Spatially Sparse Precoding in Millimeter Wave MIMO Systems

Transcription

1 Spaially Sparse Precoding in Millimeer Wave MIMO Sysems Omar El Ayach, Sridhar Rajagopal, Shadi Abu-Surra, Zhouyue Pi, and Rober W. Heah, Jr. 1 arxiv: v1 [cs.it] 11 May 2013 Absrac Millimeer wave (mmwave) signals experience orders-of-magniude more pahloss han he microwave signals currenly used in mos wireless applicaions. MmWave sysems mus herefore leverage large anenna arrays, made possible by he decrease in wavelengh, o comba pahloss wih beamforming gain. Beamforming wih muliple daa sreams, known as precoding, can be used o furher improve mmwave specral efficiency. Boh beamforming and precoding are done digially a baseband in radiional muli-anenna sysems. The high cos and power consumpion of mixed-signal devices in mmwave sysems, however, make analog processing in he RF domain more aracive. This hardware limiaion resrics he feasible se of precoders and combiners ha can be applied by pracical mmwave ransceivers. In his paper, we consider ransmi precoding and receiver combining in mmwave sysems wih large anenna arrays. We exploi he spaial srucure of mmwave channels o formulae he precoding/combining problem as a sparse reconsrucion problem. Using he principle of basis pursui, we develop algorihms ha accuraely approximae opimal unconsrained precoders and combiners such ha hey can be implemened in low-cos RF hardware. We presen numerical resuls on he performance of he proposed algorihms and show ha hey allow mmwave sysems o approach heir unconsrained performance limis, even when ransceiver hardware consrains are considered. I. INTRODUCTION The capaciy of wireless neworks has hus far scaled wih he increasing daa raffic, primarily due o improved area specral efficiency (bis/s/hz/m 2 ) [1]. A number of physical layer Omar El Ayach and Rober Heah are wih The Universiy of Texas a Ausin, Ausin, TX USA ( oelayach, rheah@uexas.edu). Sridhar Rajagopal, Shadi Abu-Surra, and Zhouyue Pi are wih Samsung Research America - Dallas, Richardson, TX, USA ( {sasurra, srajagop, zpi}@sa.samsung.com) This work was done while he firs auhor was wih Samsung Research America - Dallas. The auhors a The Universiy of Texas a Ausin were suppored in par by he Army Research Laboraory conrac W911NF and Naional Science Foundaion gran This work has appeared in par a he 2012 IEEE Inernaional Communicaions Conference (ICC).

2 2 enhancemens such as muliple anennas, channel coding, and inerference coordinaion, as well as he general rend oward nework densificaion have all been insrumenal in achieving his efficiency [1], [2]. Since here seems o be lile scope for furher gains a he physical layer, and since he widespread deploymen of heerogeneous neworks is no wihou challenges [3], hese echniques alone may no be sufficien o mee fuure raffic demands. As a resul, increasing he specrum available for commercial wireless sysems, poenially by exploring new less-congesed specrum bands, is a promising soluion o increase nework capaciy. Millimeer wave (mmwave) communicaion, for example, has enabled gigabi-per-second daa raes in indoor wireless sysems [4], [5] and fixed oudoor sysems [6]. More recenly, advances in mmwave hardware [7] and he poenial availabiliy of specrum has encouraged he wireless indusry o consider mmwave for oudoor cellular sysems [8], [9]. A main differeniaing facor in mmwave communicaion is ha he en-fold increase in carrier frequency, compared o he curren majoriy of wireless sysems, implies ha mmwave signals experience an orders-ofmagniude increase in free-space pahloss. An ineresing redeeming feaure in mmwave sysems, however, is ha he decrease in wavelengh enables packing large anenna arrays a boh he ransmier and receiver. Large arrays can provide he beamforming gain needed o overcome pahloss and esablish links wih reasonable signal-o-noise raio (SNR). Furher, large arrays may enable precoding muliple daa sreams which could improve specral efficiency and allow sysems o approach capaciy [10], [11]. While he fundamenals of precoding are he same regardless of carrier frequency, signal processing in mmwave sysems is subjec o a se of non-rivial pracical consrains. For example, radiional muliple-inpu muliple-oupu (MIMO) processing is ofen performed digially a baseband, which enables conrolling boh he signal s phase and ampliude. Digial processing, however, requires dedicaed baseband and RF hardware for each anenna elemen. Unforunaely, he high cos and power consumpion of mmwave mixed-signal hardware precludes such a ransceiver archiecure a presen, and forces mmwave sysems o rely heavily on analog or RF processing [7], [8]. Analog precoding is ofen implemened using phase shifers [7], [8], [12] which places consan modulus consrains on he elemens of he RF precoder. Several approaches have been considered for precoding in such low-complexiy ransceivers [13] [28]. The work in [13] [15] considers anenna (or anenna subse) selecion which has he advanage of replacing phase shifers wih even simpler analog swiches. Selecion, however, provides

3 3 limied array gain and performs poorly in correlaed channels such as hose experienced in mmwave [16]. To improve performance over correlaed channels, he work [17] [20] considers beam seering soluions in which phase shifers or discree lens arrays are used o opimally orien an array s response in space, poenially based on saisical channel knowledge. The sraegies in [17] [20], however, are in general subopimal since beam seering alone canno perfecly capure he channels dominan eigenmodes. The work in [21] [26] develops ieraive precoding algorihms for sysems ha leverage analog processing, and [27] furher proposes simple analyical soluions. Furher hardware limiaions have also been considered in [28], for example, which focuses on analog receiver processing wih only quanized phase conrol and finie-precision analog-o-digial converers. The work in [21] [28], however, is no specialized o mmwave MIMO sysems wih large anenna arrays. Namely, he work in [21] [28] does no leverage he srucure presen in mmwave MIMO channels and adops models ha do no fully capure he effec of limied mmwave scaering and large ighly-packed arrays [29] [31]. In his paper, we focus on he precoding insigh and soluions ha can be derived from joinly considering he following hree facors: (i) precoding wih RF hardware consrains, (ii) he use of large anenna arrays, and (iii) he limied scaering naure of mmwave channels. We consider single-user precoding for a pracical ransceiver archiecure in which a large anenna array is driven by a limied number of ransmi/receive chains [8], [10], [11], [32]. In such a sysem, ransmiers have he abiliy o apply high-dimensional (all) RF precoders, implemened via analog phase shifers, followed by low-dimensional (small) digial precoders ha can be implemened a baseband. We adop a realisic clusered channel model ha capures boh he limied scaering a high frequency and he anenna correlaion presen in large ighly-packed arrays [29] [31]. We exploi he sparse-scaering srucure of mmwave channels o formulae he design of hybrid RF/baseband precoders as a sparsiy consrained marix reconsrucion problem [33] [38]. Iniial resuls on his precoding approach were presened in [39]. In his paper, we formalize he mmwave precoding problem and show ha, insead of direcly maximizing muual informaion, near-opimal hybrid precoders can be found via an opimizaion ha resembles he problem of sparse signal recovery wih muliple measuremen vecors, also known as he simulaneously sparse approximaion problem [40] [43]. We hus provide an algorihmic precoding soluion based on he concep of orhogonal maching pursui [34], [36], [44]. The algorihm akes an

