A Coordinated Approach to Channel Estimation in Large-Scale Multiple-Antenna Systems

264 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 2, FEBRUARY 2013

A Coordinated Approach to Channel Estimation inLarge-Scale Multiple-Antenna SystemsHaifan Yin, David Gesbert Fellow, IEEE, Miltiades Filippou, and Yingzhuang Liu

AbstractThis paper addresses the problem of channel es-timation in multi-cell interference-limited cellular networks. Weconsider systems employing multiple antennas and are interestedin both the finite and large-scale antenna number regimes (so-called massive MIMO). Such systems deal with the multi-cellinterference by way of per-cell beamforming applied at each basestation. Channel estimation in such networks, which is knownto be hampered by the pilot contamination effect, constitutes amajor bottleneck for overall performance. We present a novelapproach which tackles this problem by enabling a low-ratecoordination between cells during the channel estimation phaseitself. The coordination makes use of the additional second-orderstatistical information about the user channels, which are shownto offer a powerful way of discriminating across interfering userswith even strongly correlated pilot sequences. Importantly, wedemonstrate analytically that in the large-number-of-antennasregime, the pilot contamination effect is made to vanish com-pletely under certain conditions on the channel covariance.Gains over the conventional channel estimation framework areconfirmed by our simulations for even small antenna array sizes.

Index Termsmassive MIMO, pilot contamination, channelestimation, scheduling, covariance information.

I. INTRODUCTION

FULL reuse of the frequency across neighboring cells leadsto severe interference, which in turn limits the quality ofservice offered to cellular users, especially those located at thecell edge. As service providers seek some solutions to restoreperformance in low-SINR cell locations, several approachesaimed at mitigating inter-cell interference have emerged inthe last few years. Among these, the solutions which exploitthe additional degrees of freedom made available by the useof multiple antennas seem the most promising, particularly soat the base station side where such arrays are more affordable.In an effort to solve this problem while limiting the re-

quirements for user data sharing over the backhaul network,coordinated beamforming approaches have been proposed inwhich 1) multiple-antenna processing is exploited at each basestation, and 2) the optimization of the beamforming vectors atall cooperating base stations is performed jointly. Coordinated

Manuscript received 1 February 2012; revised 15 June 2012. This workwas supported in part by National Natural Science Foundation of Chinaunder grant No. 60972015 and 61231007, by the European research projectsSAPHYRE and HARP under the FP7 ICT Objective 1.1 - The Network of theFuture, and by the French national ANR-VERSO funded project LICORNE.H. Yin was with Huazhong University of Science and Technology, Wuhan,

430074 China. He is now with EURECOM, 06410 Biot, France (e-mail:[email protected]).D. Gesbert and M. Filippou are with EURECOM, 06410 Biot, France (e-

mail: [email protected], [email protected]).Y. Liu is with Huazhong University of Science and Technology, Wuhan,

430074 China (e-mail: [email protected]).Digital Object Identifier 10.1109/JSAC.2013.130214.

beamforming does not require the exchange of user messageinformation (e.g., in network MIMO). Yet it still demandsthe exchange of channel state information (CSI) across thetransmitters on a fast time scale and low-latency basis, makingalmost as challenging to implement in practice as the abovementioned network MIMO schemes.

Fortunately a path towards solving some of the essentialpractical problems related to beamforming-based interferenceavoidance was suggested in [1]. In this work, it was pointedout that the need for exchanging Channel State Information atTransmitter (CSIT) between base stations could be alleviatedby simply increasing the number of antennas, M , at eachtransmitter (so-called massive MIMO). This result is rooted inthe law of large numbers, which predicts that, as the number ofantennas increases, the vector channel for a desired terminalwill tend to be more orthogonal to the vector channel of arandomly selected interfering user. This makes it possible toreject interference at the base station side by simply aligningthe beamforming vector with the desired channel (MaximumRatio Combining or spatial matched filter). Hence in theory,a simple fully distributed per-cell beamforming scheme canoffer performance scaling (withM ) similar to a more complexcentralized optimization.

Unfortunately, the above conclusion only holds withoutpilot-contaminated CSI estimates. In reality, channel informa-tion is acquired on the basis of finite-length pilot sequences,and crucially, in the presence of inter-cell interference. There-fore, the pilot sequences from neighboring cells would con-taminate each other. It was pointed out in [1] that pilot contam-ination constitutes a bottleneck for performance. In particular,it has been shown that pilot contamination effects [2], [3],[4] (i.e., the reuse of non-orthogonal pilot sequences acrossinterfering cells) cause the interference rejection performanceto quickly saturate with the number of antennas, therebyundermining the value of MIMO systems in cellular networks.

In this paper, we address the problem of channel esti-mation in the presence of multi-cell interference generatedfrom pilot contamination. We propose an estimation methodwhich provides a substantial improvement in performance.It relies on two key ideas. The first is the exploitation ofdormant side-information lying in the second-order statisticsof the user channels, both for desired and interfering users. Inparticular, we demonstrate a powerful result indicating that theexploitation of covariance information under certain subspaceconditions on the covariance matrices can lead to a completeremoval of pilot contamination effects in the largeM limit. Wethen turn to a practical algorithm design where this conceptis exploited. The key idea behind the new algorithm is the

