Low Complexity Hybrid Sparse Precoding and Combining in Millimeter Wave MIMO Systems

Transcription

1 1 Low Complexiy Hybrid Sparse Precoding and Combining in Millimeer Wave MIMO Sysems Crisian Rusu, Roi Méndez-Rial, Nuria González-Prelcic, and Rober W. Heah Jr. Universidade de Vigo, The Universiy of Texas a Ausin, rheah@uexas.edu Absrac Millimeer wave (mmwave) muliple-inpu mulipleoupu (MIMO) communicaion wih large anenna arrays has been proposed o enable gigabi per second communicaion for nex generaion cellular sysems and local area neworks. A key difference relaive o lower frequency soluions is ha in mmwave sysems, precoding/combining can no be performed enirely a digial baseband, due o he high cos and power consumpion of some componens of he radio frequency (RF) chain. In his paper we develop a low complexiy algorihm for finding hybrid precoders ha spli he precoding/combining process beween he analog and digial domains. Our approach explois sparsiy in he received signal o formulae he design of he precoder/combiners as a compressed sensing opimizaion problem. We use he properies of he marix conaining he array response vecors o find firs an orhonormal analog precoder, since sparse approximaion algorihms applied o orhonormal sensing marices are based on simple compuaions of correlaions. Then, we propose o perform a local search o refine he analog precoder and compue he baseband precoder. We presen numerical resuls demonsrae subsanial improvemens in complexiy while mainaining good specral efficiency. I. INTRODUCTION Millimeer wave (mmwave) is he new specral fronier for nex generaion cellular neworks and wireless local area neworks [1], [2], [3], [4]. An imporan requiremen in mmwave sysems is he use of large arrays a he ransmier and receiver o provide a reasonable link budge. The anennas form a muliple-inpu muliple-oupu (MIMO) communicaion link ha can be configured for differen objecives. The de faco approach is spaial direciviy, which provides beamforming gain needed o achieve a reasonable signal-o-noise raio (SNR) a he receiver. MmWave channels hough also have he abiliy o suppor spaial muliplexing of muliple daa sreams due o scaering and polarizaion [5], [6], [7], [8]. Unforunaely, power and cos requiremens in he mmwave analog fronend make i challenging o implemen he ypical MIMO precoding ransceiver found in lower frequency sysems, which is implemened in enirely in baseband. A soluion is he hybrid precoding framework, where he precoding/combining process is divided beween analog and digial domains [9], [10], [11]. This work was parially funded by he Spanish Governmen and he European Regional Developmen Fund (ERDF) under projecs TACTICA and COMPASS (TEC C2-1-R) by he Galician Regional Governmen and AlanTIC. This maerial is based upon work suppored in par by he Naional Science Foundaion under Gran No. NSF-CCF A popular design of hybrid precoders for mmwave channels based on variable phase shifers was proposed in [9] for a paricular mmwave sysem model incorporaing: i) he consrains on he analog precoder/combiner, ii) presence of large anenna arrays, and iii) he limied scaering naure of he mmwave channel. The design of he precoders and combiners is formulaed as a sparsiy seeking opimizaion problem wih hardware consrains. I resembles he problem of sparse signal recovery via muliple measuremen vecors (MMV), also known as he simulaneous sparse recovery problem (S OMP) [12]. The approach in [9] is elegan ye solving for he precoders sill resuls in high complexiy. A limiaion of he work in [9], is ha perfec channel sae informaion is assumed a he receiver. This has been overcome in work on adapive channel esimaion [10], where he mmwave channel esimaion problem is formulaed as a compressed sensing problem, so ha he channel parameers are esimaed using sandard CS ools. Training beamforming and combining vecors during he channel esimaion phase are designed using a muli-resoluion codebook. The main limiaion of his work is ha i assumes known array geomeries for boh he ransmier and receiver. Furher invesigaion is also needed o obain lower complexiy soluions o boh he channel