On the Number of RF Chains and Phase Shifters, and Scheduling Design with Hybrid Analog-Digital Beamforming

Transcription

1 IEEE TRASACTIOS O WIRELESS COMMUICATIOS, TO APPEAR On the umber of RF Chans and Phase Shfters, and Schedulng Desgn wth Hybrd Analog-Dgtal Beamformng Tadlo Endeshaw Bogale, Member, IEEE, Long Bao Le, Senor Member, IEEE Afshn Haghghat and Luc Vandendorpe Fellow, IEEE Abstract Ths paper consders hybrd beamformng (HB) for downlnk multuser massve multple nput multple output (MIMO) systems wth frequency selectve channels. The proposed HB desgn employs sets of dgtally controlled phase (fxed phase) pared phase shfters (PSs) and swtches. For ths system, frst we determne the requred number of rado frequency (RF) chans and PSs such that the proposed HB acheves the same performance as that of the dgtal beamformng (DB) whch utlzes (number of transmtter antennas) RF chans. We show that the performance of the DB can be acheved wth our HB just by utlzng r t RF chans and r t( r t + ) PSs, where r t s the rank of the combned dgtal precoder matrces of all subcarrers. Second, we provde a smple and novel approach to reduce the number of PSs wth only a neglgble performance degradaton. umercal results reveal that only 0 PSs per RF chan are suffcent for practcally relevant parameter settngs. Fnally, for the scenaro where the deployed number of RF chans ( a) s less than r t, we propose a smple user schedulng algorthm to select the best set of users n each subcarrer. Smulaton results valdate theoretcal expressons, and demonstrate the superorty of the proposed HB desgn over the exstng HB desgns n both flat fadng and frequency selectve channels. Index Terms Hybrd Analog-Dgtal Beamformng, Massve MIMO, Mllmeter wave, Phase shfter, RF chan I. ITRODUCTIO Multple nput multple output (MIMO) s one of the promsng technques for mprovng the spectral effcency of wreless channels. To explot the full potental of MIMO, one can leverage the conventonal dgtal beamformng (DB). There are many DB desgn approaches developed n the past couple of decades. However, these approaches are desgned manly for few number of antennas (around 0) [], []. It s shown that deployment of the massve number of antennas at the transmtter and/or recever (massve MIMO) can sgnfcantly enhance the spectral and energy effcency of mcrowave and mllmeter wave (mmwave) wreless networks [3], [4]. In DB, one rado frequency (RF) chan s requred for each antenna element at the base staton (BS) and user equpment (UE) where an RF chan ncludes low-nose amplfer, down-converter, dgtal to analog converter (DAC), analog Part of ths work has been presented n IEEE ICC 05 conference. Tadlo Endeshaw Bogale and Long Bao Le are wth the Insttute atonal de la Recherche Scentfque (IRS), Unversté du Québec, Montréal, Canada and Afshn Haghghat s wth the Interdgtal, Montreal, Canada, and Luc Vandendorpe s wth the Unversty Catholque de Louvan, Belgum. Emal: {tadlo.bogale, long.le}@emt.nrs.ca, Afshn.Haghghat@nterdgtal.com and luc.vandendorpe@uclouvan.be to dgtal converter (ADC) and so on [5], [6]. Thus, when the number of BS antennas s very large, the hgh cost and power consumptons of mxed sgnal components, lke hgh-resoluton ADCs and DACs, mply that dedcaton of a separate RF chan for each antenna s hghly neffcent [3], [7] []. For these reasons, beamformng desgn wth lmted number of RF chans has recently receved sgnfcant attenton. One approach of achevng ths goal s to deploy beamformng at both the dgtal and analog domans,.e., hybrd beamformng (HB). In the dgtal doman, beamformng can be realzed at the baseband frequency whereas, n the analog doman, beamformng can be mplemented by usng low cost phase shfters (PSs) at the RF frequency [] [4]. Dfferent mplementaton aspects of the HB archtecture can be found n [5] and [6]. In [7] [9], a HB archtecture s suggested for sngle user massve MIMO systems where matchng pursut (MP) algorthm s utlzed [5]. In [0], a codebook based HB s proposed for wdeband mmwave wreless networks. The codebooks are desgned symmetrcally for mtgatng the possble beam shft due to the large dfferences of wave lengths at dfferent sub-bands. In [], a low complexty codebook based RF beamformng based on mult-level RF beamformng and level-adaptve antenna selecton s consdered. In [5], two types of sub-array HB archtecture are consdered; nterleaved and localzed sub-arrays. In the nterleaved array, antenna elements n each sub-array scatter unformly over the whole array whereas, n a localzed array, antenna elements are adjacent to each other. In [], HB desgns utlzng nterleaved and sde-by-sde sub-arrays (.e., lke n [5]) s proposed. Ths desgn s used for adaptve angle of arrval (AOA) estmaton and beamformng by utlzng dfferental beam trackng and beam search algorthms. In [3], hybrd precodng scheme for multuser massve MIMO systems s consdered. The paper employs the zero forcng (ZF) hybrd precodng where t s desgned to maxmze the sum rate of all users. In [8], a beam algnment technque usng adaptve subspace samplng and herarchcal beam codebooks s proposed for mmwave cellular networks. A mult-beam selecton precodng approach whle explotng the sparse characterstcs of mmwave channels s employed n [4]. In [5], a beam doman reference sgnal desgn for downlnk channel wth HB archtecture s proposed to maxmze the gan n a certan drecton around the man beam. In [6], a HB desgn usng convex optmzaton s proposed

2 for power mnmzaton and maxmzaton of the worst case sgnal to nterference plus nose rato (SIR) problems. A HB desgn for GHz applcaton utlzng planar antenna arrays s consdered to analyze the SIR of all user equpments (UEs) n [7]. The work of [8] consders a HB archtecture and focuses on solvng the mnmzaton of the transmt power subject to the SIR constrants. umercal algorthm based on sem defnte programmng (SDP) relaxaton s proposed for examnng the optmzaton problem. In [9], the sum rate maxmzaton problem for the downlnk massve MIMO systems s studed usng the MP soluton approach. In [], a beam tranng (or beam steerng) problem for the GHz mmwave communcatons s formulated as a numercal optmzaton problem such that the receved sgnal s maxmzed. The paper ams at dentfyng the optmal beam par from a prescrbed codebook wth lttle overhead usng numercal search approach. Recently, the jont optmzaton of the analog and dgtal beamformng matrces are consdered n [3] to maxmze the achevable rate wth dfferent practcal constrants under the condton that each antenna can only be connected to a unque RF chan. Moreover, low complexty hybrd precodng scheme wth sub-array archtecture has been proposed to optmze the channel capacty n [3]. The consdered scheme leverages the dea of teratve successve nterference cancellaton (SIC) whch allows parallelzaton. Motvaton: The DB always acheves optmal performance for any desgn crteron and channel. Therefore, any HB desgn cannot acheve better performance than that of DB [], [5], [7]. The aforementoned HB desgns examne ther performance for specfc desgn crteron(crtera) and/or channel model(s). For nstance, the desgns of [7], [3], [9] consder sum rate maxmzaton problem and t s not clear how the performance of these HB desgns behave for other desgn crtera. Ths lmtaton also arses n all the aforementoned HB desgns. Ths s manly due to the fact that these HB desgns are problem dependent, and are not able to quantfy the performances of ther desgns n terms of that of the DB for any desgn crtera even f PSs have suffcently hgh (.e., theoretcally nfnte) resoluton. From these explanatons, we can understand that desgnng a HB ensurng the same performance as that of DB for any desgn crtera and channel has not been addressed n the aforementoned HB desgns. Objectves: In a multuser setup, whch s the focus of ths paper, the problem of desgnng a HB whle achevng the same performance as that of the DB can be addressed for the uplnk or downlnk channels. Snce there s a dualty between these channels, examnng the above problem for ether of the channels wll be suffcent whch motvates the current work to consder downlnk channel []. For ths channel, each user can have ether sngle antenna or multple antennas. In the current paper, we assume each user to be equpped wth sngle antenna. Moreover, n a typcal macro BS, transcevers are desgned to operate n a consderable range of bandwdths. The large bandwdth and multpath nature of wreless chan- In the followng, we wll use the phrase nfnte resoluton PS to reflect that the resoluton of the correspondng PS s suffcently hgh such that the effect of phase error due to quantzaton can be neglected. nels n a cellular system motvates us to consder frequency selectve channels. In addton, we assume that perfect channel state nformaton s avalable at the BS. Under these setups and assumptons, the current paper consders the followng problems: P For arbtrary transcever optmzaton crtera and channel matrx, frst we consder the desgn of a HB archtecture whle ensurng the same performance as that of the DB desgn. Specfcally, we perform a study on the number of requred RF chans and PSs under an ntutve desgn by assumng that PSs have nfnte resoluton (P A ). Then, we examne realzng the analog beamformng part of the HB desgned n P A usng practcal constant phase PSs (CPPSs) and swtches only wth neglgble performance loss (P B )? P Snce the consdered system model s a multuser wdeband massve MIMO system, each of the users may use only part of the avalable spectrum. Ths motvates us to consder the user schedulng problem usng the HB archtecture desgned n P B. The exstng works on hybrd beamformng desgn employ dfferent archtectures. In general, three archtectures are commonly adopted; partally connected archtecture as n [3] [33], fully connected archtecture as n [5], [7], [3], [9] and an archtecture utlzng dgtally controlled pared PSs as n [5], [6]. In the current paper, we have employed the modfed verson of hybrd archtecture suggested n [6] as t s sutable to address the above two problems (the detals of ths modfcaton s provded n Secton IV). We would lke to emphasze here that the current paper addresses P and P only sub-optmally. Therefore, ensurng global optmalty for these problems s stll an open research topc. On the other hand, these problems are addressed for general channel matrx and carrer frequences. Hence, the results of the current paper are vald both for mcrowave and mmwave massve MIMO applcatons. The current paper has the followng man contrbutons: ) We propose a HB desgn and determne the number of RF chans and PSs for multuser and multcarrer massve MIMO systems such that the performances of the proposed HB desgn and the DB desgn are the same by assumng that PSs have nfnte resoluton. In partcular, we show that the performance of the DB can be acheved wth the proposed HB just by utlzng p = r t ( r t + ) nfnte resoluton PSs and r t RF chans, where s the number of antennas at the BS and r t s the rank of the combned dgtal precoder matrces of all sub-carrers (B d ). Then, we provde a novel and smple approach to realze the HB by employng r t RF chans and cp p CPPSs per each RF chan. Partcularly, for fnte analog precodng matrx precson level of 0 p (detaled n Secton IV-B), only 80p CPPSs are requred per each RF chan. As wll be clear n the smulaton secton, accuracy s suffcent for practcally relevant desgn problems. Thus, n practce only 0 CPPSs are requred per each RF chan.