4 4 opimal unconsrained precoder as inpu and approximaes i as linear combinaion of beam seering vecors ha can be applied a RF (and combined digially a baseband). Furher, we exend his sparse precoding approach o receiver-side processing and show ha designing hybrid minimum mean-square error (MMSE) combiners can again be cas as a simulaneously sparse approximaion problem and solved via basis pursui [45], [46]. We argue ha, in addiion o providing pracical near-opimal precoders, he proposed framework is paricularly amenable for limied feedback operaion and is hus no limied o genie-aided sysems wih perfec ransmier channel knowledge [47]. The generaed precoders can be efficienly compressed using simple scalar quanizers (for he argumens of he beam seering vecors) and low-dimensional Grassmannian subspace quanizers (used o quanize he baseband precoder) [47] [49]. We briefly describe he consrucion of he limied feedback codebooks required, bu defer he analysis of limied feedback performance o fuure work. Finally, we presen simulaion resuls on he performance of he proposed sraegy and show ha i allows mmwave sysems o approach heir unconsrained performance limis even when pracical ransceiver consrains are considered. We use he following noaion hroughou his paper: A is a marix; a is a vecor; a is a scalar; A (i) is he i h column of A; ( ) T and ( ) denoe ranspose and conjugae ranspose respecively; A F is he Frobenius norm of A, r(a) is is race and A is is deerminan; a p is he p-norm of a; [A B] denoes horizonal concaenaion; diag(a) is a vecor formed by he diagonal elemens of A; I N is he N N ideniy marix; 0 M N is he M N all-zeros marix; CN (a; A) is a complex Gaussian vecor wih mean a and covariance marix A. Expecaion is denoed by E[ ] and he real par of a variable is denoed by R { }. II. SYSTEM MODEL In his secion, we presen he mmwave signal and channel model considered in his paper. A. Sysem Model Consider he single-user mmwave sysem shown in Fig. 1 in which a ransmier wih N anennas communicaes N s daa sreams o a receiver wih N r anennas [32]. To enable mulisream communicaion, he ransmier is equipped wih N RF N RF ransmi chains such ha N s N. This hardware archiecure enables he ransmier o apply an N RF precoder F BB using is N RF ransmi chains, followed by an N N RF N s baseband RF precoder F RF using

5 5 analog circuiry. The discree-ime ransmied signal is herefore given by x = F RF F BB s where s is he N s 1 symbol vecor such ha E [ss ] = 1 N s I Ns. Since F RF is implemened using analog phase shifers, is elemens are consrained o saisfy (F (i) RF F(i) RF ) l,l = N 1, where ( ) l,l denoes he l h diagonal elemen of a marix, i.e., all elemens of F RF have equal norm. The ransmier s oal power consrain is enforced by normalizing F BB such ha F RF F BB 2 F hardware-relaed consrains are placed on he baseband precoder. = N s; no oher For simpliciy, we consider a narrowband block-fading propagaion channel as in [10], [19], [26], [32], which yields a received signal y = ρhf RF F BB s + n, (1) where y is he N r 1 received vecor, H is he N r N channel marix such ha E [ H 2 F ] = N N r, ρ represens he average received power, and n is he vecor of i.i.d CN (0, σ 2 n) noise. In wriing (1), we implicily assume perfec iming and frequency recovery. Moreover, o enable precoding, we assume ha he channel H is known perfecly and insananeously o boh he ransmier and receiver. In pracical sysems, channel sae informaion (CSI) a he receiver can be obained via raining [17], [50] and subsequenly shared wih he ransmier via limied feedback [47]; an efficien limied feedback sraegy is presened in Secion V. Techniques for efficien mmwave channel esimaion, and a rigorous reamen of frequency selecive mmwave channels, are sill an ongoing opic of research. The receiver uses is N s N RF r pos-processing received signal where W RF N r RF chains and is analog phase shifers o obain he ỹ = ρw BBW RFHF RF F BB s + W BBW RFn, (2) is he N r N RF r RF combining marix and W BB is he N RF r N s baseband combining marix. Similarly o he RF precoder, W RF is implemened using phase shifers and herefore is such ha (W (i) RF W(i) RF ) l,l = N r 1. When Gaussian symbols are ransmied over he mmwave channel, he specral efficiency achieved is given by [51] R = log 2 ( I Ns + ρ N s R 1 n W BBW RFHF RF F BB F BBF RFH W RF W BB ), (3) where R n = σ 2 nw BB W RF W RFW BB is he noise covariance marix afer combining.

6 6 Digial Analog Analog Digial RF Chain RF Chain N s Baseband Precoder F BB N RF F RF N N r W RF Nr RF Baseband Combiner W BB N s RF Chain RF Chain spliers Fig. 1. Simplified hardware block diagram of mmwave single user sysem wih digial baseband precoding followed by consrained radio frequency precoding implemened using RF phase shifers. B. Channel Model The high free-space pahloss ha is a characerisic of mmwave propagaion leads o limied spaial seleciviy or scaering. Similarly, he large ighly-packed anenna arrays ha are characerisic of mmwave ransceivers lead o high levels of anenna correlaion. This combinaion of ighly packed arrays in sparse scaering environmens makes many of he saisical fading disribuions used in radiional MIMO analysis inaccurae for mmwave channel modeling. For his reason, we adop a narrowband clusered channel represenaion, based on he exended Saleh-Valenzuela model, which allows us o accuraely capure he mahemaical srucure presen in mmwave channels [29], [30], [52], [53]. Using he clusered channel model, he marix channel H is assumed o be a sum of he conribuions of N cl scaering clusers, each of which conribue N ray propagaion pahs o he channel marix H. Therefore, he discree-ime narrowband channel H can be wrien as N N N cl N r ray H = α il Λ r (φ r N cl N il, θil)λ r (φ il, θil)a r (φ r il, θil)a r (φ il, θil), (4) ray i=1 l=1 where α il is he complex gain of he l h ray in he i h scaering cluser, whereas φ r il (θr il ) and φ il (θ il ) are is azimuh (elevaion) angles of arrival and deparure respecively. The funcions Λ (φ il, θ il ) and Λ r(φ r il, θr il ) represen he ransmi and receive anenna elemen gain a he corresponding angles of deparure and arrival. Finally, he vecors a r (φ r l, θr l ) and a (φ il, θ il )

