MIMO IBC beamforming with combined channel estimate and covariance CSIT

MIMO IBC Beamforming with Combined Channel Estimate and Covariance CSIT

Wassim Tabikh^¶‡ Dirk Slock^¶ Yi Yuan-Wu^‡

¶EURECOM, Sophia-Antipolis, France, Email:{tabikh,slock}@eurecom.fr

‡Orange Labs, Chˆatillon, France, Email:{wassim.tabikh,yi.yuan}@orange.com

Abstract—This work deals with beamforming for the MIMO Interfering Broadcast Channel (IBC), i.e. the Multi-Input Multi- Output (MIMO) Multi-User Multi-Cell downlink (DL). The novel beamformers are here optimized for the Expected Weighted Sum Rate (EWSR) for the case of Partial Channel State Information at the Transmitters (CSIT). Gaussian (Posterior) partial CSIT can optimally combine channel estimate and channel covariance information. We introduce the first large system analysis for optimized beamformers with partial CSIT, here for the Massive MISO (MaMISO) case. In the case of Gaussian partial CSIT, the beamformers only depend on the means and covariances of the channels. The large system analysis furthermore allows to predict the EWSR performance on the basis of the channel statistics only.

Index Terms—Massive MIMO; multi-user; multi-cell; sum rate; beamforming; partial CSIT; large system analysis

I. INTRODUCTION

In this work, Tx may denote transmit/transmitter/

transmission and Rx may denote receive/receiver/reception.

Interference is the main limiting factor in wireless transmis- sion. Base stations (BSs) disposing of multiple antennas are able to serve multiple User Equipments (UEs) simultaneously.

Two popular approaches for maximum Weighted Sum Rate (WSR) beamformer (BF) optimization are the exploitation of the WSR-WSMSE (Weighted Sum MSE) correspondence as in [10] or the concave WSR minorization approach as in [9].

In the MISO case, the BF are proportional in both cases, but the stream powers are optimized with an interference aware waterfilling algorithm in [9] whereas [10] alternates between DL and dual UL MMSE updates for Rx/Tx. In the MIMO case, the BF in [9] are found as generalized eigenvectors resulting from optimal Signal-to-Leakage-plus- Noise (SLNR) considerations, whereas the DL/UL ping-pong MMSE updates in [10] correspond to power method iterations for these generalized eigenvectors.

However, Multi-User (MU) systems have precise require- ments for CSIT which is more difficult to acquire than CSI at the Rx (CSIR). Hence we focus here on the more challenging downlink (DL) and we consider maximizing the Expected WSR (EWSR) with partial CSIT. Earlier works have attempted optimal partial CSIT designs for MU MIMO, e.g. the Expected WSMSE (EWSMSE) approach applied in [1] for MIMO IBC beamformer (BF) design, based on an extension of [10] to the partial CSIT case. However, the EWSMSE approach is suboptimal and cannot even be used in the zero channel mean case (case of covariance CSIT only). In spite of that,

it has been mistakenly considered as optimal as recently as in [2]. It is also desirable to have deterministic alternatives to the cumbersome (though optimal) stochastic approximation solution of [3]. We treat the Gaussian CSIT case, optimally combining mean (channel estimate) and (channel) covariance information. The Gaussian model allows to exploit both mean (channel estimate) and covariance information, which are actually the only statistics that matter in the large system limit, regardless of actual channel (estimate) didstribution. The goal here is to go beyond the extreme of Zero-Forcing (ZF) and to introduce a meaningful BF design at finite SNR and with partial CSIT. Over the last year or so, a number of research works have proposed to exploit the channel hardening in Massive MIMO (MaMIMO) to reduce global instantaneous CSIT requirements to local instantaneous CSIT plus global statistical CSIT. This only works for MISO systems [4] in which all the work needs to be done by the transmitters. We may remark that in the case of MIMO, in which UEs possess a limited number of antennas which contribute actively (e.g. to Zero-Forcing (ZF)/Interference Alignment (IA) at high SNR), the interference subspace and hence the receiver at the UEs does not harden.