3 ) From contrbuton (), we can notce that the number of RF chans s stll r t (.e., rank of B d ) whch depends on many factors such as the number of UEs and ther channel matrces, and precoder desgn crtera (e.g., sum rate, max mn rate []). Due to ths fact, the number of RF chans deployed at the BS ( a ) could be less than the rank of B d. In such a case, the DB cannot be realzed wth our HB. For these reasons, we examne P and provde performance analyss by consderng sum rate maxmzaton problem whle ensurng rank(b d ) a. Specfcally, under the commonly used unform lnear array (ULA) channel model and ZF precodng, we have shown that the performance acheved by the HB and DB desgns are the same when the angle of departure (AOD) of the channels of the scheduled users have some specal structure whch wll be clear n Lemma. 3) We perform extensve numercal smulatons to valdate the theoretcal results. We have also studed the effects of dfferent parameters such as number of RF chans, BS antennas, PSs and total scheduled users on the performance of our desgn. Computer smulatons also demonstrate that the proposed HB desgn acheves sgnfcantly better performance than those of the exstng ones both for flat fadng and frequency selectve channels. Furthermore, the proposed desgn s convenent for practcal realzaton of massve MIMO. Ths paper s organzed as follows. Secton II dscusses the HB system model. In Secton III, a summary of Raylegh fadng and ULA channel models, and the conventonal DB s provded. The proposed HB desgn s detaled n Secton IV. In Sectons V and VI, the proposed user schedulng and sub-carrer allocaton algorthm, and ts performance analyss s presented. Smulaton results are provded n Secton VII. Fnally, we conclude the paper n Secton VIII. otatons: In ths paper, upper/lower-case boldface letters denote matrces/column vectors. X (,j), X T, X H and E(X) denote the (, j)th element, transpose, conjugate transpose and expected value of X, respectvely. dag(.), blkdag(.),., x,, I and C M (R M ) denote dagonal, block dagonal, two norm, nearest nteger greater than or equal to x, an szed vector of ones, approprate sze dentty matrx and M complex (real) entres, respectvely. The acronym s.t and..d denote subject to and ndependent and dentcally dstrbuted, respectvely. II. SYSTEM MODEL Ths secton dscusses the proposed HB for a downlnk multuser and multcarrer massve MIMO system whch s shown n Fg.. As we can see from ths fgure, the BS and each of the UEs are equpped wth and antenna, respectvely. We employ block based multuser orthogonal frequency doman multple access (OFDMA) transmsson where each block has f sub-carrers. At each symbol perod, the BS broadcasts K symbols, where K s the number of served UEs. Thus, n each OFDMA block, K f symbols wll be transmtted. For convenence, let us represent the transmtted symbols n each OFDMA block by D = [d, d, d f ], where d = [d,, d K ] T and d k s the kth UE th sub-carrer symbol. Snce we have employed OFDMA transmsson, D s the symbol matrx n frequency doman. The precodng and decodng operatons of ths frequency doman nput data s explaned as follows. Let us defne h nk (0), h nk (L p ) as the multpath channel coeffcents between the nth BS antenna and kth UE, and L p s the number of multpath channel taps between the BS and all UEs. For ths model, the kth UE receved sgnal s gven as [34] [36] r H k =[h H kab d,, h H f kab f d f ]F H + ñ H k () where B C a K s the dgtal precoder matrx of the th sub-carrer, A C a s the analog precoder matrx, F H s the nverse fast Fourer transform (FFT) matrx of sze f, ñ H k C f s the nose vector at the kth UE and h k s the channel matrx of kth UE th sub-carrer whch s gven as h k = [D hk (), D hk (),, D hk ()] T wth D H hnk = dag(λ nk({} f =0 )) as a dagonal matrx of sze f and λ nk () = L p s f s=0. At the kth UE, the h π nk (s)e j tme doman sgnal wll be transformed to frequency doman by employng FFT operaton. It follows r H k =r H k F =[h H kab d,, h H f kab f d f ] + ñ H k F. () The recovered sgnal of the kth UE s th sub-carrer can now be expressed as ˆd k = h H kab d + n k,, k (3) where n k = ñ H k f s the kth UE th sub-carrer nose sample whch s assumed to be..d zero mean crcularly symmetrc complex Gaussan (ZMCSCG) random varable wth unt varance. The current paper assumes that A s realzed wth unty modulus PSs only as n [7], [3], [9]. ote that snce the current paper assumes a sngle antenna UE, the UE s operaton s the same as that of the conventonal DB (.e., HB s not requred at the UE sde). III. CHAEL MODEL AD DIGITAL BEAMFORMIG For better exposton of the paper, ths secton summarzes the geometrcal channel model and conventonal DB. A. Channel Model To model the th sub-carrer channel between the BS and kth UE, we consder the most wdely used geometrc channel model wth L s scatterers. Under ths assumpton, hk (q) = [ h k (q), h k (q),, h k (q)] T can be expressed as [8], [5], [7], [0], [9] h k (q) = L s ρ k L s m= c km (q) τ k (θ km ) = τ k c k (q) (4) where ρ k s the dstance dependent pathloss between the BS and kth UE, c km s the complex channel coeffcent of the kth UE mth path wth E{ c km } =, θ km [0, π] s the AOD, τ k (.) s the antenna array response vector of the

