Up-Link Multi-User MIMO Capacity in Low-PowerRegime

Pasquale Memmolo, Marco Lops, Antonia M. Tulino, Reinaldo A. Valenzuela

Abstract—This paper studies the up-link of a multi-userMultiple-Input Multiple-Output (MIMO) system under a fairlygeneral channel model, subsuming a number of situations ofrelevant practical interest. Concerning the available prior in-formation, we consider the case of Prior Channel State Infor-mation at the Transmitter (PCSIT) wherein only a statisticalchannel characterization is available before transmission, whilewe assume perfect Channel State Information at the Receiver(CSIR). Under this assumptions, we characterize the sum-capacity-achieving input covariance matrix for such a system andwe also examine the low-power regime in terms of both minimumenergy contrast and multiple access slope region, validating ourtheoretical findings through a set of numerical results.

Index Terms—Multi-user MIMO, capacity, UIU model, inputoptimization, antenna correlation, optimality of beam-forming,low-power regime.

I. INTRODUCTION

Following the seminal paper [19], where capacity of single-

user MIMO systems has been determined for a channel matrix

with independent and identically distributed (i.i.d.) entries

under perfect CSIR and different instances of Channel State

Information (CSI) available at the transmitter, a number of

results, that focus on the evaluation of the capacity-achieving

input covariance matrix for Gaussian channels with corre-

lated zero-mean ([24], [7], [5], [8]) and uncorrelated with

arbitrary mean ([24], [21], [4]) entries, have appeared in the

information/communication theory literature, mainly assuming

instantaneous CSI available at the receiver but PCSIT only

available at the transmitter. A complete characterization of

the limiting covariance, valid for arbitrary channels, has been

given in [23] in the low signal-to-noise ratio (SNR), and in

[11] for arbitrary SNR. The analysis in [11] determined both

the eigenvectors (signalling directions) and the eigenvalues

(transmission powers), with arbitrary numbers of antennas

under a broad class of channel models, and was extended in

[20] to a large number of receiving and transmitting antennas.

The far more challenging scenario of multi-user MIMO

systems has instead received less attention so far. In the

context of perfect CSI at the receiver and at the transmitter the

problem of finding the capacity-achieving input distribution

of a Gaussian multi-user MIMO has been considered in [14]

and [26]. Interesting results, mainly concerning optimality of

beam-forming for low SNR’s under specific channel models

Pasquale Memmolo is with Universita Degli Studi di Napoli, Federico II,80125 Napoli, Italy.

Marco Lops is with Institut National Polytechnique Toulouse/IRIT &DAEIMI- University of Cassino.

Antonia M. Tulino and Reinaldo A. Valenzuela are with Bell Laboratories(Alcatel-Lucent), Holmdel, NJ07733, USA.

and different degrees of CSI at the transmitter have been

established in [15], [18] and [10]. In the context of perfect

CSIR and PCSIT, the purpose of the present paper is two-

fold: on one hand, indeed, it aims at determining the MIMO

multi-user sum capacity for more general scenarios than those

already considered in literature; on the other, it characterizes

the capacity region in terms of minimum energy contrast and

slope region under a fairly general channel model, subsuming

a number of situations of relevant theoretical and practical

interest. A number of numerical results are also produced,

validating our theoretical results.

The paper is organized as follows. Next section contains

the problem statement and a detailed outline of the channel

model, while Section 3 illustrates the results concerning multi-

user MIMO sum-capacity, whose the proofs are given in [12]

and omitted here due to space limitations. Section 4 contains

a general study of the capacity region in low-power regime,

along with a number of interesting special cases, while Section

5 is devoted to the numerical results. Conclusions and hints

for future developments form the object of Section 6.

