3704 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009

On the Multicell Processing Capacity of theCellular MIMO Uplink Channel in

Correlated Rayleigh Fading EnvironmentSymeon Chatzinotas, Student Member, IEEE, Muhammad Ali Imran, Member, IEEE,

and Reza Hoshyar, Member, IEEE

Abstract—In the context of cellular systems, it has beenshown that multicell processing can eliminate inter-cell inter-ference and provide high spectral efficiencies with respect totraditional interference-limited implementations. Moreover, ithas been proved that the multiplexing sum-rate capacity gainof multicell processing systems is proportional to the numberof Base Station (BS) antennas. These results have been alsoestablished for cellular systems, where BSs and User Terminals(UTs) are equipped with multiple antennas. Nevertheless, acommon simplifying assumption in the literature is the uncorre-lated nature of the Rayleigh fading coefficients within the BS-UT MIMO links. In this direction, this paper investigates theergodic multicell-processing sum-rate capacity of the GaussianMIMO Cellular Multiple-Access Channel in a correlated fadingenvironment. More specifically, the multiple antennas of bothBSs and UTs are assumed to be correlated according to theKronecker product model. Furthermore, the current systemmodel considers Rayleigh fading, uniformly distributed UTs overa planar coverage area and power-law path loss. Based on freeprobabilistic arguments, the empirical eigenvalue distributionof the channel covariance matrix is derived and it is used tocalculate both Optimal Joint Decoding and Minimum MeanSquare Error (MMSE) Filtering capacity. In addition, numericalresults are presented, where the per-cell sum-rate capacity isevaluated while varying the cell density of the system, as wellas the level of fading correlation. In this context, it is shownthat the capacity performance is greatly compromised by BS-sidecorrelation, whereas UT-side correlation has a negligible effecton the system’s performance. Furthermore, MMSE performanceis shown to be greatly suboptimal but more resilient to fadingcorrelation in comparison to optimal decoding.

Index Terms—Information theory, Information rates, mul-tiuser channels, MIMO systems, channel correlation, land mobileradio cellular systems, eigenvalues and eigenfunctions.

I. INTRODUCTION

IN the short history of wireless cellular systems, there hasbeen an intense evolutionary process trying to optimize the

multiple-access and coding schemes in order to provide thedesired quality of service. In spite of the constant improve-ment, one characteristic of cellular communication remained,namely its interference-limited nature. Considering the factthat the current cellular architectures are approaching their

Manuscript received July 11, 2008; revised November 7, 2008 and January19, 2009; accepted April 18, 2009. The associate editor coordinating thereview of this paper and approving it for publication was H. Dai.

The authors are with the Centre for Communication Systems Research,University of Surrey, United Kingdom, GU2 7XH (e-mail: {S.Chatzinotas,M.Imran, R.Hoshyar}@surrey.ac.uk).

Digital Object Identifier 10.1109/TWC.2009.080922

limit, the interest of both research and industry turned tocooperative techniques, such as BS cooperation, relaying andUT conferencing. In this paper, we focus on cooperatingBSs which are interconnected through ideal links to a centralprocessor, which has perfect Channel State Information (CSI).As a result, the received signals from UTs in multiple cells canbe jointly processed (multicell processing). In the context ofthis paper, the multicell processing can be either optimal jointdecoding or MMSE joint filtering, followed by single-userdecoding. The capacity enhancement due to BS cooperationhas been extensively studied and has been shown to growlinearly with the number of BS receive antennas [1], [2].This result also applies to the case where BSs and/or UTsare equipped with multiple antennas [3], [4], [5]. However,the majority of related results have been produced based onthe simplifying assumption that the fading coefficients of theMIMO subchannels are completely uncorrelated. In reality,this is not the case, since fading correlation may appear dueto inadequate antenna separation and/or poor local scattering[6]. In a typical macrocellular scenario, the inadequate antennaseparation mainly affects the UTs, as the components of theantenna array may be separated by a distance less than half ofthe communication wavelength due to their size limitations.On the other hand, poor local scattering affects mainly theBSs, as the number of local scatterers is insufficient due totheir elevated position. On these grounds, this paper studiesthe effect of MIMO fading correlation on the capacity perfor-mance of a multicell processing system.

In this direction, it has been shown that the correlatedchannel matrix of the point-to-point MIMO channel can beexpressed in terms of the separable variance profile, whichdepends on the eigenvalues of the correlation matrices [7],[8]. In parallel, the channel matrix of a cellular Multiple-Access (MAC) channel can be expressed in terms of thepath-loss variance profile, which depends on the consideredUT distribution, cell size and path loss exponent [2]. Themain objective of this study is to determine the eigenvaluedistribution of the channel covariance matrix, which deter-mines the optimal and the MMSE sum-rate capacity. For thecase of point-to-point correlated MIMO channel, the objectivehas been accomplished by exploiting the separability of thevariance profile [7], [8]. Similarly, for the case of the cellularMAC channel the asymptotic eigenvalue distribution wasdetermined by exploiting the row-regularity of the variance

1536-1276/09$25.00 c© 2009 IEEE

CHATZINOTAS et al.: ON THE MULTICELL PROCESSING CAPACITY OF THE CELLULAR MIMO UPLINK CHANNEL . . . 3705

profile respectively [2]. Nevertheless, the channel matrix ofa correlated cellular MAC channel – expressed as Hadamardproduct of a separable and a row-regular variance profile – isneither separable nor row-regular and hence a new approachis needed. In this context, the main contributions of this papercan be summarized as follows:

1) A cellular MIMO uplink channel model is introduced,accommodating distributed UTs, a continuous path-lossmodel and Kronecker-correlated antennas.

2) Based on a recent Random Matrix Theory result, thesum-rate capacity calculation problem is transformedto a non-linear programming problem, which can beutilized to efficiently calculate the optimal capacity forfinite cellular systems.

3) Furthermore, the asymptotic eigenvalue distributionof this channel model is analyzed based on free-probabilistic arguments and closed-forms are derivedfor the per-cell sum-rate capacity of the optimal jointdecoder and the MMSE decoder.

4) Based on the derived closed-forms, it is shown thatantenna correlation at the UT-side has no effect on theperformance, while antenna correlation at the BS-sidecompromises the multiplexing gain of the system.

5) For a set of practical parameters, the agreement ofanalytical closed-forms and Monte Carlo simulations isestablished and the effect of BS-side antenna correlationis evaluated.

The remainder of this paper is structured as follows: SectionII provides a detailed review of the MIMO correlation andmulticell processing uplink channel models. Section III definesthe considered channel model and describes the derivationof the optimal and MMSE capacity closed-forms. Section Vverifies the accuracy of the analysis by comparing with MonteCarlo simulations and presents the practical results obtainedfor a typical macrocellular scenario. Section VI concludes thepaper.