esimaion and he hybrid analog/digial precoding design problems. Hybrid precoding srucures based on he use of variable phase shifers have been proposed earlier for general MIMO archiecures in [13], bu do no ake ino accoun he characerisics of millimeer wave propagaion or leverage sparsiy of he received signal. A relaed concep called beamspace MIMO communicaion has been proposed in [14], which uses a high-resoluion discree lens array for analog spaial beamforming. This avoids he need for phase shifers bu does no have uniform performance across a broad range of angles. In his paper we propose a low-complexiy soluion o he hybrid precoding opimizaion problem posed in [9]. We ake ino accoun he full srucure of he opimizaion problem by exploiing he semi-uniary opimum precoder (opimum in he absence of hardware consrains). This srucure reduces significanly he search space in he array manifold and hus leads o a lower complexiy procedure versus ha found in [9]. The reducion in complexiy is due o an orhogonal maching sep ha fis he opimum precoder wih he closes semiuniary srucure in he array manifold ha emulaes is behavior. The orhogonal maching sep eliminaes he need for he, slow, greedy maching pursui seps deployed in he previous

2 2 approach [9]. This sep is hen followed by a local search ha furher improves he soluion by using eiher a fas oneby-one selecion procedure or a full maching pursui search bu boh only on a reduced secion of he array manifold, around he semi-uniary soluion previously found. Numerical resuls show ha he compuaional advanage comes wih no significan performance degradaion in he proposed mehod as compared o previous resuls. A. Sysem model II. PROBLEM FORMULATION Consider he mmwave sysem shown in Figure 1. The ransmier sends N s daa sreams using N anennas o he receiver, which has N r anennas. The ransmier has N RF RF ransmi chains. Due o he high cos and power consumpion associaed wih providing each anenna wih an RF chain and a digial-o-analog converer (DAC) capable of handling he high frequencies and bandwidhs of mmwave sysems, N s N RF N. In he hybrid precoding approach, he ransmier applies wo precoders: he N RF N s digial baseband precoder F BB and he N N RF analog precoder F RF. The digial precoder F BB is designed assuming infinie precision while he analog precoder F RF is assumed o have elemens of equal norm assuming only phase shifing is performed in he analog domain. The oal power consrain is enforced by normalizing F BB such ha F RF F BB 2 F = N s. While he analog precoder F RF may be applied a some inermediae frequency or a he RF frequency, we represen i using is complex baseband equivalen. Assuming narrowband operaion as in [9], he ransmied signal is x = F RF F BB s, where s represens he symbol vecor. In his paper, he inpu symbol vecor is normalized such ha E[ss H ] = 1 N s I Ns. Using ρ o denoe he average received power, H he N r N channel marix, and n a vecor wih IID CN (0, σ 2 ) enries, he received complex baseband signal of dimension N r 1 is y = ρhf RF F BB s + n. (1) MmWave channels are expeced o have limied scaering wih only a few scaering clusers. To incorporae his fac, we use a clusered channel model wih N cl scaering clusers, each of which conribue N ray propagaion pahs. The clusered model is widely used as a MIMO channel model [15], [16], [17] and is also used in mmwave [18], [2]. Wih a clusered model, he channel marix is N N N cl N r ray H = α i,l a r (φ r N cl N i,l)a (φ i,l) H. (2) ray i=1 l=1 where α i,l is he complex gain of he l h ray in he i h cluser, whereas a (φ i,l ) and a r(φ r i,l ) are he anenna array response vecors a he ransmier and receiver evaluaed a he l h pah i h cluser azimuh angles of deparure or arrival (we assume uniform linear arrays, whose responses do no depend on he elevaion angle). The receiver applies he N r Nr RF analog combining marix W RF and he Nr RF N s baseband combining marix W BB. Fig. 1: Block diagram of a mmwave single user sysem wih hybrid precoding: baseband precoding and radio frequency precoding wih RF phase shifers. Dimensions follow N s N RF N and N s Nr RF N r. The dimensions saisfy N s