Notation: In the following, X†, and (X)𝑘,𝑖 denote the

Hermitian transpose, and the element at the 𝑘th row

and 𝑖th column of the matrix X ∈ ℂ𝑚×𝑛, respec-

tively. X = (x1, . . . ,x𝑛) ∈ ℂ𝑚×𝑛 is a matrix contain-

ing the 𝑚−dimensional vectors x1, . . . ,x𝑛 as its columns.

diag(X1, . . . ,X𝑘) ∈ ℂ𝑘𝑚×𝑘𝑛 is the block-diagonal matrix

containing the matrices X1, . . . ,X𝑘 ∈ ℂ𝑚×𝑛 on the main

diagonal. Tr(A) and det(A) denote the trace and the de-

terminant of a square matrix A. I𝑚 indicates the identity

matrix of order 𝑚, while 𝜆𝑖(𝑨) denotes the 𝑖th eigenvalue of

a generic Hermitian matrix 𝑨. Finally, 𝔼 [⋅] denotes statistical

expectation.

II. THE MODEL

We consider a multi-user MIMO system consisting of 𝐾users, equipped with 𝑀1, . . . ,𝑀𝐾 antennas, respectively, and

a base station (BS) with 𝐿 receive antennas. We assume perfect

channels state information at the receiver while we assume

PCSIT only, i.e. the transmitter only knows the prior density

of the channel coefficients. The 𝑘th user is linked to the BS

through the 𝐿 ×𝑀𝑘 channel matrix H(𝑘). Denoting by x(𝑘)

the 𝑀𝑘-dimensional input vector from the 𝑘th user to the BS,

the 𝐿-dimensional signal received at the BS is given by:

y =

𝐾∑𝑘=1

√𝑔𝑘 H

(𝑘)x(𝑘) + n. (1)

ISIT 2010, Austin, Texas, U.S.A., June 13 - 18, 2010

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with {√𝑔𝑘}𝐾𝑘=1 the slow-fading component assumed inde-

pendent from {H(𝑘)}𝐾𝑘=1 and independent from user to user.

Similar independence assumptions are made on the fast fading

components. Finally {√𝑔𝑘 H(𝑘)}𝐾𝑘=1 are independent of both

{x(𝑘)}𝐾𝑘=1 and n.

As to n, it is an 𝐿-dimensional white Gaussian vector

modeling the receiver noise, with one-sided spectral density:

𝑁0 = 1𝐿𝔼[∥n∥2], (2)

and normalized spatial covariance:

Ψn ≜ 1𝑁0

𝔼[nn†] = I. (3)

The input vectors are assumed to be zero-mean Gaussian,

with covariance matrices Φ𝑘 ≜ 𝔼[x(𝑘)x(𝑘)†], with 𝑘 =1, . . . ,𝐾, satisfying the power constraint:

Tr {Φ𝑘} ≤ 𝑃𝑘 = SNR𝑘𝑁0. (4)

Denoting 𝑛T =∑𝐾

𝑘=1𝑀𝑘 and stacking the transmitted

𝐾 signal vectors x(k) in the 𝑛T-dimensional vector x =[x(1)† . . .x(𝐾)† ]†, (1) becomes:

y = Hx + n, (5)

where H is the 𝐿× 𝑛T channel matrix given by:

H =[√𝑔1 H(1), . . . ,

√𝑔𝐾 H(𝐾)

]. (6)

Note that the covariance of x is:

Φ = diag (Φ1, . . . ,Φ𝐾) , (7)

while, defining 𝑃T = 1𝐾

∑𝐾𝑘=1 𝑃𝑘,

SNR ≜ 𝑃T1𝐿𝔼[∥n∥2]

=1

𝐾

𝐾∑𝑘=1

SNR 𝑘, (8)

corresponds to the average SNR per user per receive antenna

when the channel entries are zero-mean i.i.d. and 𝑔𝑘 = 1 with

𝑘 = 1, . . . ,𝐾. In the above framework, one of our goals is to

determine the ergodic sum-capacity under PCSIT.

