arXiv:1811.02948v1 [cs.IT] 7 Nov 2018 nryefiin n ul clbei em ftenme of number the of terms in scalable fully antennas. unified and transmit hig a efficient are antennas TA develop energy fully- and u RA we that conventional architectures, show MIMO architectures, results conventional matching with simulation Our MIMO antennas model. orthogonal consumption hybrid TA power compare the and and fairly on to digital RA based order of in antennas performance Furthermore, TA preco algorithm. efficient and pursuit an design RA we meet channels, for mmWave to e M of and have conventional sparsity account antennas the for into architectures, TA constraints those these and MIMO Taking to architectures. 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MI energy hybrid (PC) reduce partially-connected hence issue, this and with power deal In To of [4]. con amounts which antennas connections, divid huge line RF of and of the shifters, comprised numbers phase of is combiners, network large consumption feed analog for power of the particular, excessive is network gain the MIMO feed to hybrid analog full due antennas. FC the scalable Unfortunately, all realize not arrays. to to hy connected able (FC) massive fully-connected is is as and chain to MIMO referred RF is each architecture This of output signal the [3]. the [2], of domain some analog moving the by the into chains operations reduce RF processing tremendously of anal number which hybrid required architectures, consider consumption to MIMO energy researchers digital motivated and has This cost [1]. analog-to-digital/digital-to-analog high dedicated resolution prohibitively high a of the to syste mmWave to connected for infeasible is due are chain, antenna (RF) frequency each radio which smal in and systems, conven- multiple-outpu scattering, However, multiple-input limited (FD) frequencies. fully-digital mmWave loss, recei tional the path at and/or high apertures transmitter the antenna with the with equipped at cope be antennas to of to array assumed large typically a are [ requirement networks systems communication rate These wireless data [2]. of high generation next the the meet of to candidates promising yial,i yrdMM ytm,i sasmdthat assumed is it systems, MIMO hybrid in Typically, are systems communication (mmWave) wave Millimeter Abstract clbeadEeg-fcetMliee Massive Millimeter Energy-Efficient and Scalable Hbi nlgdgtlacietrsaeconsidered are architectures analog-digital —Hybrid eetAryadTasi-ra Antennas Transmit-Array and Reflect-Array ai aai noi .Tln,GogFshr afM¨ulle Ralf Fischer, Georg Tulino, M. Antonia Jamali, Vahid .I I. NTRODUCTION IOArchitectures: MIMO converters (MIMO) t xploiting phase , wever, ation, sume for n IMO nlike We . sis es MO brid ers, og- der red ver hly the ms ys- 1], s f l ul clbei em ftenme ftasi nens W antennas. transmit of number the of the terms and architectures, in energy-efficient highly scalable MIMO are fully PC architectures MIMO and TA FC, and RA FD, c in conventional that amplifi the show power results to simulation and Our architectures. processing and hybrid all digital PC RA for and the the FC for the and for air architectures, network over feed loss RF the the model architectures, of TA consumption impacts power the and includes unified RA which a MIM of develop hybrid performance we and TA the fully-digital architectures, conventional and compare with fairly RA antennas to TA for order Furth in precoder (OMP). pursuit more, sparsity matching a orthogonal the on design based exploiting antennas we and channels, account mmWave into constraints Tak architectures. these MIMO conventional for const those different to meet compared to have precoders corresponding their t paper, and this precoder. streams in corresponding data the contrast, several of multiplexing In design in gain. interested beamforming are the we of charac typically terms is Th in architectures been [6]–[10]. these have literature of the antennas performance in the TA communit available and whereas antennas are RA and prototypes mirror microwave and lens. curved the a a in to investigated to widely analogous analogous is is TA RA a an optics, from h aksd fteary[6]. o t array the for the introdu On system of position control antennas. side TA the feed back in of the the exist placement not the RA, facilitate does in systems issue instance, this whereas For area other. each to is signal phase-delayed the direction TA, forward the in the to in whereas shift refle transmitted phase then array is desired signal the a phase-delayed from the adds RA, and In signal. antennas overall t active (over transmitted the elem signals passive by the each of illuminate Therefore, superposition a and antennas. receives chain passive RF of acti dedicated array Each a the antennas). with horn equipped (usually is antennas antenna active ele Bot few antenna reflect- 1. a passive of Fig. and namely array see large antennas, paper, a (TA) comprise architectures this transmit-array novel and in two (RA) consider array architectures we MIMO systems, massive MIMO a massive in mmWave antennas of of MIMO. PC number hybrid of the FC consumption as with power manner the scales similar paper, still this MIMO in hybrid shown Neverthe be reduced. are will antennas o lines as numbers the RF the of and and subset needed, phase-shifters is a required combiner only RF to no Thereby, connected [5]. o is [1], the chain where RF literature each the of in considered were architectures 1 nti ae,w td AadT nensadso that show and antennas TA and RA study we paper, this In nodrt mrv h clblt n energy-efficiency and scalability the improve to order In AadT nenshv eea datgsdsdatgsw advantages/disadvantages several have antennas TA and RA ,adRbr Schober Robert and r, 1 orwn nanalogy an Borrowing . epaesitr on shifters phase he e blocking a ces hrhn,RA hand, ther t respect ith ontrast terized eair) he ereby, ments raints utput less, cted ing ent ers er- he ve of O h y e s f Reflect-Array Antenna Architecture Transmit-Array Antenna Architecture J M each antenna. Furthermore, H C × is the channel matrix, which assuming the Saleh-Valenzuela∈ model, is given by [5]