Nr RF N r o use a limied number of RF chains and a low dimensional digial combiner following he analog-o-digial converer (ADC). The posprocessed received signal afer he hybrid combining srucure is ỹ = ρwbbw H RFHF H RF F BB s + WBBW H RFn. H (3) B. The precoder and combiner design problem There are many poenial merics o be used in he design of he hybrid precoding and combing marices. In his paper, we are ineresed in designs ha maximize he achievable sum rae wih low compuaional complexiy. In he convenional precoding paradigm, a single opimal RF precoder F op and combiner W op would be applied leading o ỹ = ρw H ophf op s + W H opn. (4) The muual informaion maximizing soluion (assuming Gaussian signaling) is given by he usual waer filling sraegy, where F op consiss of weighed columns ha correspond o he righ singular values of H. The design of he combining marix in his framework is flexible: i akes he form of he lef singular vecors of H muliplied by any nonsingular marix on he lef [13]. Maximizing he specral efficiency in he hybrid precoding case requires maximizing R = log 2 I Ns + ρ Rn 1 W N BBW H RFHF H RF F BB s F H BBF H RFH H W RF W BB, over all precoding and combining marices, being R n = σ 2 nw H BB W H RF W RFW BB he noise covariance marix afer combining. Since maximizing he muual informaion involves a join opimizaion of four marix variables wih non-convex consrains for F RF and W RF, finding global opima is very difficul. The soluion for he design of he analog and digial precoders and combiners in [9] simplifies he join ransmierreceiver opimizaion problem by decoupling he design ino he ransmier and he receiver, solving similar opimizaion problems. The objecive is o obain an equally good soluion in erms of specral efficiency bu wih lower compuaional complexiy. Because we propose a low complexiy soluion

3 3 o he algorihm in [9], we summarize he key seps of he algorihm here. a) Transmier: The goal is he design of F RF F BB o maximize he muual informaion achieved wih Gaussian signaling over he mmwave channel: log 2 I Ns + ρ N s σn 2 HF RF F BB FBBF H RFH H H. (5) In he absence of hardware limiaions and considering equal power allocaion across sreams, he opimum precoder ha maximizes (5) is given by: F op = V (:, 1 : N s ) where H = UΣV H is he singular value decomposiion. F op is he se of he N s columns in V associaed wih he highes singular values in Σ. I has been shown in [9] ha he problem of finding he precoder ha maximizes he muual informaion wih he hardware consrains associaed o mmwave archiecures can be well approximaed by solving arg min A, F BB subjec o F op A F BB F diag( F BB F H BB) 0 = N RF A F BB 2 F = N s, where 0 is he l 0 pseudo-norm accouning for he number of non-zero elemens, A of size N N cl N ray is he marix of array response vecors and F BB of size N cl N ray N s has only N RF non-zero rows (he sparsiy consrain) and heir produc has an energy consrain. We denoe by F BB he N RF N s marix F BB resriced o he rows ha are non-zero and wih F RF he subse of columns from A ha paricipae in he soluion. This problem consiss in finding N RF array response vecors and heir opimal baseband combinaion. The A marix conains he se of feasible RF precoders, i.e., he seering vecors. b) Receiver: Assuming F RF F BB fixed, design W RF W BB o minimize he mean squared error (MMSE) beween ransmied and processed received signals. The opimum MMSE combiner is well known [19] as WMMSE H = 1 ( ) 1 F H ρ BBFRFH H H HF RF F BB + σ2 nn s ρ I N s (6) F H BBF H RFH H. (7) The soluion proposed in [9] consiss of finding hybrid combiners ha minimize arg min E [ yy H] 1/2 (WMMSE W RF W BB ) F W RF,W BB subjec o W RF W RF, where W RF is he se of N r Nr RF marices wih consan-gain phase-only enries. Noe ha he opimizaion problems in (6) and (8) o be solved a he ransmier and receiver, respecively, are similar. The soluion proposed in [9] involves a greedy sraegy based on he Orhogonal Maching Pursui (OMP) approach, which can be ulimaely seen as a varian of Simulaneous (8) OMP (SOMP) [12], and conneced o he problem of sparse represenaions of muliple-measuremen vecors (MMV) [20]. In