A. Notation

Throughout the formulations of this paper, R is the cellradius, N is the number of BSs, K is the number of UTs percell and η is the power-law path loss exponent. Additionally,nBS and nUT are the number of multiple antennas at eachBS and each UT respectively. E[·] denotes the expectation,(·)∗ denotes the complex conjugate, (·)† denotes the conjugatetranspose matrix, � denotes the Hadamard product, ⊗ denotesthe Kronecker product and � denotes asymptotic equivalenceof the eigenvalue distributions. The norm of a complex scalaris denoted by |·| , whereas the Frobenius norm of a matrix orvector is denoted by ‖·‖. The inequality A � B, where A,Bare positive semidefinite matrices, denotes that A−B is alsopositive semidefinite.

II. RELATED WORK & PRELIMINARIES

A. Correlated MIMO Channel Models

Focusing on a point-to-point MIMO link, the channel matrixcan be expressed in general as [9]:

H = R1/2R GRR1/2

H GTR1/2T , (1)

where GR and GT are Gaussian matrices, whereas RR,RH and RT are deterministic or slow-varying matrices. Thematrices RR and RT , also known as the receive and transmitcorrelation matrix, depend on the angle spread, the antennabeamwidth and the antenna spacing at the receive and thetransmit end respectively. The matrix RH introduces thenotion of the keyhole or pinhole channel, which appears whenRH is a low-rank matrix. In cases where there is adequatescattering to prevent the keyhole effects (i.e. RH is full-rank),the channel matrix can be written as:

H = R1/2R GR1/2

T , (2)

where G is a Gaussian matrix. This channel matrix representsthe Kronecker correlation model [10], since the covariance ofthe vectorized channel matrix can be written as the Kroneckerproduct of the receive and transmit correlation matrix, namely:

cov (vec (H)) = RR ⊗ RT (3)

or equivalently

E

[(H)pq (H)∗rs

]= (RR)pr (RT )qs , (4)

where (X)ij is the (i, j)th element of matrix X. Accordingto the Kronecker correlation model, the correlation betweentwo subchannels equals to the product of the correspondingtransmit and receive correlation (c.f. Equation (4)). From aphysical point-of-view, the Kronecker model appears when theantennas are arranged in regular arrays and the correlationvanishes fast with distance [7]. In this point, it is worthmentioning that according to [11], [12] a MIMO channelwith a large number of keyholes converges to the KroneckerMIMO model. An interesting property of the Kronecker modelis its equivalency to the separable correlation model [7],[8], while studying the eigenvalue distribution of the channelcovariance matrix HH†. More specifically, if RR = UDRU†

and RT = VDT V† are the eigenvalue decompositions ofthe receive and transmit correlation matrices respectively, then-based on the isotropic behavior of Gaussian matrices- theeigenvalue distribution of HH† = R1/2

R GRTG†R1/2R is

equivalent to the one of D1/2R GDTG†D1/2

R . In this direction,the equivalent MIMO channel matrix can be written as:

H � D1/2R GD1/2

T . (5)

This equivalency is going to be very useful in the derivationsof Section III.

Let us now focus on the structure of the correlation matrix.A common model often used to effectively quantify the levelof spatial correlation is the exponential correlation model [13],[14], [15] . More specifically, according to the exponentialmodel, the receive/transmit correlation matrix can be con-structed utilizing a single coefficient ρe ∈ C with |ρe| ≤ 1as follows:

Rij =

⎧⎨⎩

(ρe)abs(j−i)

, i ≤ j((ρe)

abs(j−i))∗

, i > j(6)

where abs(·) denotes the absolute value. It has been shownthat the exponential model can approximate the correlation ina uniform linear array under rich scattering conditions [16].

3706 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009

Similar correlation models, such as the square exponential andthe tridiagonal model can be found in [17].

B. Point-to-point MIMO channel capacity

The already existing approaches for the point-to-pointMIMO channel can be classified in two main categories:exact analysis and asymptotic analysis. In the exact analysis,the probability distributions of finite-dimension matrices areinvestigated, resulting in closed forms which can produceexact results. On the other hand, in the asymptotic analysisa single or both dimensions of the random channel matrixgrow infinitely large in order to allow approximations andsimplifications due to the law of large numbers. Althoughthe asymptotic analysis may seem less accurate, it has beenwidely shown that asymptotic closed forms are able to produceaccurate results even for finite dimensions [18]. What is more,the asymptotic analysis is ideal for studying cases wherethe system size is of no importance, since it reveals theeffect of normalized parameters and provides insights intothe system’s performance [19]. In the category of asymptoticanalysis, the majority of the approaches consider the genericsetting where correlation affects both transmit and receiveend and the numbers of both transmit and receive antennasgrow large together while preserving a fixed ratio. Althoughthe asymptotic eigenvalue distribution analysis comprises anapproximation for matrices of finite dimensions, it is oftenemployed in order to isolate the effect of specific physicalparameters and to produce analytical closed forms. This settingis particularly suitable for studying the uplink channel of mul-ticell processing cellular systems, since the ratio of transmitand receive antennas is a constant proportional to the per-cellnumber of UTs K .

The performance of multi-antenna channels was originallyinvestigated in [20], [21] and it was shown that the capac-ity grows linearly with min (nr, nt), where nr and nt arethe number of receive and transmit antennas respectively.However, the correlated fading amongst the multiple anten-nas compromises the capacity performance with respect tothe independent fading case. This phenomenon is widelyestablished in various regimes and settings; the capacity ofthe Kronecker correlated (a.k.a. doubly correlated) MIMOchannel is expressed as a fixed-point equation based on theSteltjes’ transform [7] of the limiting eigenvalue distributionof HH†. In the same direction, authors in [22] study thecapacity of the Kronecker correlated MIMO channel based onthe principles of Random Matrix Theory [18]. The derivationresults in a fixed-point equation including functionals of theSINR and MMSE. In [23] and [8], the expectation and thevariance of the capacity are evaluated using closed formsbased on the solution of 2 × 2 equation systems. In [24], theprinciples of majorization theory [25] are applied in order toshow that the average mutual information is a Schur-concavefunction with respect to the ordered eigenvalue vector of thecorrelation matrix. In addition, the doubly correlated MIMOchannel for Toeplitz correlation matrices is analyzed in [17]based on the concept of linear spectral statistics. Finally, in[26], [27] the performance of Kronecker correlated MIMOchannels is studied using the replica method, which originatesin theoretical physics.

It should be noted that the aforementioned results specif-ically focus on the point-to-point correlated MIMO channel.In the following paragraph, we describe the channel character-istics of a multiple-access channel which is the information-theoretic basis of the cellular uplink channel.