7 7 represen he normalized receive and ransmi array response vecors a an azimuh (elevaion) angle of φ r il (θr il ) and φ il (θ il ) respecively. In Secion VI, we assume ha α il are i.i.d. CN (0, σ 2 α,i) where σ 2 α,i represens he average power of he i h cluser. The average cluser powers are such ha N cl i=1 σ2 α,i = γ where γ is a normalizaion consan ha saisfies E [ H 2 F ] = N N r [30]. The N ray azimuh and elevaion angles of deparure, φ il and θ il, wihin he cluser i are assumed o be randomly disribued wih a uniformly-random mean cluser angle of φ i and θ i respecively, and a consan angular spread (sandard deviaion) of σ φ and σ θ respecively. The azimuh and elevaion angles of arrival, φ r il and θ r il, are again randomly disribued wih mean cluser angles of (φr i, θ r i) and angular spreads (σ φ, σ θ r). While a variey of disribuions have been proposed for he angles of arrival and deparure in clusered channel models, he Laplacian disribuion has been found o be a good fi for a variey of propagaion scenarios [54], and will hus be adoped in he numerical resuls of Secion VI. Similarly, a number of paramerized mahemaical models have been proposed for he funcions Λ (φ il, θ il ) and Λ r(φ r il, θr il ). For example, if he ransmier s anenna elemens are modeled as being ideal secored elemens [55], Λ (φ il, θ il ) would be given by 1 φ Λ (φ il, θil) il = [φ min, φ max], θil [θ min, θmax], 0 oherwise, where we have assumed uni gain over he secor defined by φ l [φ min, φ max] and θ l [θmin, θmax] wihou loss of generaliy. The receive anenna elemen gain Λ r (φ r il, θr il ) is defined similarly over he azimuh secor φ r il [φr min, φ r max] and elevaion secor θ r il [θr min, θ r max]. The array response vecors a (φ il, θ il ) and a r(φ r l, θr l ) are a funcion of he ransmi and receiver anenna array srucure only, and are hus independen of he anenna elemen properies. While he algorihms and resuls derived in he remainder of his paper can be applied o arbirary anenna arrays, we give he following wo illusraive examples of commonly-used anenna arrays for compleeness. For an N-elemen uniform linear array (ULA) on he y-axis, he array response vecor can be wrien as [56] where k = 2π λ a ULAy (φ) = 1 N [ 1, e jkd sin(φ), e j2kd sin(φ),..., e j(n 1)kd sin(φ)] T, (6) and d is he iner-elemen spacing. Noe ha we do no include θ in he argumens of a ULAy as he array s response is invarian in he elevaion domain. In he case of a uniform (5)

8 8 planar array (UPA) in he yz-plane wih W and H elemens on he y and z axes respecively, he array response vecor is given by [56] a UPA (φ, θ) = 1 N ( [1,..., e jkd(m sin(φ) sin(θ)+n cos(θ)),...,..., e jkd((w 1) sin(φ)sin(θ)+(h 1) cos(θ)) ] ) T, where 0 m < W and 0 n < H are he y and z indices of an anenna elemen respecively and he anenna array size is N = W H. Considering uniform planar arrays is of ineres in mmwave beamforming since hey (i) yield smaller anenna array dimensions, (ii) faciliae packing more anenna elemens in a reasonably-sized array, and (iii) enable beamforming in he elevaion domain (also known as 3D beamforming). (7) III. SPATIALLY SPARSE PRECODING FOR THE SINGLE USER MMWAVE CHANNEL We seek o design hybrid mmwave precoders (F RF, F BB ) ha maximize he specral efficiency expression in (3). Direcly maximizing (3), however, requires a join opimizaion over he four marix variables (F RF, F BB, W RF, W BB ). Unforunaely, finding global opima for similar consrained join opimizaion problems is ofen found o be inracable [57], [58]. In he case of mmwave precoding, he non-convex consrains on F RF and W RF makes finding an exac soluion unlikely. To simplify ransceiver design, we emporarily decouple he join ransmierreceiver opimizaion problem and focus on he design of he hybrid precoders F RF F BB. Therefore, in lieu of maximizing specral efficiency, we design F RF F BB informaion achieved by Gaussian signaling over he mmwave channel o maximize he muual ( I(F RF, F BB ) = log 2 I Ns + ρ ) HF N s σn 2 RF F BB F BBF RFH. (8) We noe here ha absracing receiver operaion, and focusing on muual informaion insead of he specral efficiency expression in (3), effecively amouns o assuming ha he receiver can perform opimal neares-neighbor decoding based on he N r -dimensional received signal y. Unforunaely, such a decoder is impossible o realize wih pracical mmwave sysems in which decoders do no have access o he N r -dimensional signal. In pracical mmwave sysems, received signals mus be combined in he analog domain, and possibly in he digial domain, before any deecion or decoding is performed. For his reason, we revisi he problem of designing pracical mmwave receivers in Secion IV.

9 9 Proceeding wih he design of F RF F BB, he precoder opimizaion problem can be saed as ( (F op RF, Fop BB ) = arg max log 2 I Ns + ρ ) HF F RF, F BB N s σn 2 RF F BB F BBF RFH, s.. F RF F RF, (9) F RF F BB 2 F = N s, where F RF is he se of feasible RF precoders, i.e., he se of N N RF marices wih consanmagniude enries. To he exen of he auhors knowledge, no general soluions o (9) are known in he presence of he non-convex feasibiliy consrain F RF F RF. Therefore, we propose o solve an approximaion of (9) in order o find pracical near-opimal precoders ha can be implemened in he sysem of Fig. 1. We sar by examining he muual informaion achieved by he hybrid precoders F RF F BB and rewriing (8) in erms of he disance beween F RF F BB and he channel s opimal unconsrained precoder F op. To do so, define he channel s ordered singular value decomposiion (SVD) o be H = UΣV where U is an N r rank(h) uniary marix, Σ is a rank(h) rank(h) diagonal marix of singular values arranged in decreasing order, and V is a N rank(h) uniary marix. Using he SVD of H and sandard mahemaical manipulaion, (8) can be rewrien as ( I(F RF, F BB ) = log 2 I rank(h) + ρ ) Σ 2 V F N s σn 2 RF F BB F BBF RFV. (10) Furher, defining he following wo pariions of he marices Σ and V as Σ = Σ Σ 2, V = [V 1 V 2 ], (11) where Σ 1 is of dimension N s N s and V 1 is of dimension N N s, we noe ha he opimal unconsrained uniary precoder for H is simply given by F op = V 1. Furher noe ha he precoder V 1 canno in general be expressed as F RF F BB wih F RF F RF, and hus canno be realized in he mmwave archiecure of ineres. If he hybrid precoder F RF F BB can be made sufficienly close o he opimal precoder V 1, however, he muual informaion resuling from F op and F RF F BB can be made comparable. In fac, o simplify he forhcoming reamen of I(F RF, F BB ), we make he following sysem assumpion. Approximaion 1: We assume ha he mmwave sysem parameers (N, N r, N RF, N RF ), as r