his paper, we find a mehod ha produces good resuls when solving hese opimizaion problems (6) and avoids he use of he, slow, greedy seps. High complexiy reducion is achieved wihou performance degradaion, as shown in Secion IV. III. LOW COMPLEXITY HYBRID PRECODING/COMBINING A. The proposed mehod SOLUTIONS In his secion we explain he proposed low complexiy precoding algorihm in deail; he derivaion of he combiner is similar and is omied for breviy. The key idea o reduce compuaional complexiy when solving he opimizaion problem in (6) is o reduce he process of searching columns of he overcomplee marix A o searching columns of orhonormal marices, subses of his overcomplee marix, such ha simple correlaions replace he maching pursui ieraions. Recall ha he N elemen seering vecors of an uniform linear array (ULA) ake he form a ULA (φ) = 1 [ 1 e jkd sin φ... e jkd(n 1) sin φ] T. N (9) The do producs, in absolue value, beween any wo such disinc vecors, assuming he sine erms are uniformly disribued in N poins in he inerval [ 1, 1) (i.e., sin φ l = 1 + 2l/N, for l = 0,..., N 1) are given by: a ULA (φ l ) H a ULA (φ i ) = 1 (l i)n 2jπ 1 e N (l i) N 1 e2jπ N = 1 N sin(π(l i)n /N) sin(π(l i)/n). (10) The parameer N is he angular resoluion. In general, he objecive is o solve he problem on a grid as fine as possible, i.e. N as large as possible. I is clear ha he do producs only depend on he disance beween he wo seering vecors (l i) his acually leads o a circulan Hermiian Gram marix G = A H A. Because of his, when α = N/N is an ineger i is possible o consruc α orhonormal marices of size N N by choosing columns of A equally spaced wih disance N (modulo wrapping). In his way A is viewed as a concaenaion of α orhonormal marices. We are ineresed in hese orhonormal marices since F op is semiuniary (since his marix has orhonormal columns bu is no square, i.e., FopF H op = I Ns bu F op Fop H I N ). Therefore, is approximaion needs o obey (F RF F BB ) H (F RF F BB ) I Ns and hus FRF HF RF and FBB H F BB need o closely approximae ideniy marices. By his argumen we search for F RF among he α orhonormal marices from A. We move o solve (6) wih an addiional consrain ha he soluion F RF needs o be an orhonormal marix (bu sill composed of seering vecors). The proposed wo-sep process is described nex. The proposed algorihm is divided ino wo disinc seps. In sep A we propose o design an orhonormal RF precoder F RF, wihou any concern o he baseband coder. This simplifies

4 4 Algorihm 1 Hybrid Analog-Digial Design by Orhogonal Maching and Local Search (OM+LS). Inpu: he opimal unconsrained precoder F op, he resoluion N, he sparsiy arge N RF and he local search lengh (L) Oupu: F RF and F BB such ha he objecive funcion of (6) is as low as possible under he given consrains. Sep A. Compue an orhonormal F RF : Compue all correlaions R = A H F op. Wih α = N/N, for l = 1,..., α: Build orhogonal marix Q l by selecing N columns α apar (mod N) of A saring from index l. Compue correlaions r l = norms(q H l F op), where he funcion compues he 2 norm for each row of he resuling produc. r l is of size N. Compue he overall energy conribuion of Q l o: c l = N s k=1 r l 2 [k] where r l [k] represens he k h larges enry of r l, in absolue value. Selec l max = arg max c l. l Se F RF as he N RF columns of Q lmax ha have he maximum enries in r lmax, sored according o hese values in descending order. Sep B. Based on he orhonormal F RF wih suppor S on he columns of A (i.e., F RF = (A ) S ) consruc he new pair (F RF, F BB ) based on a local search procedure: For l = 1,..., N RF : Remove he l h column from he suppor S, o produce he suppor S. Consruc he soluion (F RF, F BB ) on he suppor S : F RF = (A ) S, F BB = (FRF HF RF)\FRF HF op. Compue he normalized residual F res = F op F RF F BB, F res = F res F res 1 F. Add o he suppor S a new index from he se S L = {S(l) L,..., S(l) + L} (modulo N) of columns from A ha maximally correlaes wih he curren residual F res i.e., maximum enry in norms((a ) H S L F res ). Compue final soluion (F RF, F BB ) on he suppor S. grealy he compuaional complexiy needed since sparse approximaion algorihms (like OMP) applied o orhonormal sensing marices are reduced o he compuaion of he correlaions and he selecion of N RF columns of A ha produce he highes correlaions. Since we are no ineresed in imposing explicily he orhogonal consrain on F RF, in sep B we furher reduce he objecive funcion by saring a local search, of lengh L, around he seleced seering vecors from A (he suppor se S) o find a beer suppor for he RF precoder. The baseband precoder is compued his ime a every sep, similarly o he OMP soluion. The whole approach is similar o a block learning mechanism presened in [21]. Noice anoher immediae approach also follows here. Insead of sep B of he proposed mehod, suppose ha we apply he SOMP approach using he frequency dicionary A resriced o he full index se S J = N RF j=1 S j of size (2L+1)N RF, and no on he full resoluion N, as described in sep B. The full correlaions sill need o be compued, bu he subsequen seps involve a working dimension (2L+1)N RF N. Generally N is large since we wan o produce resuls under a good resoluion. In his approach, sep A acs like a grid reducion sep. We call his approach OM+SOMP, since i differs from he iniial approach by replacing sep B wih an SOMP approach and no a sequenial aom updae. In erms of complexiy OM+LS is simpler han OM+SOMP bu in erms of performance we expec he laer o do beer. The same discussion applies a he receiver for he design of he pair (W RF, W BB ) by solving arg min A r, W BB E[yy H ] 1/2 (W MMSE A r W BB ) F subjec o diag( W BB W H BB) 0 = N RF r (11) using he same wo sep approach of orhogonal maching followed by a local search sraegy depiced in Algorihm 1 for he new objecive funcion. B. Compuaional complexiy Equipped wih he wo proposed sraegies, orhogonal maching o reduce he array manifold space followed by eiher a fas local search (OM+LS) or a slower full maching pursui search (OMP+SOMP) in he reduced manifold, we now esablish how hey improve on he previous soluion o problem (6) described in [9]. The compuaional complexiy of he proposed mehod is dominaed by he compuaion of he correlaions R = A H F op. The oher compuaions of sep A only represen fas manipulaions of he elemens of his marix noe ha all producs Q H l F op are conained in R. Sep B is dominaed by correlaion compuaions of subses of 2L + 1 columns from A wih he curren residual F res. The correlaions in sep B are compued only for (2L + 1) imes. The local search parameer should obey 1 L α/2 and generally akes low values. All correlaions of he ype FRF HF op are compued for boh algorihms only once when he full correlaions A H F op are found. Considering ha seering vecors (insead of he full N), again N RF α < N RF N N, wih N = αn he wo mehods (we omi OM+LS for breviy) ake approximaely C SOMP 2αN RF N 2 N s (N + 1) C OM+SOMP 2αN s N 2 + 2αN s N + 2αN + 2N s N RF (N RF 1)(α + 1)(N + 1) (12) number of operaions. The speed up comes from he fac ha he full correlaion marix in he proposed mehod is compued only once (in sep A) while for he SOMP approach is compuaion needs o ake place N RF imes. Noice ha boh compuaionally expensive procedures of sep A are included in he firs sep of he regular SOMP, hus he (N RF -1) facor for he run of he maching pursui seps of OM+SOMP. We now leverage he speed-up provided by OM+SOMP o increase he resoluion and produce beer soluions. Consider ha SOMP runs for a resoluion N 1 = α 1 N while OM+SOMP runs for a differen resoluion N 2 = α 2 N, we would like o know how large can α 2 be such ha complexiy

5 5 S N = 32 N = 64 N = 128 Specral Efficiency (bis/s/hz) Opimum SOMP OM+SOMP Ns=4 Ns=2 Ns= RF Number of RF chains N Fig. 2: Evoluion of speedup S when OM+SOMP and SOMP are used wih he same angular resoluion N = 4N for fixed N s = 4. of he proposed mehod does no surpass ha of SOMP. Considering he local search lengh L = α 1 /d, d 2, in he case of OM+SOMP we analyze and compare he complexiies in (12) o reach: α 2 dn RF N s (N + 1)(α 1 N N RF ) N RF (N RF 1)N s (N + 1) + dn (N s (N + 1) + 1) (13) Keeping he value of α 2 under his bound guaranees ha he compuaional complexiy of he proposed will be lower han ha of SOMP. Alernaively, consider