0733-8716/13/$31.00 c 2013 IEEE

YIN et al.: A COORDINATED APPROACH TO CHANNEL ESTIMATION IN LARGE-SCALE MULTIPLE-ANTENNA SYSTEMS 265

use of a covariance-aware pilot assignment strategy withinthe channel estimation phase itself. While diversity-basedscheduling methods have been popularized for maximizingvarious throughput-fairness performance criteria [5], [6], [7],[8], the potential benefit of user-to-pilot assignment in thecontext of interference-prone channel estimation has receivedvery little attention so far.More specifically, our contributions are the following: We

first develop a Bayesian channel estimation method makingexplicit use of covariance information in the inter-cell inter-ference scenario with pilot contamination. We show that thechannel estimation performance is a function of the degreeto which dominant signal subspaces pertaining to the desiredand interference channel covariance overlap with each other.Therefore we exploit the fact that the desired user signals andinterfering user signals are received at the base station with(at least approximately) finite-rank covariance matrices. Thisis typically the case in realistic scenarios due to the limitedangle spread followed by incoming paths originating fromstreet-level users [9]. Finally, we propose a pilot sequenceassignment strategy based on assigning carefully selectedgroups of users to identical pilot sequences. The gains areshown to depend on system parameters such as the typicalangle spread measured at the base station and the number ofbase station antennas. Performance close to the interference-free channel estimation scenario is obtained for moderatenumbers of antennas and users.The notations adopted in the paper are as follows. We use

boldface to denote matrices and vectors. Specifically, IM de-notes the MM identity matrix. Let (X)T , (X), and (X)Hdenote the transpose, conjugate, and conjugate transpose ofa matrix X respectively. E {} denotes the expectation, Fdenotes the Frobenius norm, and diag{a1, ..., aN} denotes adiagonal matrix or a block diagonal matrix with a1, ..., aN atthe main diagonal. The Kronecker product of two matrices Xand Y is denoted by XY. is used for definitions.

II. SIGNAL AND CHANNEL MODELS

We consider a network of L time-synchronized1 cells, withfull spectrum reuse. Estimation of (block-fading) channels inthe uplink is considered,2 and all the base stations are equippedwithM antennas. To simplify the notations, we assume the 1stcell is the target cell, unless otherwise notified. We assume thepilots, of length , used by single-antenna users in the samecell are mutually orthogonal. As a result, intra-cell interferenceis negligible in the channel estimation phase. However, non-orthogonal (possibly identical) pilots are reused from cell tocell, resulting in pilot contamination from L 1 interferingcells. For ease of exposition, we consider the case where asingle user per cell transmits its pilot sequence to its servingbase. The pilot sequence used in the l-th cell is denoted by:

sl = [ sl1 sl2 sl ]T . (1)1Note that assuming synchronization between uplink pilots provides a worst

case scenario from a pilot contamination point of view, since any lack ofsynchronization will tend to statistically decorrelate the pilots.2Similar ideas would be applicable for downlink channel estimation,

provided the UE is equipped with multiple antennas as well, in which casethe estimation would help resolve interferences originating from neighboringbase stations.

The powers of pilot sequences are assumed equal such that|sl1|2 + + |sl |2 = , l = 1, 2, . . . , L.The channel vector between the l-th cell user and the target

base station is hl. Thus, h1 is the desired channel while hl, l >1 are interference channels. All channel vectors are assumedto beM 1 complex Gaussian, undergoing correlation due tothe finite multipath angle spread at the base station side [10]:

hl = R1/2l hWl, l = 1, 2, . . . , L, (2)

where hWl CN (0, IM ) is the spatially white M 1 SIMOchannel, and CN (0, IM ) denotes zero-mean complex Gaus-sian distribution with covariance matrix IM . In this paper, wemake the assumption that covariance matrix Rl E{hlhHl }can be obtained separately from the desired and interferencechannels (see Section VI for how this could be done inpractice).During the pilot phase, the M signal received at the

target base station is

Y =

Ll=1

hlsTl +N, (3)

where N CM is the spatially and temporally white ad-ditive Gaussian noise (AWGN) with zero-mean and element-wise variance 2n.

III. COVARIANCE-BASED CHANNEL ESTIMATION

A. Pilot Contamination

Conventional channel estimation relies on correlating thereceived signal with the known pilot sequence (referred hereas Least Squares (LS) estimate for example). Hence, using themodel in (3), an LS estimator for the desired channel h1 is

hLS1 = Ys1(s1T s1)

1. (4)

The conventional estimator suffers from a lack of orthogo-nality between the desired and interfering pilots, an effectknown as pilot contamination [2], [11], [12]. In particular,when the same pilot sequence is reused in all L cells, i.e.,s1 = = sL = s, the estimator can be written as

hLS1 = h1 +Ll =1

hl +Ns/ . (5)

As it appears in (5), the interfering channels leak directlyinto the desired channel estimate. The estimation performanceis then limited by the signal to interfering ratio at the basestation, which in turns limits the ability to design an effectiveinterference-avoiding beamforming solution.