s sQ 1 L BBP 1 r r H t t H = hlhr(θl , φl )ht (θl , φl ), (2)

RFC

RFC RFC PS 1 PS m PS M √ L l

n =1

N PSfrag replacements 1 X where L is the number of effective channel paths corresponding to a limited number of scatterers and hl C is the channel h ∈t t h r r RFC coefficient of the l-th path. Moreover, t(θ , φ ) ( r(θ , φ )) RFC l l l l

RFC

N 1 denotes the transmitter (receiver) response vector

n PS 1 PS m PS M t r at elevation angle θl [0, π] (θl [0, π]) and azimuth angle BBP φt [0, 2π] (φr [0, 2∈π]). For a uniform∈ planar transmit array, s s l l RFC: RF chain, PS: phase shifter, BBP: Baseband precoder 1 Q ∈ ∈ t t we can obtain ht(φl , θl ) as [5] Fig. 1. Schematic illustration of the considered reflect-array and transmit-array t t massive MIMO architectures. ht(θl , φl )= 2πd t t t j λ ((m1 1) sin(θl ) sin(φl )+(m2 1) cos(θl )) vec e − − , (3) m1,m2 note that the recent paper [11] also studied RA antennas where h i  a precoder was designed based on alternating optimization where λ denotes the wavelength, and d is the distance between the array antenna elements. Assuming a uniform planar receiver (AO). We employ this precoder as a benchmark and show that r r array, hr(θl , φl ) can be modeled in a similar manner as the proposed OMP-based precoder outperforms the AO-based t t precoder in [11]. Moreover, the focus of this paper is mainly the ht(θl , φl ) in (3). scalability and energy-efficiency of RA and TA MIMO systems B. Transmit Signal and Power Consumption Models which was not studied in [11]. Furthermore, in this paper, a Q 1 Let s C × denote the vector of Q independent data more detailed model for the channel between the active and ∈ passive antennas (which affects the model for the precoder streams that we wish to transmit. Assuming linear precoding, structure) is considered compared to [11]. the relation between x and s is as follows Notations: Bold capital and small letters are used to denote T H x = PtxFs, (4) matrices and vectors, respectively. A F , A , and A denote k k CM Q the Frobenius norm, transpose, and Hermitian of matrix A, where F × is the precoderp and Ptx denotes the transmit ∈ H 2