ha OM+SOMP and SOMP runs wih he same angular resoluion N = αn. In his case we show he achieved speed-up (S = C SOMP /C OM+SOMP ). Figure 2 show he evoluion of S for various dimensions. IV. NUMERICAL RESULTS In his secion we provide several Mone Carlo simulaion resuls o illusrae he performance of he hybrid precoder/combiner soluion presened in he previous secion. We consider he narrowband clusered channel model in (2) wih N cl = 6 clusers and N ray = 8 propagaion pahs per cluser. For purposes of simulaions, he enries of H denoed by α i,l are assumed o be IID CN (0, σα,i 2 ) where σ2 α,i is he average power of he i h cluser. We se σα 2 i,l = σα, 2 all clusers wih equal power saisfying he normalizaion consrain E[ H 2 F ] = N N r. The scaling facor in fron of he summaion in (2) is used o ensure ha E[ H 2 F ] = N N r. The angles of deparure and arrival φ i,l, φr i,l are normal random disribued wih mean cluser angle φ i, φr i uniformly random disribued in [0, 2π]. The angle spread is se o σφ = σr φ = 7.5. Two ULAs wih N = 32/64 and N r = 32/64 are considered for ransmission and recepion. The same oal power consrain is fixed for all precoders wih equal power allocaion per sream and he signal o noise raio is given by SNR = ρ σ. n 2 Fig. 3 shows he specral efficiency achieved by he hybrid analog digial precoders SOMP and OM+SOMP, ogeher wih SNR(dB) Fig. 3: Specral efficiency achieved by differen precoders for ULA sysem wih 32 ransmi/receive anennas. The cluser channel has N cl = 6, N ray = 8 wih an angular spread of 7.5. N RF = 4, Nr RF = 4 RF chains are considered and N s {1, 2, 4} daa sreams. Specral Efficiency (bis/s/hz) Opimum SOMP OM+SOMP SNR(dB) Ns=6 Ns=2 Ns=1 Fig. 4: Specral efficiency achieved by differen precoders for ULA sysem wih 64 ransmi/receive anennas. The cluser channel has N cl = 6, N ray = 8 wih an angular spread of 7.5. N RF = 6 RF chains are considered and N s {1, 2, 6} daa sreams. he he opimum unconsrained soluion for differen SNR values. We assume a sysem wih N = N r = 32 ULAs for he ransmier and receiver, N RF = 6 RF chains and N s {1, 2, 4} daa sreams. Fig. 4 plos again he specral efficiency for a differen seup: N RF = 6 and N s {1, 2, 6}. We see ha he proposed precoder OM+SOMP achieves specral efficiencies ha are very closed o hose achieved by SOMP for all he se ups. For low number of sreams OM+SOMP almos perfecly maches SOMP, while he gap beween boh mehods increases for high SNR and high number of daa sreams. A he same ime, he efficiencies obained by boh hybrid precoders are similar o he opimum unconsrained soluion for low number of daa sreams. The differences beween he opimum unconsrained soluion and he unconsrained ones, however, becomes non-negligible when he number of sreams

6 6 Specral Efficiency (bis/s/hz) Opimum SOMP OM+SOMP Angle Spread (degrees) Ns=4 Ns=2 Ns=1 Fig. 5: Specral efficiency as a funcion of he Angle Spread. We assume an ULA sysem wih 32 ransmi/receive anennas, N RF = 4 RF chains and N s {1, 2, 4} daa sreams. The cluser channel has N cl = 6, N ray = 8 and varying angular spread. We have he SNR= 0 db. equals he number of available RF chains. Fig. 5 shows he specral efficiency as a funcion of he angle spread of he channel scaerers. We assume a sysem wih N = N r = 32 anennas, N RF = 6 RF chains and N s {1, 2, 4} daa sreams. The signal o noise raio is fixed o SNR= 0 db. We see ha he performance of he hybrid precoders decreases wih an increase of he angle spread, while he gap beween OM+SOMP and SOMP remains consan. In all cases, OM+SOMP and SOMP run wih he same angular resoluion N and hus OM+SOMP achieves a considerable speedup wihou sacrificing he performance in erms of he specral efficiency. V. CONCLUSIONS In his paper we developed a new opimizaion algorihm for he design of hybrid precoders and combiners for mmwave MIMO sysems. Our wo soluions incorporae consrains ha accoun for he pracical hardware limiaions a hese frequencies: analog beamforming based on quanized variable phase shifers and he use of a limied number of RF chains. The main innovaion in our work is o exploi he array geomery in a way ha allows us o reduce he