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MASSIVE multiple-input multiple-output (MIMO) systems. Joint Optimization of Hybrid Beamforming for Multi-User Massive MIMO Downlink

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Joint Optimization of Hybrid Beamforming for

Multi-User Massive MIMO Downlink

Zheda Li, Student Member, IEEE, Shengqian Han, Member, IEEE, Seun Sangodoyin, Student Member, IEEE, Rui Wang, Student Member, IEEE, and Andreas F. Molisch, Fellow, IEEE

Abstract—Considering the design of two-stage beamformers for the downlink of multi-user massive multiple-input-multiple- output (MIMO) system in frequency division duplexing (FDD) mode, this paper investigates the case where both link ends are equipped with hybrid digital/analog (HDA) beamforming structures. A virtual sectorization is realized by channel-statistics- based user grouping and analog beamforming, where the user equipment (UE) only needs to feed back its intra-group effective channel, and the overall cost of channel state information (CSI) acquisition is significantly reduced. Under the Kronecker channel model assumption, we first show that the strongest eigenbeams of the receive correlation matrix form the optimal analog combiner to maximize the intra-group signal to inter-group interference plus noise ratio. Then, with the partial knowledge of instanta- neous CSI, we jointly optimize the digital precoder and combiner by maximizing a lower bound of the conditional average net sum-rate. Simulations over the propagation channels obtained from geometric-based stochastic models, ray tracing results, and measured outdoor channels, demonstrate that our proposed beamforming strategy outperforms state-of-the-art methods.

Index Terms—Massive MIMO, FDD, training overhead, hybrid beamforming, virtual sectorization.

I. INTRODUCTION

M

ASSIVE multiple-input multiple-output (MIMO) sys- tems will be an important component of fifth- generation (5G) cellular systems, because they provide a tremendous increase of spectral efficiency by employing very large arrays at the base station (BS) [2, 3]. However the inherent constraints of cost and energy make it impractical to build a complete radio frequency (RF) chain for each antenna element. A promising solution to this dilemma are hybrid digital/analog (HDA) beamforming structures, which uses a combination of analog beamformers in the RF domain, together with a smaller number of RF chains. This concept was first proposed in [4,5]. While formulated originally for MIMO with arbitrary number of antenna elements, the approach is particularly applicable to massive MIMO, and in that context interest in HDA transceivers has surged over the past years, e.g., [6–12] and the references in [13].

This work was supported in part by Intel, in part by the National Science Foundation, and in part by the National High Technology Research and Development Program of China under Grant 2014AA01A705. Part of this work was presented at the International Conference on Communications (ICC) 2016 [1].

Z. Li, A. F. Molisch, S. Sangodoyin, and R. Wang are with the Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089 (e-mail: {zhedali, molisch, sangodoy, wang78}@usc.edu). S. Han is with the School of Electronics and Information Engineering, Beihang University, Beijing, 100191, P. R. China (e-mail: sqhan@buaa.edu.cn).

The performance advantages of massive MIMO systems have been established under the assumption of perfect channel state information (CSI) at the BS [2,3], which is also assumed by most hybrid beamformer optimizations [6–12]. To alleviate the burden of training overhead, [14,15] propose efficient beam training schemes for HDA systems with subarray structure and fully-connected structure, respectively, while [16] designs a precoder purely based on the channel statistics. Without the requirement of CSIT (CSI at the transmitter), orthogonal space-time block codes are generalized to massive MIMO system in [17, 18]. However, [14, 15, 17, 18] only consider the single-user MIMO scenario, meanwhile [16–18] do not incorporate the constraint of the HDA beamforming structure. To exploit the potentially significant multiplexing gains of large antenna arrays, we aim to investigate the multiuser (MU)-MIMO [19, 20] scenario with HDA structure, which is also explored in [8, 10–12, 21]. However, most of them require instantaneous CSI acquisition with large antenna ar- rays, which is not straightforward for MU-MIMO. Meanwhile, configuration of an HDA structure with multiple RF chains also at the user equipment (UE), which is another central aspect of this paper, has not yet been widely explored in the literature. A joint interleaved training and transmission design is proposed in [22], which achieves the same performance as the traditional fully digital system with significantly reduced training overhead. However, [22] only considers scenarios where the UE is equipped with a single antenna.