4 Source Tx (Dgtal part) RF Chan Tx (Analog part) Freq. Dom. data source D (,:) D (,:) D (K,:) Freq. Dom. BF a F H IFFT (row) & add CP a RF RF analog analog d,, d f B,, B f analog Analog RF a A BF Analog BF Analog BF H Dscard CP & take FFT Dscard CP & take FFT K F Dscard CP & take FFT ˆd ˆd ˆd K Decode ˆd,, ˆd f Decode ˆd,, ˆd f ˆd k = h H kab d + n k Decode ˆd K,, ˆd f K Fg.. System model of the proposed HB for multuser and multcarrer systems. kth UE, τ k = [ τ k (θ k ), τ k (θ k ),, τ k (θ kls )] and c k (q) = L sρ k [c k (q), c k (q),, c kls (q)] T. For performance analyss (dscussed n Secton VI), ths paper adopts the most wdely used Raylegh fadng and ULA channel models. The model (4) turns out to be Raylegh fadng channel when L s s very large, and (4) turns out to be ULA channel when τ k (.) s modeled as [8] τ k (θ) = [, e j π λ d sn (θ),, e j( ) π λ d sn (θ) ] T (5) where j =, λ s the transmsson wave length and d s the antenna spacng. B. Dgtal Beamformng For better understandng of the proposed HB desgn, ths subsecton provdes a bref summary on the structure of the DB matrx whch s obtaned by employng RF chans. Assume that we have employed DB approach to get the precoder matrces of all sub-carrers for an arbtrary desgn crtera. Wth the DB, the recovered data ˆd k can be expressed as ˆd d k = h H kb d d + n k,, k (6) where B d s the dgtal precoder matrx of sub-carrer. By takng the QR decomposton of the combned precoder matrx B d = [B d, B d,, B d f ], one can get B d = Q d B d, where Q d C rt s a untary matrx whch satsfes (Q d ) H Q d = I rt, B d C rt (K f ) s an upper trangular matrx and r t s the rank of the matrx B d. Hence, ˆd k can be equvalently expressed as ˆd d k = h H kq d B d d + n k,, k (7) where B d s the sub-matrx of B d correspondng to sub-carrer. By agan computng the QR decomposton of (Q d ) H and after dong some mathematcal manpulatons, one can get (see (8) of [37] for the detals) ˆd d k =h H kã B B d d + n k,, k (8) where à = [G, G] H, G = dag(g, g,, g rt ), G C rt ( rt) and B C rt rt, wth each elements of G and G has a maxmum ampltude of. ote that for a massve MIMO applcaton, low complexty QR decomposton algorthms can be appled [38]. IV. HYBRID BEAMFORMIG DESIGS In ths secton, we descrbe the desgn of analog and dgtal precoder matrces of (3). As n the conventonal DB, the entres of A and B of ths equaton can be optmzed by consderng dfferent desgn objectves such as sum rate maxmzaton, SIR balancng etc. However, the rows of B depends on the avalable number of RF chans a whch s fxed a pror n the producton stage of the BS. Furthermore, snce A s realzed usng electronc components (.e., PSs), the number of PSs P S to realze A s agan fxed durng the producton stage of the BS. Ths secton determnes a and P S such that the HB desgn s able to mantan the same performance as that of the DB (whch uses RF chans) for any desgn crtera. Indeed, ths can be met whenever a and P S are determned whle ensurng ˆd k = ˆd d k (.e., the receved sgnal wth HB s the same as that of the DB) wthout consderng a desgn crtera. In the followng, we examne P by explotng ths dea. Specfcally, frst we provde the proposed HB desgn where the dgtal precodng part of Fg. s realzed usng mcroprocessors whereas, ts analog precodng employs nfnte resoluton PSs (P A ). Then, we extend ths result to handle P B. A. Hybrd Beamformng Desgn for P A Ths secton dscusses the proposed HB desgn for P A. One can notce that (3) and (8) have the same mathematcal structure. Hence, one may thnk of drectly settng A = à and B = B B d to address P A. However, snce the ampltude of each of the elements of à s not necessarly one, t s not clear how one can realze ths matrx usng PSs only. Hence, such drect plug-n wll not help to desgn the HB archtecture. For ths reason, the authors of [6] frst come up wth a novel and clever method to represent any vector x C as x = Wz (see Theorem of [6]), where W C and z C wth W (,j) =,, j whch leads them to conclude that the performance of any DB can be acheved wth the HB f the number of RF chans are at least two tmes that of the number of data streams (.e., two tmes the rank of B d ) [5], [6]. ow f we utlze ths technque to our HB archtecture, r t RF chans and r t dgtally controlled PSs (DCPSs) are needed to acheve the same performance as that of the DB desgn. In the followng, we propose new and smple method to reduce the number of RF chans and DCPSs compared to [6]. To ths end, we consder the followng theorem.

5 Theorem : Gven any real number x wth x, t can be shown that x =e j cos ( x ) + e j cos ( x ) (9) jx =e j sn ( x ) + e j(π sn ( x )) (0) where j =. Proof: When x, we wll have e j cos ( x ) + e j cos ( x ) = cos(cos ( x )) + j sn(cos ( x )) + cos( cos ( x )) + j sn( cos ( x )) = x. Smlar to ths expresson, one can prove that e j sn ( x ) + e j(π sn ( x )) = jx. The (m, n)th element of à can also be rewrtten as ã mn e jφmn, where 0 ã mn. By applyng (9) of Theorem, we can express à (m,n) as à (m,n) =ã mn e jφmn () =e j(cos ( ãmn )+φ mn) + e j(cos ( ãmn ) φ mn). From ths equaton, we can notce that each element of à can be equvalently expressed as a sum of two DCPSs. As the maxmum number of non-zero elements of à s r t ( r t +) (.e., from (8)), the soluton obtaned n DB can be acheved by employng r t ( r t + ) DCPSs and r t RF chans. Ths leads us to get the HB archtecture of Fg..(a) by settng A = Ã, B = B B d. () From ths result, the followng deas can be noted: The mathematcal manpulaton on ˆd d k alone does not brng new result. However, by utlzng ˆd d k and Theorem, we are able reduce the number of RF chans (by half) and DCPSs (slght dfference) compared to [6]. On the other hand, the result of ths theorem also helps us to come up wth a practcal scheme to realze A wth lmted number of CPPSs whch s dscussed n the next Secton. We would lke to emphasze here that the DCPSs of ths secton are assumed to have nfnte resoluton whch may not be realstc. Thus, the result of ths secton can be used as a benchmark for future theoretcal results (or other practcal desgns) n the HB research. B. Realzng A wth Lmted umber of CPPSs (P B ) The HB desgn approach of the above subsecton has lmtatons whch arses on how to realze A usng practcal PSs. Ths subsecton provdes a smple approach to realze A just by usng CPPSs and swtches only (.e., P ). In ths regard, t s consdered that cp pars of CPPSs are shared by each RF chan and all BS antennas wth the help of swtches as shown n Fg..(b). In fact, any swtch can be represented by 0 (dsconnected) or (connected). By denotng the swtchng matrx between the CPPSs of the th RF chan and antennas as S, the analog We have learned that recently [39] ensures the same performance as that of the DB for sngle user case. The authors of [39] bascally come up wth the same result as (9) whch s proven n a dfferent method. Thus, the work of [39] can be used for the setup of the current paper. However, stll t s not clear how to apply the approach of [39] to examne P B and P. precodng correspondng to the th RF chan of the HB can be expressed as S d cp, where S {0, } s an cp szed swtchng matrx and d cp s an cp szed vector whose entres are the scalar values correspondng to a par of CPPSs. In ths desgn, t s assumed that each par of CPPSs can be connected to a maxmum of L antennas. Furthermore, each antenna wll receve sgnals from a maxmum of L CPPSs correspondng to each RF chan. These constrants are equvalent to ensurng that the maxmum sum of each column (row) of S to be L( L) as shown n Fg..(b). Wth these constrants, one approach of examnng P B s by frst solvng mn S,d cp, L, L r t = S d cp à (:,) (3) s.t S (:,m) L, S (m,:) L, S (m,n) {0, } and then settng A (:,) = S d cp where à (:,) s as defned n (). The above explanaton suggests that each entry of d cp s a scalar value correspondng to a par of CPPSs. Furthermore, L and L correspond to the number of avalable swtches. Snce d cp, L and L are determned a pror n the producton stage of PSs and swtches, they cannot be optmzed for each realzatons of à (:,). The above problem can therefore be solved by applyng a two step approach; determnaton of d cp, L and L (for general à (:,) ), and optmzaton of S (for each realzaton of à (:,) ) whle keepng d cp, L and L constant. To ths end, we consder two cases where the frst (second) case allows the swtchng matrces S to be desgned by enablng asymmetrc (symmetrc) sgnal flow n between RF chans and antennas. ) Case : Asymmetrc sgnal flow: For such a case, we suggest to examne the above problem such that A (:,) à (:,) ɛ s ensured for arbtrary desred error tolerance ɛ > 0. One approach of mantanng ths nequalty s by desgnng d cp, S, L and L whle ensurng R{A(j,) } R{à (j,) } ɛ and I{A (j,) } I{à (j,) } ɛ. To ths end, let us consder a smple example to llustrate our soluton approach when ɛ = 0. The accuracy ɛ = 0 means that a number n between 0 and s represented by decmal places only. For example, 0.46 s represented as 0.4. Furthermore, wth ths accuracy level, any number n between 0 and can be represented as a sum of two values taken from F = [0., 0.,, ] and F = [0.00, 0.0, 0.0,, 0.09] (for nstance, 0.4 = ). Ths shows that for an accuracy of ɛ = 0 p, only 0p numbers are requred to represent any scalar value n between 0 and. We employ ths number representaton n our HB desgn. That s, for the accuracy level of ɛ = 0, usng the result of Theorem, the followng CPPSs are requred to ensure R{A (j,) } R{à (j,) } ɛ n each RF chan As dscussed above, each of the real (complex) entres of à (:,) are n the range of [, ]. Thus, to realze each of these entres wth accuracy ɛ = 0, four CPPSs taken from the above sets are requred. For example, f the real part of the nth element of à (:,) s.64, t can be represented by usng four PSs (.e., e j cos (0.80) + e j cos (0.80) + e j cos (0.0) +