C. Cellular uplink models

This section focuses on the evolution of channel modellingin the area of BS cooperation. The description starts withsingle-antenna cellular systems and concludes with the ex-tension of the channel model for multiple-antennas at bothtransmit and receive ends. The Gaussian Cellular Multiple-Access Channel (GCMAC) has been the starting point forstudying the Shannon-theoretic limits of cellular systems. Itall began with Wyner’s model [28], which assumes that allthe UTs in the cell of interest have equal channel gains, whichare normalized to 1. It considers interference only from theUTs of the two neighboring cells, which are all assumed tohave a fixed channel gain, also known as interference factorα, which ranges in [0, 1]. Assuming that there is a power-lawpath loss model which affects the channel gain, Wyner hasmodeled the case where the UTs of each cell are collocatedwith the cell’s BS, since no distance-dependent degradation ofthe channel gain is considered. The same assumption is madeby Somekh-Shamai [1], which have extended Wyner’s modelfor flat fading environment. In both [28] and [1], a singleinterference factor α is utilized to model both the cell densityand the path loss. The interference factor α ranges in [0, 1] ,where α = 0 represents the case of perfect isolation amongthe cells and α = 1 represents the case of BSs’ collocation,namely a MIMO MAC channel. Subsequently, the models in[29], [4] were presented, which differ from the aforementionedmodels in the sense that they consider interference from allthe cells of the system (i.e. multiple-tier interference). In [4],the multiple-tier interference model is combined with multipleantennas and the asymptotic performance of optimal and groupMMSE decoders is derived for orthogonal intra-cell UTs. In[29], an interference coefficient is defined for each BS-UTlink based on the power-law path loss model. Although theauthor in [29] takes into account a more realistic structureof the path loss effect, the UTs of each cell have still equalchannel gain and this refers to the case where the UTs ofeach cell are collocated with the cell’s BS. Nevertheless, thismodel is more detailed than the previously described models,since it decomposes the interference factor α, so that the celldensity/radius and the path loss exponent can be modelled andstudied separately. Finally, the model used in [30] extends theprevious models by considering that the UTs are no longercollocated, but they can be (uniformly) distributed across thecell’s coverage area. In this point, it should be noted thatfor all the aforementioned Gaussian multiple-access channelmodels the optimal capacity-achieving transmission strategyis superposition coding over the available bandwidth [31],[1]. In other words, the ensemble of system UTs transmitssimultaneously over the same bandwidth.

In the latter model [30], by assuming power-law path loss,flat fading and uniformly distributed UTs, the received signal

CHATZINOTAS et al.: ON THE MULTICELL PROCESSING CAPACITY OF THE CELLULAR MIMO UPLINK CHANNEL . . . 3707

at cell n, at time index i, is given by:

yn[i] =N∑

m=1

K∑k=1

ςnmk gnm

k [i]xmk [i] + zn[i], (7)

where xmk [i] is the ith complex channel symbol transmitted by

the kth UT of the mth cell and {gnmk } are independent, strictly

stationary and ergodic complex random processes in the timeindex i, which represent the flat fading processes experiencedin the transmission path between the nth BS and the kth UT inthe mth cell. The fading coefficients are assumed to have unitpower, i.e. E[|gnm

k [i]|2] = 1 for all (n, m, k) and all UTs aresubject to an average power constraint, i.e. E[|xm

k [i]|2] ≤ Pfor all (m, k). The interference factors ςnm

k in the transmissionpath between the mth BS and the kth UT in the nth cell arecalculated according to the “modified” power-law path lossmodel [29], [32]:

ςnmk =

(1 + dnm

k

)−η/2. (8)

Dropping the time index i, the aforementioned model can bemore compactly expressed as a vector memoryless channel ofthe form:

y = Hx + z. (9)

The channel matrix H can be written as,

H = Σ � G, (10)

where Σ is a N × KN deterministic matrix and G is aGaussian N ×KN matrix with complex circularly symmetric(c.c.s.) independent identically distributed (i.i.d.) elements ofunit variance, comprising the corresponding Rayleigh fadingcoefficients. The entries of the Σ matrix are defined by thevariance profile function

ς(u, v

)=(1 + d (u, v)

)−η/2, (11)

where u ∈ [0, 1] and v ∈ [0, K] are the normalized indicesfor the BSs and the UTs respectively and d (u, v) is thenormalized distance between BS u and user v. In the case ofmultiple UT and/or BS antennas (nUT and nBS respectively),the channel matrix H can be written as,

H = ΣM � GM , (12)

where GM is a standard complex Gaussian NnBS×KNnUT

matrix with elements of unit variance, comprising the Rayleighfading coefficients between the KNnUT transmit and theNnBS receive antennas. Similarly, ΣM is a NnBS×KNnUT

deterministic matrix, comprising the path loss coefficients be-tween the KNnUT transmit and the NnBS receive antennas.Since the multiple antennas of each UT / BS are collocated,ΣM can be written as a block matrix based on the varianceprofile matrix Σ of Equation (10)

ΣM = Σ⊗ J, (13)

where J is a nBS × nUT matrix of ones.

III. CHANNEL MODEL & ASSUMPTIONS

Let us assume that K UTs are uniformly distributed in eachcell of a planar cellular system (Fig. 1) comprising N basestations and that each BS and each UT are equipped with nBS

and nUT antennas respectively.

Fig. 1. Ground plan of the cellular system comprising of BSs with multipleantennas and UTs distributed on a uniform hexagonal grid.

Under conditions of correlated flat fading, the receivedsignal at cell n, at time index i, is given by:

yn[i] =N∑

m=1

K∑k=1

ςnmk (RR

nmk )

12 Gnm

k [i] (RTnmk )

12 xm

k [i]

+ zn[i], (14)

where xmk [i] is the ith complex channel symbol vector nUT ×1

transmitted by the kth UT of the mth cell and {Gnmk } is a

nBS×nUT random matrix with independent, strictly stationaryand ergodic complex random elements in the time index i.According to the Kronecker correlation model, RT

nmk and

RRnmk are deterministic transmit and receive correlation ma-

trices of dimensions nUT ×nUT and nBS ×nBS respectively.In this context, the following normalizations are consideredin order to ensure that the correlation matrices do not affectthe path loss gain of the BS-UT links: tr (RT

nmk ) = nUT

and tr (RRnmk ) = nBS for all (n, m, k). The matrix product

(RRnmk )

12 Gnm

k [i] (RTnmk )