A significant push for large system analysis in MaMIMO systems appeared in [5]. It allows to obtain deterministic (instead of fast fading channel dependent) expressions for various scalar quantities, facilitating the analysis and design of wireless systems. E.g. it may allow to evaluate beamforming performance without computing explicit beamformers. The analysis in [5] allowed e.g. the determination of the optimal regularization factor in Regularized ZF (R-ZF) BF, both with perfect and partial CSIT. A little known extension appeared in [6] for optimal beamformers, but only for the perfect CSIT MISO BC case. Some other extensions appeared recently in [7] or [8] where MISO IBC is considered with perfect CSIT and weighted R-ZF BF, with two optimized weight levels, for intracell or intercell interference. The contributions of this paper are:

• A novel optimal (max EWSR) beamforming design for MISO IBC with partial CSIT, an extension of the concave WSR minorization approach in [9] to the partial CSIT case. Whereas this approach finds BF’s as generalized eigenvectors of higher rank matrices, also in the case of MISO with partial CSIT, we propose Jacobi iterations for reduced computational complexity and simplified large system analysis.

• A novel large system analysis of the proposed BF design, that constitutes the first large system analysis of opti- mized BF design with partial CSIT. Furthermore, three types of channel estimate are considered (deterministic, quantized, LMMSE). The analysis also applies to subop- timal EWSMSE and Naive EWSR designs.

II. THEIBC SIGNALMODEL

In the rest of this paper we shall consider a per stream approach (which in the perfect CSI case would be equivalent to per user). In an IBC formulation, one stream per user can be expected to be the usual scenario. In the development below, in the case of more than one stream per user, we shall treat each stream as an individual user. So, consider again an IBC with C cells with a total of K users. We shall consider a system- wide numbering of the users. Userk is served by BSb_k. We shall initially consider users equipped with N_k antennas but the partial CSIT developments and associated large system analysis will be valid only for the MISO case (N_k = 1). The N_k×1received signal at user kin cell b_k is

yk=Hk,b_kgkxk

| {z } signal

+X

i6=k

b_i=b_k

Hk,b_kgixi

| {z } intracell interf.

+X

j6=b_k

X

i:b_i=j

Hk,jgixi

| {z }

intercell interf.

+vk

(1) wherexk is the intended (white, unit variance) scalar signal stream,H_k,bk is the N_k×M_bk channel from BS b_k to user k. BS b_k serves K_bk = P

i:b_i=b_k1 users. We considering a noise whitened signal representation so that we get for the noise v_k ∼ CN(0, I_Nk). The M_bk ×1 spatial Tx filter or beamformer (BF) isg_k. Treating interference as noise, userk will apply a linear Rx filterfk to maximize the signal power (diversity) while reducing any residual interference that would not have been (sufficiently) suppressed by the BS Tx. The Rx filter output isxbk=f_k^Hyk

xb_k = f_k^HH_k,bkg_kx_k+

K

X

i=1,6=k

f_k^HH_k,big_ix_i+f_k^Hv_k

= f_k^Hh_k,kx_k+X

i6=k

f_k^Hh_k,ix_i+f_k^Hv_k

(2)

wherehk,i =Hk,b_igi is the channel-Tx cascade vector.

III. MAXWSRWITHPERFECTCSIT

Consider as a starting point for the optimization the weighted sum rate (WSR)

W SR=W SR(g) =

K

X

k=1

u_k ln 1 ek

(3) wheregrepresents the collection of BFsgk, theek=ek(g)are the Minimum Mean Squared Errors (MMSEs) for estimating

the xk: 1

e_k=1+g^H_k H_k,b^H

kR⁻¹

k Hk,b_kgk= (1−g_k^HH_k,b^H

kR⁻¹_k Hk,b_kgk)⁻¹ R_k =H_k,bkQ_kH_k,b^H

k+R_k, Q_i =g_ig^H_i , R_k =X

i6=k

Hk,b_iQiH_k,b^H_i+IN_k.

Rk,R_k are the total and interference plus noise Rx covariance(4) matrices resp. and ek is the MMSE obtained at the output xbk=f_k^Hyk of the optimal (MMSE) linear Rx fk,

fk =R⁻¹_k Hk,b_kgk . (5) The WSR cost function needs to be augmented with the power

constraints X

k:b_k=j

tr{Qk} ≤Pj. (6)

In a classical difference of convex functions (DC program- ming) approach, Kim and Giannakis [9] propose to keep the concave signal terms and to replace the convex interference terms by the linear (and hence concave) tangent approxi- mation. Note that it is more appropriate to consider this DC programming as an instance of a minorization approach because whereas the linearization is carried out w.r.t. the Tx covariance matricesQ_k, the resulting problem then gets repa- rameterized in terms of the BF’s g_k, and the minorization is insensitive to the parameterization. More specifically, consider the dependence of WSR on Qk alone. Then