10 10 well as he parameers of he mmwave propagaion channel (N cl, N ray,...), are such ha he hybrid precoders F RF F BB can be made sufficienly close o he opimal uniary precoder F op = V 1. Mahemaically, his closeness is defined by he following wo equivalen approximaions: 1) The eigenvalues of he marix I Ns V 1F RF F BB F BB F BB V 1 are small. In he case of mmwave precoding, his can be equivalenly saed as V 1F RF F BB I Ns. 1 2) The singular values of he marix V 2F RF F BB are small; alernaively V 2F RF F BB 0. This approximaion is similar o he high-resoluion approximaion used o simplify he analysis of limied feedback MIMO sysems by assuming ha codebooks are large enough such ha hey conain codewords ha are sufficienly close o he opimal unquanized precoder [49]. In he case of mmwave precoding, his approximaion is expeced o be igh in sysems of ineres which include: (i) a reasonably large number of anennas N, (ii) a number of ransmi chains N s < N RF N, and (iii) correlaed channel marices H. Funcionally, Approximaion 1 allows us furher simplify I(F RF, F BB ). To do so, we use he pariions defined in (11) and furher define he following pariion of V F RF F BB F BB F RF V as V F RF F BB F BBF RFV = V 1F RF F BB F BB F RF V 1 V1F RF F BB F BB F RF V 2 = Q 11 Q 12 V2F RF F BB F BB F RF V 1 V2F RF F BB F BB F RF V, 2 Q 21 Q 22 which allows us o approximae he muual informaion achieved by F RF F BB as ( I(F RF, F BB ) = log 2 I rank(h) + ρ ) Σ 2 V F N s σn 2 RF F BB F BBF RFV = log 2 I rank(h) + ρ Σ2 1, 0 Q 11 Q 12 N s σn 2 0 Σ 2 2 Q 21 Q 22 ( (a) = log 2 I Ns + ρ ) Σ 2 N s σ 1Q n 2 11 ( + log 2 I + ρ ( Σ 2 N s σ 2Q n 2 22 ρ2 N 2 Σ 2 s σ 2Q n 4 21 I Ns + ρ ) ) 1 Σ 2 N s σ 1Q n 2 11 Σ 2 1Q 12 ( (b) log 2 I Ns + ρ ) Σ 2 N s σ 1V1F n 2 RF F BB F BBF RFV 1, (12) 1 For he eigenvalues of I Ns V 1F RFF BBF BBF BBV 1 o be small, we need V 1F RFF BB Ψ where Ψ is any N s N s uniary marix (no necessarily I Ns ). However, if F RFF BB is a valid precoder wih V 1F RFF BB Ψ, hen so is he roaed precoder F RF FBB = F RFF BBΨ for which we have V 1F RF FBB I Ns. Since F BB can be arbirarily roaed, he condiions V 1F RFF BBF BBF BBV 1 I Ns and V 1F RFF BB I Ns can be considered equivalen in his case wihou loss of generaliy.

11 11 where (a) is a resul of using he Schur complemen ideniy for marix deerminans and (b) follows from invoking Approximaion 1 which implies ha Q 12, Q 21 and Q 22 are approximaely zero. Using (12), muual informaion can be furher simplified by wriing ( I(F RF, F BB ) (a) log 2 I Ns + ρ ) Σ 2 N s σn 2 1 ( ( + log 2 I Ns I Ns + ρ ) ) 1 Σ 2 ρ N s σn 2 1 Σ 2 N s σn 2 1 (I Ns V1F RF F BB F BBF RFV 1 ) ( (b) log 2 I Ns + ρ ) Σ 2 N s σn 2 1 ( ( r I Ns + ρ ) ) 1 Σ 2 ρ N s σn 2 1 Σ 2 N s σn 2 1 (I Ns V1F RF F BB F BBF RFV 1 ) ( (c) log 2 I Ns + ρ ) Σ 2 N s σn 2 1 r (I Ns V1F RF F BB F BBF RFV 1 ) (13) ( = log 2 I Ns + ρ ) Σ 2 N s σn 2 1 ( ) N s V1F RF F BB 2 F, (14) where we noe ha (a) is exac given (12), and (b) follows from Approximaion 1 which implies ha he eigenvalues of he marix X = (I Ns + ρ N sσ 2 n ρ Σ 2 1) 1 N sσ Σ 2 n 2 1 (I Ns V1F RF F BB F BB F RF V 1) are small and hus allows us o use he following approximaion log 2 I Ns X log 2 (1 r(x)) r(x). Finally (c) follows from adoping a high effecive-snr approximaion which implies ha (I + ρ N sσ 2 n ρ Σ 2 1) 1 N sσ Σ 2 n 2 1 I Ns and yields he final resul in (14). 2 We noice ha he firs erm in (14) is he muual informaion achieved by he opimal precoder F op = V 1 and ha he dependence of I(F RF, F BB ) on he hybrid precoder F RF F BB is now capured in he second and final erm of (13) and (14). Assuming F RF F BB is made exacly uniary, we noe ha he second erm in (13) and (14) is nohing bu he squared chordal disance beween he wo poins F op = V 1 and F RF F BB on he Grassmann manifold. Since Approximaion 1 saes he hese wo poins are close, we can exploi he manifold s locally Euclidean propery o replace he chordal disance by he Euclidean disance F op F RF F BB F [59]. Therefore, near-opimal hybrid precoders ha approximaely maximize I(F RF, F BB ) can be found by insead minimizing F op F RF F BB F. 2 Noe here ha i is no he nominal SNR ρ N sσ 2 n ha is assumed o be high. This would be a problemaic assumpion in mmwave sysems. I is, however, only he effecive-snrs in he channel s dominan N s subspaces ha are assumed o be sufficienly high. This is a reasonable assumpion since hese effecive SNRs include he large array gain from mmwave beamforming.

