MME-FC serves as an upper bound o he capaciy achieved by any linear fronend deecor. However, selecion is a non-linear operaion, and by appropriaely choosing he anenna subse, he ne inerference seen by he desired daa sream can be reduced.
TABLE I ERGODIC CAPACITY IN BIT//HZ) FOR MME-ML RECEIVER. 0.9,),3),3) ECK FC α.0036.006.0036 - - C MME-sel.07.48.05.5.33 TABLE II ERGODIC CAPACITY IN BIT//HZ) FOR ML RECEIVER.,),3),3) ECK FC γ.5 64.5.5 - - C ML-sel 6.49 7.55 6.54 7.55 8.47 0.8 0.7 0.6 0.5 0.4 0.3,) or,3),3) Full Complexiy ECK 0. Therefore, in a muli-user sysem, he link capaciy of MME- ML receiver can increase by processing he signal from a subse of anennas. This is in conras o a single-user sysem, where selecion a he receiver necessarily reduces he capaciy. VI. IMULATION REULT For he purpose of simulaions, we assume a uniform linear array ULA) wih anenna spacing d = 0.4λ, where λ is he wavelengh. The angle of arrival AoA) of user k follows normal disribuion N θ rk, σ rk ), where θ rk and σ rk are he mean AoA and he RM angle spread for user k. We use he characerizaion of R given in []. We compare he performance of various selecion marices for =, N r = 3 anennas and L = demod chains. The simulaion parameers are chosen as follows: θ r = 60, θ r = 75, σ r = 0, σ r = 0, ρ = 0dB and ρ = 0dB. A. MME Deecor Table I compares he ergodic capaciy for differen subses of anennas. Also shown are he values of α for hese subses. We see ha he opimal anenna subse,3) gives a 0.4 bis/s/hz selecion gain. We define selecion gain as he difference in he ergodic capaciy beween he bes and he wors anenna subses. Noe ha opimal CK-based selecion comes wihin 0.03 bis/s/hz of he ECK-based selecion. From Table I, we also see ha opimal anenna selecion gives a. bis/s/hz advanage over full complexiy receiver, which uses he signals received a all he anennas. This conforms wih he inuiion given in ecion V ha C MME-sel can be greaer han C MME-FC. We also plo he CDF of he capaciy Fig. ), which provides a complee characerizaion of capaciy, unlike ergodic capaciy, which provides informaion on only he firs momen. B. ML Deecor Table II compares he ergodic capaciy and γ for differen subses of anennas for ML receiver. We see ha we ge significan selecion gain of abou bis/s/hz. Noe ha he capaciy of FC receiver is greaer han ha of anenna selecion. Figure plos he CDF of capaciy for ML receiver. Noe ha his definiion is differen from he one used in [5]. elecion gain is defined in [5] as he difference in he required NR, a some arge symbol error rae, beween he bes and he wors anenna subses. 0. 0 0 3 4 5 6 7 Fig.. CDF of he capaciy MME-ML receiver We see ha he curves for opimal CK-based selecion and ECK-based selecion are indisinguishably close. VII. CONCLUION In his paper, we have derived he opimal receiver anenna selecion crieria for a muliple access sysem. We considered wo receiver archiecure, which employ differen muliuser deecors, and showed ha he opimal channel saisics-based selecion gives significan selecion gains. For a sufficienly correlaed channel, we showed ha he saisics-based soluion achieves capaciy close o ha of exac channel knowledgebased selecion. When MME fron end is employed o suppress muli-user inerference, opimally selecing a subse of anennas a he receiver can increase he capaciy of he muliuser sysem as compared o a full complexiy receiver ha uses all he available anennas. This is he main difference beween he behavior of muli-user anenna selecion and single-user anenna selecion. REFERENCE [] G. J. Foschini, M. J. Gans, On he Limis of Wireless Communicaions in a Fading Environmen When Using Muliple Anennas, Wireless Pers. Commun., Vol. 6, No. 3, pp. 3 335, Mar. 998. [] A. F. Molisch, M. Z. Win MIMO sysems wih Anenna elecion, IEEE Microwave Mag., Vol. 5, No., pp. 46 56, Mar. 004. [3] R.. Blum, J. H. Winers, On Opimum MIMO wih Anenna elecion, Proc. IEEE In. Conf. Commun., pp. 30 305, Anchorage, AK, May 003. [4] R.. Blum, MIMO Capaciy wih Anenna elecion and Inerference, Proc. ICAP, Vol. 4, Hong Kong, Apr. 03. [5] D. A. Gore, A. Paulraj, MIMO Anenna ubse elecion wih pace- Time Coding, IEEE Trans. ignal Processing, Vol. 50, No. 0, pp. 580 588, Oc. 00. [6] A. Gorokhov, D. Gore, A. Paulraj, Receive Anenna elecion for MIMO Fla-Fading Channels: Theory and Algorihms, IEEE Trans. Inform. Theory, Vol. 49, No. 0, pp. 687 696, Oc. 003. [7] A. Gorokhov, D. Gore, A. Paulraj, Receive Anenna elecion for MIMO paial Muliplexing: Theory and Algorihms, IEEE Trans. ignal Processing, Vol. 5, No., pp. 796 807, Nov. 003. [8] R. U. Nabar, D. Gore, A. J. Paulraj, Opimal elecion and Use of Transmi Anennas in Wireless ysems, Inernaional Conf. on Telecommunicaions, Acapulco, Mexico, May 000.
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