6 RF Chan DCPS Combner Ant. RF Chan CPPS Swtch Ant. RF Chan CPPS Swtch Ant. RF RF rt+ r t+ RF RF cp RF RF cp RF rt RF rt rt+ RF rt RF cp Asymmetrc RF rt RF cp (a) (b) (c) Fg.. Detaled HB desgn wth: (a) DCPSs (b): CPPSs and Swtches (Asymmetrc) (c): CPPSs and Swtches (Symmetrc). { ± cos Real = (.00),, ± cos ( 0.0), ± cos (0.0),, ± cos (.00) ± cos ( 0.09),, ± cos ( 0.0), ± cos (0.0),, ± cos, (0.09) { sn (±0.0),, sn (±.00), π sn (±0.0),, π sn (±.00) Imag = sn (±0.0),, sn (±0.09), π sn (±0.00),, π sn (±0.09). (4) e j cos (0.0). Usng ths result, t can be shown that to acheve 0 p accuracy, approxmately p shared pars of CPPSs are requred per each RF chan. As A (j,) can be realzed by at most 4p pars of PSs where each of these pars are unquely obtaned from at most 0 CPPSs, the complexty of searchng these 4p pars of CPPSs s neglgble. Also, the phases of these CPPSs are not necessarly spaced unformly (for example, cos (0.0), cos (0.0) and cos (0.) are 84., 78.4, 7.5 and 66.4, respectvely). Thus, wth ths desgn, we have cp = p, L =, L = 4p, d cp = d k cp = d cp,, k are taken from pars of CPPSs as n (4) and computng S s straghtforward. In ths desgn, each par of CPPSs can be connected to a maxmum of antennas. In such a case, there could be a scenaro where a par of CPPSs can be connected to one antenna only whereas, the other pars are connected to all the antennas (.e., S (m,:) L = ). Such a phenomena may lead to asymmetrc sgnal flow n the archtecture as shown n Fg..(b) whch may not be desrable n practce 3. Ths motvates us to consder Case n the followng. ) Case : Symmetrc sgnal flow: For ths case, the determnaton of d cp, L and L, and optmzaton of S whle ensurng a prescrbed error tolerance ɛ s not trval. In the followng, we provde smple method to address the problem consdered n ths case. To ths end, we utlze d cp and L as n Case (.e., cp = p, L = 4p) but modfy L as. Wth these settngs, we round-off each of the sgnfcant L dgts of the real (magnary) components of A (:,) such that the rounded-off vector wll le to ts correspondng sgnfcant 3 In scenaro where asymmetrc sgnal flow does not have an mpact, the approach dscussed up to now can be utlzed. dgts 4. By dong so, we have observed neglgble performance loss compared to Case (see Secton VII-D for the detals). We would lke to recall here that the desgn approach of ths subsecton s only a partcular HB archtecture and hence t may not be the global optmal desgn. By utlzng the proposed desgn, however, we acheve the followng advantages compared to those of the exstng ones: ) The exstng HB approaches (for example [7], [8], [3], [9]) utlze quantzed DCPSs whereas, the proposed approach employs CPPSs. Ths puts our desgn to be advantageous over those of the exstng ones as the prce and energy consumpton of DCPSs are much hgher than those of CPPSs especally when the DCPSs have hgh resoluton []. ) We are able to provde an nsght on the relaton between the number of CPPSs and the accuracy of analog beamformng matrx whch s vald for any desgn crtera and channel when asymmetrc sgnal flow s allowed. Ths helps us to desgn the analog beamformng matrx whle mantanng symmetrc sgnal flow whch s practcally useful. Wreless communcatons channels have a non zero coherence tme T c where the channel s assumed to be almost constant. Thus, both the dgtal and analog beamformng matrces may need to be updated every T c seconds (for example, T c 0.5 mllsecond n long term evoluton (LTE) network [4], [4]). Hence, for mcrowave frequency bands, the proposed HB can utlze electronc swtches whch needs to be updated once every T c [4]. Furthermore, mmwave swtches are capable of swtchng at a fracton of nanosecond 4 Here p s selected such that L s satsfed (.e., at least one swtch for each of the real and magnary term of A (j,) ).

7 speeds where smart swtches have also been used for GHz applcaton at the recever sde n [43]. On the other hand, accordng to the detaled study of [], the prce and energy consumptons of swtches are nsgnfcant compared to those of PSs [], [44]. In some cases, reasonable performance can be obtaned just by desgnng the beamformng matrces based on the long term channel statstcs as n [45] where the beamformng matrces are kept constant for the duraton much larger than T c. For these reasons, we beleve that the ntroducton of swtches wll ncur neglgble delay, cost and energy consumpton n the proposed HB desgn [], [44]. ote that we have provded three dfferent hybrd archtecture mplementaton aspects where each of them requres dfferent sets of PSs and/or swtches. As wll be demonstrated n the smulaton secton, the performances of all these archtectures are very close to each other. As mentoned above, the prce of dgtally controlled PSs s much hgher than that of PSs havng fxed phases. On the other hand, the dgtally controlled swtches can be realzed as descrbed n [43] and ths desgn can be customzed to have ether fully connected (partally) connected HB archtecture as suggested n [33]. Due to these reasons, we beleve that the archtecture proposed n Fg..(c) s cost effectve, smple to mplement and can acheve the desred performance. Havng sad ths, however, detaled comparson of ths desgn compared to those of Fg..(a), Fg..(b), and the exstng hybrd archtectures n terms of mplementaton cost, energy consumpton and performance requres sgnfcant effort and tme, and t s left for future research. The HB desgn approach dscussed n ths secton employs r t = rank(b d ) RF chans. However, for an arbtrary channel matrx of all sub-carrers and K, the number of deployed RF chans ( a n Fg. ) may be less than that of the rank of B d. In such a case, the DB cannot be mplemented usng the proposed HB archtecture. The followng secton provdes the proposed user schedulng and sub-carrer allocaton algorthm whle ensurng rank(b d ) a. V. PROPOSED USER SCHEDULIG AD SUB-CARRIER ALLOCATIO (P) Ths secton provdes the proposed user schedulng. One can understand from the above secton that the soluton of the scheduler can be realzed usng the proposed HB desgn (.e., wth the desred accuracy) f the HB archtecture has a RF chans and rank(b d ) a. Hence, one can examne the schedulng problem to optmze B d whle ntroducng ths constrant. In practce a scheduler s usually desgned to optmze some performance crtera. To ths end, we examne maxmzaton of the sum rate of all sub-carrers wth a per sub-carrer power constrant as max B d f K log( + γ k ), = k= s.t tr{(b d ) H B d } P, rank(b d ) a (5) where K (P ) s the number of UEs served (avalable power) n sub-carrer and γ k s SIR of the kth UE n sub-carrer h (.e., γ k = H k bd k j k wth B d hh k bd j +σ = [bd, bd,, bd K ]). In [46], t s shown that the ZF precodng approach together wth user schedulng acheves the capacty regon of a multuser system when the total number of scheduled users K t are very large. Furthermore, n a massve MIMO setup wth suffcent number of scatterers, a smple precodng approach such as ZF precodng technque can acheve the optmal sum rate [4]. Due to these reasons, we utlze ZF precodng to desgn B d of problem (5). When a = (.e., DB scenaro), the rank constrant of (5) s satsfed mplctly and the above problem can be examned ndependently for each sub-carrer as max B d K k= log( + γ k ) f(b d ), s.t tr{(b d ) H B d } P,. (6) However, when a <, the soluton of (6) may not necessarly satsfy the rank constrant of (5). In the followng, we dscuss the proposed user schedulng and sub-carrer allocaton algorthm to solve (5) whch s summarzed n Algorthm I. As we can see, our algorthm employs two phases whch are explaned as follows. In the frst phase, we examne (5) by droppng ts rank constrant. Ths rank relaxed problem (.e., (6)) s solved teratvely by ncreasng ts sum rate and number of served users smultaneously for each sub-carrer. Then, we compute rank(b d ) and, f rank(b d ) a, as the constrant of (5) s satsfed, we consder ths B d as our hybrd precoder. However, f rank(b d ) > a, the constrant of (5) s volated and we wll execute the second phase. In ths phase, frst, we compute Q d from the sngular value decomposton (SVD) of the precoders of the frst S sub-carrers havng the maxmum sum rate, where S s the mnmum number of sub-carrers ensurng rank([b d, B d,, B d S]) a. Then, for fxed Q d, we re-express (7) as ˆd d k = h H k B d d + n k,, k (7) where h H k = hh k Qd. Fnally, we perform Phase I for the system (7) and set B d as Bd = Qd B d. From ths explanaton, we can understand that a gven user may or may not be scheduled to use all of the avalable subcarrers. We would lke to menton here that Algorthm I can also be extended straghtforwardly for other desgn crtera and precodng method. Convergence of Algorthm I: As can be seen from Algorthm I, the proposed schedulng algorthm employs a two step approach where the number of scheduled UEs s ncreased sequentally. From step 5 of ths algorthm, one can notce that these UEs are selected sequentally whle ensurng a non-decreasng total sum rate. Furthermore, as the BS has fnte avalable power, the sum rate acheved by the proposed algorthm s fnte. For these reasons, the proposed algorthm s guaranteed to converge to a fnte total sum rate. However, Algorthm I may not necessarly converge to the global optmal soluton, and we beleve that the development of global optmal user schedulng and sub-carrer allocaton algorthm under the proposed hybrd beamformng desgn s not trval, and t s stll an open research problem.