∠ power. Here, we assume E ss I and F . respectively. E represents expectation and is the angle = Q F = 1 a { } k k of the complex{·} number a in polar coordinates. Moreover, In order to fairly compare the power consumptions of the (µ, Σ) denotes a complex normal random variable (RV) conventional MIMO architectures, i.e., FD, FC, and PC, and withCN mean vector µ and covariance matrix Σ. Furthermore, the proposed new MIMO architectures, i.e., RA and TA, a 0n and 0n m denote a vector of size n and a matrix of size power consumption model that accounts for digital baseband × n m, respectively, whose elements are all zeros. Moreover, In processing, the RF network, and the power amplifiers is needed. is× the n n identity matrix and C represents the set of complex Baseband Circuitry: The circuit power consumption com- numbers.× vec(A) denotes the vectorized version of matrix prises the power consumed for baseband processing, denoted A. Moreover, [a(m, n)]m,n represents a matrix with element by Pbb, and by each RF chain (i.e., by the digital-to-analog a(m, n) in its m-th row and n-th column. Am,n and an denote converter, local oscillator, and mixer), denoted by Prfc. Note the element in the m-th row and n-th column of matrix A that although in principle Pbb may vary as a function M, in and the n-th element of vector a, respectively. Finally, vec(A) the remainder of this paper, we assume Pbb is constant since returns a vector whose elements are the stacked columns of its impact is typically much smaller than that of Prfc [12]. matrix A. RF Network: In this paper, we assume an RF network with passive phase shifters, dividers, and combiners which II. SYSTEM, CHANNEL, SIGNAL, AND POWER introduce insertion loss. For large RF networks, the insertion CONSUMPTION MODELS loss may easily exceed 30 dB which makes its precompensation infeasible due to amplifier nonlinearity at high gains [4]. In In this section, we present the system and channel models as practice, to compensate for this insertion loss, several stages well as the signal and power consumption models. of power amplification are implemented throughout the RF network to ensure that a minimum power is delivered to drive A. System and Channel Models the power amplifiers before transmission via the antennas. For We consider a point-to-point MIMO system where the trans- instance, for the gain-compensation amplifier design in [4], each mitter and receiver have M and J antennas, respectively. The amplifier has up to 15 dB gain at 40 mW power consumption. input-output MIMO model is given by The number of required intermediate power amplifiers, denoted

y = Hx + z, (1) 2In this paper, we consider a constraint on the maximum power radiated from the passive array which is typically enforced by regulations. Alternatively, M 1 J 1 where x C × and y C × are the transmit and one can consider a constraint on the power radiated from the active antennas. ∈ ∈ J 1 Although our derivations in Section III and the proposed precoder in Section IV received vectors, respectively. Moreover, z C × denotes ∈ can be applied under both power constraints, we focus on the former power the additive white Gaussian noise vector at the receiver, i.e., constraint for RA and TA antennas since this enables a more straightforward 2 2 z (0J , σ IJ ) where σ denotes the noise variance at comparison with conventional MIMO architectures. ∼ CN by Namp, depends on the specific RF network architecture. further amplification is needed within each stage of power Motivated by the experimental design in [4] and for analytical division/combining. However, for simplicity, we neglect the tractability, we assume three stages of power amplification additional power consumption in this paper. where the signal is pre-amplified once before being fed to 3) Partially-Connected MIMO Architectures: As can be seen the power divider, passive phase shifter, and power combiner, from (8), a huge challenge of the FC hybrid structure is respectively, to compensate the losses incurred in each stage. scalability with respect to the number of antennas M. To Power Amplifiers: The power consumed by the power address this issue, the PC hybrid MIMO structure has been amplifiers is commonly modeled as Prd/ρpa where Prd is the proposed in the literature [5]. The signal model for the PC radiated output power and ρpa denotes the power amplifier architecture is identical to that in (7), i.e., F = RB, with the efficiency [4], [12]–[14]. difference that R is now a block-diagonal matrix In summary, the total power consumption, denoted by Ptot, r 0r 0r is obtained as 1 1 ··· 1 0r r2 0r  2 ··· 2  (5) R = . . . . , (9) Ptot = Pbb + NPrfc + NampPamp + Prd/ρpa, . . .. .   where P is the power consumed by each RF power amplifier 0rN 0rN rN  amp  ···  used in the RF network. Note that for conventional MIMO rn 1   where rn A is the RF precoder vector which connects architectures, P is identical to P , whereas for RA and TA × rd tx the output∈ of the n-th RF chain to r antennas, and 0 is a antennas, is the power radiated by the active antennas which n n Prd vector of length n with all elements being equal to one. Note is not necessarily the same as the power radiated by the N Ptx that r = M has to hold. In the simplest case, all RF passive array, cf. Section III-B for details. n=1 n chains are connected to the same number of antennas, i.e., rn = P III. MATHEMATICAL CHARACTERIZATION OF DIFFERENT M/N, n, where we assume that N is a divisor of M. Since ∀ MIMO ARCHITECTURES the PC architecture does not require power combiners, only N amplifiers are needed in front of the power dividers and In this section, we characterize the constraints that different M amplifiers are needed in front of the phase shifters, i.e., there MIMO systems impose on precoder matrix F and the corre- are N + M M amplifiers in total. Therefore, the total power sponding total power consumption P as a function of M and tot consumption≈ for the PC MIMO architecture is given by N. P = P + NP + MP + P /ρ . (10) A. Conventional MIMO Architectures tot bb rfc amp tx pa In the following, we study the conventional FD, FC, and PC B. Reflect-Array and Transmit-Array MIMO Architectures MIMO architectures. 1) Fully-Digital MIMO Architectures: Here, we have N = In the following, we first model the constraints that the RA M RF chains which enable FD precoding, i.e., F = B where B and TA architectures impose on the precoder. Subsequently, we is referred to as the digital precoder. Moreover, since we do not quantify the total power consumptions of these architectures. 1) Constraints on the Hybrid Precoder: For the considered have an analog RF network, we obtain Namp = 0. Therefore, the total consumed power is given by RA and TA architectures, we assume that each active feed antenna is connected to a dedicated RF chain, i.e., there are Ptot = Pbb + MPrfc + Ptx/ρpa. (6) N active antennas. Moreover, we assume that the passive array 2) Fully-Connected Hybrid MIMO Architectures: In the FC comprises M antenna elements. To facilitate presentation, we hybrid architecture, we have N RF chains whose outputs are characterize the positions of the passive antenna elements by connected to all M antennas via analog dividers, phase shifters, (rm,n, φm,n, θm,n) in N different spherical coordinate systems corresponding to the locations of the active antennas such that and combiners. Typically, the relation Q N M holds. For this MIMO architecture, the precoder has≤ structure≪ each active antenna is the origin of one coordinate system and the z-axis is the direction of the main lobe of the antenna F = RB, (7) pattern. Note that the values of (rm,n, φm,n, θm,n) depend on the specific configuration of the feed antenna and the array where B CN Q denotes the digital precoder and R × antennas. We further assume that the passive antennas have an AM N represents∈ the analog RF precoder where A = x x ∈ × omni- pattern. C and x =1 . Based on the model introduced in Section II-B,| ∈ Proposition 1: Assuming r λ, the precoder F for the to compensate| | the losses incurred in the RF network, N m,n RA and TA antennas has the form≫ amplifiers are needed in front of the power dividers, MN λ√ρary amplifiers are needed in front of the phase shifters, and MN F = DTB, (11) amplifiers are needed in front of the power combiners. In total, 4π we need N(1 + 2M) 2MN amplifiers when M 1. where ρary denotes the passive array power efficiency, B ≈ ≫ N Q M M ∈ Therefore, the total power consumption is obtained as C × is the digital baseband precoder, D C × is a diagonal matrix which controls the phase shifters∈ and is (8) Ptot = Pbb + NPrfc +2MNPamp + Ptx/ρpa. given by