Concerning the channel matrix H, we consider the general

model:

H(𝑘) = H(𝑘) + U(𝑘)R H(𝑘)U

(𝑘)†

T , (9)

where U(𝑘)R and U

(𝑘)T are (𝐿×𝐿) and (𝑀𝑘×𝑀𝑘) deterministic

unitary matrices and H(𝑘) is an (𝐿×𝑀𝑘) random matrix

with independent columns, the distribution of whose entries

is jointly symmetric with respect to zero, i.e., zero-mean and

arbitrary in amplitude. The variances of the entries of H(𝑘)

can be assembled into another (𝐿×𝑀𝑘) matrix, G(𝑘), such

that

(G(𝑘))𝑖,𝑗 = 𝔼[∣(H(𝑘))𝑖,𝑗 ∣2]. (10)

Finally H(𝑘) is a (𝐿×𝑀𝑘) random matrix, such that

H(𝑘)†H(𝑘) ∼ Γ𝑚(𝛼,Ω). Recall here that:

Definition 1 The 𝑚 × 𝑚 random matrix M is a complexGamma matrix with scalar parameter 𝛼 (𝛼 ≥ 𝑚) and matrix

parameter Ω, (M ∼ Γ𝑚(𝛼,Ω)), if the joint distribution of itsentries can be written as [13]:

𝑓M(A) =det (A)

𝛼−𝑚det (Ω)

𝛼

Γ𝑚(𝛼)exp [−Tr (ΩA)] . (11)

where Γ𝑝(𝑞), 𝑝 ≤ 𝑞, is the complex multivariate Gammafunction [6] given by:

Γ𝑝(𝑞) = 𝜋𝑝(𝑝−1)

2

𝑝∏ℓ=1

Γ(𝑞 − ℓ+ 1).

Note that, for integer 𝛼,𝑀 reduces to a central Wishart matrix[6], with 𝛼 degrees of freedom and parameter matrix Ω−1.

While not universal, this representation, introduced in [11]

and [20], encompasses most channels of interest. For the

reader’s sake, we summarize these in what follows, distin-

guishing between H(𝑘) ∕= 0 and H(𝑘) = 0.

First notice that, if H(𝑘) is deterministic and H(𝑘) Gaussian

with i.i.d. entries (whereby U(𝑘)R and U

(𝑘)T are immaterial),

then (9) reduces to a mere Ricean channel, while yielding a

shadowed-Rice (SR) model if, under the same conditions for

H(𝑘), H(𝑘) is complex-Gamma: SR is a credited model for the

land mobile satellite (LMS) channel. Moreover, if U(𝑘)R and

U(𝑘)T are identity matrices and H(𝑘) still retains independent

entries, then (9) models the use of antennas with different

polarizations and/or radiation patterns, wherein the channel

coefficients, albeit weakly correlated, may exhibit significantly

different marginal statistics: in this case, the crosspolar and

pattern discrimination factors are absorbed into H(𝑘) and

G(𝑘).

The companion case H(𝑘) = 0 and H(𝑘) Gaussian with i.i.d.

entries corresponds to the model proposed in [9] and [25] and

experimentally validated in the latter.

Note in particular that, if U(𝑘)R and U

(𝑘)T are constrained

to be Fourier matrices, then (9) yields the virtual repre-sentation proposed in [16]. 1 In this case, the columns

of U(𝑘)R and U

(𝑘)T can be interpreted as steering vectors

launching and receiving energy on specific spatial directions.

If (G(𝑘))𝑖,𝑗=𝜆𝑖(Θ(𝑘)R )𝜆𝑗(Θ

(𝑘)T ), with Θ

(𝑘)R and Θ

(𝑘)T repre-

senting the correlation matrices at the receiver and at the

transmitter, respectively, so that U(𝑘)R and U

(𝑘)T denote the

corresponding eigenvector, then (9) reduces to the separablecorrelation model (also termed Kronecker product) of [17]. It

is finally worth noting in passing that not subsumed by (9) is

the keyhole (or pinhole) channel [1], [2], [11], namely:

H(𝑘) = cRc†T, (12)

where cR and cT are column vectors, each having independent

random entries, whose distribution is symmetric with respect

to zero, i.e., zero-mean and arbitrary in amplitude.

1In this representation, the independence of the entries of H(𝑘) is onlyapproximate except for 𝑀𝑘, 𝐿→∞.