12 12 In fac, even wihou reaing F RF F BB as a poin on he Grassmann manifold, Approximaion 1 implies ha V1F RF F BB 2 F, and consequenly (14), can be approximaely maximized by insead maximizing r (V 1F RF F BB ). 3 Since maximizing r (V 1F RF F BB ) is again equivalen o minimizing F op F RF F BB F, he precoder design problem can be rewrien as (F op RF, Fop BB ) = arg min F BB,F RF F op F RF F BB F, s.. F RF F RF, (15) F RF F BB 2 F = N s, which can now be summarized as finding he projecion of F op ono he se of hybrid precoders of he form F RF F BB wih F RF F RF. Furher, his projecion is defined wih respec o he sandard Frobenius norm 2 F. Unforunaely, he complex non-convex naure of he feasible se F RF makes finding such a projecion boh analyically (in closed form) and algorihmically inracable [62] [65]. To provide near-opimal soluions o he problem in (15), we propose o exploi he srucure of he mmwave MIMO channels generaed by he clusered channel model in Secion II-B. Namely, we leverage he following observaions on mmwave precoding: 1) Srucure of opimal precoder: Recall ha he opimal uniary precoder is F op = V 1, and ha he columns of he uniary marix V form an orhonormal basis for he channel s row space. 2) Srucure of clusered mmwave channels: Examining he channel model in (4), we noe ha he array response vecors a (φ il, θ il ), i, l, θ il ) also form a finie spanning se for he channel s row space. In fac, when N cl N ray N, we noe ha he array response vecors a (φ il, θ il ) will be linearly independen wih probabiliy one and will hus form anoher minimal basis for he channel s row space when N cl N ray min(n, N r ). Noe: To esablish he linear independence of he vecors a (φ il, θ il ), consider he case of uniform linear arrays. When ULAs are considered, he N N cl N ray marix formed by he collecion of vecors a (φ il ) i, l will be a Vandermonde marix which has full rank whenever he angles φ il are disinc. This even occurs wih probabiliy one when φ il 3 This is since he magniude of V 1F RFF BB s off-diagonal enries is negligible and all V 1F RFF BB s diagonals mus be made close o one. Thus V 1F RFF BB 2 F, i.e., he l2 norm of V 1F RFF BB s diagonals, can be maximized by opimizing r (V 1F RFF BB), i.e., he l1 norm of he diagonals [38], [60], [61].

13 13 are generaed from a coninuous disribuion. Linear independence can be esablished in he case of UPAs by wriing heir response vecors as a Kronecker produc of wo ULA response vecors [66]. 3) Connecion beween F op and a (φ il θ il ): Regardless of wheher N cln ray N or no, observaion 1 implies ha he columns of he opimal precoder F op = V 1 are relaed o he vecors a (φ il, θ il ) hrough a linear ransformaion. As a resul, he columns of F op can be wrien as linear combinaions of a (φ il, θ il ), i, l. 4) Vecors a (φ il θ il ) as columns of F RF: Recall ha he vecors a (φ il, θ il ) are consanmagniude phase-only vecors which can be applied a RF using analog phase shifers. Therefore, he mmwave ransmier can apply N RF of he vecors a (φ il, θ il ) a RF (via he RF precoder F RF ), and form arbirary linear combinaions of hem using is digial precoder F BB. Namely, i can consruc he linear combinaion ha minimizes F op F RF F BB F. Therefore, by exploiing he srucure of H, we noice ha near-opimal hybrid precoders can be found by furher resricing F RF o be he se of vecors of he form a (φ il, θ il ) and solving (F op RF, Fop BB ) = arg min F op F RF F BB F, s.. F (i) RF { a (φ il, θ il) 1 i N cl, 1 l N ray }, F RF F BB 2 F = N s, (16) which amouns o finding he bes low dimensional represenaion of F op using he basis vecors a (φ il, θ i,l ). We noe here ha he se of basis vecors can be exended o include array response vecors a (, ) in direcions oher han {(φ il, θ il ) 1 i N cl, 1 l N ray }, hough he effec of his basis exension is ypically negligible. In any case, he precoding problem consiss of selecing he bes N RF Finally, we noe ha he consrain of F (i) RF array response vecors and finding heir opimal baseband combinaion. objecive o obain he following equivalen problem F op BB = arg min F op A FBB F, F BB can be embedded direcly ino he opimizaion s.. diag( F BB F BB ) 0 = N RF, A FBB 2 F = N s, (17)

14 where A = [ ] a (φ 1,1, θ1,1),..., a (φ N cl,n ray, θn cl,n ray ) is an N N cl N ray 14 marix of array response vecors and F BB is an N cl N ray N s marix. The marices A and F BB ac as auxiliary variables from which we obain F op RF diag( F BB F BB ) 0 = N RF and Fop BB respecively. Namely, he sparsiy consrain saes ha F BB canno have more han N RF non-zero rows. When only N RF rows of F BB are non zero, only N RF columns of he marix A are effecively seleced. As a resul, he baseband precoder F op BB F op BB and he RF precoder Fop RF will be given by he N RF non-zero rows of will be given by he corresponding N RF columns of A. Essenially, we have reformulaed he problem of joinly designing F RF and F BB ino a sparsiy consrained marix reconsrucion problem wih one variable. Alhough he underlying moivaion differs, and so does he inerpreaion of he differen variables involved in (17), he resuling problem formulaion is idenical o he opimizaion problem encounered in he lieraure on sparse signal recovery. Thus, he exensive lieraure on sparse reconsrucion can now be used for hybrid precoder design [34], [36]. To see his more clearly, noe ha in he simples case of single sream beamforming, (17) simplifies o op f BB = arg min f op A fbb F, fbb s.. f BB 0 = N RF, A fbb 2 F = N s, (18) in which he sparsiy consrain is now on he vecor f BB. This beamforming problem can be solved, for example, by relaxing he sparsiy consrain and using convex opimizaion o solve is l2 l1 relaxaion. Alernaively, (18) can be solved using ools from [34] [37], [44]. In he more general case of N s > 1, he problem in (17) is equivalen o he problem of sparse signal recovery wih muliple measuremen vecors, also known as he simulaneously sparse approximaion problem [40] [43]. So, for he general case of N s 1, we presen an algorihmic soluion based on he well-known concep of orhogonal maching pursui [34], [36], [44]. The pseudo-code for he precoder soluion is given in Algorihm 1. In summary, he precoding algorihm sars by finding he vecor a (φ il, θ il ) along which he opimal precoder has he maximum projecion. I hen appends he seleced column vecor a (φ il, θ il ) o he RF precoder F RF. Afer he dominan vecor is found, and he leas squares soluion o F BB is calculaed in sep 7, he conribuion of he seleced vecor is removed in sep 8 and he algorihm proceeds o find he column along which he residual precoding marix F res has he larges projecion. The