Note that for large M, the loss caused by the power dividers M D = diag ej2πβ1 , ... , ej2πβ , (12) and power combiners may exceed the maximum gain that the M N power amplifiers can provide without introducing non-linear where βm [0, 1], and T C × is a fixed ma- distortions, see [4, Fig. 7] for an example setup. Thereby, trix which depends∈ on the antenna∈ configuration, namely TABLE I COMPARISON OF DIFFERENT MIMO ARCHITECTURES, NAMELY FULLY DIGITAL (FD), FULLY CONNECTED (FC), PARTIALLY CONNECTED (PC), REFLECT-ARRAY (RA), AND TRANSMIT-ARRAY (TA).

Architecture Precoder F Constraint Total Power Consumption Ptot M Q FD B B C × Pbb + MPrfc + Ptx/ρpa N ∈Q M N FC RB B C × , R A × P + NP +2MNP + P /ρ ∈ ∈ bb rfc amp tx pa CN Q Arn 1 N PC RB B × , R = diag r1, ... , rN , rn × , n=1 rn = M Pbb + NPrfc + MPamp + Ptx/ρpa N Q M ∈N { } ∈ RA & TA cDTB B C × , T C × is a fixed matrix (see (13)), D = diag d1, ... , dM , dm A, c = √ρaryλ/(4π) Pbb + NPrfc + Ptx/(ρrtaρpa) ∈ ∈ { P } ∈