W SR=ukln det(R⁻¹

k Rk) +W SR_k, W SR_k=PK

i=1,i6=ku_iln det(R⁻¹

i R_i) (7)

whereln det(R⁻¹

k R_k)is concave inQ_kandW SR_k is convex in Q_k. Since a linear function is simultaneously convex and concave, consider the first order Taylor series expansion inQ_k around Qˆ (i.e. all Qˆ_i) with e.g.Rˆ_i=R_i( ˆQ), then

W SR_k(Qk,Q)ˆ ≈W SR_k( ˆQk,Q)ˆ −tr{(Qk−Qˆk) ˆAk} Aˆk=− ∂W SR_k(Q_k,Q)ˆ

∂Qk

_Qˆ

k,Qˆ

=

K

X

i6=k

uiH_i,b^H_k( ˆR⁻¹

i −Rˆ⁻¹_i )Hi,b_k

(8) Note that the linearized (tangent) expression forW SR_k con- stitutes a lower bound for it. Now, dropping constant terms, reparameterizing theQ_k=g_kg^H_k, performing this linearization for all users, and augmenting the WSR cost function with the constraints, we get the Lagrangian W SR(g,ˆg, λ)

=

C

X

j=1

λjPj+

K

X

k=1

ukln(1+g_k^HBˆkgk)−g^H_k ( ˆAk+λb_kI)gk (9) where Bˆk=H_k,b^H_kRˆ⁻¹

k Hk,b_k . (10)

The gradient (w.r.t. g_k) of this concave WSR lower bound is actually still the same as that of the original WSR criterion!

And it allows an interpretation as a generalized eigenvector condition

Bˆ_kg_k =1 +g_k^HBˆ_kg_k uk

( ˆA_k+λ_bkI)g_k (11)

or hence g_k⁰ = Vmax( ˆBk,Aˆk +λb_kI) is the (normalized)

”max” generalized eigenvector of the two indicated matrices, with max eigenvalueσ_k=σ_max( ˆB_k,Aˆ_k+λ_bkI). Letσ_k⁽¹⁾= g_k^0HBˆ_kg⁰_k,σ_k⁽²⁾=g⁰_k^HAˆ_kg_k⁰. The advantage of formulation (9) is that it allows straightforward power adaptation: introducing stream powers p_k ≥ 0 and substituting g_k =√

p_kg_k⁰ in (9) yields

W SR=

C

X

j=1

λjPj+

K

X

k=1

{ukln(1 +pkσ_k⁽¹⁾)−pk(σ⁽²⁾_k +λb_k)}

which leads to the following interference leakage (σ_k⁽²⁾) aware water filling

pk= u_k σ_k⁽²⁾+λb_k

− 1 σ⁽¹⁾_k

!⁺

(12) where the Lagrange multipliers are adjusted to satisfy the power constraintsP

k:b_k=jp_k=P_j. Withσ⁽²⁾_k = 0this would be standard waterfilling. The minorization approach is crucial for this waterfilling power optimization.

IV. JOINTMEAN ANDCOVARIANCEGAUSSIANCSIT In this section we drop the user indexkfor simplicity. The separable MIMO correlation model is

H=H+C_r^1/2H Ce _t^1/2 (13) whereH= EH, andCr^1/2,C_t^1/2are Hermitian square-roots of the Rx and Tx side covariance matrices

E(H−H)(H−H)^H=tr{C_t}C_r E(H−H)^H(H−H) =tr{Cr}Ct

(14) and the elements of He are i.i.d. ∼ CN(0,1). It is also of interest to consider the total Tx side correlation matrix

St= EH^HH=H^HH+tr{Cr}Ct. (15) V. MAMIMO LIMIT

If the number of Tx antennas M becomes very large, we get a convergence for any quadratic term of the form

HQH^{H M}−→^→∞ E

H|HbHQH^H =HQH^H+tr{QCt}Cr.(16) and hence we get the following MaMIMO limit matrices

R˘_k=I_Nk+

K

X

i=1

n

H_k,biQ_iH^H_k,b

i+tr{Q_iC_t,k,bi}C_r,ko R˘_k=IN_k+

K

X

i=1,6=k

n

Hk,b_iQiH^H_k,bi+tr{QiCt,k,b_i}Cr,k

o

(17) B˘k =H^H_k,bkR˘⁻¹

k Hk,b_k+tr{Cr,kR˘⁻¹

k }Ct,k,b_k (18) A˘_k =

K

X

i6=k

u_ih H^H_i,b

k

R˘⁻¹

i −R˘⁻¹_i H_i,bk

+tr{

R˘⁻¹

i −R˘⁻¹_i

Cr,i}Ct,i,b_k

i .