B. Bayesian Estimation

We hereby propose an improved channel estimator with theaim of reducing the pilot contamination effect, and takingadvantage of the multiple antenna dimensions. We suggestto do so by exploiting side information lying in the secondorder statistics of the channel vectors. The role of covariancematrices is to capture structure information related to thedistribution (mainly mean and spread) of the multipath anglesof arrival at the base station. Due to the typically elevatedposition of the base station, rays impinge on the antennas with


a finite angle-of-arrival (AOA) spread and a user location-dependent mean angle. Note that covariance-aided channelestimation itself is not a novel idea, e.g., in [13]. In [14],the authors focus on optimal design of pilot sequences andthey exploit the covariance matrices of desired channels andcolored interference. The optimal training sequences weredeveloped with adaptation to the statistics of disturbance. Inour paper, however, the pilot design is shown not having animpact on interference reduction, since fully aligned pilotsare transmitted. Instead, we focus on i) studying the limitingbehavior of covariance-based estimates in the presence ofinterference and large-scale antenna arrays, and ii) how toshape covariance information for the full benefit of channelestimation quality.Two Bayesian channel estimators can be formed. In the first,

all channels are estimated at the target base station (includinginterfering ones). In the second, only h1 is estimated. Byvectorizing the received signal and noise, our model (3) canbe represented as

y = Sh+ n, (6)

where y = vec(Y), n = vec(N), and h CLM1 is obtainedby stacking all L channels into a vector. The pilot matrix Sis defined as

S [s1 IM sL IM

]. (7)

Applying Bayes rule, the conditional distribution of thechannels h given the received training signal y is

p(h|y) = p(h)p(y|h)p(y)

. (8)

We use the multivariate Gaussian probability density function(PDF) of the random vector h and assume its rows h1, ,hLare mutually independent, giving the joint PDF:

p(h) =

exp

(

Ll=1

hHl R1l hl

)LM (detR1 detRL)M

. (9)

Note that we derive this Bayesian estimator under the stan-dard condition of covariance matrix invertibility, although weshow later this hypothesis is actually challenged by realityin the large-number-of-antennas regime. Fortunately, our finalexpressions for channel estimators completely skip the covari-ance inversion.Using (6), we may obtain:

p(y|h) =exp

((y Sh)H(y Sh)/2n

)(2n)

M. (10)

Combining the equations (9) and (10), the expression of (8)can be rewritten as

p(h|y) = exp (l(h))AB

, (11)

whereA p(y)(2n)M , B LM (detR1 detRL)M =LM (detR)M , and

l(h) hHRh+ (y Sh)H(y Sh)/2n, (12)in which R diag(R1, ,RL), R R1.

Using the maximum a posteriori (MAP) decision rule, theBayesian estimator yields the most probable value given theobservation y [15]:

h = arg maxhCLM1

p(h|y)= arg min

hCLM1l(h)

= (2nILM +RSHS)1RSHy. (13)

Interestingly, the Bayesian estimate as shown in (13) coin-cides with the minimum mean square error (MMSE) estimate,which has the form

hMMSE = RSH(SRSH + 2nIM )1y. (14)

(13) and (14) are equivalent thanks to the matrix inversionidentity (I+AB)1A = A(I +BA)1.

C. Channel Estimation with Full Pilot Reuse

Previously we have given expressions whereby interferingchannels are estimated simultaneously with the desired chan-nel. This could be of use in designing zero-forcing type re-ceivers. Even though it is clear that Zero-Forcing (ZF) type (orother sophisticated) receivers would give better performanceat finite M (see [3] for an analysis of this problem), in thispaper, however, we focus on simple matched filters, sincesuch filters are made more relevant by the users of massiveMIMO. Matched filters require the knowledge of the desiredchannel only, so that interference channels can be consideredas nuisance parameters. For this case, the single user channelestimation shown below can be used. For ease of exposition,the worst case situation with a unique pilot sequence reusedin all L cells is considered:

s = [ s1 s2 s ]T . (15)Similar to (7), we define a training matrix S s IM . Notethat SH S = IM . Then the vectorized received training signalat the target base station can be expressed as

y = S

Ll=1

hl + n. (16)

Since the Bayesian estimator and the MMSE estimator areidentical, we omit the derivation and simply give the expres-sion of this estimator for the desired channel h1 only:

h1 = R1SH

(S

(Ll=1

Rl

)SH + 2nIM

)1y (17)

= R1

(2nIM +

Ll=1

Rl

)1SHy. (18)

Note that the MMSE channel estimation in the presence ofidentical pilots is also undertaken in other works such as [3].In the section below, we examine the degradation caused

by the pilot contamination on the estimation performance. Inparticular, we point out the role played by the use of covari-ance matrices in dramatically reducing the pilot contaminationeffects under certain conditions on the rank structure.


We are interested in the mean squared error (MSE) ofthe proposed estimators, which can be defined as: M E{h h2F }, or for the single user channel estimate M1 E{h1 h12F }.The estimation MSE of (13) is

M = trR

(ILM +

SH S

2nR

)1 . (19)Specifically, when identical pilots are used in all cells, theMSEs are

M = tr{R

(ILM +

JLL IM2n

R

)1}, (20)

M1 = trR1 R21

(2nIM +

Ll=1

Rl

)1 , (21)where JLL is an L L unit matrix consisting of all 1s. Thederivations to obtain M and M1 use standard methods andthe details are omitted here due to lack of space. However,similar methods can be found in [16]. Of course, it is clearfrom (20) and (21) that the MSE is not dependent on thespecific design of the pilot sequence, but on the power of it.We can readily obtain the channel estimate of (18) in an

interference-free scenario, by setting interference terms tozero:

hno int1 = R1(2nIM + R1

)1SH(Sh1 + n), (22)

where the superscript no int refers to the no interferencecase, and the corresponding MSE:

Mno int1 = tr{R1

(IM +

2nR1

)1}. (23)

D. Large Scale Analysis

We seek to analyze the performance for the above estimatorsin the regime of large antenna numberM . For tractability, ouranalysis is based on the assumption of uniform linear array(ULA) with supercritical antenna spacing (i.e., less than orequal to half wavelength).Hence we have the following multipath model3

hi =1P

Pp=1

a(ip)ip, (24)

where P is the arbitrary number of i.i.d. paths, ip CN (0, 2i ) is independent over channel index i and path indexp, where i is the i-th channels average attenuation. a() isthe steering vector, as shown in [17]

a()