(rm,n, φm,n, θm,n), m, n, and is given by signal transmitted over active antennas. In fact, due to the ∀ 3 2πrm,n aforementioned power losses , i.e., ρS, ρP , and ρA, the power j e− λ radiated by the active antennas P is not identical to the power T = G(θm,n, φm,n) . (13) rd rm,n radiated by the passive antennas P . Therefore, the total power " #m,n tx q loss is obtained as Proof: The proof is given in Appendix A. 2 Proposition 1 states that both RA and TA antennas have Ptot= Pbb + NPrfc + Ptx B F /ρpa identical precoder structures, as given by (11), except that k k ρary Pbb + NPrfc + Ptx/(ρrtaρpa), (14) may assume different values for RA and TA antennas, see ≈ Section III-B2 for details. Therefore, the proposed precoder in where ρrta = ρSρary. As can be seen from (14), unlike for Section IV can be applied to both RA and TA antennas. conventional MIMO architectures, the total power consumption 2) Power Consumption and Losses: We assume passive of the RA and TA antennas does not explicitly change with arrays for the RA and TA architectures, i.e., there is no increasing number of passive antennas M which may lead to an improved energy-efficiency and scalability. The constraints signal amplification and Namp = 0. Nevertheless, we have several power losses due to propagation over the air and other imposed on the precoder and the total power consumption of the inefficiencies which are discussed in detail in the following: different massive MIMO architectures discussed in this paper Spillover loss: Since the effective area of the array is finite, are summarized in Table I. some of the power radiated by the active antenna will not be captured by the passive antennas, resulting in a spillover loss IV. PRECODING DESIGN [8]. We define the efficiency factor ρS to take into account the In this section, we propose an efficient linear precoder design spillover. for RA and TA antennas exploiting the sparsity of the mmWave Taper loss: In general, the density of the received power channel. Ideally, we would determine the optimal precoder differs across the passive antennas due to their corresponding which maximizes the achievable rate, denoted by R, based on different values of G(θm,n, φm,n) and rm,n. For single-stream P H H transmission, it is well-known that the non-uniform power maximize R = log I + tx HFF H F 2 J σ2 distribution across the passive antennas leads to a reduction of ∈F 2   the achievable and is referred to as taper loss [15, C1: F F 1, (15) k k ≤ Chapter 15]. For multiple-stream transmission, taper loss leads where C1 enforces the transmit power constraint and is the to a reduction of the achievable rate. We define the efficiency set of feasible precoders which depends on the adoptedF MIMO factor ρT to account for this loss. architecture. For instance, for FC hybrid MIMO, we have Aperture loss: Ideally, for RA antennas, the total power N Q M N = F = RB B C × and R A × . Unfortunately, captured by the aperture will be reflected. In practice, however, problemF { (15) is| not∈ tractable for hybrid∈ MIMO} architectures a certain fraction of the captured power may be absorbed by the (including the considered RA and TA) since set is not convex RA. Similarly, for TA antennas, the aperture may not be able due to modulo-one constraint on the elementsF of the analog to fully forward the captured power and some of the power precoder, cf. (7), (9), and (11). Let Fopt denote the optimal may be reflected in the backward direction or be absorbed by unconstrained precoder for the FD MIMO architecture, i.e., the TA. The aperture power efficiency is taken into account by M Q opt = C × . Instead of (15), minimization of F F F introducing the efficiency factor ρA. Fis commonly adopted in the literature as designk criterion− fok r Phase shifters: Each phase shifter introduces a certain loss constrained hybrid precoders [2], [3], [11], [12]. Therefore, we which is captured by the efficiency factor ρP . For TA antenna, consider the following optimization problem for the RA and the received signal passes through the phase shifter once before TA hybrid MIMO architectures being forwarded whereas for RA antennas, the signal passes 2 through the phase shifter twice before being reflected. Hence, minimize Fopt cDTB B ,D − F the overall phase shifter efficiency factors for RA and TA ∈B ∈D 2 2 C1: cDTB 1, (16) antennas are ρP and ρP , respectively. k kF ≤ Note that the effects of the spillover and taper losses are λ√ρary CN Q where c = 4π , = B × , and = D included in matrix T in (13). Therefore, the array efficiency CM M B ∈ A D ∈ 2 × D = diag d1, ... , dM , dm . Note that (16) is for RA and TA antennas is obtained as ρary = ρP ρA and still non-convex| due{ to multiplication } of∈ D and B as well as the ρary = ρP ρA, respectively, which accounts for the combined non-convexity of set . Nevertheless, (16) allows us to design effects of the aperture and phase shifter losses. The transmit D H 2 an efficient suboptimal solution in the following. power is E x x = P F = P where x = √P Fs and { } txk kF tx tx F F = 1 whereas the power radiated by the active antennas 3Note that taper loss reduces the achievable rate but does not constitute a k k H 2 is P = E x¯ x¯ = P B where x¯ = √P Bs is the power loss. rd { } txk kF tx 1) Rationale Behind the Proposed Precoder: Let H = Algorithm 1 OMP-based Precoder Design H res opt n UΣV denote the singular value decomposition (SVD) of 1: initialize: F0 = F