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III. UP-LINK CAPACITY

We assume an ergodic-channel regime whereby the fun-

damental operational limit is the ergodic (or time-averaged)

capacity. As anticipated we focus on the PCSIT case, namely

wherein the transmitter only knows the prior density of the

channel coefficients. Under this scenario, assuming perfect

CSIR, the transmission matrix Φ∗ that achieves the ergodic

sum-capacity is:

Φ∗ = arg maxΦ

𝔼

[log2 det

(I +

SNR

𝑃THΦH†

)],

where the maximum is over the block-diagonal matrices Φ =diag (Φ1, . . . ,Φ𝐾) with Φ𝑘 satisfying the power constraint

Tr{Φk} = 𝑃𝑘.

Under the general channel model in (9), the solution to the

above optimization problem is given by following theorem :

Theorem 1 [12] Let us consider the scenarios:∙ The channel of the generic 𝑘th user with 𝑘 = 1, . . . ,𝐾

is either described by (12), or by (9) with H(𝑘)=0;∙ The channel of the generic 𝑘th user with 𝑘 = 1, . . . ,𝐾

is either described by (12), or by (9) with H(𝑘) havingzero-mean i.i.d. Gaussian entries;

∙ There is at most one user, such that its channel isdescribed by (9) with H(𝑘) deterministic and H(𝑘) havingzero-mean i.i.d. Gaussian entries, while the channels ofall other users are either described by Definition (12) orby (9) with U

(𝑘)𝑅 =I and H(𝑘) = 0.

Then, the sum-capacity-achieving input covariance matricesΦ∗

𝑘, with 𝑘 = 1, . . . ,𝐾, are:

Φ∗𝑘 = U

(𝑘)T Λ𝑘U

(𝑘)†T , (13)

where:∙ U

(𝑘)T with 𝑘 = 1, . . . ,𝐾 is the eigenvector matrix of

𝔼[H(𝑘)†H(𝑘)

];

∙ Λ𝑘 with 𝑘 = 1, . . . ,𝐾 is a diagonal matrix whose 𝑗thdiagonal element is the positive solution to:

𝜆𝑘,𝑗 = 0,SNR𝔼

[𝑍

(𝑘)𝑗

]

𝑃T≤ 𝜗𝑘 (14)

𝜆𝑘,𝑗 =1− MSE𝑘,𝑗

𝜗𝑘, otherwise

(15)

where

MSE𝑘,𝑗 = 𝔼

[1

1 + SNR𝑃T𝜆𝑘,𝑗𝑍

(𝑘)𝑗

], (16)

with

𝑍(𝑘)𝑗 = 𝑔𝑘h

(𝑘)†𝑗 B−1

𝑘,𝑗h(𝑘)𝑗 , (17)

and

B𝑘,𝑗 = I + SNR𝑃T

𝐾∑ℓ=1

𝑔ℓH(ℓ)ΛℓH

(ℓ)†

−SNR𝑃T𝜆𝑘,𝑗𝑔𝑘h

(𝑘)𝑗 h

(𝑘)†𝑗 . (18)

In (17), h(𝑘)𝑗 denotes the 𝑗th column of the transformed

channel matrix H(𝑘) = H(𝑘)U(𝑘)†

T .

The quantity 𝜗𝑘 in the above theorem is defined as

𝜗𝑘 =1

𝑃𝑘

𝑀𝑘∑𝑗=1

(1− MSE𝑘,𝑗) ,

and represents the normalized signal-to-noise-plus interference

ratio for the 𝑘th user.

It is worth noticing that the key information the mobile

terminal needs to implement the optimal solution of Theorem

1 is the diagonal of the matrix 𝔼[B𝑘,𝑗 ] defined in (18), which

we may assume to be feedback from the base station.

Beyond some asymptotic regimes (low- and high-SNR), the

power profile satisfying Theorems 1 cannot, in general, be

found explicitly. Note, however, that conditions (15) represent

a set of coupled equations that begets an iterative algorithm

as described in [12].