1

ej2D cos()

...

ej2(M1)D

cos()

, (25)3Note that the Gaussian model (2) can well approximate the multipath

model (24) as long as there are enough paths. Since the number of elementarypaths is typically very large, we have P 1 this assumption is valid inpractice.

where D is the antenna spacing at the base station and is the signal wavelength, such that D /2. ip [0, ]is a random AOA. Note that we can limit angles to [0, ]because any [, 0] can be replaced by giving thesame steering vector.Below, we momentarily assume that the selected users

exhibit multipath AOAs that do not overlap with the AOAs forthe desired user, i.e., the AOA spread and user locations aresuch that multipath for the desired user are confined to a regionof space where interfering paths are very unlikely to exist.Although the asymptotic analysis below makes use of thiscondition, it will be shown in Section IV how such a structurecan be shaped implicitly by the coordinated pilot assignment.Finally, simulations reveal in Section V the robustness withrespect to an overlap between AOA regions of desired andinterference channels (for instance in the case of GaussianAOA distribution).Our main result is as follows:

Theorem 1. Assume the multipath angle of arrival yieldingchannel hj , j = 1, . . . , L, in (24), is distributed according toan arbitrary density pj() with bounded support, i.e., pj() =0 for / [minj , maxj ] for some fixed minj maxj [0, ] . Ifthe L 1 intervals [mini , maxi ] , i = 2, . . . , L are strictlynon-overlapping with the desired channels AOA interval4

[min1 , max1 ], we have

limM

h1 = hno int1 . (26)

Proof: From the channel model (24), we get

Ri =2iP

Pp=1

E{a(ip)a(ip)H} = 2iE{a(i)a(i)H},

where i has the PDF pi() for all i = 1, . . . , L. The proof ofTheorem 1 relies on three intermediate lemmas which exploitthe eigenstructures of the covariance matrices. The proofs ofthe lemmas are given in the appendix. The essential ingredientis to exhibit an asymptotic (at large M ) orthonormal vectorbasis for Ri constructed from steering vectors at regularlysampled spatial frequencies.

Lemma 1. Define (x) [ 1 ejx ej(M1)x ]Tand A span{(x), x [1, 1]}. Given b1, b2 [1, 1] andb1 < b2, define B span{(x), x [b1, b2]}, then

dim{A} = M dim{B} (b2 b1)M/2 when M grows large.Proof: See Appendix A.

Lemma 1 characterizes the number of dimensions a linearspace has, which is spanned by (x), in which x plays therole of spatial frequency.

Lemma 2. With a bounded support of AOAs, the rank ofchannel covariance matrix Ri satisfies:

rank(Ri)

M di, as M ,

4This condition is just one example of practical scenario leading tonon-overlapping signal subspaces between the desired and the interferencecovariances, however, more general multipath scenarios could be used.


where di is defined as

di (cos(mini ) cos(maxi )

) D.

Proof: See Appendix B.Lemma 2 indicates that for large M , there exists a null

space null(Ri) of dimension (1di)M . Interestingly, relatedeigenstructure properties of the covariance matrices wereindependently derived in [18] for the purpose of reducing theoverhead of downlink channel estimation and CSI feedback inmassive MIMO for FDD systems.

Lemma 3. The null space null(Ri) includes a certain set ofunit-norm vectors:

null(Ri) span{a()M

, / [mini , maxi ]}, as M .

Proof: See Appendix C.This lemma indicates that multipath components with AOA

outside the AOA region for a given user will tend to fall inthe null space of its covariance matrix in the large-number-of-antennas case.We now return to the proof of Theorem 1. Ri can be

decomposed intoRi = UiiU

Hi , (27)

where Ui is the signal eigenvector matrix of size M mi, inwhichmi diM . i is an eigenvalue matrix of sizemimi.Due to Lemma 3 and the fact that densities pi() and p1()have non-overlapping supports, we have

UHi U1 = 0, i = 1, as M . (28)Combining the channel estimate (18) and the channel model

(16), we obtain

h1 = R1

(2nIM +

Ll=1

Rl

)1SH

(S

Li=1

hi + n

).

According to (28), matrices R1 andLl=2

Rl span orthogonal

subspaces in the large M limit. Therefore we place ourselves

in the asymptotic regime for M , when Ll=2

Rl can be eigen-

decomposed according to

Ll=2

Rl = WWH , (29)

where W is the eigenvector matrix such that WHW = Iand span {W} is included in the orthogonal complement ofspan {U1}. Now denote V the unitary matrix correspond-ing to the orthogonal complement of both span {W} andspan {U1}, so that the M M identity matrix can now bedecomposed into:

IM = U1UH1 +WW

H +VVH . (30)

Thus, for large M ,

h1 U11UH1(2nU1U

H1 +

2nVV

H + 2nWWH

+U11UH1 +WW

H)1( L

i=1

hi + SHn

).

Due to asymptotic orthogonality between U1, W and V,

h1 U11(2Im1 + 1)1UH1 (L

i=1

hi + SHn)

U11(2Im1+1)1(UH1 h1+L

i=2

UH1 hi+SHn1

).

However, since hi span{a(), [mini , maxi ]

}, we have

from Lemma 3 thatUH1 hiUH1 h1 0, for i = 1 when M .

Therefore

limM

h1 = U11(2nIm1 + 1

)1(UH1 h1 +

SHn

),

which is identical to hno int1 if we apply the EVD decomposi-tion (27) for R1 in (22). This proves Theorem 1.We also believe that, although antenna calibration is needed

as a technical assumption in the theorem, orthogonality of co-variances signal subspaces will occur in non-tightly calibratedsettings provided the AOA regions do not overlap.