8 Algorthm I: User schedulng and sub-carrer allocaton algorthm. Input: Users to schedule {,,, K t }, f, K and a. Phase I: for = : f do for n = : K do leftmargn=0cm ) Set K m = K {m}, m K t, where denotes unon. ) Compute f m (B d m ), where f m(b d m the objectve functon of (6) wth K ) s m users. 3) Compute m = arg max{f m (B d m ), m} 4) Set B d = B d m and f(b d )new = f m (B d m ) 5) f f(b d )new f(b d )old then Update K = K { m }, K t = 6) K t \{ m } and f(b d )old = f(b d )new. else Break 7) end f end for end for 8) Stack the precoders B d = [B d, B d,, B d f ] and f rank(b d ) a then Employ ths B d as the HB precoder. else Go to Phase II end f Phase II: ) Sort f(b d ), n decreasng order f(bd ) f(b d ),, f(b d f ). ) Compute T = [B d,, B d S], where S s mnmum number of sub-carrers wth rank(t) a. 3) Compute SVD(T) = UΛV H wth decreasng order of the dagonal elements of Λ. 4) Set Q d of (7) as the frst a columns of U. 5) For fxed Q d, perform Phase I for the system (7) and set B d as Bd = Qd B d. Output: The precoders of all sub-carrers B d, B d,, B d f and correspondng scheduled users. VI. PERFORMACE AALYSIS In ths secton we provde performance analyss of the proposed user schedulng and sub-carrer allocaton algorthm. Massve MIMO system can be realzed both at the mcrowave and mmwave frequency bands. Recently several feld measurements have been conducted to examne the characterstcs of massve MIMO channels. For the practcally relevant number of antenna elements, t has been demonstrated that the channel covarance matrces of each UE experences hgh rank n a typcal outdoor envronment n a mcrowave massve MIMO system [47]. These results also suggest that despte the statstcal dfference between the measured channels and the..d. channels, most of the theoretcal conclusons made under the ndependence assumpton (.e...d Raylegh fadng channel) are stll vald for the real massve MIMO channels n the mcrowave frequency bands [47], [48]. On the other hand, the ULA antenna array patterns are commonly employed for wreless applcatons where mmwave (mcrowave) frequency bands typcally have very low (hgh) number of scatterers [7], [45], [49], []. These reasons motvate us examne the performance of the proposed user schedulng and sub-carrer allocaton algorthm for the Raylegh fadng and ULA channel models. By combnng the ZF precodng and Algorthm I, problem (5) can be solved and realzed by the followng three possble approaches. ) Antenna Selecton Beamformng Approach: When the beamformng matrx B d has effectve sze a K matrx (.e., when the remanng entres of B d are set to 0 a prory), the rank constrant of (5) s satsfed mplctly. Thus, for such a settng, ths problem can be solved ndependently for each sub-carrer just by employng the ZF precodng and Phase I of Algorthm I. As ths approach mplctly selects a antennas from avalable antennas, we refer to ths approach as an antenna selecton beamformng (ASB). We would lke to menton here that such an approach s wdely known n the exstng lterature [5]. Hence, the ASB approach can be treated as an exstng approach. ) Proposed Hybrd Beamformng Approach: In ths approach, we utlze the proposed HB archtecture of Fg.. Here we apply the ZF precodng to desgn the precoders B d and Algorthm I to schedule the served users and sub-carrers. We refer to ths as the proposed HB approach. 3) Dgtal Beamformng Approach: The upper bound soluton of problem (5) s acheved when we have number of RF chans whch corresponds to the conventonal DB approach. In the followng, we analyze the performances of these three approaches for the Raylegh fadng and ULA channel models. A. Raylegh Fadng Channel In ths subsecton, we examne the above approaches by assumng that the channel coeffcents h H k, k of (4) are..d Raylegh fadng. Lemma : Under ZF beamformng, Raylegh fadng channel h H k and large K t, we can have where K ASB R HB (R ASB R ASB ) and K HB when K HB = K ASB (R HB ) are the served set of users (acheved sum rate) n sub-carrer usng the exstng ASB and proposed HB approaches, respectvely. Proof: See Appendx A. From Lemma, we understand that the proposed HB acheves the same sum rate as that of the DB one when

9 K HB = K DB. However, n general, the set of served users (obtaned by Algorthm I) of the HB and DB approaches may not be necessarly the same for all channel realzatons. Ths motvates us to examne the performances of the aforementoned three approaches for the case where K ASB K HB for some. For such a case, we are not able to quantfy K DB for each channel realzaton. Thus, here we compare the performances of these three approaches by examnng ther acheved average rates under ZF beamformng wth equal power allocaton strategy as follows. Theorem : Under ZF beamformng wth equal power allocaton, P = P, K = K and a unt varance..d Raylegh the relaton between R ASB, R HB and R DB fadng channel h H k, we can have the followng average rates. ( E{R ASB } K f log + P ) K E{χa K+ max (K g )} ( E{R DB } K f log + P ) K E{χ K+ max (K g )} (8) ( E{R HB } K S log + P ) K E{χ K+ max (K s )} + ( K( f S) log + P ) K E{χa K+ max (K g )} where S, K g = Kt K, K s = Kt f K a and the notaton E{χ M max(l)} denotes the expected value of the maxmum of L ndependent Ch-square dstrbuted random varables each wth M degrees of freedom. For the smulaton, we employ smple trapezod numercal ntegraton approach of Matlab to compute E{χ M max(l)}. As wll be demonstrated n the smulaton secton, the bound derved n ths theorem s tght 5. Proof: See Appendx B. B. Unform Lnear Array (ULA) Channel From the proof of Theorem, we can observe that the proposed HB approach acheves lower average sum rates than that of the DB approach. Ths performance loss occurs due to the rank constrant of B d of (5). For the ZF precodng of ths paper, B d has the same rank as that of the combned channels of all users. Thus, the proposed HB approach acheves the same performance as that of the DB one f the combned channel of all of the K t users has approxmately a maxmum rank of a. In ths regard, we consder the followng lemma. Lemma : When d = λ and the AOD of the K t users satsfy sn (θ km ) n sn (θ)[, ], n =,,, a, where θ s an arbtrary angle, we can acheve K HB = K DB Proof: See Appendx C. and R HB = R DB,. VII. SIMULATIO RESULTS Ths secton presents smulaton results. We have used f = 64, L p = 8 (.e., 8 tap channel), ρ k =, k and K max = 8. The sgnal to nose rato (SR) whch s defned as SR = f P K maxσ s controlled by varyng P = P whle 5 ote that a closed form expresson for E{χ M max(l)} has been provded n [37] when M s set to. keepng the nose power at mw. We have used the Raylegh and ULA fadng channel models as defned n Secton III-A. All of the plots are generated by averagng over 000 channel realzatons and ASR denotes average sum rate. In Sectons VII-A - VII-C, the analog beamformng part of the proposed HB desgn approach s desgned by consderng asymmetrc sgnal flow wth ɛ = 0. A. Raylegh Fadng Channel In ths subsecton, we provde smulaton results for the scenaro where the channel (4) s taken from..d Raylegh fadng channel model. ) Verfcaton of Theoretcal Rates: In ths smulaton, we examne the tghtness of the upper bound rates gven n (8) under equal power allocaton polcy. To ths end, we take = 64, a = 6, K = K max and K t = 8. Fg. 3 shows the rates acheved by smulaton and theory. As can be seen from ths fgure, the bound derved n (8) s very tght. Furthermore, as expected the rate acheved by the proposed HB approach s hgher than that of the exstng ASB approach, and superor performance s acheved by the DB approach. ) Effect of Power Allocaton and umber of Users (K ): As can be observed from Secton VI, the theoretcal average sum rate expressons of (8) s derved by assumng that K s fxed a pror and Fg. 3 s plotted for fxed K = K max. However, when we employ Algorthm I, the number of served users per sub-carrer s updated adaptvely. Hence the number of served users per sub-carrer may vary from one channel realzaton to another. Furthermore, from fundamentals of MIMO communcatons, ZF precodng wth water fllng power allocaton acheves better performance than that of the equal power allocaton. Ths smulaton demonstrates the jont benefts of the ZF precodng wth water fllng power allocaton and Algorthm I (.e., choosng K adaptvely). To ths end, we set K K max, K t = 6 and = 64. Fg. 4 shows the performances of the exstng ASB, proposed HB and DB approaches for these parameter settngs. As we can see from ths fgure, for all approaches, performng power allocaton wth adaptve K s advantageous whch s expected 6. In the subsequent smulatons, we employ ZF precodng wth water fllng power allocaton and Algorthm I (.e., the number of served users of sub-carrer K K max s chosen adaptvely). 3) Comparson of Proposed HB and Exstng ASB Approaches: In ths smulaton, we examne and compare the performances of the proposed HB and exstng ASB approaches for dfferent parameter settngs. Fg. 5 shows the average sum rate acheved by these approaches for dfferent SR and K t. From ths fgure, we can observe that ncreasng K t ncreases the average sum rate of both approaches (for all SR values) slghtly up to some K t. Ths s expected because lm Kt K to E{χ L max(k t )} c, K to for fxed L. ext we evaluate the effect of the number of RF chans on the performances of these approaches when K t = 3 as shown n Fg. 6. From ths fgure, one can observe that ncreasng a ncreases the average sum rate. Fnally, we examne the effect of the number of transmtter antennas when K t = 3 as shown 6 ote that the complexty of water fllng power allocaton s almost the same as that of the equal power allocaton.