From Theorem 1, it immediately follows

Theorem 2 [12] If:

SNR𝜗𝑘 𝑃T

𝔼

[𝑍

(𝑘)𝑖

]< 1 𝑖 = 2, . . . ,𝑀𝑘 (19)

where 𝑍(𝑘)𝑖 is given as in Theorem 1, then the 𝑘th user should

undertake beam-forming, allocating all the useful power on theeigenvector of 𝔼

[H(𝑘)†H(𝑘)

]corresponding to the maximum

eigenvalue. If such eigenvalue has multiplicity 𝜇𝑘 > 1 , thenthe transmitter should evenly split the power along the 𝜇𝑘multiple directions corresponding to the maximum eigenvalue.

IV. LOW-SNR ANALYSIS

The key performance measures in the low-SNR regime

(the wide-band regime) are the minimum energy contrast

per information bit, 𝐸𝑏

𝑁0 minsay, required to reliably convey

any positive rate, and 𝑆0 the capacity slope, measured in

bits/s/Hz/(3 dB). We recall here that, for a single user channel,

the first-order approximation of the capacity, is:

𝐶

(𝐸𝑏

𝑁0

)= 𝑆0

𝐸𝑏

𝑁0

∣∣∣dB− 𝐸𝑏

𝑁0 min

∣∣∣dB

3 dB+ 𝜖 (20)

with 𝜖 = 𝑜(

𝐸𝑏

𝑁0− 𝐸𝑏

𝑁0 min

), whereby sustaining a rate 𝑅 with

power 𝑃 requires a bandwidth:

𝐵 =𝑅

𝐶(

𝑃𝑅𝑁0

) (21)

≈ 𝑅

𝑆0

3 dB

𝑃𝑅𝑁0

∣∣∣dB− 𝐸𝑏

𝑁0 min

∣∣∣dB

With reference to our multi-user setting, we define

𝑅𝑘(SNR 𝑘) the rate of the 𝑘th user, with SNR 𝑘 defined in (4),

whereby the 𝑘th user transmitted energy contrast is

𝐸𝑘

𝑁0=

SNR 𝑘

𝑅𝑘(SNR 𝑘), 𝑘 = 1, . . . ,𝐾. (22)

ISIT 2010, Austin, Texas, U.S.A., June 13 - 18, 2010

2310

The object of interest, here, is the minimum energy contrast

per information bit 𝐸𝑘

𝑁0 min, which is the limit of (22) for

vanishing small transmit power (or equivalently increasingly

large bandwidth). In what follows, we extensively resort to

the concept of capacity per unit cost for multi-access channels

developed in [22]. Defining:

Φ∗𝑘 = V𝑘V

†𝑘, 𝑘 = 1, . . . ,𝐾, (23)

with V𝑘 consisting of the ℓ𝑘 orthonormal eigenvectors of

the maximal-eigenvalue eigenspace of 𝔼[H(𝑘)H(𝑘)† ], we can

prove the subsequent results:

Theorem 3 For any rate set 𝑅1, . . . , 𝑅𝐾 compatible withcondition (22), the minimum energy contrast is:

𝐸𝑘

𝑁0min

=log𝑒 2

𝑔𝑘𝐺∗𝑘

, (24)

where

𝐺∗𝑘 = 𝔼

[Tr

(H(𝑘)Φ

∗𝑘H

(𝑘)†)]

= 𝜆max(𝔼[H(𝑘)†H(𝑘)]).

with Φ∗𝑘 given in (23). Otherwise stated, defining the average

received energy contrast

𝐸𝑘

𝑁0

𝑟

=SNR 𝑘/𝑃𝑘

𝑅𝑘(SNR 𝑘)𝑔𝑘𝔼

[Tr

(H(𝑘)Φ𝑘H

(𝑘)†)], (25)

we have:𝐸𝑘

𝑁0

𝑟

min

= log𝑒 2. (26)

Define now 𝑆0𝑘 the slope of the 𝑘th user capacity and 𝜌𝑘,𝑗 :

𝜌𝑘,𝑗 =𝑅𝑘

𝑅𝑗

for all 𝑗 ∕= 𝑘 = 1, . . . ,𝐾.