The instantaneous CSI at the transmitter, i.e., the BS for the downlink, can be retrieved from uplink training according to channel reciprocity [23] in time division duplexing (TDD) systems. Nevertheless, if the size of the ensemble of antenna elements of the UEs served in MU-MIMO fashion is on the same order as that of the number of BS antennas, the resulting training overhead significantly affects spectral efficiency, in particular when the coherence time is small. This might impede uplink and downlink training at the antenna-level for both TDD and frequency division duplexing (FDD) mode [24], especially at millimeter (mm)-wave frequencies where the channel varies fast.

In order to implement massive MIMO under these circum- stances, a scheme called “Joint Spatial Division Multiplexing” (JSDM) was proposed in [25], which provides a two-stage pre- coder naturally fitting the HDA structure. The first-stage (ana- log) beamforming is based on the slowly-varying second order channel statistics, which significantly reduces the dimension of the effective channel that needs to be trained and fed back within each coherent fading block. A thorough optimization

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on the covariance-based analog beamforming is investigated in [26], which assumes the capacity-achieving digital beam- formers. To further alleviate the downlink training and uplink feedback burden, UEs with similar transmit channel covariance are grouped together and inter-group interference is suppressed by a block diagonalization (BD) based analog precoder, which creates multiple “virtual sectors”. The virtual sectorization is different from the physical/geometric sectorization. The physical/geometric sectorization is usually realized by splitting a single cell into multiple sectors over the angular domain from the perspective of the BS. However, within each virtual sector consisting of users with similar channel covariance, due to the effect of multiple MPC clusters, the angular span is not necessarily contiguous as that of the geometric sector. With this virtual sectorization, downlink training can be conducted in different virtual sectors in parallel, and each UE only needs to feed back the intra-group channels, leading to the reduction of both training and feedback overhead proportional to the number of formed virtual sectors. Extending this idea to the multicell scenario in [27, 28], covariance-based analog pre- coders not only offer beamforming gains to intra-cell desired signals but also suppress inter-cell interference.

In practice, however, to maintain the orthogonality between virtual sectors, JSDM often conservatively groups UEs into few groups, because UEs’ transmit channel covariances tend to be partially overlapped with each other. This limits the reduction of training and feedback overhead. Once grouping UEs into more virtual sectors violates the orthogonality condi- tion, JSDM is not able to combat the inter-group interference. To solve the problem, [24] proposed to strike overlapped- eigenbeams of UEs in different groups, which however sacri- fices some beamforming gains. Another limitation of JSDM is that it was designed only for the case that the UE has a single antenna. Concerning scenarios with many antennas at the UE, [29] extends [25] by selecting the strongest eigenbeam of UE- side channel covariance to form an analog combiner, whereas fixing a single RF chain for each UE, it neglects the potential that each UE may receive multiple symbols simultaneously.

In this paper, we generalize the JSDM scheme to support non-orthogonal virtual sectorization and HDA structures at both BS and UEs. With multiple RF chains at both link ends, spatial multiplexing at each individual UE is explored to enhance the overall system performance. To reduce the problem complexity, we decouple designs of analog and digital beamformers, and propose multi-layer beamforming strategies with different time scales. In the first layer, analog precoders/combiners are designed based on the second order channel statistics, while digital precoder optimization utilizes the mixed intra-group channel and long-term CSI in the second layer. Since each UE is able to evaluate the instant covariance of its received interference in the data transmission phase, digital combiners are based on both intra and inter-group instantaneous effective channels in the third layer.

Another set of problems that bears some formal resemblance to our task is multi-cell digital beamforming optimization, where each UE either needs to feed back the instantaneous channels from all cooperative BSs, e.g., in coordinated multi- point (CoMP) [30], or the instantaneous channel from its serv-

ing BS together with instant covariance of interference plus noise, e.g., in [31, 32]. However, the problem we are solving is distinct, and actually more challenging, since we assume that the BS does not know the instantaneous information of inter-group interference. The main contributions of this paper are thus as follows.

We investigate the design of hybrid beamformers at both link ends for multiuser massive MIMO systems. For the analog beamformer based on second order statistics, we show that under the Kronecker channel model, the optimization of analog combiners and precoders that max- imize the intra-group signal to inter-group interference plus noise ratio can be decoupled, where the optimal com- biner of each UE consists of the dominant eigenvectors of its receive correlation matrix.