12 represents the multiple-antenna

correlated flat fading processes experienced in the transmissionpath between the nBS receive antennas of the nth BS andthe nUT transmit antennas of the kth UT in the mth cell.The fading coefficients are assumed to have unit power, i.e.Ei[Gnm

k [i]Gnmk [i]†] = I for all (n, m, k) and all UTs are sub-

ject to a power constraint P , i.e. Ei[xmk [i]xm

k [i]†] PnUT

InUT

for all (m, k). The vector zn[i] represents the AWGN noise atthe receiver with E[zn[i]] = 0, E[zn[i]zn[i]†] = σ2I. To sim-plify notations, the parameter γ = P/σ2 is defined as the UTtransmit power normalized by the receiver noise power. Thevariance coefficients ςnm

k in the transmission path betweenthe mth BS and the kth UT in the nth cell are calculatedaccording to the “modified” power-law path loss model (cf.(8)). Dropping the time index i, the aforementioned model canbe more compactly expressed as a vector memoryless channel

3708 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009

of the formY = HX + Z, (15)

where Y = [y(1)... y(N)]T with y(n) = [y1... ynBS ] repre-senting the received signal vector by the nBS antennas of thenth BS, X = [x(1)

(1) . . .x(1)(K)x

(2)(1) . . . . . .x(N−1)

(K) x(N)(1) . . .x(N)

(K)]T

with x(n)(k) = [x1... xnUT ] representing the transmit signal

vector by the nUT antennas of the kth UT in the nth cell andZ=[z(1)... z(N)]T with z(n) = [z1... znBS ] being i.i.d c.c.s.random variables representing AWGN. In order to simplifythe notations, it is assumed that all BSs/UTs are characterizedby identical receive RR and transmit RT correlation matrices.However, it should be noted that the following analysis can bestraightforwardly generalized to encompass the more realisticcase of different correlation matrices for each BS/UT. Thechannel matrix H can be written as

H = ΣM �((

IN ⊗ RR12

)GM

(IKN ⊗ RT

12

)), (16)

where GM is a NnBS × KNnUT Gaussian matrix withi.i.d. c.s.s. elements of unit variance. As explained before,the Kronecker correlation model is equivalent to a separablevariance profile model in terms of its eigenvalue distribution.Based on this equivalence, the channel matrix can be rewrittenas follows:

H = ΣM �((

IN ⊗ RR12

)GM

(IKN ⊗ RT

12

))� ΣM �

(D

12RGMD

12T

)= ΣM �

(d†

RdT

) 12 � GM (17)

where DR and DT are the diagonal eigenvalue matrices ofIN×N ⊗ RR and IN×N ⊗ RT respectively and dR and dT

are row vectors containing the diagonal elements of DR andDT respectively. As it can be seen, the MIMO correlationmodel has been transformed into an uncorrelated model witha variance profile Ω = ΣM� (d†

RdT)12 , which is neither row

regular nor separable.

IV. EIGENVALUE DISTRIBUTION ANALYSIS & CAPACITY

RESULTS

A. A Random Matrix Theory approach

On the basis of a recent result in Random Matrix Theory[33, Theorem 2.4 and Theorem 4.1] the optimal per-cell sum-rate capacity of the derived channel model is given by:

Copt(γ, N, nBS , K, nUT ) =1N

(log det

(γ

nUTT−1

)

+ log det(

γ

nUTT−1

)− 1

KNγ

∥∥∥Ω � (tT t

) 12∥∥∥2 )

(18)

where T and T are given as the solution of the followingNnBS + KNnUT equations:

ti =γ

1 + 1KNnUT

tr(ΩiT

) for i = 1 . . .NnBS (19)

tj =γ

1 + 1KNnUT

tr (ΩjT)for j = 1 . . .KNnUT (20)

with the unknown variables

T = diag (t) and t = [t1 . . . tNnBS ]

T = diag(t)

and t = [t1 . . . tKNnUT ]

and

Ωj = diag (ωj)2

where ωj = [ω1j . . . ωNnBSj ] is the jth column of Ω

Ωi = diag (ωi)2

where ωi = [ωi1 . . . ωiKNnUT ] is the ith row of Ω

This result simplifies the capacity computation in large sys-tems by converting the original problem to a non-linearprogramming problem. Hence, this approach can be utilizedto efficiently calculate the optimal capacity for finite cellularsystems. However, the size of the problem i.e. the number ofequations still depends on the size of the system N and thusthis solution cannot provide asymptotic results.

B. A Free Probability Approach

This section describes a free probability approach which canbe utilized to derive a closed form for the probability densityfunction of the asymptotic eigenvalue distribution. Firstly,the uncorrelated model is studied, followed by the transmitand receive single-side correlation model. Subsequently, theproduced results for the single-side case are utilized to deducethe solution for the double-side case. In this point, it shouldbe noted that free probability theory was established byVoiculescu [34] and it has been also used in [14], [15] to in-vestigate the case of point-to-point MIMO channels correlatedon a single side according to the exponential model.

1) Uncorrelated Point-to-point Channel: In this case, thereis no variance profile or equivalently the variance profile ismatrix of ones. Therefore, considering a Gaussian channelmatrix G ∼ CN (0, I), the empirical eigenvalue distributionof 1

N G†G converges almost surely (a.s.) to the non-randomlimiting eigenvalue distribution of the Marcenko-Pastur law[35], whose Shannon transform is given by

V 1N G†G(y) a.s.−→ VMP(y, β) (21)

where VMP (y, β) = log(

1 + y − 14φ (y, β)

)

+1β

log(

1 + yβ − 14φ (y, β)

)− 1

4βyφ (y, β)

φ (y, β) =

(√y(1 +

√β)2

+ 1 −√

y(1 −

√β)2

+ 1

)2

and η-transform is given by [36, p. 303]

ηMP (y, β) = 1 − φ (y, β)4βy

(22)

where β is the ratio of the horizontal to the vertical dimensionof the G matrix. The transforms of the Marcenko-Pastur laware going to be useful in the capacity derivations of theuncorrelated case.

CHATZINOTAS et al.: ON THE MULTICELL PROCESSING CAPACITY OF THE CELLULAR MIMO UPLINK CHANNEL . . . 3709

2) Uncorrelated Cellular Channel: In this case, there is arow-regular path-loss variance profile and thus the channelmatrix is written as H = ΣM � GM. For the sake ofcompleteness, we include the derivation of the asymptoticeigenvalue distribution of 1

N HH† based on the analysis in[29]. In this direction, 1

N H†H can be written as the sum ofKNnUT × KNnUT unit rank matrices, i.e.