Let us first consider the case that, for all 𝑘 = 1, . . . ,𝐾,

𝜆max(𝔼[H(𝑘)†H(𝑘)]) has unit multiplicity and only two users

are present. Define (𝛼, 1− 𝛼) the density of the time-sharing

variable. For future reference, we also introduce the dispersion

of an (𝑛× 𝑛) Hermitian random matrix A as:

𝜁(A) = 𝑛𝔼[Tr(A2)]

𝔼2[Tr(A)]. (27)

We have the following theorem:

Theorem 4 For constant 𝜌 = 𝑅1

𝑅2and vanishingly small rates,

the optimum multi-access slope region is the closure of theconvex hull of the set:{(𝑆01, 𝑆02) :

1

2𝑆01≤ 2𝐿

𝜁(H(1)Φ∗1H

(1)†), 𝑆02 ≤ 2𝐿

𝜁(H(2)Φ∗2H

(2)†)

𝑆01 ≤ 2𝐿𝜌

𝜁(H(1)Φ∗1H

(1)†)(𝜌+ 2− 2𝛼),

𝑆02 ≤ 2𝐿𝜌

𝜁(H(2)Φ∗2H

(2)†) (1 + 2𝛼𝜌)

1

2

}(28)

over all the admissible densities (𝛼, 1− 𝛼).

The more general situation of a 𝐾 users system can be

dealt with by introducing a time sharing variable over a 𝐾-

dimensional alphabet with probability mass {𝛼𝑖}𝐾𝑖=1. Define

u = [1, . . . ,𝐾] the 𝐾-dimensional vector containing the user

identities and by 𝜋𝑗 the 𝑗th of its 𝐾! permutations: with a

slight notational abuse we denote 𝜋𝑗(𝑘) the pointer to the

location of the 𝑘th user in the 𝑗th index permutation. We can

state the following

Theorem 5 For constant 𝜌𝑘,𝑗 and vanishingly small rates, theoptimum multi-access slope region is the closure of the convexhull of the set:{

1

2(𝑆01, . . . , 𝑆0𝐾) : 𝑆0𝑘 ≤ 2𝐿

𝜁(H(𝑘)Φ∗𝑘H

(𝑘)†),

𝑆0𝑘 ≤ 22𝐿

𝜁(H(𝑘)Φ∗𝑘H

(𝑘)†)

1

1 +

𝐾!∑𝑖=1

2𝛼𝑖

𝐾∑ℓ=𝜋𝑖(𝑘)+1

1

𝜌𝑘,ℓ

1

2

}(29)

over all the admissible {𝛼𝑖}𝐾𝑖=1, with Φ∗𝑘 given in (23).

Implicit in (29) is the convention that the summation,∑𝐾ℓ=𝜋𝑖(𝑘)+1

1𝜌𝑘,ℓ

= 0, is zero as the lower index 𝜋𝑖(𝑘) + 1

exceeds 𝐾 namely when 𝜋𝑖(𝑘) = 𝐾.

The proof, omitted here due to the lack of space, hinges

upon the extreme points representation of the convex hull of

compact convex sets. In particular, convex hulls having a finite

number of extreme points can be described through convex

combinations thereof [3]: for the case at hand the capacity

region of the considered MIMO multi-user system admits

indeed 𝐾! such points, which is also an intuitive justification

of the mathematical form of (29).

Let us now move to the case that the maximum eigenvalue

of the matrix 𝔼[H(𝑘)†H(𝑘)] has multiplicity 𝜇𝑘 > 1. Using

the results of [12] where the optimal transmission policy for

the 𝑘th user has been shown to be uniform power allocation

over its best 𝜇𝑘 eigenmodes, we obtain the following

Theorem 6 For constant 𝜌𝑘,𝑗 and vanishingly small rates, theoptimum multi-access slope region for the 𝐾 users channel isthe closure of the convex hull of the set:{