IV. COORDINATED PILOT ASSIGNMENT

We have seen from above that the performance of thecovariance-aided channel estimation is particularly sensitiveto the degree with which the signal subspaces of covariancematrices for the desired and the interference channels overlapwith each other. In the ideal case where the desired andthe interference covariances span distinct subspaces, we havedemonstrated that the pilot contamination effect tends tovanish in the large-antenna-array case. In this section, wemake use of this property by designing a suitable coordinationprotocol for assigning pilot sequences to users in the Lcells. The role of the coordination is to optimize the useof covariance matrices in an effort to try and satisfy thenon-overlapping AOA constraint of Theorem 1. We assumethat in all L cells, the considered pilot sequence will beassigned to one (out of K) user in each of the L cells. LetG {1, . . . ,K}, then Kl G denotes the index of the userin the l-th cell who is assigned the pilot sequence s. The setof selected users is denoted by U in what follows.We use the estimation MSE (21) as a performance metric

to be minimized in order to find the best user set. (20) isan alternative if we take the estimates of interfering channelsinto consideration. For a given user set U , we define a networkutility function

F(U) |U|j=1

Mj(U)tr {Rjj(U)} , (31)

where |U| is the cardinal number of the set U . Mj(U) isthe estimation MSE for the desired channel at the j-th basestation, with a notation readily extended from M1 in (21),where this time cell j is the target cell when computingMj .Rjj(U) is the covariance matrix of the desired channel at thej-th cell.The principle of the coordinated pilot assignment consists

in exploiting covariance information at all cells (a total ofKL2 covariance matrices) in order to minimize the sum MSEmetric. Hence, L users are assigned an identical pilot sequence


TABLE IBASIC SIMULATION PARAMETERS

Cell radius 1 kmCell edge SNR 20 dBNumber of users per-cell 10Distance from a user to its BS 800 mPath loss exponent 3Carrier frequency 2 GHzAntenna spacing /2Number of paths 50Pilot length 10

when the corresponding L2 covariance matrices exhibit themost orthogonal signal subspaces. Note that the MSE-basedcriterion (31) implicitly exploits the property of subspaceorthogonality, e.g., at high SNRs, the proposed MSE-based cri-terion will be minimized by choices of users with covariancematrices showing maximum signal subspace orthogonality,thereby implicitly satisfying the conditions behind Theorem1. In view of minimizing the search complexity, a classicalgreedy approach is proposed:1) Initialize U = 2) For l = 1, . . . , L do:Kl = argmin

kGF(U {k})

U U {Kl}EndThe coordination can be interpreted as follows: To minimize

the estimation error, a base station tends to assign a givenpilot to the user whose spatial feature has most differenceswith the interfering users assigned the same pilot. Clearly,the performance will improve with the number of users, as itbecomes more likely to find users with discriminable second-order statistics.

V. NUMERICAL RESULTS

In order to preserve fairness between users and avoid havinghigh-SNR users being systematically assigned the consideredpilot, we consider a symmetric multicell network where theusers are all distributed on the cell edge and have the samedistance from their base stations. In practice, users withgreater average SNR levels (but equal across cells) can beassigned together on a separate pilot pool. We adopt the modelof a cluster of synchronized and hexagonally shaped cells.Some basic simulation parameters are given in Table I. Wekeep these parameters in the following simulations, unlessotherwise stated.The channel vector between the u-th user in the l-th cell

and the target base station is

hlu =1P

Pp=1

a(lup)lup, (32)

where lup and lup are the AOA and the attenuation of thep-th path between the u-th user in the l-th cell and the targetbase station respectively. Note that the variance of lup, p is2lu, which includes the distance-based path loss lu betweenthe user and the target base station (which can be anyone ofthe L base stations):

lu =

dlu, (33)

where is a constant dependent on the prescribed averageSNR at cell edge. dlu is the geographical distance. is thepath-loss exponent.Two types of AOA distributions are considered here, a non-

bounded one (Gaussian) and a bounded one (uniform):1) Gaussian distribution: For the channel coefficients hlu,

the AOAs of all P paths are i.i.d. Gaussian random variableswith mean lu and standard deviation . Here we supposeall the desired channels and interference channels have thesame standard deviation of AOA. Note that Gaussian AOAdistributions cannot fulfill the conditions of non-overlappingAOA support domains in Theorem 1, nevertheless the useof the proposed method in this context also gives substantialgains as 2 decreases.2) Uniform distribution: For the channel hlu, the AOAs

are uniformly distributed over [lu , lu+ ], where luis the mean AOA.Two performance metrics are used to evaluate the proposed

channel estimation scheme. The first one is a normalizedchannel estimation error

err 10log10

L

j=1

hjj hjj2F

Lj=1

hjj2F

, (34)where hjj and hjj are the desired channel at the j-thbase station and its estimate respectively. Note that we onlyconsider the estimation error of the desired channel. Thesecond performance metric is the per-cell rate of the downlinkobtained assuming standard MRC beamformer based on thechannel estimates. The beamforming weight vector of the j-th base station is wMRCj = hjj . We define the per-cell rate asfollows:

C

Lj=1

log2(1 + SINRj)