Given the analog beamformers, the digital precoders are optimized by maximizing a lower bound of the net conditional average data rate, where the BS knows only instantaneous intra-group channels and second-order statistics. We develop a block descent algorithm to solve the problem by establishing the equivalence betweem the problem and a weighted average mean square error minimization (WAMMSE) problem.

We illustrate the performance of the proposed scheme by various simulation results, based on a synthetic one- ring cluster channel model, and ray tracing data of an outdoor campus environment at28 GHz, which show that orthogonal user grouping is not necessarily optimal when taking the feedback overhead into account.

Systematic simulations are also conducted over real channel matrices obtained from a massive MIMO mea- surement campaign with center frequency 2.53 GHz in Cologne [33], which demonstrate advantages of our proposed scheme also at low frequencies.

A part of this work was presented in a conference paper [1]. The material in this journal version is substantially extended: new items are the proofs for a number of theorems, simulations results based on ray tracing channel profiles and the use of real measurements to validate advantages of the proposed method. Notations: (·), (·)T, (·), and (·)−1 stand for Hermitian transpose, transpose, conjugate, and pseudo inverse operators, respectively, E[·] (E[·|·]) represents expectation (conditional), tr(·) and | · | denote matrix trace and determinant, respectively,

⊗ denotes Kronecker product, Inis the n-by-n identity matrix, CN (m, K) indicates the circularly symmetric complex Gaus- sian distribution with mean vector m and covariance matrix K, vec(·) vectorizes a matrix by stacking its columns, and k · k is the Frobenius norm of a matrix.

II. SYSTEM ANDSPATIALCHANNELMODEL

Consider the downlink transmission of a single-cell massive MIMO FDD system, where the BS is equipped with M antennas and b RF chains to serve K UEs, and each UE has N antennas and aUE RF chains. To reduce the complexity of the digital precoder at the BS and also the channel feedback overhead at the UEs, the concept of virtual sectorization proposed in [25] is employed as shown in Fig. 1. We partition

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Digital Precoder

Digital Precoder

RF Chain

RF Chain . . .

RF Chain

RF Chain . . . . . .

+

+ . . . . . . .

. .

. . .

. . .

. . .

. . .

. . .

Digital Combiner RF Chain

RF chain

. . .

. . . . . .

+ +

+ . . . .

. . .

Fig. 1: Logic structure of hybrid beamforming with virtual sectorization

the RF chains at the BS into G groups, where the g-th group serves kg UEs with its digital precoder Vg. Let dg and bg

denote the number of data streams and RF chains of the BS assigned to the g-th group, respectively, where dg ≤ bg and PG

g=1bg = b. Then, for the g-th group, the digital precoder Vg has the dimension of bg × dg because data streams of each group are processed separately. Therefore, given the total number of RF chains at the BS, the dimension of Vgdecreases with G, leading to reduced computational complexity. Note that Fig. 1 exhibits the logic structure of virtual sectorization realized by the proposed beamforming strategy, which is not in conflict with typical hybrid beamforming structures, e.g., shared-array structures [13].

By assuming flat fading within the coherence time, the received signal at the i-th UE in the g-th group (denoted by UEgi) is expressed by

ˆsgi=FgiWgiHgiBgVgisgi

| {z }

Desired signals

+FgiW

giHgiBg

kg

X

j=1, j,i

Vgjsgj

| {z }

Intra-group interference

+ FgiW

giHgi

G

X

z=1,z,g kz

X

l=1

BzVzlszl

| {z }

Inter-group interference

+ FgiW

gingi

| {z }

Noise

, (1)

where Vg = [Vg1, ..., Vgkg] ∈ Cbg×dg is the digital precoder of the g-th group, Vgi ∈ Cbg×dgi is the digital precoder for UEgi, dgi is the number of data streams assigned to UEgi with

Pkg

i=1dgi = dg, Fgi ∈ CaUE×dgi represents the digital combiner of UEgi, Bg ∈ CM ×bg indicates the analog precoder for group g, Wgi ∈ CN ×aUE is the analog combiner at UEgi, Hgi ∈ CN ×M denotes the channel matrix of UEgi, and ngi ∼ CN (0, δ2giIN) denotes the additive white Gaus- sian noise vector.