1N

H†H =NnBS∑

i=1

h†ihi, (23)

where hi ∼ CN (0,Vi) denotes the ith 1 × KNnUT rowvector of 1√

NH, since the term 1

N has been incorporated inthe unit rank matrices. The covariance matrix equals Vi =1N (diag(σi))

2, where diag(σi) stands for a diagonal matrix

with the elements of vector σi across the diagonal with σi

being the ith row of ΣM. The unit-rank matrices Wi = h†ihi

constitute complex singular Wishart matrices with one degreeof freedom and their density according to [37, Theorem 3-4]is

fVi(Wi) = B−1Vi

det (Wi)1−KnUT N

e−tr(V−1i

Wi)

BVi = πKnUT N−1det (Vi) . (24)

If h†i = QiSi is a singular value decomposition, then the

density can be written as

fVi(Wi) = B−1Vi

det(SiS

†i

)1−KnUT N

e−tr(V−1i

QiSiS†iQ†

i ).(25)

It can be easily seen that if Vi = I, the matrices would beunitarily invariant [38, Definition 17.7] and therefore asymp-totically free [39]. Although in our case Vi = 1

N (diag(σi))2,

we assume that the asymptotic freeness still holds. Similar ap-proximations have been already investigated in an information-theoretic context, providing useful analytical insights andaccurate numerical results [40], [41]. In this context, the R-transform of each unit rank matrix [18, Example 2.28] is givenby

Rhi†hi

(w) =1

KnUT N

‖hi‖2

1 − w ‖hi‖2 (26)

and the asymptotic R-transform of H†H is equal to the sumof the R-transforms of all the unit rank matrices [18, Theorem2.64]

limN→∞

R 1N H†H(w) � lim

N→∞

NnBS∑i=1

Rhi†hi

(w)

= limN→∞

1KnUT N

NnBS∑i=1

‖hi‖2

1 − w ‖hi‖2

(27)

Since the variance profile function of Equation (11) definesrectangular block-circulant matrix with 1 × K blocks whichis symmetric about v = Ku, the channel matrix H isasymptotically row-regular [18, Definition 2.10] and thus theasymptotic norm of hi converges to a deterministic constantfor every BS, i.e ∀i

limN→∞

‖hi‖2 = limN→∞

1N

KNnUT∑j=1

ς2ij =

∫ KnUT

0

ς2(u, v)dv

(28)

where ςij is the (i, j)th element of the ΣM matrix. In addition,based on the row-regularity it can be seen that ∀v

nBS

∫ KnUT

0

ς2(u, v)dv =∫ nBS

0

∫ KnUT

0

ς2(u, v)dudv.

(29)Therefore, Equation (27) can be simplified to [18, Theorem2.31, Example 2.26]

limN→∞

R 1N H†H(w) (30)

� 1KnUT

∫ nBS

0

∫ KnUT

0 ς2(u, v)dv

1 − w∫ KnUT

0 ς2(u, v)dvdu

=1

KnUT

∫ nBS

0

∫ KnUT

0ς2(u, v)dudv

nBS − w∫ nBS

0

∫ KnUT

0 ς2(u, v)dudv

= q(ΣM)1

1 − KnUT

nBSwq(ΣM)

= Rq(ΣM) 1N GM

†GM(w). (31)

where

q(ΣM ) � ‖ΣM‖2/(KN2nUT nBS

)(32)

is the Frobenius norm of the ΣM matrix ‖ΣM‖ �√tr{ΣM

†ΣM} normalized with the matrix dimensions and

‖ΣM‖2 = tr{Σ†

MΣM

}= tr

{(Σ ⊗ J)† (Σ ⊗ J)

}= tr

{(Σ† ⊗ J†) (Σ ⊗ J)

}= tr

{Σ†Σ⊗ J†J

}= tr

{Σ†Σ

}tr{J†J

}= tr

{Σ†Σ

}nUT nBS

= ‖Σ‖2nUT nBS. (33)

Using Equations (32) and (33), it can be seen that

q(ΣM ) = q(Σ) = ‖Σ‖2/(KN2

)(34)

In the asymptotic case, q(Σ) is given by

limN→∞

q(Σ) =1K

∫ K

0

ς2(u, v)dv. (35)

The probability density function (p.d.f.) of the limiting eigen-value distribution of 1

N H†H follows a scaled version ofthe Marcenko-Pastur law and hence the Shannon transformof the limiting eigenvalue distribution of 1

N H†H can beapproximated by

V 1N H†H

(γ

KnUT

)� VMP

(q(Σ)

γ

KnUT,KnUT

nBS

). (36)

3) UT-side Correlated Cellular Channel : Assuming thatthere is no receive correlation at the BS side i.e RR = I, thechannel matrix of Equation (17) can be rewritten as follows:

1√N

H =(W

(IKN ⊗ RT

12

))�(W

(IKN ⊗ DT

12

)), (37)

3710 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009

where W = 1√N

ΣM � GM and therefore

1N

H†H =NnBS∑

i=1

h†ihi

�NnBS∑

i=1

(IKN ⊗ DT

12

)w†

iwi

(IKN ⊗ DT

12

)

=NnBS∑

i=1

((1KN ⊗ λ

12T

)� wi

)† ((1KN ⊗ λ

12T

)� wi

),

(38)

where wi denotes the ith 1 × KNnUT row vector of W,1KN is a 1×KN row vector of ones and λT is a row vectorcontaining the eigenvalues of RT. Hence, the R-transform canbe written as

limN→∞

R 1N H†H(w) = lim

N→∞

NnBS∑i=1

Rhi†hi

(w)

= limN→∞

1KnUT N

NnBS∑i=1

‖hi‖2

1 − w ‖hi‖2

=q (Ω)

1 − KnUT

nBSwq (Ω)

= Rq(Ω) 1N GM

†GM(ω), (39)

where

q (Ω) =‖hi‖2

KNnUT=

∥∥∥(1KN ⊗ λ12T

)wi

∥∥∥2

KNnUT

=1

nUT

nUT∑j=1

λT(j) · 1K

∫ K

0

ς2(u, v

)dv

=1K

∫ K

0

ς2(u, v

)dv. (40)

It can be seen that the scaling of the Marcenko-Pastur law isidentical for the cases of uncorrelated and UT-side correlatedantennas, i.e. q (Σ) = q (Ω). As a result, the per-cell capacityfor UT-side correlation is given by (50) which coincides withthe case of uncorrelated multiple antennas. Therefore, wecan conclude for large values of K (K � nUT ) UT-sidecorrelation has no effect on the system’s performance. Thisascertainment is expected, since the capacity scaling is dictatedby the rank of the channel matrix H, which depends only onthe number of BS antennas in a cellular scenario.