L,

where SINRj is the received signal-to-noise-plus-interferenceratio (SINR) by the scheduled user in the j-th cell.Numerical results of the proposed channel estimation

scheme are now shown. In the figures, LS stands for conven-tional LS channel estimation. CB denotes the Covariance-aided Bayesian estimation (without coordinated pilot as-signment), and CPA is the proposed Coordinated PilotAssignment-based Bayesian estimation.We first validate Theorem 1 in Fig. 1 with a 2-cell network,

where the two users positions are fixed. AOAs of desiredchannels are uniformly distributed with a mean of 90 degrees,and the angle spreads of all channels are 20 degrees, yieldingno overlap between desired and interfering multipaths. The pi-lot contamination is quickly eliminated with growing numberof antennas.In Fig. 2 and Fig. 3, the estimation MSEs versus the

BS antenna numbers are illustrated. When the AOAs haveuniform distributions with = 10 degrees, as shown in Fig.2, the performance of CPA estimator improves quickly withM from 2 to 10. In the 2-cell network, CPA has the abilityof avoiding the overlap between AOAs for the desired andinterference channels. For comparison, Fig. 3 is obtained with


0 20 40 60 80 100 120 140 160 180 20040

35

30

25

20

15

10

Antenna Number

Est

imat

ion

Err

or [d

B]

Conventional LS EstimationCovarianceaided Bayesian (CB) Estimation LS Interference Free ScenarioCB Interference Free Scenario

Fig. 1. Estimation MSE vs. BS antenna number, 2-cell network, fixedpositions of two users, uniformly distributed AOAs with = 20 degrees,non-overlapping multipath.

0 5 10 15 20 25 30 35 40 45 5040

35

30

25

20

15

10

5

Antenna Number

Est

imat

ion

Err

or [d

B]

Conventional LS EstimationCovarianceaided Bayesian EstimationCoordinated Pilot Assignmentbased EstimationLS Interference Free ScenarioCB Interference Free Scenario

Fig. 2. Estimation MSE vs. antenna number, uniformly distributed AOAswith = 10 degrees, 2-cell network.

Gaussian AOA distribution. We can observe a gap remainsbetween the CPA and the interference-free one, due to thenon-boundedness of the Gaussian PDF. Nevertheless, the gainsover the classical estimator remain substantial.We then examine the impact of standard deviation of

Gaussian AOAs on the estimation. Fig. 4 shows that theestimation error is a monotonically increasing function of .In contrast, an angle spread tending toward zero will causethe channel direction to collapse into a deterministic quantity,yielding large gains for covariance-based channel estimation.Figs. 5 and 6 depict the downlink per-cell rate achieved

by the MRC beamforming strategy and suggest large gainswhen the Bayesian estimation is used in conjunction withthe proposed coordinated pilot assignment strategy and in-termediate gains when it is used alone. Obviously the rateperformance almost saturates with M in the classical LS case

0 5 10 15 20 25 30 35 40 45 5040

35

30

25

20

15

10

5

Antenna Number

Est

imat

ion

Err

or [d

B]

LS Estimation

CB Estimation

CPA Estimation

LS Interference Free

CB Interference Free

Fig. 3. Estimation MSE vs. antenna number, Gaussian distributed AOAswith = 10 degrees, 2-cell network.

0 10 20 30 40 50 60 70 80 9035

30

25

20

15

10

5

0

Standard Deviation of AOA [degree]

Est

imat

ion

Err

or [d

B]

Conventional LS EstimationCovarianceaided Bayesian EstimationCoordinated Pilot Assignmentbased Estimation

Fig. 4. Estimation MSE vs. standard deviation of Gaussian distributed AOAswith M = 10, 7-cell network.

(due to pilot contamination) while it increases quickly withM for the proposed estimators, indicating the full benefits ofmassive MIMO systems are exploited.

VI. DISCUSSIONS

In this paper, we assumed the individual covariance matricescan be estimated separately. This could be done in practiceby exploiting resource blocks where the desired user andinterference users are known to be assigned at different times.In future networks, one may imagine a specific trainingdesign for learning second-order statistics. Since covarianceinformation varies much slower than fast fading, such trainingmay not consume a substantial amount of resources.The proposed coordinated estimation method would intro-

duce information exchange between base stations. Although


0 5 10 15 20 25 30 35 40 45 504

6

8

10

12

14

16

18

Antenna Number

Perc

ell R

ate

[bits

/sec

/Hz]


Fig. 5. Per-cell rate vs. antenna number, 2-cell network, Gaussian distributedAOAs with = 10 degrees.

the second-order statistics vary much slower than the instan-taneous CSI, base stations still have to update the covarianceinformation every now and then so as to maintain performance.Clearly, the overhead depends on the degree of user mobility.

VII. CONCLUSIONS

This paper proposes a covariance-aided channel estimationframework in the context of interference-limited multi-cellmultiple antenna systems. We develop Bayesian estimators anddemonstrate analytically the efficiency of such an approach forlarge-scale antenna systems, leading to a complete removalof pilot contamination effects in the case covariance matricessatisfy a certain non-overlapping condition on their dominantsubspaces. We suggest a coordinated pilot assignment strategythat helps shape covariance matrices toward satisfying theneeded condition and show channel estimation performanceclose to interference-free scenarios.

ACKNOWLEDGMENT

Discussions with Dirk Slock and Laura Cottatellucci aregratefully acknowledged.

APPENDIX

A. Proof of Lemma 1:

Define the series

xi 1 + 2(i 1)M

, i = 1, . . . ,M,

and

i (xi)

M.

Then we have i A, i = 1, . . . ,M and

Hk i =1 ej2(ik)

M(1 e j2(ik)M )= 0, k = i.