A general double directional channel description is consid- ered as following:

Hi=1

Li

PPi

p=1gi pa(θi p)bi p), 1LiAiiGiB

i, (2)

where Pi is the number of multipath components (MPC) con- necting the BS and UEi, Lidenotes the overall large scale loss,

including path loss plus shadowing, and a(·) and b(·) are steer- ing vectors of the antenna array with respect to direction of arrival (DOA) θ = [θaz, θel] and direction of departure (DOD) φ = [φaz, φel], respectively. Note that both DOD and DOA contain components in azimuth and elevation domain, which are distinguished by subscripts “az” and “el”, respectively. Ai and Bi consist of stacked steering vectors of DOA and DOD, respectively, i.e., Ai , [a(θi1), a(θi2), ..., a(θi Pi)] and Bi , [b(φi1), b(φi2), ..., b(φi Pi)], ∀i. gi p reflects the small- scale fading of UEi’s p-th MPC with zero mean and variance σi p2 , ∀i, p. Note that the assumption of zero mean might not be fully compatible with the assumption of infinitely resolving massive MIMO antennas in a mathematical sense. However, in practice multipath components occur in clusters, and if the MIMO array can resolve between the clusters, but not within, the effective channel could fulfill the conditions described above. Irrespective of these considerations, the model we use is widely used in the literature, e.g., [6, 12]. Since only a few MPCs distinguished in the angular resolution of the large array, the Rayleigh fading effect may not be satisfied, and we do not assume particular distributions for variables [gi p]. A performance analysis in realistic (ray tracing or measured) channels that do not follow any of these constraints is given in Section V. σi p2 can be viewed as the average power of UEi’s p-th MPC normalized by the large scale loss. Assuming that different MPCs exhibit independent fading, we have PPp=1i σ2i p = 1. Diagonal matrices ∆i and Gi have

diagonal entries equal to the standard deviations[σi p]Pp=1i and normalized path amplitudes [gi p = σgi pi p]Ppi=1 respectively, i.e.,

, diag(σi1, σi2, ..., σi Pi) and Gi , diag(gi1, gi2, ..., gi Pi). Since the angular power spectrum remains the same within the stationarity region of the channel statistics, which can be equivalent to tens or hundreds of coherence blocks, the nor- malized cost for acquisition of long-term CSI, including[Ai], [∆i], [Bi], and [Li], is thus negligible. Defining the M N -by- M N full covariance of Hgi as Kgi , E[vec(Hgi)vec(Hgi)],

∀g, i, we can obtain its closed-form expression by averaging over small scale fading realizations:

Kgi = 1 Li

Pi

X

p=1

σ2i p(bi p)bTi p)) ⊗ (a(θi p)ai p)). (3)

As analyzed in [25], to reduce the downlink training overhead and considering the practical constraints on analog hardware [5], the analog beamformers [Bg] and [Wgi] need

to be designed based on long-term channel information, i.e., [Ai, ∆i, Bi, Li], while the digital beamformers [Vg] and [Fgi]

can be designed based on reduced-dimensional instantaneous channels. Given the analog beamformers [Bg] and [Wgi],

we consider that orthogonal downlink training is employed for the estimation of the analog-precoded-combined effective channel at UEs. Defining the effective channel from the z- th group to UE gi as ¯Hgi,z , WgiHgiBz ∈ CaUE×bz, we assume perfect effective channel estimation at the UEs, which is widely considered in related works, e.g., [8, 11], since the downlink training can exploit the analog beamforming gains from both link ends to achieve relatively high SNR for channel estimation. The effective channels are required for the

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optimization of digital precoders at the BS, which can be fed back by the UEs in FDD systems. Under virtual sectorization the feedback overhead of each UE can be reduced by limiting that each UE only feeds back the intra-group effective channel, i.e., UEgi feeds back ¯Hgi,g.

Consider that the UEs employ an orthogonal analog feed- back scheme [34] to feed back the effective channels. Then, the overhead of UEgi to feed back ¯Hgi,g is bg channel uses, and the total feedback overhead can be obtained as τfb = β PGg=1kgbg channel uses [34], where β indicates the required number of repetition transmissions for robustness. We can see the impact of the number of groups G on the feedback overhead clearly under the special case where the b RF chains of the BS are evenly assigned to the G groups. In this case, we have τfb= βKb/G, which is reduced by a factor of G.