10 Per Subcarrer ASR (bps/hz) Smulaton Theory (Upper bound rate) DB Proposed HB 0 Exstng ASB SR (db) Per subcarrer ASR (bps/hz) SR=dB SR=6dB SR=6dB Exstng ASB Proposed HB SR= 0.dB K t Fg. 3. Comparson of theoretcal and smulated ASR of the exstng ASB, proposed HB and DB approaches under ZF precodng and equal power allocaton. Fg. 5. K t. The ASR of exstng ASB and proposed HB approaches for dfferent Per subcarrer ASR (bps/hz) Equal power wth K =K max Water fllng wth adaptve K K max DB Exstng ASB Proposed HB SR (db) Per subcarrer ASR (bps/hz) SR=dB SR=6dB Exstng ASB Proposed HB SR=6dB SR= 0.dB umber of RF chans ( a ) Fg. 4. Comparson of ASRs acheved by ZF precodng wth equal power and K = K max versus ZF precodng wth water fllng power allocaton and adaptve K. Fg. 6. a. The ASR of exstng ASB and proposed HB approaches for dfferent n Fg, 7. From ths fgure, we also observe that ncreasng ncreases the average sum rate of the proposed HB approach whch s n lne wth the theoretcal result. However, the average sum rate of the exstng ASB approach does not ncrease wth. Ths s due to the fact that the exstng ASB approach employs only the frst a antennas. From Fgs. 3-7, one can notce that the proposed HB approach acheves better performance than that of the exstng ASB approach. B. Unform Lnear Array Channel Ths subsecton provdes smulaton results for the ULA channel model. To ths end, we set L s = 8, K K max, K t = 3 and = 64. Under such settngs, we plot the sum rates obtaned by exstng ASB, proposed HB, and DB approaches for the followng two cases. Case I: In ths case, we examne the average rates when θ km, m, k are taken randomly from a unform dstrbuton U[0, π] as shown n Fg. 8. As we can see from ths fgure, the proposed HB approach acheves sgnfcantly better performance than that of the exstng ASB approach and superor performance s acheved by the DB approach whch s expected. Case II: For ths case, we examne the sum rates of the aforementoned three approaches when θ km, m, k are selected as n the condtons stated by Lemma (Fg. 9). As we can see from ths fgure, the proposed HB approach acheves the same performance as that of DB and nferor performance s acheved by the exstng ASB approach whch s n lne wth Lemma. The effects of and a on the performances of the exstng ASB and proposed HB for ULA channels can be studed lke n the above subsecton. The detals are omtted for concseness. C. Effect of the umber of Phase Shfters Up to now, we employ the number of PSs as derved n Secton IV-A. However, as motvated prevously, t s practcally nterestng to realze the proposed HB archtecture wth lmted number of CPPSs as n Secton IV-B. Ths smulaton examnes the sum rate of the proposed HB for = 8 for dfferent number of CPPSs per each RF chan (.e., dfferent

11 Fg. 7.. Per subcarrer ASR (bps/hz) Exstng ASB (All ) Proposed HB (=3) Proposed HB (=64) Proposed HB (=96) SR (db) The ASR of exstng ASB and proposed HB approaches for dfferent Persubcarrer ASR (bps/hz) Exstng ASB Proposed HB DB SR (db) Fg. 9. The ASR of ASB, proposed HB, and DB for ULA channels wth AODs are as n Lemma wth θ = π. Per subcarrer ASR (bps/hz) Exstng ASB Proposed HB DB SR (db) Fg. 8. The ASR of exstng ASB, proposed HB, and DB for ULA channels wth AOD are taken from U[0, π]. levels of ɛ) as shown n Fg. 0. As can be seen from ths fgure, the average sum rate saturates after a certan number of CPPSs whch s around for our setup. Ths demonstrates that the proposed HB can be realzed wth qute small number of CPPSs (.e., from 0 to CPPSs per RF chan) and hence t s sutable for practcal mplementaton. When the number of CPPSs are zero, the proposed HB yelds the same average sum rate as that of the exstng ASB whch s expected. D. Comparson of the Proposed and Exstng Approaches for Flat fadng Channel As detaled n the ntroducton secton, a number of HB approaches are proposed where most of them employ MP algorthms. Ths motvates us to compare the performances of the proposed approach wth those of [7] and [9]. The work of [7] proposes a HB for sngle user massve MIMO system wth flat fadng channel. The algorthm of ths paper can be extended easly for multuser setup when each recever has sngle antenna by utlzng approprate DB. Also n [9], a HB algorthm s proposed for flat fadng multuser massve MIMO setup. Ths smulaton compares the algorthms of these pa- Per subcarrer ASR (bps/hz) 0 SR=6dB SR=6dB Proposed HB (K t =8), ULA Proposed HB (K t =3), ULA Proposed HB (K t =8), Raylegh Proposed HB (K t =3), Raylegh umber of phase shfters per RF Chan Fg. 0. The ASR of the proposed HB wth dfferent number of CPPSs per RF chan (Asymmetrc sgnal flow). pers, the exstng ASB and the proposed HB algorthms. To ths end, we take = 64, ρ k =, K = 6 (.e., the number of served users) and employ ZF precoder for all algorthms (.e., the proposed algorthm, and those of [7] and [9]). Fg. shows the performances of these algorthms for ULA channel wth dfferent number of scatterers L s and RF chans a. As can be seen from ths fgure, the performances of [9] and [7] are better than that of the ASB algorthm. However, the sum rates acheved by the algorthms of [9] and [7] are sgnfcantly lower than that of the DB especally when L s s large. For the asymmetrc (Asy) sgnal flow case, the proposed HB acheves the same performance as that of the DB for both a = 6 and 4 when P S 3 (.e., less than P S of [9] and [7]). Ths fgure also confrms that deployng only P S = 6 CPPSs per RF chan s stll suffcent both for Asy and symmetrc (Sym) sgnals flows. Hence, the proposed HB desgn s also cost effcent. ote that the algorthms presented n Fg. have almost the same computatonal complexty.