(19)

It suffices now to replace the matrices Ak, Bk in the DC programming approach of Section III by the matricesA˘k,B˘k

above to get a maximum EWSR design.

VI. THEMISO CASE: 3 CHANNELMODELS

In this caseC_r= 1and we shall denote the matricesR,H^H as the scalar r and the vector h. The channelh, its estimate bhand estimation errorehhave covariance matrices Θ =Chh, Θ =b C

bhbh andC

eheh.

(i) In the case of a deterministic channel estimate, we have bh=h+ehwherehandehare independent.

(ii) The channel quantization error model considered in [5]

is a variation on the deterministic model in which we have a quantized channel of the form bh⁰ = ηh+eh with again eh independent from h. Letbh=η⁻¹bh⁰.

(iii) In the case of aLMMSE channel estimate, we haveh= bh+ehin whichbhandehare decorrelated and hence independent in the Gaussian case. In the partial CSIT case, the termhh^H of the perfect CSIT case gets replaced by its estimate S = E_h|bhhh^H =bhbh^H+Θe where in the deterministic case, Θ =e −C

eheh=−σ²

ehI, in the quantized caseΘ =e −η⁻²C

eheh=

−_1−τ^τ2₂C_hh and actuallyΘ =b _1−τ^1+τ2₂C_hh, and in the LMMSE caseΘ =e C

eheh is the posterior covariance Θ = Θe −Θ(Θ +σ²

ehI)⁻¹Θ.

VII. MAXEWSR BFIN THEMAMISO LIMIT

The EWSR represents two rounds of averaging over the partial CSIT

EW SR= E

bhmaxgEW SR(g) EW SR(g) = E_h|bhW SR(g) =PK

k=1uk E_h|bhln(rk/r_k)

(a)=PK

k=1uk[ln(rk)−ln(r_k)]

(20) where transition (a)represents the MaMISO limit and

r_k= 1 +X

i6=k

E_h|bh|h^H_k,bigi|²= 1 +X

i6=k

g_i^HSk,b_igi

rk=r_k+g^H_k Sk,b_kgk.

(21) By adding the Lagrange terms for the BS power constraints, PC

c=1λc(Pc− X

k:b_k=c

||gk||²), to the EWSR in (20), we get the gradient

∂EW SR

∂g_k^∗ =αkSk,b_kgk−h P

i6=kβiSi,b_k+λb_kIi gk= 0 α_k= ^u_r^k

k , β_k =u_k(_r¹

k

−_r¹

k).

(22) This leads to the iterative (power method like) solution

g_k⁰(λb_k) =



 X

i6=k

βibhi,b_kbh^H_i,bk+Θek+λb_kI





−1

bhk,b_kαkbh^H_k,b

kgk

gk =ξcg⁰_k, ξc =q Pc/P

k:b_k=c||g⁰_k(λb_k)||² Θek=P

i6=kβiΘei,b_k −αkΘek,b_k.

(23) The BF scale factors ξ_c are introduced because instead of a bisection method to force satisfaction of the power constraints, the Lagrange multipliers (if non-zero) can be adapted analyt- ically as in [1] by exploiting P

k:bk=cg^H_k ∂EW SR

∂g_k^∗ = 0and

X

k:bk=c

||gk||²=Pc. Then from (22) we get λ_c=

λ⁰_c , ζc> Pc

0 , ζc≤Pc

(24) where

λ⁰_c =_P¹

c

P

k:b_k=c[αkg^H_k Sk,cgk−P

i6=kβig_k^HSi,cgk] ζ_c=P

k:b_k=c||g⁰_k(0)||² (25) Indeed, in the case of multiple power constraints, not all constraints are necessarily satisfied with equality.