Based on (1), we can obtain the net data rate of UEgi as R¯gi=(1−τtrfb

Td ) ·log2|Idgi+F

giH¯gi,gVgiV

giH¯

gi,gFgi

−1 gi|,

(4) where τtr = b denotes the overhead of orthogonal downlink training in term of channel uses, Td is the total number of channel uses in a coherence block assigned to an individual UE, and Ωgi can be expressed as

gi=Fgi ¯Hgi,g

kg

X

j,i

VgjVgjH¯gi,g+

G

X

z,g kz

X

l=1

gi,zVzlVzlH¯gi,z+

WgiWgiδ2giFgi,FgigiFgi, (5) where Ωgi is the instantaneous covariance of interference plus noise at the UEgi. To serve all potential UEs, we consider that the coherence block with size T is split into α orthogonal time slots, i.e., Td=Tα, where UEs assigned with the same time slot are simultaneously served by MU-MIMO.

III. ANALOGBEAMFORMERS ANDDIGITALCOMBINER

OPTIMIZATION

In this section, we optimize the analog beamformers and digital combiners at UEs based on channel covariance and instant effective channels, respectively. Since different channel information is exploited for the design of analog and digital beamformers, joint optimization for them is very challenging. We resort to decoupled optimization by designing the analog beamformers to mitigate the inter-group interference while designing the digital combiner to maximize the net data rate of each UE.

A. Analog Beamformers

We begin with the design of the analog combiner of UEgi by maximizing the received “intra-group signal to inter-group interference plus noise” ratio, which is defined under the assumption of equal power allocation over groups as

γ¯gi=

Pt

G kBgk2E[k ¯Hgi,g

k2] PG

z=1,z,g Pt

G kBzk2E[k ¯Hgi,zk 2]+δ2

gikWgik2

, (6)

where Pt is the total transmit power. The nominator rep- resents the average received signal energy from the analog

precoder of the desired group, while the first term of de- nominator indicates the combined interference from other groups, and the second term is the variance of analog- combiner-projected noise. Let the analog combiner Wgi satisfy the semi-unitary property, i.e., WgiWgi = IaUE, which is also assumed in [8]. Therefore, given the analog precoders [Bg]Gg=1, the optimal analog combiner Wgi that maximizes γ¯gi can be readily found by solving a generalized Rayleigh quotient problem [35], which consists of aUE dominant eigenvectors of the matrix PGz=1,z,g Pt

G kBzk2E[HgiBzB

zHgi]+ δ2giIN−1 Pt

G kBgk2E[HgiBgB

gHgi]. Then, given the analog combiner of UEgi, we can obtain that the analog-combined effective channel WgiHgi as vec(WgiHgi) ∼ CN (0, (IMWgi)Kgi(IM ⊗ Wgi)), based on which existing analog pre- coder design methods, e.g., BD and eigen-beamforming (EB) schemes [25] can be employed.

Since the analog beamformers are coupled with each other, iterative updates of analog precoders and combiners are re- quired in general. Nevertheless, we next show that the iteration between analog precoders and combiners can be avoided if the Kronecker channel model [36] is valid, i.e., the channel covariance Kgi can be expressed as

Kgi = Σt,gi ⊗ Σr,gi, (7) where Σt,gi and Σr,gi are transmit and receive correlation matrices of UEgi, respectively.

Proposition 1:Under the Kronecker channel model (7), the optimal semi-unitary analog combiner that maximizes γ¯gi is independent from analog precoders [Bg]Gg=1 and consists of aUEdominant eigenvectors of receive correlation matrix Σr,gi. Detailed proof can be found in Appendix A. It is also intuitively pleasing, as the Kronecker model implies that the angular spectrum at the BS does not change when the one at the UE is modified, and vice versa. According to Proposition 1, as long as the Kronecker channel model is satisfied, each UE can optimize its own analog combiner individually by selecting its strongest receive eigenmodes, and then exist- ing analog precoder design methods can be applied based on the analog-combined effective channel without requiring iterations. With the validity of Proposition 1, decoupling the designs of analog precoder and combiner significantly reduces the problem complexity. Section V reveals comparisons of different covariance-based analog precoders, i.e., BD and EB. Although the Kronecker channel model [37] is widely used for the conventional cellular communication at low carrier frequency, e.g.,2 GHz, this model is not universally applicable to the mm-wave channel, when directional characteristics are highly dependent at both link ends [38]. However, in the later simulation part, we will demonstrate that this decoupled design still performs well even in the mm-wave channel in Section V-B.