4) BS-side Correlated Cellular Channel: Assuming thatthere is no transmit correlation at the UT side i.e. RT = I, thechannel matrix of Equation (17) can be rewritten as follows:

1√N

H =((

IN ⊗ RR12

)W)

�((

IN ⊗ DR12

)W)

(41)

and therefore

1N

H†H =1N

N∑i=1

H†iHi

=N∑

i=1

W†iDRWi =

nBS∑j=1

λR(j)N∑

i=1

w†iwi, (42)

where Hi and Wi are submatrices of H and W respectivelywith dimensions nBS ×KNnUT and λR is a row vector con-taining the eigenvalues of RR. Based on the previous analysis,the asymptotic eigenvalue distribution of A =

∑Ni=1 w†

iwi

follows a scaled version of the Marcenko-Pastur law. Hence,the R-transform of A can be written as

RA(w) � Rq(Σ) 1N G†G(w) =

q(Σ)1 − KnUT wq(Σ)

(43)

where G is a N × KNnUT matrix distributed as CN (0, I)and

q(Σ) =‖wi‖2

KNnUT=

1K

∫ K

0

ς2(u, v

)dv (44)

The R-transform of 1N H†H is calculated based on [18,

Theorems 2.31 and 2.64]

R 1N H†H(w) =

nBS∑j=1

λR(j)RA(λR(j)w). (45)

The asymptotic eigenvalue pdf (AEPDF) of 1N H†H is ob-

tained by determining the imaginary part of the Cauchytransform G for real arguments

f∞1N H†H(x) = lim

y→0+

1π

I{G 1

N H†H(x + jy)}

(46)

considering that the Cauchy transform is derived from the R-transform [42] as follows

G−11N H†H(w) = R 1

N H†H(−w) − 1w

. (47)

The AEPDF of 1N HH† can be also derived as follows:

nBS

KnUTf∞

1N H†H(x) + (1 − nBS

KnUT)δ(x) = f∞

1N H†H(x) (48)

since the matrices 1N HH† and 1

N H†H have the same non zeroeigenvalues, but their sizes differ by a factor of nBS/KnUT .

5) Double-side Correlated Cellular Channel: By combin-ing the two previous cases, it can be easily seen that the a.e.d.for the double-side Kronecker correlation model coincideswith the BS-side correlation case, since UT-side correlationhas no effect on the asymptotic eigenvalue distribution of1N H†H. Figure 2 illustrates the AEPDF of 1

N H†H varyingthe level of correlation at the BS antennas ρR. As it can beseen, by increasing the level of fading correlation, the plot ofthe eigenvalue distribution is gradually decomposing into twosegments.

C. Optimal Capacity

According to [18], the per-cell asymptotic Optimal JointDecoding sum-rate capacity Copt assuming a very large num-ber of cells and no CSI available at the UT-side (e.g. uniform

CHATZINOTAS et al.: ON THE MULTICELL PROCESSING CAPACITY OF THE CELLULAR MIMO UPLINK CHANNEL . . . 3711

0 2 4 6 8 10 12 14 16 18 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08f∞

(x)

x

ρR

=0

ρR

=0.25

ρR

=0.5

ρR

=0.75

ρR

=1

Fig. 2. Asymptotic Eigenvalue Probability Distribution Function (AEPDF) of1N

H†H (omitting the zero eigenvalues) while varying the level of correlationat the BS antennas ρR . Parameters: K = 4, nUT = 2, nBS = 2, η = 2, γ =10.

power allocation), is given by:

Copt(γ, N, nBS , K, nUT )

= limN→∞

1N

I (x;y | H )

= limN→∞

1N

E

[log det

(I +

γ

nUTHH†

)]

= limN→∞

E

[1N

NnBS∑i=1

log(

1 +γ

KnUTλi

(1N

HH†))]

= nBS

∫ ∞

0

log(

1 +γ

KnUTx

)f∞

1N HH†(x)dx

= KnUT

∫ ∞

0

log(

1 +γ

KnUTx

)f∞

1N H†H(x)dx, (49)

where γ = KNγ is the system transmit power normalizedby the receiver noise power respectively and λi (X) denotesthe eigenvalues of matrix X. Equation (49) can be utilizedin combination with Equation (46) and (47) for the case ofcorrelated BS antennas. For uncorrelated BS antennas, theoptimal per-cell sum-rate capacity is given by:

Copt(γ, N, nBS , K, nUT )= nBSV 1

N HH† (γ/KnUT )

= nBSKnUTV 1N H†H (γ/KnUT )

�nBSKnUTVMP

(q (Σ)

γ

KnUT,KnUT

nBS

), (50)

where VMP is calculated based on Equation (21). It should benoted that if CSI is available at the UT-side, multiuser iterativewaterfilling [43] can be employed to optimize the transmitterinput and thus the produced capacity.

D. MMSE Capacity

A global joint decoder will be extremely demanding interms of computational load as the complexity of symbol-by-symbol multiuser detection increases exponentially as the

number of users to be detected in the system increases[36]. However, for a coded system MMSE in combinationwith Successive Interference Cancellation(SIC) yields linearcomplexity in the number of users, or at least polynomialif one considers that the computation of the MMSE filters,matrix-vector multiplications and subtraction are quadratic orcubic in the number of users [44, Chap. 8]. Based on thisargument, the following equations describe the sub-optimalcapacity achieved by a linear MMSE filter followed by single-stream decoding. Based on the arguments in [18, Equation1.9][36], [45], [46], the MMSE and the Signal to Interferenceand Noise Ratio (SINR) for the kth date stream, assuming noCSI available at the UT-side (e.g. uniform power allocation),can be written as:

mmsek =

[(IKNnUT +

γ

nUTH†H

)−1]

k,k

,

1 + SINRk = 1 +1 − mmsek

mmsek= mmse−1

k . (51)

Considering single-stream decoding, the per-cell asymptoticMMSE capacity is given by the mean individual streamrate multiplied by the number of streams per cell (Equation(52) at the top of the next page) which can be utilized incombination with Equation (46) and (47) for the case ofcorrelated BS antennas. For uncorrelated BS antennas, theasymptotic MMSE capacity is given by:

Cmmse(γ, N, nBS , K, nUT )

= −KnUT log(

η 1N H†H

(γ

KnUT

))

= −KnUT log(

ηMP

(q (Σ)

γ

KnUT,KnUT

nBS

)), (53)

where ηMP is calculated based on Equation (22). In thispoint, it should be noted that MMSE filtering exhibits aninterference-limited behavior, when the number of transmittersis larger than the number of receive antennas [4]. Morespecifically, in the previous transmission strategies the signalsof all system UTs have been superpositioned on the sharedtime-frequency medium, which is sensible if optimal decodingis in place. However, if MMSE filtering is applied, theperformance can be enhanced by orthogonalizing the intra-cell UTs so that only a single UT per cell transmits using theshared medium. This scenario resembles to cellular systemsemploying intra-cell TDMA, FDMA or orthogonal CDMAand its performance is evaluated in section V-A by meansof Monte Carlo simulations.