0 10 20 30 40 50 60 70 80 902.5

3

3.5

4

4.5

5

5.5

6

6.5

Standard Deviation of AOA [degree]

Perc

ell R

ate

[bits

/sec

/Hz]


Fig. 6. Per-cell rate vs. standard deviation of AOA (Gaussian distribution)with M = 10, 7-cell network.

Thus {i |i = 1, . . . ,M } forms an orthogonal basis of A, andtherefore

dim{A} = M.

Define

B {i

i Z [M(b1+1)2 +1+ 1, M(b2 + 1)2 +11]}

,

where x and x are rounded-above and rounded-belowoperators respectively. Then B is part of an orthogonal basisof the space B, which indicates dim{B} |B|. By countingvectors in B, we have that

dim{B} M(b2 + 1)2

+ 1 M(b1 + 1)2

+ 1 1

= M(b2 + 1)2

M(b1 + 1)2

1. (35)

Now we define

C {i

i Z and i [1, M(b1 + 1)2 + 1]

[M(b2 + 1)

2+ 1,M

]}.

Then C is part of an orthogonal basis of A. Furthermore,

|C| = M(b1 + 1)2

+ 1+M M(b2 + 1)2

+ 1+ 1

= M M(b2 + 1)2

+ M(b1 + 1)2

+ 1.

Consider the equivalent form of B

B ={ b2

b1

f(x)(x)dx

|f(x)| < , x [b1, b2]}.


Taking any vector i C , we have

Hi

b2b1

f(x)(x)dx =1M

b2b1

f(x)(xi)H(x)dx

=1M

b2b1

f(x)1 ejM(xxi)1 ej(xxi) dx.

Since i C , we can observe xi / [b1, b2], thus

limM

Hi

b2b1

f(x)(x)dx = 0.

Therefore C B when M . Hence we havedim{B} = M dim{B} |C|

dim{B} M(b2 + 1)2

M(b1 + 1)2

1. (36)

Combining (35) and (36), we can easily obtain

dim{B} M(b2 + 1)2

M(b1 + 1)2

1

M(b2 b1)2

+ o(M),

and Lemma 1 is proved.

B. Proof of Lemma 2:

We define

b(x) a(cos1(x

D)

), x [D

,D

]. (37)

It is clear from Lemma 1 that b(x) = (2x). Hence, for anyinterval [xmin, xmax] in [ 12 , 12 ],

dim{span

{b(x), x [xmin, xmax]}}

(xmax xmin)M when M is large. (38)Additionally, for i = 1, . . . , L, we have

span {Ri} = span{

0

a()a()Hpi()d

},

Thus, due to the bounded support of pi(), we can obtain

span {Ri}= span{ maxi

mini

a()a()Hpi()d

}

= span

{ maximini

b(D

cos())bH(

D

cos())pi()d

}.

Then, by interpreting the integral as a (continuous) sum, wehave

span {Ri}span{b(x), x [D

cos(maxi ),

D

cos(mini )]

}.

From (38), we obtain

rank(Ri) (cos(mini ) cos(maxi )

) DM,

for large M , and Lemma 2 is proved.

C. Proof of Lemma 3:

Take an angle / [mini , maxi ] and define

u a()M

.

Then we have

uHRiu =1

Ma()HRia()

=1

MaH()E

{a()aH ()

}a()

=1

ME

{aH()a()2}=

1

ME

M1m=0

e2j(m1)D (cos()cos())

2

=

maximini

1MM1m=0


2

pi()d.

According to the well-known result on the sum of geometricseries, we can easily obtain

limM

1MM1m=0


2

= 0,

since = , [mini , maxi ]. Thuslim

MuHRiu = 0,

which proves Lemma 3.

REFERENCES

[1] T. L. Marzetta, Noncooperative cellular wireless with unlimited num-bers of base station antennas, IEEE Trans. Wireless Commun., vol. 9,no. 11, pp. 35903600, Nov. 2010.

[2] J. Jose, A. Ashikhmin, T. L. Marzetta, and S. Vishwanath, Pilot contam-ination problem in multi-cell TDD systems, in Proc. IEEE InternationalSymposium on Information Theory (ISIT09), Seoul, Korea, Jun. 2009,pp. 21842188.

[3] J. Hoydis, S. ten Brink, and M. Debbah, Massive MIMO: How manyantennas do we need? in Proc. 2011 49th Annual Allerton Conferenceon Communication, Control, and Computing (Allerton), Sep. 2011, pp.545 550.

[4] J. Jose, A. Ashikhmin, T. L. Marzetta, and S. Vishwanath, Pilotcontamination and precoding in multi-cell TDD systems, IEEE Trans.Wireless Commun., vol. 10, no. 8, pp. 26402651, Aug. 2011.

[5] R. Knopp and P. A. Humblet, Information capacity and power controlin single-cell multiuser communications, in Proc. IEEE InternationalConference on Communications, vol. 1, Seattle, WA, USA, Jun. 1995,pp. 331335.

[6] P. Viswanath, D. N. C. Tse, and R. Laroia, Opportunistic beamformingusing dumb antennas, IEEE Trans. Inf. Theory, vol. 48, no. 6, pp. 12771294, Jun. 2002.

[7] J. Jang and K. B. Lee, Transmit power adaptation for multiuser OFDMsystems, IEEE J. Sel. Areas Commun., vol. 21, no. 2, pp. 171178, Feb.2003.

[8] D. Gesbert, M. Kountouris, R. Heath, C. Chae, and T. Salzer, Shiftingthe MIMO paradigm, IEEE Signal Process. Mag., vol. 24, no. 5, pp.3646, Sep. 2007.