B. Digital Combiner

Given the analog beamformers[Bg]Gg=1 and Wgi as well as the digital precoder[Vg]Gg=1, the optimal digital combiner Fgi can be obtained by maximizing the net data rate ¯Rgi given in

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(4). It is not hard to show that the optimal Fgi is the linear minimum mean square error (MMSE) combiner, which is

Fgi = ( ¯Hgi,gVgiVgiH¯gi,g+Ωgi)−1H¯gi,gVgi, (8) where Ωgiis defined in (5).

IV. DIGITALPRECODEROPTIMIZATION

Upon substituting the optimal linear MMSE combiner Fgi into (4), the net data rate of UEgi can be rewritten as

Rgi=Td−τtr−τfb

Td log2|IaUE+ ¯Hgi,gVgiV

giH¯

gi,g−1gi|, (9)

where Ωgi is defined in (5).

Since each UE, say UEgi, only feeds back the intra-group effective channel ¯Hgi,gbut not the inter-group effective chan- nels ¯Hgi,zfor z , g, we need to optimize the digital precoders [Vg]Gg=1 by maximizing the sum rate of UEs averaged over the uncertainties. By noting the correlation between ¯Hgi,gand gi,z, both of which are determined by the same propagation channel Hgi, we formulate the digital precoder optimization problem aimed at maximizing the conditional-average net sum rate of UEs as

[Vmaxgi]

PG

g=1Pki=1g E[Rgi| ¯Hgi,g] (10a)

s.t. PGg=1Pi=1kg tr(BgVgiV

giB

g) ≤ Pt. (10b)

Problem (10) is difficult to solve because it is hard to find an explicit expression for the conditional average data rate. To tackle this difficulty, we first define the mean square error (MSE) matrix Egi of UEgi in the virtual downlink system, which can be obtained from (1) by replacing [Fgi] with[ ˜Fgi] as

Egi , E[(ˆsgi− sgi)(ˆsgi − sgi)]

= ˜Fgi( ¯Hgi,gVgiV

giH¯

gi,g+Ωgi) ˜Fgi + Idgi

VgiH¯gi,gF˜gi− ˜FgiH¯gi,gVgi, (11) where ˜Fgi is an auxiliary variable, representing the analog combiner of UEgi designed by the BS in the second layer, while Fgi will be eventually optimized in the third layer by UEgi as exhibited in Section III-B. Then, we develop the following theorem.

Theorem 1: Let Agi 0 be a dgi-by-dgi weight matrix for UEgi. Then, the solution of the following weighted conditional average mean square error minimization (WAMMSE) problem provides a lower bound of the maximal conditional average net sum rate obtained from (10)

min

[Agi],[ ˜Fgi],[Vgi]

PG g=1P

kg

i=1tr(AgiE˜gi) −log2|Agi| (12a)

s.t. PGg=1Pki=1g tr(BgVgiV

giB

g) ≤ Pt, (12b)

where ˜Egi is the conditional expectation of the MSE matrix Egi of UEgi’s data streams given ¯Hgi,g, i.e., ˜Egi= E[Egi| ¯Hgi,g]= F˜gi˜JgiF˜gi + Idgi − VgiH¯gi,gF˜gi − ˜FgiH¯gi,gVgi, and ˜Jgi is defined as:

˜Jgi , ¯Hgi,gPkjg=1VgjVgjH¯gi,g+ PG

z,gP kz

l=1E[ ¯Hgi,zVzlV

zlH¯

gi,z| ¯Hgi,g]+ WgiWgiδ 2gi. (13)

Detailed proof can be found in Appendix B. Although the objective function (12a) is not jointly convex with respect to [Agi], [ ˜Fgi], and [Vgi], it is respectively convex for every group of variables if the others are fixed. Based on this fact, we utilize a block descent algorithm [31] to find a suboptimal solution to problem (12) in Section IV-A, which is equivalent to maximizing the lower bound of problem (10).