A. Min EWSMSE and Naive Max EWSR

The minimum EWSMSE BF design of [1] can be obtained from the EWSR design above by setting Θek,b_k = 0 in (21)- (25) and hence Sk,b_k = bhk,b_kbh^H_k,b

k. On the other hand, the naive EWSR approach, which ignores the covariance of ehin any occurrence, is obtained from the EWSR design above by settingΘek= 0in (21)-(25) and settingSi,c=bhi,cbh^H_i,cfor any (i, c). Of course, these simplifications should be carried out in the BF design from (21)-(25), but not in the EWSR evaluation in (20)-(21).

VIII. LARGESYSTEMAPPROXIMATION OF THEEWSR Due to the law of large numbers, the scalars rk,r_k, ak = bh^H_k,b

kg_k⁰,||g_k⁰(λb_k)||²,g_k^HSk,cgk,g_k^HSi,cgk and henceαk,βk, ξ_c,λ⁰_c andζ_c converge to deterministic limits asM, K→ ∞ at fixed ratio β = M/K. We shall perform a large system analysis to determine these deterministic limits, which will also provide the limiting value for

EW SR=

K

X

k=1

u_k ln(r_k/r_k) =

K

X

k=1

u_k ln(1 +γ_k) (26) where theγ_k are the limiting SINRs. The deterministic limits will follow the same iterations as the BF design algorithm for which we can rewrite iterationj as

µ^(j)_k,i= ˘g_k^(j−1)HSi,b_k˘g_k^(j−1), ∀i, k r^(j)

k = 1 +P

i6=kξ^2,(j−1)_b

i µ^(j)_i,k, r^(j)_k =r^(j)

k +ξ_b^2,(j−1)

k µ^(j)_k,k α^(j)_k = ^uk

r^(j)_k , β_k^(j)=uk( ¹

r^(j)

k

− ¹

r^(j)_k ) a^(j−1)_k =bh^H_k,b

k˘g^(j−1)_k Θe^(j)_k =P

i6=kβ_i^(j)Θei,b_k −α^(j)_k Θek,b_k

ψ_k^(j)(0) =bh^H_k,b

k



 X

i6=k

β_i^(j)bhi,b_kbh^H_i,b

k+Θe^(j)_k





−2

bhk,b_k

ν_k^(j)=α^(j)_k ξ_b^(j−1)

k a^(j−1)_k ζc^(j)=ξc^2,(j−1)P

k:b_k=cψ_k^(j)(0)ν_k^2,(j)

˘λ^(j)c = _P¹

c

X

k:b_k=c

[α^(j)_k ξ_b^2,(j−1)

k µ^(j)_k,k−X

i6=k

β_i^(j)ξ_b^2,(j−1)

k µ^(j)_k,i] λ^(j)c =

( λ˘^(j)c , ζc^(j)> Pc

0 , ζc^(j)≤Pc

(27)

˘ g_k^(j)=



 X

i6=k

β_i^(j)bh_i,bkbh^H_i,b

k+Θe^(j)_k +λ^(j)_b

k I





−1

bh_k,bkν_k^(j)

ψ^(j)_k (λ^(j)_b

k) =bh^H_k,b

k



 X

i6=k

β_i^(j)bh_i,bkbh^H_i,b

k+Θe^(j)_k +λ^(j)_b

kI





−2

bh_k,bk

ξc^(j)= q

Pc/P

k:b_k=cψ_k^(j)(λ^(j)_b

k)ν_k^2,(j) g_k^(j)=ξ^(j)c g˘^(j)_k

(28) where we used the short-hand notation for e.g. ν_k^2,(j) = (ν_k^(j))². Note that a^(j)_k can be computed recursively by in- troducing

φ^(j)_k =bh^H_k,b

k



 X

i6=k

β_i^(j)bhi,b_kbh^H_i,bk+Θe^(j)_k +λ^(j)_b

k I





−1

bhk,b_k,

⇒ a^(j)_k =φ^(j)_k ν_k^(j)=φ^(j)_k α^(j)_k ξ_b^(j−1)

k a^(j−1)_k .

(29) A. Large System Analysis

In the large system analysis, we do not compute the BFsgk. Instead, deterministic limits are determined for the following quantities: φ^(j)_k ,ψ_k^(j)(0),ψ_k^(j)(λ^(j)_b

k),µ^(j)_k,i.