V. NUMERICAL RESULTS

The analytical results (Equations (49),(50),(52),(53)) havebeen verified by running Monte Carlo simulations over 100random instances of the system and by averaging the pro-duced capacity results. More specifically, for each systeminstance the complex matrix (IN ⊗ RR

12 )GM is constructed

by randomly generating correlated fading coefficients accord-ing to the exponential model with ρR being the BS-sidecorrelation coefficient. UT-side correlation is not consideredin the numerical results, since it does not have an effect oncapacity for large K . Subsequently, the variance profile matrix

3712 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009

Cmmse(γ, N, nBS, K, nUT ) = limN→∞

KnUT E

[log

(1

NKnUT

NKnUT∑k=1

(1 + SINRk)

)]

(50)= − lim

N→∞KnUT E

⎡⎣log

⎛⎝ 1

NKnUT

NKnUT∑k=1

[(INKnUT +

γ

nUTH†H

)−1]

k,k

⎞⎠⎤⎦

≥ − limN→∞

KnUT log

(1

NKnUTE

[Tr

{(INKnUT +

γ

nUTH†H

)−1}])

= − limN→∞

KnUT log

⎛⎝E

⎡⎣ 1

NKnUT

KNnUT∑j=1

11 + γ

KnUTλj

(1N H†H

)⎤⎦⎞⎠

= −KnUT log

(∫ ∞

0

11 + γ

KnUTx

f∞1N H†H(x)dx

)

= −KnUT log

(∫ ∞

0+

11 + γ

KnUTx

f∞1N H†H(x)dx + 1 − nBS

KnUT

)(52)

Σ is constructed by randomly placing the UTs according toa uniform distribution in the planar coverage area and bycalculating the variance profile coefficients using Equation(11). It should be noted that the simulated system includesN = 7 BSs, which is adequately large to converge with theasymptotic analysis results. In the context of the mathematicalanalysis, the distance dnm

k can be calculated assuming that theUTs are positioned on a uniform planar grid as in Fig. 1 [47].The numerical results presented in this section refer to theoptimal and MMSE per-cell sum-rate capacity averaged overa large number of fading realizations and UT positions. Afterconstructing the channel matrix H, the optimal per-cell sum-rate capacity is calculated by evaluating the formula in [20]

Copt =1N

E

[logdet

(INnBS +

γ

nUTHH†

)], (54)

while the MMSE per-cell capacity is calculated by summingall the individual stream rates and normalizing by the numberof cells [18]

Cmmse = (55)

− 1N

E

⎡⎣NKnUT∑

k=1

log

[(IKNnUT +

γ

nUTH†H

)−1]

k,k

⎤⎦ ,

where [X]k,k denotes the kth diagonal element of the Xmatrix. In this context, Figures 3 and 4 depict the optimaland MMSE per-cell sum-rate capacity respectively versusthe normalized cell radius R varying the level of receivecorrelation ρR = [0, 0.9, 0.99, 1]. As it can be observed inboth cases, the BS-side correlation decreases the degrees offreedom due to the multiple receive antennas and thereforecompromises the capacity performance of the system. In theno-correlation extreme ρR = 0, the optimal capacity curve isidentical to the curve derived in [3] for multicell processingcellular systems with multiple antennas. In the full-correlationextreme ρR = 1, the capacity curve degrades to the single-antenna capacity [2], since no multiplexing gain is achievedby the multiple BS antennas. In the MMSE-receiver case,

10−2

10−1

100

101

102

103

0

5

10

Opt

imal

Cap

acity

in n

ats/

Hz/

cell

Cell Radius R

10−2

10−1

100

101

102

1030

5

10

15

20

Opt

imal

Cap

acity

in b

its/s

ec/H

z/C

ell

ρR=0

ρR=0.9

ρR=0.99

ρR=1

Fig. 3. Optimal per-cell sum-rate capacity vs. the normalized cell radiusR varying the level of BS-side correlation ρR = [0, 0.9, 0.99, 1] in a planarcellular system with uniformly distributed UTs. Analysis curve and simulationpoints are marked using a solid line and circle points respectively. Parameters:K = 16, γ = 10, nBS = 2, nUT = 1, η = 2.

it can be seen that the achieved capacity is much lowerthan the optimal due to the lack of interference-suppressingdimensions, but the effect of correlation is less grave especiallyfor short cell radii. It should be noted that in Figures 3 and 4the analysis curve and the simulation points are marked usinga solid line and circle points respectively in order to verifytheir close agreement.

Subsequently, Figure 5 illustrates the per-cell sum-ratecapacity versus the level of BS-side correlation for a fixed cellsize. It can be observed that the optimal capacity degradationbecomes detrimental for high correlation levels, whereas theMMSE receiver appears to be much more resistant to fadingcorrelation. Finally, Figure 6 depicts the per-cell sum-ratecapacity versus the normalized cell radius R varying thenumber of BS antennas nBS for two values of correlationρR = [0, 0.8]. By observing the figure, it becomes clearthat the linear capacity scaling with the number of receive

CHATZINOTAS et al.: ON THE MULTICELL PROCESSING CAPACITY OF THE CELLULAR MIMO UPLINK CHANNEL . . . 3713

10−1

100

101

102

103

0

0.5

1

1.5

2

2.5M

MSE

Cap

acity

in n

ats/

Hz/

cell

Cell Radius R

10−1

100

101

102

1030

0.5

1

1.5

2

2.5

3

3.5

MM

SE C

apac

ity in

bits

/sec

/Hz/

Cel

l

ρR=0

ρR=0.9

ρR=0.99

ρR=1

Fig. 4. MMSE per-cell sum-rate capacity vs. the normalized cell RadiusR varying the level of BS-side correlation ρR = [0, 0.9, 0.99, 1] in a planarcellular system with uniformly distributed UTs. Analysis curve and simulationpoints are marked using a solid line and circle points respectively. Parameters:K = 16, γ = 10, nBS = 2, nUT = 1, η = 2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

5

10

Cap

acity

in n

ats/

Hz/

cell

BS Correlation Level ρR

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

Cap

acity

in b

its/s

ec/H

z/C

ellMMSE

Optimal

Fig. 5. Optimal and MMSE per-cell sum-rate capacity vs. the level of BS-sidecorrelation ρR for a fixed-radius cellular system with uniformly distributedUTs. Parameters: K = 16, γ = 10, nBS = 2, nUT = 1, η = 2.

antennas nBS remains in spite of the degrading effect of fadingcorrelation.

A. Practical Results

This section aims at denormalizing the cellular systemparameters employed in the analysis in order to present morepractical numerical results. These results can be used toevaluate the capacity enhancement which BS cooperation canprovide in the context of real-world cellular infrastructure. Inthis direction, if L0 is the power loss at the reference distanced0, the scaled variance profile function is given by

ς(d(t)) =

√L0

(1 + d(t)/d0

)−η

. (56)

The values of L0 and η have been fitted to the path loss modeldefined in the “Urban Macro” scenario of [48]. Furthermore,

10−2

10−1

100

101

102

103

0

20

Cap

acity

in n

ats/

Hz/

cell

Cell Radius R

10−2

10−1

100

101

102

1030

10

20

Cap

acity

in b

its/s

ec/H

z/C

ell

nBS

=4

nBS

=3

nBS

=2

Fig. 6. Optimal per-cell sum-rate capacity vs. the normalized cell Radius Rvarying the number of BS antennas nBS for two values of receive correlationρR = [0, 0.8] (solid and dashed line respectively) in a planar cellular systemwith uniformly distributed UTs. Parameters: K = 16, γ = 10, nUT =1, η = 2.