15 15 Algorihm 1 Spaially Sparse Precoding via Orhogonal Maching Pursui Require: F op 1: F RF = Empy Marix 2: F res = F op 3: for i N RF 4: Ψ = A F res do 5: k = arg [ max l=1,..., ] Ncl N ray (ΨΨ ) l,l 6: F RF = F RF A (k) 7: F BB = (F RF F RF) 1 F RF F op 8: F res = Fop F RFF BB F op F RF F BB F 9: end for 10: F BB = N s F BB F RF F BB F 11: reurn F RF, F BB process coninues unil all N RF beamforming vecors have been seleced. A he end of he N RF ieraions, he algorihm would have (i) consruced an N N RF RF precoding marix F RF, and (ii) found he opimal N RF N s baseband precoder F BB which minimizes F op F RF F BB 2 F. Sep 10 ensures ha he ransmi power consrain is exacly saisfied. To gain more inuiion abou he proposed precoding framework, Fig. 2 plos he beam paerns generaed by a ransmier wih a 256-elemen planar array for an example channel realizaion using (i) he channel s opimal unconsrained precoder, (ii) he proposed precoding sraegy wih N RF = 4, and (ii) he beam seering vecor in he channel s dominan physical direcion. We observe ha in pracical mmwave channels, opimal precoders do in fac generae spaially sparse beam paerns and hus may be accuraely approximaed by a finie combinaion of array response vecors. Furher, Fig. 2 indicaes ha Algorihm 1 succeeds in generaing beam paerns which closely resemble hose generaed by F op. Therefore, Algorihm 1 succeeds in selecing he bes N RF seering direcions and forming appropriae linear combinaions of he seleced response vecors. This beam paern similariy will ulimaely resul in favorable specral efficiency performance as shown in Secion VI. Having presened he proposed precoding framework, we conclude his secion wih he following design remarks. Remark 2: We noe ha he mmwave erminals need no know he exac angles (φ il, θ il ) ha make up he channel marix H, and need no use he marix A as defined earlier. We have

16 16 (a) Beam Paern of Opimal Beamforming Vecor (b) Beam Paern wih Proposed Soluion (c) Beam Paern of Opimal Seering Vecor Fig. 2. Beam paern generaed a 256-elemen square array in an example channel realizaion wih 6 scaering clusers using (a) opimal unconsrained beamforming, (b) he proposed sparse precoding soluion wih 4 RF chains, and (c) he beam seering vecor in he channel s dominan physical direcion. The proposed algorihm is shown o resul in beam paerns ha closely resemble he paerns generaed by opimal beamforming; his beam paern similariy will ulimaely resul in similar specral efficiency. For illusraion purposes, he channel s angle spread is se o 0 in his figure. only used his finie basis for simpliciy of exposiion. In general, he mmwave erminals can insead selec basis vecors of he form a (φ, θ) using any finie se of represenaive azimuh and elevaion direcions (such as a se of equally spaced angles for example). This approach avoids having o decompose H ino is geomeric represenaion and is naurally suied for limied feedback operaion. This approach will be discussed furher in Secion V. Remark 3: I may be advanageous in some cases o impose he addiional consrain ha F BB be uniary. Uniary precoders can be more efficienly quanized and are hus more aracive in limied feedback sysems. Wih his addiional consrain, (17) can be solved again via Algorihm

17 17 1 by replacing he leas squares soluion for F BB in sep 7, by he soluion o he corresponding orhogonal Procruses problem [67]. This is given by F BB = Û ˆV where Û and ˆV are uniary marices defined by he singular value decomposiion of F RF F op, i.e., F RF F op = Û ˆΣ ˆV [67]. Remark 4: In he limi of large anenna arrays (N, N r ) in very poor scaering environmens for which N cl N ray = o(min(n, N r )), he resuls of [18], [66] indicae ha simple RF-only beam seering becomes opimal, i.e., i becomes opimal o simply ransmi each sream along one of he N s mos dominan vecors a (φ il, θ il ). For arrays of pracical sizes, however, Secion VI shows ha here can be significan gains from more involved precoding sraegies such as he one presened in his secion. IV. PRACTICAL MILLIMETER WAVE RECEIVER DESIGN In Secion III, we absraced receiver-side processing and focused on designing pracical mmwave precoders ha maximize muual informaion. Effecively, we assumed ha he mmwave receiver can opimally decode daa using i N r -dimensional received signal. Such a decoder can be of prohibiively high complexiy in muli-anenna sysems, making lower-complexiy receivers such as he commonly used linear MMSE receiver more appealing for pracical implemenaion. In fac, in mmwave archiecures such as he one shown in Fig. 1, such opimal decoders are impossible o realize since received signals mus be linearly combined in he analog domain before any deecion or decoding is performed. In his secion, we address he problem of designing linear combiners for he mmwave receiver in Fig. 1, which uses boh analog and digial processing before deecion. Assuming he hybrid precoders F RF F BB are fixed, we seek o design hybrid combiners W RF W BB ha minimize he mean-squared-error (MSE) beween he ransmied and processed received signals. The combiner design problem can herefore be saed as (W op RF, Wop BB ) = arg min E [ s WBBW RFy 2 ] 2, W RF, W BB s.. W RF W RF, (19) where W RF is he se of feasible RF combiners, i.e., W RF is he se of N r N RF r marices wih consan-gain phase-only enries. In he absence of any hardware limiaions ha resric he se

18 18 of feasible linear receivers, he exac soluion o (19) is well known [68] o be W MMSE = E [sy ] E [yy ] 1 = (a) = 1 ( ρ ρ N s F BBF RFH F BBF RFH HF RF F BB + σ2 nn s ρ ( ρ HF RF F BB F N BBF RFH + σni 2 Nr s ) 1 I N s F BBF RFH, ) 1 (20) where (a) follows from applying he marix inversion lemma. Jus as in he precoding case, however, his opimal unconsrained MMSE combiner WMMSE need no be decomposable ino a produc of RF and baseband combiners WBB W RF wih W RF W RF. Therefore WMMSE canno be realized in he sysem of Fig. 1. Furher, jus as in he precoding case, he complex nonconvex consrain W RF W RF makes solving (19) analyically impossible and algorihmically non-rivial. To overcome his difficuly, we leverage he mehodology used in [45], [46] o find linear MMSE esimaors wih complex srucural consrains. We sar by reformulaing he problem in (19) by expanding MSE as follows E [ s WBBW RFy 2] 2 = E [(s W BB WRFy) (s WBBW RFy)] = E [r ((s W BBW RFy) (s W BBW RFy) )] = r (E [ss ]) 2R {r (E [sy ] W BBW RF)} (21) + r (W BBW RFE [ss ] W BBW RF). We now noe ha since he opimizaion problem in (19) is over he variables W RF W BB, we can add any erm ha is independen of W RF and W BB o is objecive funcion wihou changing he oucome of he opimizaion. Thus, we choose o add he consan erm r (W MMSE E [yy ] W MMSE ) r (E [ss ]) and minimize he equivalen objecive funcion J (W RF, W BB ) = r (W MMSE E [yy ] W MMSE) 2R {r (E [sy ] W RF W BB )} + r (W BBW RFE [ss ] W BBW RF) (a) = r (W MMSE E [yy ] W MMSE) 2R {r (W MMSEE [yy ] W RF W BB )} + r (W BBW RFE [ss ] W BBW RF) = r ((W MMSE W BBW RF) E [yy ] (W MMSE W BBW RF) ) = E [yy ] 1/2 (W MMSE W RF W BB ) 2 F, (22) and