The following results can now be obtained by applying the principles developed in [5]. For (with asymptotic equalities)

φk=bh^H_k,b

k[P

i6=kβibhi,b_kbh^H_i,b

k+Θek+λb_kI]⁻¹bhk,b_k

=tr{Θbk,b_k[P

i6=kβibhi,b_kbh^H_i,b

k+Θek+λb_kI]⁻¹}, (30) the deterministic limitφk=ek,b_k(λb_k)can be computed from the C sets of implicit equations

e_k,c(λ_c) =tr{Θb_k,c(X

i6=k

βi

1 +β_ie_i,c(λ_c)Θb_i,c+Θee_k,c+λ_cI

| {z }

T_k,c(λc)

)⁻¹}

(31) where e

Θek,c=P

i6=kβiΘei,c−αkΘek,c, henceΘek= e

Θek,b_k. For ψk(λ) =ψk,b_k(λ), we get

ψk,b_k(λ) =bh^H_k,b

k[P

i6=kβibhi,b_kbh^H_i,b

k+Θek+λ I]⁻²bhk,b_k

=tr{Θbk,b_k[P

i6=kβibhi,b_kbh^H_i,b

k+Θek+λ I]⁻²}

=−_dλ^d tr{Θb_k,bkT_k,b⁻¹

k(λ)}=−e⁰_k,b

k(λ)

(32) The ψk,b_k(λb_k)can be solved from the linear equations

ψk,c(λc) =tr{Θbk,cT_k,c⁻¹(λc)T_k,c⁰ (λc)T_k,c⁻¹(λc)}

withT_k,c⁰ (λc) =P

i6=k

β_iψ_i,c(λ_c)

(1+β_ie_i,c(λ_c))²Θbi,c+I . (33) The C sets of equations (30) for the C valuesλ_c have to be augmented with a set C+1for the e_k(0). And also (33) has to be considered forψ_k(0).

Forµk,i we introduce µk,i =ν_k²bµk,i+ν_k²eµk,i withµbk,i =

˘

g^H_kbhi,b_kbh^H_i,b

kg˘k/ν_k²andµek,i= ˘g_k^HΘei,b_k˘gk/ν_k². We shall obtain µek,i asµek,i =e⁰_k,b

k,i,b_k= _∂z^∂ ek,b_k,i,b_k(λb_k,0) where ek,c,i,d(λc, z) =tr{Θbk,cT_k,c,i,d⁻¹ (λc, z)} with T_k,c,i,d(λ_c, z) =X

j6=k

βjΘbj,c

1 +β_je_j,c,i,d(λ_c, z)+Θee_k,c+λ_cI−zΘe_i,d (34)

We can then obtain the e⁰_k,c,i,d as the solution of the linear equations

e⁰_k,c,i,d=tr{Θbk,cT_k,c,i,d⁻¹ (λc,0)T_k,c,i,d⁰ (λc,0)T_k,c,i,d⁻¹ (λc,0)}

T_k,c,i,d⁰ (λc,0) =X

j6=k

βje⁰_j,c,i,d(λc)

(1 +βjej,c(λc))²Θbj,c+Θei,d (35) On the other hand, we get bµk,k = a²_k = ν_k²e²_k,b

k(λb_k). For i6=k,

µbk,i =|bh^H_i,b

k[P

j6=kβjbhj,b_kbh^H_j,b

k+Θek+λb_kI]⁻¹bhk,b_k|²

= ^|bh

H i,bk[P

j6=k,iβjbh_j,bkbh^H_j,bk+eΘk+λ_bkI]⁻¹bh_k,bk|² (1+βibh^H_i,bk[P

j6=k,iβjbh_j,bkbh^H_j,bk+eΘk+λ_bkI]⁻¹bh_i,bk)²

(36) Hence µb_k,i = µbb_k,i/(1 +β_ie_i,bk(λ_bk))² where the bbµ_k,i are obtained from a system of equations similar to that for µe_k,i (as in (34), (35)), by replacing the Θe_i,bk byΘb_i,bk.

IX. NUMERICALRESULTS

We plot the performance of the proposed EWSR BF with MF initialization and compare it to the proposed large system approximation. The channel correlation matrix is modeled as in [5]. We consider the quantization error channel model.

Fig. 1 shows the performance of the precoder and its large sys- tem approximation for i.i.d. channels forC= 2cells. Monte Carlo simulations are averaged over 1000 channel realizations.

It can be observed that for i.i.d channels the approximation is very accurate which validates our asymptotic approach.