and DM = 0M M , n. × ∀ channel matrix H, where U and V are unitary matrices contain- 2: for i = 1, ... , N do H H res ing the left and right singular vectors, respectively, and Σ is a 3: li∗ = argmaxl=1,...,L (ΨiΨi )l,l where Ψi = Ht Fi 1. n − diagonal matrix containing the singular values. The optimal un- 4: Update DM using (21). Fopt v v v i n n constrained precoder is = [α1 1, α2 2, ... , αQ Q] where 5: Update Bi using (23) for Wi = c n=1 DM TM . res opt i n n vq is the right singular vector corresponding to the q-th largest 6: Update F = F c DM TM Bi. i − n=1 P singular value and αq is the power allocation factor obtained via 7: end for the water filling algorithm [16]. For the spatially sparse channel N n P 8: Return D = n=1 DM and B = BN . t t model introduced in (2), t = ht(θl , φl ), l = 1, ... , L forms a vector space forH the rows of H. In∀ addition, since P t t  i L M and (θl , φl ) is taken from a continuous distribution, D n T n B denote the residual precoder in iteration ≪ c n=1 M M i the elements of t are with probability one linearly inde- i where Bi is the baseband precoder designed in iteration i. H opt pendent. Thereby, the columns of F can be written as a InP each iteration, we project the residual matrix from the v linear combination of the transmit array response, i.e., q = previous iteration on the space defined by Ht and find the h t t where are the corresponding coefficients l cl,q t(θl , φl ) cl,q direction l∗ that has the maximum projected value. This can opt [3]. More compactly, F can be rewritten as be mathematically formulated as P opt H F = HtC, (17) li∗ = argmaxl=1,...,L (ΨiΨi )l,l, (20) where H h t t h t t CM L and C H res t = [ t(θ1, φ1), ... , t(θL, φL)] × where Ψi = H F . L Q ∈ ∈ t i 1 C contains the coefficients . The similarity of the − n n × αqcl,q Step 2–Computation of DM : DM is initialized to the structure of the optimal precoder in (17) and the hybrid precoder n zero matrix 0M M . We obtain the diagonal elements of DM F RB has motivated researchers to use the channel response × = corresponding to the indices in set n as t t M vectors ht(θl , φl ) for the columns of R. Since R has N n n columns (i.e., N RF chains), the problem in (16) can be DM = exp j ∠(HM )m,l∗ ∠Tm,n , m n.(21) m,m t i − ∀ ∈M approximated as choosing the best N columns of Ht to approx-  h i opt In other words, the passive antennas m n create a coherent imate F [1]–[3]. Unfortunately, this concept is not directly ∈M applicable to the precoder in (11) because of its different wave plane in direction li∗ for the signal illuminated by the n-th structure. Hence, we rewrite (17) in a more useful form. Let active antenna. Step 3–Computation of Bi: By defining Wi = us divide the index set of the passive antennas 1, ... , M into i n n { } c DM TM , we can formulate the following optimiza- N mutually exclusive sets n, n = 1, ... , N. Thereby, (17) n=1 M tion problem for Bi can be rewritten as P opt 2 N Bi =argmin F WiB , opt n B − F F = HtM C, (18) ∈B 2 n=1 C1 : W iB P , (22) X k kF ≤ tx n CM L M M where HtM = I n Ht × and I n 0,1 × is a which has the following well-known normalized least square M ∈ M ∈{ } diagonal matrix whose m-th diagonal entry is one if m n solution [3] and zero otherwise. Now, let us rewrite the precoder in (11)∈M as (WHW ) 1WHFopt B = i i − i . (23) N i H 1 H opt n n Wi(Wi Wi)− Wi F F F = c DM TM B, (19) k k n=1 Note that Bi effectively eliminates the interference between the X data streams. where D n I D AM N and T n I T M = n × M = n Algorithm 1 summarizes the above main steps for the pro- CM N . ComparingM (18)∈ and (19) motivates us toM choose∈ × posed OMP-based precoder design. n n n¯ n DM such that DM TM becomes similar to HtM . To do this, we have to address the following two challenges. First, V. SIMULATION RESULTS n n since DM has only M/N non-zero elements and HtM has In this section, we first describe the considered simulation n ML/N non-zero elements, HtM cannot be fully reconstructed setup and introduce the adopted benchmark schemes. Sub- n n by DM TM . Hereby, we choose to reconstruct only one sequently, we compare the performances of the considered n n n column of HtM by DM TM . The unmatched columns of mmWave massive MIMO architectures. n n DM TM are treated as interference. Fortunately, for large M, the interference approaches zero due to channel harden- A. Simulation Setup n ing. Second, we have to choose which column of HtM to We generate the channel matrices according to (2). Thereby, t r t r reconstruct. In this paper, we employ OMP to choose the best we assume that the angles θl , θl , φl , and φl are uniformly n ¯t ¯r ¯t ¯r N columns of HtM . Based on these insights, we present the distributed RVs in the intervals [0, θl ], [0, θl ], [0, φl ], and [0, φl ], ¯t ¯r ¯t ¯r proposed precoder in the following. respectively, and θl and θl (φl and φl ) are the elevation (az- 2) Proposed Precoder: Let us fix sets n, n = 1, ... , N imuth) coverage angles of the transmitter and receiver antennas, a priori. The proposed precoder employs NMiterations where in respectively. Moreover, we use a square uniform planar array each iteration, the following three steps are performed: in (3), i.e., a √M √M planar array. The channel coefficient res opt × Step 1–Choosing the Next Dimension: Let F = F for each effective path is modeled as hl = h¯lh˜l where h¯l i − p TABLE II DEFAULT VALUESOF SYSTEM PARAMETER [4], [10], [12], [13], [17].