[9] R. B. Ertel, P. Cardieri, K. W. Sowerby, T. S. Rappaport, and J. H. Reed,Overview of spatial channel models for antenna array communicationsystems, IEEE Pers. Commun., vol. 5, no. 1, pp. 10 22, Feb. 1998.

[10] A. J. Paulraj, R. Nabar, and D. Gore, Introduction to space-time wirelesscommunications. Cambridge University Press, 2003.

[11] H. Q. Ngo, T. L. Marzetta, and E. G. Larsson, Analysis of the pilotcontamination effect in very large multicell multiuser MIMO systems forphysical channel models, in Proc. IEEE International Conference onAcoustics, Speech and Signal Processing (ICASSP11), Prague, CzechRepublic, May 2011, pp. 34643467.


[12] B. Gopalakrishnan and N. Jindal, An analysis of pilot contaminationon multi-user MIMO cellular systems with many antennas, in 2011IEEE 12th International Workshop on Signal Processing Advances inWireless Communications (SPAWC), Jun. 2011, pp. 381385.

[13] A. Scherb and K. Kammeyer, Bayesian channel estimation for doublycorrelated MIMO systems, in Proc. IEEE Workshop Smart Antennas,2007.

[14] E. Bjornson and B. Ottersten, A framework for training-based esti-mation in arbitrarily correlated Rician MIMO channels with Riciandisturbance, IEEE Trans. Signal Process., vol. 58, no. 3, pp. 18071820, Mar. 2010.

[15] R. M. Gray and L. D. Davisson, An introduction to statistical signalprocessing. Cambridge University Press, 2004.

[16] S. M. Kay, Fundamentals of Statistical Signal Processing: EstimationTheory. Englewood Cliffs, NJ: Prentice Hall, 1993.

[17] J. A. Tsai, R. M. Buehrer, and B. D. Woerner, The impact of AOAenergy distribution on the spatial fading correlation of linear antennaarray, in Proc. IEEE Vehicular Technology Conference, (VTC 02),vol. 2, May 2002, pp. 933937.

[18] J. Nam, J. Ahn, and G. Caire, Joint spatial division and multiplexing:Realizing massive MIMO gains with limited channel state information,in 46th Annual Conference on Information Sciences and Systems, (CISS2012), Princeton University, NJ, USA, Mar. 2012.

Haifan Yin received the B.Sc. degree in Electricaland Electronic Engineering and the M.Sc. degreein Electronics and Information Engineering fromHuazhong University of Science and Technology,Wuhan, China, in 2009 and 2012 respectively. From2009 to 2011, he had been with Wuhan NationalLaboratory for Optoelectronics, China, working onthe implementation of TD-LTE systems as an R&Dengineer. In September 2012, he joined the Mo-bile Communications Department at EURECOM,France, where he is now a Ph.D. student. His current

research interests include channel estimation, multi-cell cooperative networks,and multiuser information theory.

David Gesbert (IEEE Fellow) is Professor andHead of the Mobile Communications Department,EURECOM, France. He obtained the Ph.D degreefrom Ecole Nationale Superieure des Telecommu-nications, France, in 1997. From 1997 to 1999 hehas been with the Information Systems Laboratory,Stanford University. In 1999, he was a foundingengineer of Iospan Wireless Inc, San Jose, Ca.,astartup company pioneering MIMO-OFDM (nowIntel). Between 2001 and 2003 he has been with theDepartment of Informatics, University of Oslo as an

adjunct professor. D. Gesbert has published about 200 papers and severalpatents all in the area of signal processing, communications, and wirelessnetworks.D. Gesbert was a co-editor of several special issues on wireless networks

and communications theory, for JSAC (2003, 2007, 2009), EURASIP Journalon Applied Signal Processing (2004, 2007), Wireless Communications Mag-azine (2006). He served on the IEEE Signal Processing for CommunicationsTechnical Committee, 2003-2008. He authored or co-authored papers winningthe 2004 IEEE Best Tutorial Paper Award (Communications Society) fora 2003 JSAC paper on MIMO systems, 2005 Best Paper (Young Author)Award for Signal Proc. Society journals, and the Best Paper Award for the2004 ACM MSWiM workshop. He co-authored the book Space time wirelesscommunications: From parameter estimation to MIMO systems, CambridgePress, 2006.

Miltiades Filippou was born in Athens, Greecein 1984. He received his Dipl. Eng. Degree inElectrical and Computer Engineering from the Na-tional Technical University of Athens in 2007. From2007 until 2009 he was occupied at the WirelessCommunications laboratory of the same school asa postgraduate researcher in the area of satellitecommunications. In May 2011, he joined the MobileCommunications Department at EURECOM wherehe is currently conducting his Ph.D. studies underthe supervision of Prof. David Gesbert. His current

research interests include: MIMO communications, cognitive radio systems,channel estimation, cooperative communications and digital signal processing.

Yingzhuang Liu is a professor of Huazhong Uni-versity of Sci.&Tech. (HUST). His main researchfield is broadband wireless communication, includ-ing LTE and IMT Advanced system, etc., especiallyits Radio Resource Management. From 2000 to2001, he was a postdoctoral researcher in ParisUniversity XI. From 2003 up to now, he has presidedover 10 national key projects, published more than80 papers and held more than 30 patents in the fieldof broadband wireless communication. He is nowthe group leader of broadband wireless research of

HUST, which has more than 10 young teachers and more than 20 PhDstudents.

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A Coordinated Approach to Channel Estimation in Large-Scale Multiple-Antenna Systems