12 ASR (bps/hz) ASR (bps/hz) 0 DB Pro ( PS =3, Asym) Pro ( PS =3, Sym) Pro ( PS =6, Asym) Pro ( PS =6, Sym) Alg n [9] ( PS =64) Alg n [7] ( PS =64) Exstng ASB umber of scatterers (L s ) (a) DB Pro ( cp =3, Asym) Pro ( cp =3, Sym) Pro ( cp =6, Asym) Pro ( cp =6, Sym) Alg n [9] ( PS =64) Alg n [7] ( PS =64) Exstng ASB umber of Scatterers (L s ) (b) Fg.. Comparson of the proposed and exstng algorthms for flat fadng ULA channel at SR = 0dB: (a) when a = 6, (b) when a = 4. In ths fgure, P S denotes the number of PSs per RF chan, SR = P σ and σ = mw. VIII. COCLUSIOS Ths paper consders hybrd beamformng for downlnk multuser massve MIMO systems n frequency selectve channels. We examne the scenaro where the transmtter equpped wth antennas s servng K decentralzed sngle antenna users. For ths scenaro, frst we quantfy the requred number of RF chans and PSs such that the proposed HB acheves the same performance as that of the DB whch utlzes RF chans. We show that the performance obtaned by the DB can be acheved wth our HB just by utlzng r t RF chans and r t ( r t +) PSs, where r t s the rank of the combned dgtal precoder matrces of all sub-carrers. Second, we provde smple and novel approach to reduce the number of PSs wth neglgble performance degradaton. From smulaton, we have found that only 0 PSs per RF chan are suffcent for most practcal parameter settngs. Fnally, for the case where the deployed number of RF chans a < r t, we propose a smple user schedulng and sub-carrer allocaton algorthm to choose the best set of served users n a sub-carrer. The performance of the proposed schedulng algorthm s examned analytcally. Extensve numercal smulatons are performed to valdate theoretcal results, and study the effects of dfferent parameters such as a, and PSs. Computer smulatons also demonstrate that the proposed HB acheves sgnfcantly better performance than those of the exstng HBs n both flat fadng and frequency selectve channels. Moreover, our HB desgn s smple and convenent for practcal mplementaton of massve MIMO systems. APPEDIX A: PROOF OF Lemma For convenence, we provde the proof of Lemma by omttng the superscrpt (.) d n B d Exstng ASB approach When the beamformng matrx of each sub-carrer B has a rows, the rank constrant of (5) s satsfed mplctly. Under ths settng, the user schedulng can be performed per sub-carrer ndependently. The remanng task s to examne ths problem for each sub-carrer. B ASB max,p ASB k K k= K k= log( + p ASB k ), s.t H H (K ASB )B ASB = I, p ASB k [(B ASB ) H B ASB ] k,k P (9) where H (K ASB ) C a K s the truncated channel matrx of the users of sub-carrer scheduled by the exstng ASB. As the total number of users K t s very large, at optmalty rank(h (K ASB )) = a s satsfed almost surely. Thus, wthout loss of generalty, we assume that H (K ASB ) s a full rank channel matrx. Under the ZF beamformng desgn, we have B ASB = H (K ASB )[H (K ASB ) H H (K ASB )]. (0) By employng B ASB and performng some mathematcal manpulatons, the power allocaton part of (9) can be reexpressed as R ASB where b ASB k = max p ASB k K,,k k= k= log ( + p ASB k gk ASB ), K s.t p ASB k = P () s the kth column of B ASB and gk ASB = b ASB k. () From ZF precodng and rank(b ASB ) satsfes column of B ASB (b ASB k ) = K, the kth h m (K ASB ) H b ASB k = δ k,m, m =,, K (3) ) s the mth col- should be orthogonal to the ) : m =,, K, m where δ k,m s the Drac delta and h m (K ASB umn of H (K ASB ). Thus, b ASB k sub-space B ASB k k}. It follows b ASB k = g ASB k = = span{h m (K ASB h H k (KASB b ASB k h H k (KASB )Γ k (KASB ) )Γ k (KASB )h k (K ASB ), = hh k(k ASB )Γ k(k ASB ) (4)

13 where Γ k (KASB ) s the orthogonal projector of Bk ASB and the thrd equalty holds due to the fact that any orthogonal projector s dempotent [5]. Proposed HB approach In the proposed approach, the combned precoder matrx B HB obtaned by Algorthm I wll have a rank of a. By applyng smlar technque as above, the rate acheved by the proposed approach can be obtaned by solvng the followng optmzaton problem where R HB = max b HB k = g HB k = p HB k K,,k k= k= log ( + p HB k gk HB ), K s.t p HB k = P (5) h H k (KHB h H k (KHB )Γ k (KHB ) )Γ k (KHB )h k (K HB ), h H k (KHB )Γ k (KHB ), (6) Γ k (KHB ) s the orthogonal projector of Bk HB = span{h m (K HB ) : m =,,, K, m k} and b HB k s the kth column of B HB. In both the proposed HB and exstng ASB approaches, the number of served users of sub-carrer s K. Furthermore, when K HB = K ASB,, (Bk HB) s a superset of (Bk ASB ) (.e., (Bk HB) (Bk ASB ), ). Ths s due to the fact that the dmenson of h m (Bk HB) ( ) s larger than that of h m (Bk ASB ) whch s a. For these reasons, we wll have g HB k g ASB k R HB R ASB, when K HB = K ASB. APPEDIX B: PROOF OF Theorem When K ASB K HB K DB for some, the relaton between R ASB, R HB and R D cannot be quantfed for each channel realzaton. Thus, we compare these three approaches by examnng ther average sum rates by assumng ZF precodng and equal power allocaton strategy. For better exposton of the proof of Theorem, let us consder the followng Lemma. Lemma C.: Let A = Z H Z be a non sngular hermtan matrx of sze K K and B = A, where Z C a K and K a. We partton A and B as [ a a A = H a A ], B = [ ] b b b B (7) where a (b ) s a scalar value, and the rest of the terms are approprate dmenson vectors or matrces. If a 0 and A s non sngular, we can express b as b = a a H A a. (8) And, f each element of Z s taken from..d ZMCSCG random varable wth varance, then b χ a K+ (9) Proof: The frst equalty (8) can be proved by applyng the well known Schur Complement theorem. The detaled dervaton can also be found from Theorem A5. of [53]. To prove (9) we note that both a and a H A a are strctly non negatve real values. And when a a H A a s a non negatve real valued term, by applyng Theorem 3..0 of [53], the probablty densty functon of b can be expressed as W ( a K +, ), where W (.) (.,.) denotes a real valued Wshart dstrbuton. It follows b W ( a K +, ) χ a K+ () where the second dstrbuton s due to the fact that W ( a K +, ) has the same dstrbuton as that of Ch-square (χ ) dstrbuton wth a K + degrees of freedom (see Corollary 3.. of [53]). As expected when a = K, b s a χ dstrbuton wth degree of feedom. In the followng we prove (8). By settng Z of Lemma 3 as Z = H (K ASB ), we get [(H (K ASB ) H H (K ASB )) ], χ. Hence a K+ x ASB [(H (K ASB ) H H (K ASB )) ] k,k χ a K+. Snce log( + x) s a concave functon, by employng Jensen s nequalty, we wll have ( E{Rk ASB } log + P ) K E{xASB } (3) The current paper employs schedulng of K t K users and when we have K t users, there are K g = Kt K ndependent groups. And the proposed approach selects a group havng maxmum sum rate whch s drectly related to x ASB. Thus, a group wll acheve the best maxmum sum rate f ts x ASB s the hghest of all of these K g groups. Therefore, E{Rk ASB } s bounded as ( E{Rk ASB } log + P ) K E{xASB max (K g )} (3) where x ASB max (K g ) = max{x ASB (), x ASB (),, x ASB (K g )} s the maxmum of K g ndependent Ch-square dstrbuted random varable wth a K + degrees of freedom. In the followng, we valuate E{x ASB max (K g )}. By applyng order statstcs, the probablty densty functon (pdf) of z max x ASB max (K g ) can be expressed as [54] where f zmax (x) = K g (F (x)) Kg f(x) (33) a K+ γ(, x F (x) = ) Γ( a K+ ), f(x) = a K+ Γ( a K+ ) x a K+ e x (34)