A. Optimization Algorithm

Given the weight matrices[Agi] and digital precoders [Vgi], the optimal digital combiners[ ˜Fgi] can be obtained based on the first-order optimality condition as

F˜gi = ˜J−1giH¯gi,gVgi, ∀ i, g, (14) where ˜Jgi defined in (13) is the instantaneous covariance of the received signal conditionally averaged over the inter-group interference channels. Similarly, we can obtain the optimal [Agi] and [Vgi] as

Agi = (Idgi − VgiH¯gi,gF˜gi)

−1, ∀ i, g, (15)

Vgi = Pkj=1g H¯gj,gF˜gjAgjF˜

gjH¯gj,g+ PG

z,gP kz l=1E[ ¯H

zl,gF˜zlAzlF˜

zlH¯zl,g| ¯Hzl,z] + µBgBg−1H¯gi,gF˜giA

gi, ∀ i, g, (16)

where µ is a Lagrange multiplier that can be found through bisection search to satisfy the power constrain (12b). To obtain F˜gi and Vgi from (14) and (16), the conditional expectation of the form E[ ¯Hgi,zQ ¯Hgi,z| ¯Hgi,g] for z , g needs to be computed. The conditional expectation taken over inter-group interference channels is correlated to the realization of intra- group channel, i.e., ¯Hgi,g, because[ ¯Hgi,z]Gz=1 are all functions of Hi, and Q is a constant matrix. We compute the conditional expectation by using the following means. First vectorize E[ ¯Hgi,zQ ¯Hgi,z| ¯Hgi,g] as

vec(E[ ¯Hgi,zQ ¯Hgi,z| ¯Hgi,g])= E[ ¯Hgi,z⊗ ¯Hgi,z| ¯Hgi,g]vec(Q), then we only need to compute the conditional expectation E[ ¯Hgi,z⊗ ¯Hgi,z| ¯Hgi,g]. Further, defining y = vec( ¯Hgi,g) and

x= vec( ¯Hgi,z), we can find that E[ ¯Hgi,z⊗ ¯Hgi,z| ¯Hgi,g] is just a reshaped version of E[xx|y]. Since x and y are joint complex Gaussian vectors, we can find the conditional second moment of x as

E[xx|y]= E[x|y]E[x|y]+ Kx x|y, (17)

where E[x|y]=KxyK−1yyy and Kx x|y=Kx x− KxyK−1yyKxy.

Kx x and Kyy are the covariance matrices of x and y respectively, and Kxy is the cross covariance matrix of x and y, all of which are functions of the channel covariance Kgi and the analog beamformer and can be obtained as

Kx x= (BTz ⊗ Wgi)Kgi(Bz⊗ Wgi), (18a) Kyy= (BTg ⊗ Wgi)Kgi(Bg⊗ Wgi), (18b) Kxy= (BTz ⊗ Wgi)Kgi(Bg⊗ Wgi). (18c)

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Algorithm 1 WAMMSE algorithm for digital precoder

1: Initialize[Vgi] such that constraint (12b) is satisfied.

2: repeat

3: update[ ˜Fgi] according to (14) and (13),

4: update[Agi] according to (15),

5: update[Vgi] according to (16),

6: until the required accuracy or the maximum number of iterations is reached.

Directly computing (18) involves the multiplication of a large-dimensional matrix Kgi, leading to high complexity. To circumvent it, we substitute (3) into (18a) and obtain

Kx x = 1 Li

Pi

X

p=1

σ2i p(BTzbi p)bTi p)Bz) ⊗ (Wgia(θi p)ai p)Wgi). (19) We can find that the complexity of matrix computations in (19) is significantly less than that of (18a), which involves matrix multiplications with dimension M N -by-M N . Similar simplifi- cations can be applied to (18b) and (18c), which alleviate the overall computational burden. Moreover, these computations occur only once within the stationarity time of second order channel statistics, whose complexity is therefore affordable.

The full algorithm is summarized in Algorithm 1. Since the objective function (12a) is convex with respect to an individual variable while fixing others, each iteration based on the first order optimality condition will generate a new set of variables, which reduce the value of the objective function (12a). Meanwhile, substituting (15) into (12a), we can observe that the objective function is reduced to PGg=1Pki=1g log2| ˜Egi|, which is lower bounded by zero. Therefore, the convergence of the proposed WAMMSE algorithm is guaranteed to a locally optimal solution. We omit the detailed proof due to lack of space, but a similar one can be found in [31].