¯t ¯r ¯t ¯r Parameter ℓ η θl , θl φl , φl L N0 NF W λ d Rr Rd κ ρP ρA Pbb Prfc Pamp ρamp Value 100 m 2 π/3 2π/3 8 174 dBm/Hz 6 dB 100 MHz 5 mm λ/2 FI:2d, PI: d√2M FI: d√M , PI: d√M 6 2 dB RA: 0.5 dB, TA: 1.5 dB 200 mW 120 mW 40 mW 0.3 − 4 √π √4π − − −

and h˜l are the path loss and the random fading components, C. Performance Evaluation respectively, and are given by In Fig. 2, we show a) the spectral efficiency R (bits/s/Hz) λ η h¯l = and h˜l = (0, 1), (24) given in (15), b) the total consumed power Ptot (Watt), and c) 4πℓ CN   the energy efficiency, defined as WR/Ptot, (Bits/Joule) versus respectively. In (24), ℓ denotes the distance between the trans- the number of transmit antennas M for N = 4, Q = 4, mitter and the receiver and η represents the path-loss exponent. J = 64, and Ptx = 10 Watt. As can be seen from Fig. 2 a), 2 The noise power at the receiver is given by σ = WN0NF the FC hybrid architecture can closely approach the spectral where W is the bandwidth, N0 represents the noise power efficiency of the FD architecture. As expected, PC hybrid spectral density, and NF denotes the noise figure. We arrange MIMO has a lower spectral efficiency compared to FC hybrid the active antennas with respect to the array of passive antennas MIMO due to the fewer degrees of freedom of PC MIMO for as follows. All active antennas have distance Rd from the beamforming as MN and M phase shifters are used in the passive array and are located on a ring of radius Rr. The line FC and PC architectures, respectively. Although the RA and that connects the center of the ring to the center of the plane TA architectures have M phase shifters, too, they achieve a is perpendicular to the array plane. Moreover, we adopt the lower spectral efficiency compared to the PC architecture since following simple class of axisymmetric feed antenna patterns, the superposition of the signals of the different active antennas which is widely used in the antenna community [8], [15], occurs over the air and cannot be fully controlled. This creates an unintended interference between the signals transmitted 2(κ + 1)cosκ(θ), if 0 θ π G(θ, φ)= ≤ ≤ 2 (25) from different active antennas. Nevertheless, this interference 0, if π <θ π, is considerably reduced for PI compared to FI which leads to ( 2 ≤ a considerable improvement in spectral efficiency. Finally, we where κ 2 is a number and normalization factor 2(κ + 1) en- observe from Fig. 2 a) that under FI, the proposed OMP-based sures that≥ 1 G(φ, θ)dΩ = 1 holds where dΩ = sin(θ)dθdφ Ω 4π precoder outperforms the AO-based precoder in [11] for large [15]. The default values of the system parameters are provided M. This might be attributed to the fact that for large M, the in Table II.R The results shown in this section are averaged over iterative AO-based algorithm in [11] is more prune to getting 103 random realization of the channel matrix. trapped in a local optimum which is avoided by the proposed OMP-based precoder which efficiently exploits the sparsity B. Benchmark Schemes of the mmWave channel. On the other hand, the proposed precoder with PI outperforms the precoder in [11] for the entire For the FD MIMO architecture, we consider the optimal considered range of M due to the reduction of interference unconstrained precoder obtained from the SVD of the channel between the signals emitted by different active antennas for and water filling power allocation. For the FC hybrid archi- partial illumination. Recall that RA and TA antennas have tecture, we considered the spatially-sparse precoder introduced identical precoder structures given in (11) but different values in [3]. We note that the precoder for the PC architecture can of ρary which affects their total power consumptions, cf. (14). be rewritten as F = DTD˜ where T˜ is a fixed matrix whose This is the reason why in Fig. 2 a), RA and TA antennas yield element in the m-th row and n-th column is one if the m-th identical spectral efficiency. antenna is connected to the n-th RF chain and zero otherwise. The main advantage of the RA and TA architectures is their Therefore, we can apply the proposed precoder design also to scalability in terms of the number of antennas M which is the PC architecture. Finally, we also use the AO-based precoder evident from Figs. 2 b) and c). In fact, for PI, RA and TA recently proposed for RA in [11] as a benchmark. Note that MIMO achieve similar performance as FD and FC MIMO if in [11], each active antenna illuminates the full passive array, they are equipped with N times more antennas, e.g., in Fig. 2 referred to as full illumination (FI). However, for the proposed a), FD and FC MIMO with M = 256 antennas and RA and TA precoder, only a part of the passive array is responsible for MIMO with M = 1024 antennas (under PI) achieve the same reflection/transmission of the signal received from a given active spectral efficiency of 64 bits/s/Hz. However, from Fig. 2 b), antenna. Therefore, in addition to FI, we also consider the case we observe that the total transmit power of the conventional where each active antenna mostly illuminates the subset of FD, FC, and PC architectures significantly increases as M passive antenna elements allocated to it, referred to as partial increases which makes their implementation quite costly or illumination (PI)4. This is achieved by proper configuration of even infeasible. On the other hand, the total power consumption the positions of active antennas with respect to the passive array of the RA and TA architectures stays almost the same as M via Rr and Rd, see Table II. increases. As a result, we observe in Fig. 2 c) that the energy efficiency of the conventional FD, FC, and PC architectures decreases as M increases whereas the energy efficiency of the 4We do not show the results for the precoder in [11] for PI since this precoder was not designed for PI and, as a result, has a poor performance in this case. proposed RA and TA architectures increases. From Figs. 2 b), Hence, the comparison would not be fair. we observe that PI yields a lower power consumption than PSfrag replacements