19 19 where (a) follows from noicing ha WMMSE = E [sy ] E [yy ] 1 which implies ha he second erm can be rewrien as r (E [sy ] W RF W BB ) = r ( E [sy ] E [yy ] 1 ) E [yy ] W RF W BB = r (WMMSE E [yy ] W RF W BB ). As a resul of (22), he MMSE esimaion problem is equivalen o finding hybrid combiners ha solve (W op RF, Wop BB ) = arg min W RF, W BB s.. W RF W RF, E [yy ] 1/2 (W MMSE W RF W BB ) F (23) which amouns o finding he projecion of he unconsrained MMSE combiner W MMSE ono he se of hybrid combiners of he form W RF W BB wih W RF W RF. Thus, he design of MMSE receivers for he mmwave sysem of ineres closely resembles he design of is hybrid precoders. Unlike in he precoding case however, he projecion now is no wih respec o he sandard norm 2 F and is insead an E [yy ]-weighed Frobenius norm. Unforunaely, as in he case of he precoding problem in (15), he non-convex consrain on W RF precludes us from pracically solving he projecion problem in (23). The same observaions ha allowed us o leverage he srucure of mmwave channels o solve he precoding problem in Secion III, however, can be ranslaed o he receiver side o solve he combiner problem as well. Namely, because of he srucure of clusered mmwave channels, near-opimal receivers can be found by furher consraining W RF o have columns of he form a r (φ, θ) and insead solving where A r = W op BB = arg min E [yy ] 1/2 W MMSE E [yy ] 1/2 A r WBB F, W BB s.. diag( W BB W BB ) 0 = Nr RF [ ] a r (φ r 1,1, θ1,1), r..., a (φ r N cl,n ray, θn r cl,n ray ) is an N r N cl N ray (24) marix of array response vecors and W BB is an N cl N ray N s marix; he quaniies A r and W BB ac as auxiliary variables from which we obain W RF and W BB in a manner similar o Secion III. 4 As a resul, he MMSE esimaion problem is again equivalen o he problem of sparse signal recovery wih muliple measuremen vecors and can hus be solved via he orhogonal maching pursui concep used in Secion III. For compleeness he pseudo code is given in Algorihm 2. Remark 5: This secion relaxes he perfec-receiver assumpion of Secion III and proposes 4 As noed in Secion III he receiver need no know he exac angles (φ r il, θ r il) and can insead use any se of represenaive azimuh and elevaion angles of arrival o consruc he marix of basis vecors A r.

20 20 Algorihm 2 Spaially Sparse MMSE Combining via Orhogonal Maching Pursui Require: W MMSE 1: W RF = Empy Marix 2: W res = W MMSE 3: for i N RF do r 4: Ψ = A re [yy ] W res 5: k = arg max [ l=1,..., Ncl ] N ray (ΨΨ ) l,l 6: W RF = W RF A (k) r 7: W BB = (W RF E [yy ] W RF ) 1 W RF E [yy ] W MMSE 8: W res = W MMSE W RF W BB W MMSE W RF W BB F 9: end for 10: reurn W RF, W BB pracical mehods o find low-complexiy linear receivers. The design of precoders and combiners, however, remains decoupled as we have assumed ha he precoders F RF F BB are fixed while designing W RF W BB (and ha receivers are opimal while designing F RF F BB ). This decoupled approach simplifies mmwave ransceiver design, and will be shown o perform well in Secion VI, however, some simple join decisions may be boh pracical and beneficial. For example, consider he case where a receiver only has a single RF chain and hus is resriced o applying a single response vecor a r (φ, θ). In such a siuaion, designing F RF F BB o radiae power in N RF differen direcions may lead o a loss in acual received power (since he receiver can only form a beam in one direcion). As a resul, i is beneficial o accoun for he limiaions of he more-consrained erminal when designing eiher precoders or combiners. To do so, we propose o run Algorihms 1 and 2 in succession according o he following rules 1. Solve for F N RF < Nr RF RF F BB using Algorihm Given F RF F BB, solve for W RF W BB using Algorihm Solve for W N RF > Nr RF RF W BB using Algorihm 2 assuming F RF F BB = F op. 2. Solve for F RF F BB for he effecive channel WBB W RF H. In summary, saring wih he more consrained side, he hybrid precoder or combiner is found using Algorihm 1 or 2. Then, given he oupu, he remaining processing marix is found by appropriaely updaing he effecive mmwave channel. Finally, we noe ha while he numerical resuls of Secion VI indicae ha his decoupled (25)

21 21 approach o mmwave ransceiver design yields near-opimal specral efficiency, a more direc join opimizaion of (F RF, F BB, W RF, W BB ) is an ineresing opic for fuure invesigaion. Similarly, while we have solved he sparse formulaion of he precoding and combining problems via orhogonal maching pursui, he problems in (17) and (24) can be solved by leveraging oher algorihms for simulaneously sparse approximaion [42]. V. LIMITED FEEDBACK SPATIALLY SPARSE PRECODING Secion III implicily assumed ha he ransmier has perfec knowledge of he channel marix H and is hus able o calculae F op and approximae i as a hybrid RF/baseband precoder F RF F BB. Since such ransmier channel knowledge may no be available in pracical sysems, we propose o fulfill his channel knowledge requiremen via limied feedback [21], [47] [49]. Namely, we assume ha he receiver (i) acquires perfec knowledge of H, (ii) calculaes F op and a corresponding hybrid approximaion F RF F BB, hen (iii) feeds back informaion abou F RF F BB o he ransmier. Since hybrid precoders are naurally decomposed ino an RF and baseband componen, we propose o quanize F RF and F BB separaely while exploiing he mahemaical srucure presen in each of hem. A. Quanizing he RF Precoder Recall ha he precoder F RF calculaed Secion III has N RF Therefore, F RF admis a naural paramerizaion in erms of he N RF columns of he form a (φ, θ). angles ha i uses. Thus, F RF can be efficienly encoded by quanizing is 2N RF For simpliciy, we propose o uniformly quanize he N RF azimuh and elevaion free variables. azimuh and elevaion angles using N φ and N θ bis respecively. Therefore, he quanized azimuh and elevaion angles are such ha { } ˆφ k C φ = φ min + φ max φ min, φ 2 N φ+1 min + 3(φ max φ min),..., φ 2 N φ+1 max φ max φ min 2 N φ+1 { } (26) ˆθ k C θ = θmin + θ max θmin, θ 2 N θ+1 min + 3(θ max θmin),..., θ 2 N θ+1 max θ max θmin 2 N θ+1 where we recall ha [φ min, φ max] and [θ min, θ max] are he secors over which Λ (φ, θ) 0. The receiver can hen quanize F RF by simply selecing he enries of C φ and C θ ha are closes in Euclidean disance o F RF s angles. Alernaively, as saed in Remark 2, Algorihm 1 can be