1

2(𝑆01, . . . , 𝑆0𝐾) : 𝑆0𝑘 ≤ 2𝐿

𝜁(H(𝑘)Φ∗𝑘H

(𝑘)†),

𝑆0𝑘 ≤ 2

𝜁(H(𝑘)Φ∗𝑘H

(𝑘)†)+Tr

(𝔼

[Tr(H(𝑘)† ˆΦ

∗𝑘H

(𝑘))

𝜆2max(𝔼[H(𝑘)†H(𝑘)])

H(𝑘)Φ∗𝑘H

(𝑘)†]A𝑘

) 1

2

}

over all the admissible {𝛼𝑖}𝐾𝑖=1, where

A𝑘 =

𝐾!∑𝑖=1

2𝛼𝑖

𝐾∑ℓ=𝜋𝑖(𝑘)+1

1

𝜌𝑘,ℓ𝔼

[H(ℓ)Φ

∗ℓH

(ℓ)†

Tr(H(ℓ)Φ∗ℓH

(ℓ)†)

].

Corollary 1 At low-SNR or equivalently in the wide-bandregime, the sum rate, 𝑅(SNR ) =

∑𝐾𝑘=1𝑅𝑘(SNR ), as function

of the 𝐸𝑏

𝑁0= SNR

𝑅(SNR )with 𝐸𝑏 denoting the “system” energy

ISIT 2010, Austin, Texas, U.S.A., June 13 - 18, 2010

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−1 −0.5 0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

1

1.2

Eb/N0 (db)

Sum

Rat

e (b

it/s/

Hz)

2 users asymptote2 users optimal4 users asymptote4 users optimal

Mi = 2 with i = 1,…,4

ΘT(1) = Θ

T(3) = I

( ΘT(2))

j,j’ = e−0.05 (j−j’)

2

( ΘT(4))

j,j’ = e−0.1 (j−j’)

2

Fig.1: Sum-rate and its linear approximation versus𝐸𝑏𝑁0

(in dB).

per bit 2 admits an affine expansion as the one given in (20),where

𝐸𝑏

𝑁0min

=log𝑒 2

𝔼[Tr(HΦ

∗H†

)]

=log𝑒 2∑𝐾

𝑘=1ℋ(𝜸𝑘), (30)

with 𝐺∗𝑘 = 𝜆max(𝔼[H(𝑘)†H(𝑘)]), and ℋ(𝜸𝑘) denoting the

harmonic mean of the elements of the 𝐾-dimensional vectorof 𝝆 = [𝑔1𝐺

∗1𝜌𝑘,1, . . . , 𝑔𝑗𝐺

∗𝑗𝜌𝑘,𝑗 , . . . , 𝑔𝐾𝐺

∗𝐾𝜌𝑘,𝐾 ], while the

slope is:

𝑆0 =2𝐿

𝜁(HΦ

∗H†

) , (31)

where Φ∗

= diag(Φ

∗1, . . . , Φ

∗𝐾

)with Φ

∗𝑘 given in (23).

V. NUMERICAL RESULTS

We consider an ideal hexagonal cell with the BS placed

at its center and uniform users distribution. The slow fading

components is generated as: 𝑔𝑘 = 𝜙𝑑−𝛾𝑘 𝜓𝑘, where 𝜙 is

the path-loss constant, 𝛾 the path-loss exponent, and 𝜓𝑘 the

shadowing term, while 𝑑𝑘 denotes the distance between the

base-station, and the 𝑘th user. In all numerical results we

assume that 𝐿 = 4 and the receive correlation matrix is given

by: (ΘR(𝑘))ℓ,ℓ′ = 𝑒−3∣𝑙−𝑙′∣𝑒𝑗2𝜋𝑓0/𝑐 cos(𝜙𝑘)(𝑙−𝑙′) where 𝜙𝑘 is

the angle that characterizes the position of the 𝑘th user in a

polar coordinate system with the base station at the pole and

the polar axis coinciding with the linear array. Fig. 1 represents

a numerical validation of Corollary 1 in the light of Theorem

1 and indeed represents the sum-capacity and its first order

behavior for two users and for four users, as a function of 𝐸𝑏

𝑁0

(in dB). The transmit correlation parameters and the number

of transmitting antennas are indicated in the figure.

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