TABLE IPARAMETERS FOR PRACTICAL CELLULAR SYSTEMS

Parameter Symbol Value/Range (units)Cell Radius R 0.1 − 3 Km

Reference Distance d0 1 mReference Path Loss L0 34.5 dBPath Loss Exponent η 3.5

Antennas per BS nBS 2BS Correlation Level ρR 0.8624

Antennas per UT nUT 2UTs per Cell K 16

UT Transmit Power PT 200 mWThermal Noise Density N0 −169 dBm/Hz

Channel Bandwidth B 5 MHz

the BS correlation level was selected according to [48] as-suming 2 degrees angle spread, 50 degrees angle of arrivaland an antenna spacing of 4λ, where λ is the communicationwavelength. Table I includes a concise list of the nominalparameter values used for producing the results in Figure 7.In addition, this section evaluates the performance of MMSEfiltering in combination with intra-cell UT orthogonalization,so that it can be compared with the aforestudied widebandtransmission cases. In this direction, a UT is randomly selectedfor each cell and their channel vectors are concatenated inorder to construct the square NnUT × NnUT matrix Horth.Subsequently, the per-cell MMSE capacity is evaluated inaccordance to equation (55):

Corthmmse = (57)

− 1N

E

⎡⎣NnUT∑

k=1

log

[(INnUT +

γ

nUTH†

orthHorth

)−1]

k,k

⎤⎦ .

It is interesting that in the considered parameter range, theeffect of both BS-side correlation and cell density on theMMSE capacity is negligible due to the interference-limitedbehavior which has also been observed in [4], [29]. On thecontrary, the optimal capacity performance is degraded by 1bit/sec/Hz due to correlation, which is acceptable considering

3714 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009

0.5 1 1.5 2 2.5 30

20

40

60C

apac

ity in

nat

s/H

z/ce

ll

Cell Radius R in Km

0.5 1 1.5 2 2.5 30

20

40

60

80

Cap

acity

in b

its/s

ec/H

z/C

ell

MMSE: No CorrelationMMSE Orth: No CorrelationOptimal: No CorrelationMMSE: Practical CorrelationMMSE Orth: Practical CorrelationOptimal: Practical Correlation

Fig. 7. Optimal and MMSE per-cell sum-rate capacity vs. the cell RadiusR in Km considering the practical parameters in Table I.

the high spectral efficiency enhancement due to multicell pro-cessing. Furthermore, it can be observed that for nBS ≥ nUT

the performance of MMSE filtering combined with intra-cell orthogonalization is no longer interference-limited, sincethere are sufficient degrees of freedom to suppress inter-cellinterference.

VI. CONCLUSION

In this paper, we have considered a multicell processingsystem with MIMO links and distributed UTs. In this context,we have investigated the effect of antenna correlation on thecapacity performance of the system. The presented results hasbeen derived considering that the variances of the Gaussianchannel gains are scaled by a generic variance profile whichincorporates both path loss and antenna correlation. In thisdirection, we have presented two analytical approaches: afinite Random Matrix Theory approach and an asymptoticFree Probability approach. The former approach is useful forreducing the complexity of capacity calculation in finite sys-tems, whereas the latter provides closed forms and interestinginsights on the system performance. The main findings canbe summarized as follows: antenna correlation degrades thecapacity performance of the system, especially if it appearson the BS side. What is more, for large number of UTs percell, the effect of UT-side correlation is negligible. Finally, it isshown that the MMSE performance is greatly suboptimal butmore resilient to fading correlation in comparison to optimaldecoding.

ACKNOWLEDGMENT

The work reported in this paper has formed part of the“Fundamental Limits to Wireless Network Capacity” Elec-tive Research Programme of the Virtual Centre of Excel-lence in Mobile & Personal Communications, Mobile VCE,www.mobilevce.com. This research has been funded by thefollowing Industrial Companies who are Members of MobileVCE - BBC, BT, Huawei, Nokia, Nokia Siemens Networks,Nortel, Vodafone. Fully detailed technical reports on this

research are available to staff from these Industrial Membersof Mobile VCE. The authors would like to thank Prof. G.Caire and Prof. D. Tse for the useful discussions.

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Symeon Chatzinotas (S’06) received the B.Eng.in Telecommunications from Aristotle University ofThessaloniki, Greece and the M.Sc. and Ph.D. inElectronic Engineering from University of Surrey,UK in 2003, 2006 and 2009 respectively. He is cur-rently a researcher in the Mobile CommunicationsResearch Group, Center of Communication SystemsResearch (CCSR), University of Surrey, workingon information-theoretic capacity limits in multicelljoint processing systems.

In the past, he has worked in numerous R&Dprojects for the Institute of Informatics & Telecommunications, NationalCenter for Scientific Research “Demokritos,” Greece and the Institute ofTelematics and Informatics, Center of Research and Technology Hellas. Hehas authored more than 30 technical papers in refereed international journals,conferences and scientific books. His research interests are on multiuserinformation theory, cooperative communications and wireless networks op-timization.

Muhammad Ali Imran received his B.Sc. degree inElectrical Engineering from University of Engineer-ing and Technology Lahore, Pakistan , in 1999, andthe M.Sc. and Ph.D. degrees from Imperial CollegeLondon, UK, in 2002 and 2007, respectively. He iscurrently a research fellow at the Centre for Commu-nication Systems Research (CCSR) at the Universityof Surrey, UK. His main research interests includethe analysis and modelling of the physical layer,resource allocation in multiuser communication net-works and evaluation of the fundamental capacity

limits of wireless networks.

Reza Hoshyar (M03) received the B.S. degreein communications engineering and the M.S. andPh.D. degrees, both in mobile communications, fromTehran University, Tehran, Iran, in 1991, 1996, and2001, respectively. He received the top and secondrank awards for his B.S. and M.S. degrees fromTehran University. From 2002 to June 2006, he hasbeen a Research Fellow in the Mobile Communica-tions Research Group, Center for CommunicationSystems Research (CCSR), University of Surrey,Surrey, U.K. During this period, he has been ac-

tively involved in IST European projects MUMOR, STRIKE, and WINNER.Currently, he is a Senior Research Fellow at CCSR and participating as thetechnical work package leader in ICT European project ROCKET, focusingon advanced digital signal processing, coding and modulation, and schedulingtechniques for relaying, and cooperative communications in OFDM/OFDMAsystems.