Although the large system analysis for the sum rate seems complex, we need to calculate it only once per given SNR (independent of channel realizations). In Fig. 2 we consider

Fig. 1. Expected sum rate comparison forC= 2, K= 3, M= 10, N= 1, Θk,b_i=IM∀i,∀kandτ²=₁₀¹.

the actual EWSR (averaged over 1000 channel realizations) for another scenario, in which the channels exhibit low channel covariance rank (few dominating multipath), in which the covariance information adds significant contributions to the channel estimate based CSIT. The EWSR curve does not sat- urate at high SNR because therank(Θk,b_i) = 2information allows covariance CSIT based ZF for 4 users. We perceive an unlimited gain for the proposed approach with respect to EWSMSE of [1] in this case of low rank correlation matrices, and of course even more w.r.t. the naive approach of perfect CSIT in which the true channels simply get replaced by their estimates.

X. CONCLUSION

In this work, we derived and presented an asymptoti- cally optimal beamforming algorithm for the case of partial

Fig. 2. Expected sum rate comparison forC= 2, K= 8, M= 8, N= 1, rank(Θk,b_i) = 2,∀i,∀kandτ²= ¹₂.

CSIT and its large system performance analysis. Important EWSR gains have been illustrated over the naive approach in which the true channels are replaced by their estimates in a perfect CSIT approach, and also over the more sophisti- cated approach which relates EWSR to EWSME. The gain over EWSMSE comes from exploiting the channel covariance information also in the signal power (and not only in the interference terms).

Acknowledgments

EURECOM’s research is partially supported by its indus- trial members: ORANGE, BMW, SFR, ST Microelectronics, Symantec, SAP, Monaco Telecom, iABG, by the projects HIGHTS (EU H2020), DUPLEX (French ANR) and MASS- START (French FUI). The research of Orange Labs is partially supported by the EU H2020 5GPPP project One5G.

REFERENCES

[1] F. Negro, I. Ghauri, and D.T.M. Slock, ”Sum Rate Maximization in the Noisy MIMO Interfering Broadcast Channel with Partial CSIT via the Expected Weighted MSE,” in IEEE Int’l Symp. on Wireless Commun.

Systems (ISWCS), Paris, France, Aug. 2012.

[2] X. Li, E. Bj¨ornson, E.G. Larsson, S. Zhou, and J. Wang, ”Multi-cell MMSE Precoder for Massive MIMO Systems and New Large System Analysis,” in IEEE Globecom, San Diego, CA, USA, 2015.

[3] M. Razaviyayn, M.S. Boroujeni, and Z.-Q. Luo, ”Stochastic Weighted MMSE Approach to Sum Rate Maximization for a MIMO Interference Channel,” in IEEE workshop on Signal Processing Advances in Wireless Communications (SPAWC), Darmstadt, Germany, June 2013.

[4] H. Asgharimoghaddam, A. Tolli, and N. Rajatheva, ”Decentralizing the Optimal Multi-cell Beamforming via Large System Analysis,” in IEEE Int’l Conf. on Communications (ICC), 2014.

[5] S. Wagner, R. Couillet, M. Debbah, and D. Slock, ”Large System Analysis of Linear Precoding in Correlated MISO Broadcast Channels Under Limited Feedback,” IEEE Trans. Info. Theory, Jul. 2012.

[6] S. Wagner and D.T.M. Slock, ”Weighted Sum Rate Maximization of Correlated MISO Broadcast Channels under Linear Precoding : a Large System Analysis,” in IEEE Int’l Workshop on Sig. Proc. Advances in Wireless Comm’s (SPAWC), San Fransisco, CA, USA, 2011.

[7] S. Lakshminarayana, M. Assaad, and M. Debbah, ”Coordinated Multi- cell Beamforming for Massive MIMO: A Random Matrix Approach,”

IEEE Trans. Information Theory, Jun. 2015.

[8] A. Muller, R. Couillet, E. Bjornson, S. Wagner, and M. Debbah,

”Interference-Aware RZF Precoding for Multi Cell Downlink Systems,”

IEEE Trans. Signal Proc., Aug. 2015.

[9] S.-J. Kim and G.B. Giannakis, ”Optimal Resource Allocation for MIMO Ad Hoc Cognitive Radio Networks,” IEEE Tr. Info. Theory, May 2011.

[10] S.S. Christensen, R. Agarwal, E. de Carvalho, and J. Cioffi, ”Weighted sum-rate maximization using weighted MMSE for MIMO-BC beam- forming design,” IEEE Trans. on Wireless Comm’s, Dec. 2008.