search complexiy and hus he overall complexiy of he algorihm. Simulaion resuls show ha he specral efficiency achieved by using he new algorihms is comparable o he unconsrained soluion, ye wih subsanially lower overall complexiy. [4] T. Rappapor, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. Wong, J. Schulz, M. Samimi, and F. Guierrez, Millimeer wave mobile communicaions for 5G cellular: I will work! IEEE Access, vol. 1, pp , [5] E. Torkildson, C. Sheldon, U. Madhow, and M. Rodwell, Millimeerwave spaial muliplexing in an indoor environmen, in IEEE GLOBE- COM Workshops, December 2009, pp [6] C. Sheldon, M. Seo, E. Torkildson, M. Rodwell, and U. Madhow, Fourchannel spaial muliplexing over a millimeer-wave line-of-sigh link, in IEEE GLOBECOM Workshops, June 2009, pp [7] T. Rappapor, F. Guierrez, E. Ben-Dor, J. Murdock, Y. Qiao, and J. Tamir, Broadband millimeer-wave propagaion measuremens and models using adapive-beam anennas for oudoor urban cellular communicaions, IEEE Transacions on Anennas and Propagaion, vol. 61, no. 4, pp , [8] S. Sun, T. Rappapor, R. W. Heah Jr., A. Nix, and S. Rangan, MIMO for millimeer wave wireless communicaions: Beamforming, spaial muliplexing, or boh? IEEE Communicaions Magazine, December [9] O. El Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. W. Heah Jr., Spaially sparse precoding in millimeer wave MIMO sysems, IEEE Transacions on Wireless Communicaions, vol. 13, no. 3, pp , March [10] A. Alkhaeeb, O. E. Ayach, G. Leus, and R. W. Heah Jr, Channel esimaion and hybrid precoding for millimeer wave cellular sysems, IEEE Journal of Seleced Topics in Signal Processing, vol. 8, no. 5, Ocober [11] W. Roh, J.-Y. Seol, J. Park, B. Lee, J. Lee, Y. Kim, J. Cho, K. Cheun, and F. Aryanfar, Millimeer-wave beamforming as an enabling echnology for 5g cellular communicaions: heoreical feasibiliy and prooype resuls, IEEE Communicaions Magazine, vol. 52, no. 2, pp , February [12] A. Tropp, A. C. Gilber, and M. J. Srauss, Algorihms for simulaneous sparse approximaion. Par I: Greedy pursui, Signal Processing (Special Issue on Sparse Approximaions in Signal and Image Processing), pp , [13] X. Zhang, A. Molisch, and S. Kung, Variable-phase-shif-based RFbaseband codesign for MIMO anenna selecion, IEEE Transacions on Signal Processing, vol. 53, no. 11, pp , [14] A. Sayeed and J. Brady, Beamspace MIMO for millimeer-wave communicaions: Sysem archiecure, modeling, analysis, and measuremens, in Proc. of 2013 IEEE Global Telecommunicaions Conference (GLOBECOM), Alana, GA, [15] J. V. Wallace and M. A. Jensen, Modeling he indoor MIMO wireless channel, IEEE Transacions on Anennas and Propagaion, vol. 50, no. 5, pp , [16] P. Almers, E. Bonek, A. Burr, N. Czink, M. Debbah, V. Degli- Esposi, H. Hofseer, P. Kysi, D. Laurenson, G. Maz, A. F. Molisch, C. Oesges, and H. Ozcelik, Survey of channel and radio propagaion models for wireless MIMO sysems, EURASIP Journal on Wireless Communicaions and Neworking, [17] A. Forenza, D. J. Love, and R. W. Heah, Simplified spaial correlaion models for clusered MIMO channels wih differen array configuraions, IEEE Transacions on Vehicular Technology, vol. 56, no. 4, pp , [18] C. Gusafson, K. Haneda, S. Wyne, and F. Tufvesson, On mm-wave mulipah clusering and channel modeling, IEEE Transacions on Anennas and Propagaion, vol. 62, no. 3, pp , [19] T. Kailah, A. H. Sayed, and B. Hassibi, Linear esimaion. Prenice Hall, [20] J. Chen and X. Huo, Theoreical resuls on sparse represenaions of muliple-measuremen vecors, IEEE Transacions on Signal Processing, vol. 54, no. 12, pp , [21] C. Rusu and B. Dumirescu, Block orhonormal overcomplee dicionary learning, in 21s European Signal Processing Conference, 2013, pp REFERENCES [1] Z. Pi and F. Khan, An inroducion o millimeer-wave mobile broadband sysems, Communicaions Magazine, IEEE, vol. 49, no. 6, pp , [2] T. Rappapor, R. W. Heah Jr., R. Daniels, and J. Murdock, Millimeer wave wireless communicaions. Prenice Hall, [3] F. Boccardi, R. Heah, A. Lozano, T. Marzea, and P. Popovski, Five disrupive echnology direcions for 5G, IEEE Communicaions Magazine, vol. 52, no. 2, pp , February 2014.