14 wth Γ(.) s the Gamma functon and γ(.) as the lower ncomplete Gamma functon. It follows E{x ASB max (K g )} = K g (F (x)) Kg f(x)dx. (35) 0 When K g =, E{x ASB max (K g )} = a K +. However, for general K g, gettng closed form soluton for ths ntegral s non trval. Due to ths reason, we utlze numercal approach to evaluate ths ntegral (for example smple trapezod numercal ntegraton approach of Matlab). Lke n (3), one can also get the followng upper bound rate for the DB approach ( E{Rk DB } log + P ) K E{xDB max (K g )} (36) where E{x DB max (K g )} s the expected value of the maxmum of K g ndependent χ K random varables. And for the proposed HB approach, we wll have the followng rates. E{Rk HB } log ( + P K E{xHB max (K s )}) S, E{Rk HB } log ( + P K E{xHB max (K g )}) > S (37) where K s = f K t ak and E{xHB max (K s ) (E{x HB max (K g )) s the expected value of the maxmum of K s (K g ) ndependent χ K+ (χ a K+ ) random varables. By substtutng (3), (36) and (37) nto the average sum rate expressons of all subcarrers, we get (8). APPEDIX C: PROOF OF Lemma 3 In the followng, we provde channel matrces that satsfy rank(h = [H, H,, H f ]) a for the ULA multpath channel models. By employng the multpath and ULA channel models (4) and (5), and after dong some mathematcal manpulatons, h k can be expressed as h k =τ k C k f (38) where C k = [c k (0), c k (),, c k (L p )], f = [, e j π j f, e π j f,, e π (L f p ) ] T, j = and τ k s as defned n (4). Usng (38), one can rewrte H = τ C F, where τ = [τ, τ,, τ Kt ], C = blkdag(c, C,, C Kt ) and F = [ F, F,, F f ] wth F = I Kt f. For any C and F, snce rank(h) rank(τ ), one can mantan rank(h) a just by ensurng rank(τ ) a. As can be seen from (4), for the gven θ km, m =,, L s, τ k (θ km ) s a Fourer vector wth resoluton. 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Cressler, On the analyss and desgn of low-loss sngle-pole double-throw W-Band swtches utlzng saturated SGe HBTs, IEEE Trans. Mcrow. Theory and Techn., vol. 6, no., pp , ov. 04. [45] M. R. Akdenz, Y. Lu, M. K. Samm, S. Sun, S. Rangan, T. S. Rappaport, and E. Erkp, Mllmeter wave channel modelng and cellular capacty evaluaton, IEEE J. Select. Areas n Commun., vol. 3, no. 6, pp , Jun. 04. [46] T. Yoo and A. Goldsmth, On the optmalty of multantenna broadcast schedulng usng zero-forcng beamformng, IEEE Trans. Sel. Area. Commun., vol. 4, no. 3, pp , Mar [47] X. Gao, O. Edfors, F. Rusek, and F. Tufvesson, Massve MIMO performance evaluaton based on measured propagaton data, vol. 4, no. 7, pp , Jul. 05. [48] H. Q. go, E. G. Larsson, and T. L. Marzetta, Energy and spectral effcency of very large multuser MIMO systems, IEEE Trans. Commun., vol. 6, no. 4, pp , Apr. 03. [49] Z. Gao, L. Da, D. M, Z. Wang, M. A. Imran, and M. Z. Shakr, Mmwave massve MIMO based wreless backhaul for 5G ultra-dense network, IEEE Wreless Commun. Mag. (To appear), Jul. 05. [] A. Manolakos, M. Chowdhury, and A. Goldsmth, Energy-based modulaton for noncoherent massve SIMO systems, Jul. 05, http: //arxv.org/abs/ [5] S. Thoen, L. Van der Perre, B. Gyselnckx, and M. Engels, Performance analyss of combned Transmt-SC/Receve-MRC, IEEE Trans. Commun., vol. 49, no., pp. 5 8, Jan. 00. [5] R. H. Horn and C. R. Johnson, Matrx Analyss, Cambrdge Unversty Press, Cambrdge, 985. [53] R. J. Murhead, Aspects of Multvarate Statstcal Theory, John Wley and Sons Inc., ew Jersey, 98. [54] A. Papouls and S. U. Plla, Probablty, Random Varables and Stochastc Processes, McGraw-Hll, 00. Tadlo Endeshaw Bogale (S 09-M 4) has receved a BSc degree n Electrcal Engneerng from Jmma Unversty, Ethopa. From 004 to 007, he was workng n Etho Telecom, Adds Ababa, Ethopa. He receved MSc and PhD degrees n Electrcal Engneerng from Karlstad Unversty, Sweden and Unversty Catholque de Louvan (UCL), Louvan la neuve, Belgum n 008 and 03, respectvely. From January 04 to October 04, Tadlo was workng as a postdoctoral researcher at the Insttut atonal de la Recherche Scentfque (IRS), Montreal, Canada. Snce ovember 04, he has been workng as a jont postdoctoral researcher wth IRS and Unversty of Western Ontaro, London, Canada. Currently, he s workng on assessng the potental technologes to enable the future 5G network. Specfcally, hs research focuses on the explotaton of massve MIMO and mllmeter wave (mmwave) technques for 5G network. Hs research nterests nclude hybrd Analog-dgtal Beamformng for massve MIMO and mmwave systems, plot contamnaton reducton for multcell massve MIMO systems, spectrum sensng and resource allocaton for cogntve rado networks, robust (non-robust) transcever desgn for multuser MIMO systems, centralzed and dstrbuted algorthms, and convex optmzaton technques for multuser systems. He has organzed a workshop on Cogntve Rado for 5G networks whch s collocated n CROWCOM 05 conference. He was actng as a sesson char for ICC, CISS and CROWCOM conferences and EWCOM# workshop. Dr. Bogale has also served as a TPC member on dfferent nternatonal conferences such as IEEE PIMRC and VTC, and EAI CROWCOM. Recently, he has delvered a tutoral n IEEE PIMRC 05 on the 5G network.

16 Long Bao Le (S 04-M 07-SM ) receved the B.Eng. degree n Electrcal Engneerng from Ho Ch Mnh Cty Unversty of Technology, Vetnam, n 999, the M.Eng. degree n Telecommuncatons from Asan Insttute of Technology, Thaland, n 00, and the Ph.D. degree n Electrcal Engneerng from the Unversty of Mantoba, Canada, n 007. He was a Postdoctoral Researcher at Massachusetts Insttute of Technology (008 00) and Unversty of Waterloo ( ). Snce 00, he has been wth the Insttut atonal de la Recherche Scentfque (IRS), Unversté du Québec, Montréal, QC, Canada where he s currently an assocate professor. Hs current research nterests nclude smart grds, cogntve rado, rado resource management, network control and optmzaton, and emergng enablng technologes for 5G wreless systems. He s a co-author of the book Rado Resource Management n Mult-Ter Cellular Wreless etworks (Wley, 03). Dr. Le s a member of the edtoral board of IEEE TRASACTIOS O WIRELESS COMMUICATIOS, IEEE COMMUICATIOS SURVEYS AD TUTORIALS, and IEEE WIRELESS COMMUICATIOS LETTERS. He has served as a techncal program commttee char/co-char for several IEEE conferences ncludng WCC, VTC and PIMRC. Afshn Haghghat (Senor Member, IEEE) receved the B.S. degree from KT Unversty of Technology, Tehran, Iran, and the M.A.Sc. and Ph.D. degrees from Concorda Unversty, Montreal, Quebec, Canada, n 99, 998 and 005, respectvely, all n electrcal engneerng. From 997 to 998, he was at SR-Telecom Inc., Montreal, Quebec, Canada, where he was nvolved n desgn of ntegrated mcrowave transcevers for pont-to-multpont wreless broadband systems. In 998, he joned Harrs Corporaton, Montreal, Quebec, Canada, where he partcpated n and led development of varous advanced modem and transcever platforms for hgh capacty backhaul mcrowave dgtal rados. Snce 005, he s at InterDgtal Inc. where he s nvolved n the development of 4G/5G cellular systems. Hs man areas of nterest and expertse are RF & Antenna Systems, Communcatons and Sgnal Processng. Luc Vandendorpe (M 93-SM 99-F 06) was born n Mouscron, Belgum, n 96. He receved the electrcal engneerng degree (summa cum laude) and the Ph.D. degree from the Unverst Catholque de Louvan (UCL), Louvan La euve, Belgum, n 985 and 99, respectvely. Snce 985, he has been wth the Communcatons and Remote Sensng Laboratory of UCL, where he rst worked n the eld of bt rate reducton technques for vdeo codng. In 99, he was a Vstng Scentst and Research Fellow at the Telecommuncatons and Trafc Control Systems Group of the Delft Techncal Unversty, The etherlands, where he worked on spread spectrum technques for personal communcatons systems. From October 99 to August 997, he was a Senor Research Assocate of the Belgan SF at UCL and an nvted Assstant Professor. He s now Full Professor wth the Insttute for Informaton and Communcaton Technologes, Electroncs and Appled Mathematcs, UCL. Hs current nterest s n dgtal communcaton systems and more precsely resource allocaton for OFDM(A)-based multcell systems, MIMO and dstrbuted MIMO, sensor networks, turbo-based communcatons systems, physcal layer securty, and UWB based postonng. Dr. Vandendorpe was corecpent of the 990 Bennal Alcatel-Bell Award from the Belgan SF for a contrbuton n the eld of mage codng. In 000, he was corecpent (wth J. Louveaux and F. Deryck) of the Bennal Semens Award from the Belgan SF for a contrbuton about lter-bank-based multcarrer transmsson. In 004, he was co-wnner (wth J. Czyz) of the Face Authentcaton Competton, FAC 004. He s or has been TPC member for numerous IEEE conferences (VTC, Globecom, SPAWC, ICC, PIMRC, WCC) and for the Turbo Symposum. He was Co-Techncal Char for IEEE ICASSP 006. He served as an Edtor for Synchronzaton and Equalzaton of the IEEE TRASACTIOS O COMMUICATIOS between 000 and 00, as ab Assocate Edtor of the IEEE TRASACTIOS O WIRELESS COMMUICATIOS between 003 and 005, and as an Assocate Edtor of the IEEE TRASACTIOS O SIGAL PROCESSIG between 004 and 006. He was Char of the IEEE Benelux jont chapter on communcatons and vehcular technology between 999 and 003. He was an elected member of the Sgnal Processng for Communcatons Commttee between 000 and 005, and an elected member of the Sensor Array and Multchannel Sgnal Processng Commttee of the Sgnal Processng Socety between 006 and 008, and between 009 and 0. He s the Edtor-n-Chef for the EURASIP Journal on Wreless Communcatons and etworkng.