B. Initialization Strategy

The performance and convergence speed of the WAMMSE algorithm largely depend on the selected initial value of[Vgi].

In order to achieve better performance, one may run the WAMMSE algorithm many times with different initializations and then keep the best result, which however leads to very high complexity. In the following we derive an initial Vgi aimed at maximizing the signal to leakage plus noise ratio (SLNR) for UEgi, ∀ i, g.

Since the BS does not know the inter-group effective interference channel, we define the SLNR in the following way

SLNRgi =tr(PS,gi)

tr( ˜PI,gi), (20) where PS,gi = ˆFgiH¯gi,gVgiV

giH¯

gi,gFˆgi is the instant covari- ance of received desired signal sgi, ˆFgi is a preliminary digital combiner which is heuristically selected as the dgi dominant left singular vectors of the effective channel ¯Hgi,g, and ˜PI,gi represents the covariance of leakage from the signal intended

for UEgi plus noise conditionally averaged over the inter-group interference channels, which can be expressed as

I,gi=PGz,gPkl=1z FˆzlE[ ¯Hzl,gVgiV

giH¯

zl,g| ¯Hzl,z] ˆFzl

+Pkj,ig FˆgjH¯gj,gVgiV

giH¯

gj,gFˆgj2giFˆ

giW

giWgiFˆgi. (21)

Since the analog combiner Wgi consists of eigenvectors as given after (6) and the preliminary digital combiner ˆFgi consists of singular vectors, we have ˆFgiWgiWgiFˆgi = Id

gi.

We choose the initial digital precoders satisfying the semi- unitary property, i.e., VgiVgi = ζgiIdgi, where ζgi is a power normalization factor, ∀g, i. Utilizing the fact that tr(XY) = tr(YX), we can respectively represent tr(PS,gi) andtr( ˜PI,gi) as

tr(PS,gi)= tr(VgiH¯

gi,gFˆgiFˆ

giH¯gi,gVgi), (22)

tr( ˜PI,gi)= tr(Vgi(PGz,gPkl=1z E[ ¯Hzl,gFˆzlFˆzlH¯zl,g| ¯Hzl,z]+ Pkg

j,iH¯

gj,gFˆgjFˆ

gjH¯gj,g+dbgig δ2giIbg)Vgi)

, tr(VgiU˜giVgi). (23)

As a result, to maximize (20) is equivalent to solving a generalized Rayleigh quotient problem [35]. We can obtain the optimal initial Vgi = pζgiVgi , where Vgi consists of dgidominant eigenvectors of the matrix ˜U−1giH¯gi,gFˆgiFˆgiH¯gi,g. Note that the conditional expectation in ˜Ugi can be eval- uated similarly by (17)∼(18). Let Pgi = PPGtdgi

g=1dg

denote the power allocated to UEgi under the assumption of equal power allocation over data streams. Since existing BD and EB analog precoders [25] satisfy BgBg = Ibg, we have Pgi = tr(VgiBgBgVgi) = tr(VgiVgi) = ζgidgi, leading to ζgi=Pdgi

gi. Therefore, we obtain the initial preliminary digital precoder Vgi=

rPgi

dgiVgi, ∀g, i.

Remark 1: Given the analog beamformers at both link ends, we can view multiple virtual sectors as multiple “small cells”, and the BS can be viewed as a “central unit” connecting to

“small cells”. However, compared with traditional multicell beamformer optimization [31], the problem in our formulation is unique: at the “central unit”, we only have the knowledge of the intra-cell CSI for every “small cell” to its UEs, and the global channel statistics. Without the knowledge of interference channel, we resort to maximize a lower bound of the net sum rate as problem (10) exhibits, and prove its equivalence to a WAMMSE problem (12). Therefore, our contribution is to generalize the weighted-minimum-mean- square-error (WMMSE) method [31] to accommodate the situation with the mixed knowledge of partial instantaneous CSI and channel statistics.

Remark 2: In conclusion, the proposed hybrid beamform- ing scheme includes three layers. In the first layer, both link ends respectively form the analog beamformer based on the second order channel statistics, or we say, the long-term CSI. The overhead of long-term CSI acquisition is negligible after normalization by the stationarity region of channel statistics, which can be equivalent to thousands of coherence blocks. In the third layer, since an individual UE is able to evaluate the instant covariance of interference plus noise, the optimal

References

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