PSfrag replacements

PSfrag replacements a) b) c) 55 350 400 FD SVD FC OMP [3] 50 300 350 PI PC OMP (Prop.) 45 RA AO [11] 300 250 RA OMP (Prop.) PI FI 40 PI TA OMP (Prop.) 250 FI FI 200 15 35 200

150 10 600 700 800 30 FD SVD 150 FD SVD FC OMP [3] 100 FC OMP [3] 25 PC OMP (Prop.) 100 PC OMP (Prop.) RA AO [11] RA AO [11] Spectral Efficiency (Bits/s/Hz) 50 Energy Efficiency (MBits/Joule) 20 RA OMP (Prop.) Total Power Consumption (Watt) 50 RA OMP (Prop.) M = 256 TA OMP (Prop.) M = 1024 TA OMP (Prop.) 15 0 0 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 Number of Antennas M Number of Antennas M Number of Antennas M

Fig. 2. a) Spectral efficiency (bits/s/Hz), b) total consumed power Ptot (Watt), and c) energy efficiency (MBits/Joule) versus number of transmit antennas M for N = 4, Q = 4, J = 64, and Ptx = 500 mWatt.

λ√ρary FI since each active antenna more efficiently uses its transmit the precoder matrix can be written as F = 4π DTB which power and illuminates mostly the part of passive array that is is given in (11) and concludes the proof. responsible for reflection/transmission of its signal. This leads REFERENCES to a higher energy efficiency of PI compared to FI in Fig. 2 c), too. From Figs. 2 b) and c), we observe that TA antennas [1] X. Gao, L. Dai, and A. M. Sayeed, “Low RF-Complexity Technologies to Enable Millimeter-Wave MIMO with Large Antenna Array for 5G have higher energy efficiency and lower power consumption Wireless Communications,” IEEE Commun. Mag., vol. 56, no. 4, pp. 211– compared to RA antennas which is due to higher array effi- 217, 2018. [2] I. Ahmed, H. Khammari, A. Shahid, A. Musa, K. S. Kim, E. De Poorter, ciency factor, i.e., [ρary]dB = 2[ρP ]dB + [ρA]dB = 4.5 dB − and I. 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