22 22 run direcly using he N 2 N φ+n θ marix of quanized response vecors A quan. = [a (φ 1, θ 1),..., a (φ i, θ l), ]..., a (φ 2, Nφ θ2 ), (27) Nθ and he index of he seleced angles can be fed back o he ransmier. While his laer approach has higher search complexiy, i has he advanage of (i) joinly quanizing all 2N RF and (ii) auomaically maching he baseband precoder F BB o he quanized angles. angles, B. Quanizing he Baseband Precoder To efficienly quanize F BB, we begin by highlighing is mahemaical srucure in mmwave sysems of ineres. Namely, we noe ha for sysems wih large anenna arrays, we ypically have ha F RF F RF I N RF. When coupled wih Approximaion 1, we have ha F BB F BB I Ns, i.e., F BB is approximaely uniary. In fac, F BB can be made exacly uniary as discussed in Remark 3. Furher, we recall ha he specral efficiency expression in (3) is invarian o N s N s uniary ransformaions of he baseband precoder. Therefore, F BB is a subspace quaniy ha can be quanized on he Grassmann manifold [47], [48]. Suiable codebooks for F BB can be designed using Lloyd s algorihm on a raining se of baseband precoders and using he chordal disance as a disance measure [69]. Since such codebook consrucion is well-sudied in he lieraure on limied feedback MIMO, we omi is deails for breviy and refer he reader o [70, Secion IV] for an in-deph descripion of he process. VI. SIMULATION RESULTS In his secion, we presen simulaion resuls o demonsrae he performance of he spaially sparse precoding algorihm presened in Secion III when combined wih he sparse MMSE combining soluion presened in Secion IV. We model he propagaion environmen as a N cl = 8 cluser environmen wih N ray = 10 rays per cluser wih Laplacian disribued azimuh and elevaion angles of arrival and deparure [30], [54]. For simpliciy of exposiion, we assume all clusers are of equal power, i.e., σα,i 2 = σα 2 i, and ha he angle spread a boh he ransmier and receiver are equal in he azimuh and elevaion domain, i.e., σφ = σr φ = σ θ = σr θ. Since oudoor deploymens are likely o use secorized ransmiers o decrease inerference and increase beamforming gain, we consider arrays of direcional anenna elemens wih a response given in (5) [8], [9]. The ransmier s secor angle is assumed o be 60 -wide in he azimuh domain and

23 23 15 Opimal Unconsrained Precoding Beam Seering Sparse Precoding & Combining N s =2 Specral Efficiency (bis/s/hz) 10 5 N s = SNR (db) Fig. 3. Specral Efficiency achieved by various precoding soluions for a mmwave sysem wih planar arrays a he ransmier and receiver. The propagaion medium is a N cl = 8 cluser environmen wih N ray = 10 and an angular spread of 7.5. Four RF chains are assumed o be available for sparse precoding and MMSE combining. 20 -wide in elevaion [8]. In conras, we assume ha he receivers have relaively smaller anenna arrays of omni-direcional elemens; his is since receivers mus be able o seer beams in any direcion since heir locaion and orienaion in real sysems is random. The iner-elemen spacing d is assumed o be half-wavelengh. We compare he performance of he proposed sraegy o opimal unconsrained precoding in which sreams are sen along he channel s dominan eigenmodes. We also compare wih a simple beam seering soluion in which daa sreams are seered ono he channel s bes propagaion pahs. 5 For fairness, he same oal power consrain is enforced on all precoding soluions and signal-o-noise raio is defined as SNR = ρ. σn 2 Fig. 3 shows he specral efficiency achieved in a sysem wih square planar arrays a boh ransmier and receiver. For he proposed precoding sraegy, boh ransmier and receiver are assumed o have four ransceiver chains wih which hey ransmi N s = 1 or 2 sreams. Fig. 5 Noe ha, when N s > 1, he bes propagaion pahs in erms of specral efficiency may no be he ones wih he highes gains. This is since, wih no receiver baseband processing, differen pahs mus be sufficienly separaed so as hey do no inerfere. In his case, he bes pahs are chosen via a cosly exhausive search. Furher, when power allocaion is considered in Fig. 5, he same waerfilling power allocaion is applied o he beam seering soluion.

24 Opimal Unconsrained Precoding Beam Seering Sparse Precoding & Combining N s =2 Specral Efficiency (bis/s/hz) N s = SNR (db) Fig. 4. Specral Efficiency achieved in a mmwave sysem wih planar arrays a he ransmier and receiver. Channel parameers are se as in Fig. 3. Six RF chains are available for sparse precoding and combining. 3 shows ha he proposed framework achieves specral efficiencies ha are essenially equal o hose achieved by he opimal unconsrained soluion in he case N s = 1 and are wihin a small gap from opimaliy in he case of N s = 2. This implies ha he proposed sraegy can very accuraely approximae he channel s dominan singular vecors as a combinaion of four seering vecors. When compared o radiional beam seering, Fig. 3 shows ha here is a non-negligible improvemen o be had from more sophisicaed precoding sraegies in mmwave sysems wih pracical array sizes. To explore performance in mmwave sysems wih larger anenna arrays, Fig. 4 plos he performance achieved in a sysem wih N RF = N RF r = 6 RF chains. Fig. 4 shows ha he proposed precoding/combining soluion achieves almos-perfec performance in boh N s = 1 and N s = 2 cases. Furher, we noe ha alhough beam seering is expeced o be opimal in he limi of large arrays, as discussed in Remark 4, he proposed soluion sill ouperforms beam seering by approximaely 5 db in his larger mmwave sysem. While Secion III focused on he design of fixed-rank precoders wih equal power allocaion across sreams, he same framework can be applied o sysems in which N s is deermined dynamically and sreams are sen wih unequal power. This configuraion allows us o compare