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February 18, 2013

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ISSSTA****, Sydney, Australia, ** Aug. - * Sep. **04

Capacity of MIMO Channels with

One-sided Correlation

Giuseppa Alfano & Antonia M. Tulino Angel lozano Sergio Verd

Universit di Napoli Federico II

a Bell Labs (Lucent Technologies) Princeton University

Naples 80125, Italy Holmdel, NJ 07733, USA Princeton, NJ 08544, USA

Email: *******@**.*********.*** Email: ****@******.*** Email: *****@*********.***

at the output of an MMSE receiver) can be also derived.

Abstract We present closed-form expressions for the marginal

density distribution of the unordered eigenvalues of H H where By explicitly incorporating the input covariance, we can

is an input covariance and H is a matrix representing a MIMO express the capacity rather than simply the mutual information.

(multi-input multi-output) Rayleigh-faded channel with one-sided

The correlation is allowed to take place at either end of

correlation at either end of the link, transmitter or receiver, with

the link, a relevant point given that it is the end with the

no constraints on the numbers of antennas therein. Using the

fewest antennas that constrains the number of spatial degrees

foregoing distribution, we then derive analytical expressions for

the capacity. The expressions found are evaluated through several of freedom.

examples conducted with correlation structures of practical Throughout the paper, we use i,j to denote the (i,j )-th

interest.

entry of a matrix and j to indicate its j -th eigenvalue.

I. I NTRODUCTION II. P ROBLEM F ORMULATION

The central idea in multi-antenna communication is to Denoted by nT and nR the number of transmit and receive

generate a MIMO (multi-input multi-output) channel by trans- antennas, we consider the complex frequency- at linear model

mitting and receiving through antenna arrays [1] [2]. The

y = g Hx + n

deployment of these arrays on small platforms (compact base

stations, mobile terminals, portable computers, etc) almost

where x and y are the input and output vectors while n

inevitably results in some degree of correlation between

is white Gaussian noise. The channel is represented by the

the constituent antennas. This has motivated a multiplicity

(nR nT ) zero-mean random matrix g H normalized such

of efforts aimed at characterizing analytically the capacity

that

and mutual information (with an isotropic input) of MIMO

E [Tr{HH }] = nR nT .

channels in the presence of correlation. These problems have

In terms of the correlation between the entries of H, we adhere

been tacked primarily by means of asymptotic analysis: for

large numbers of antennas in [3] [6] and at low signal-to- to the so-called separable model whereby [12]

noise ratio in [7]. Non-asymptotically, the mutual information 1/ 2 1/ 2

H = R W T (1)

has been characterized in integral form in [8] and explicitly

in [9] [11]. While the expression in [11] allows for simulta-

where the (nR nT ) matrix W has IID zero-mean unit-

neous correlations at both transmitter and receiver, it is not

variance complex Gaussian random entries while the entries

immediately applicable if either end exhibits no correlation.

of T and R indicate, respectively, the correlation between

This case, relevant because it models the frequent scenario

transmit and between receive antennas.

where correlation is strongly dominant at either transmitter or

The capacity-achieving input covariance, normalized by its

receiver, is addressed in [9] and [10] albeit in the latter with

energy per dimension, is denoted by

the restriction that the correlation be at the end of the link with

E [xx ]

the most antennas. All these contributions obtain the mutual

= 1

x 2]

information through the moment-generating function.

nT E [

In this paper, we focus on one-sided correlation but with

where the normalization ensures that E [Tr{ }]=nT . The

emphasis on the distribution of the eigenvalues of H H

ergodic capacity is thus

where H is the channel matrix and the input covariance.

For channels with IID (independent identically distributed) 1/ 2 1/ 2

R W T T W

C (SNR) = E log2 det I + SNR

Rayleigh-faded entries, this distribution is given in [1]. Here,

we present its counterpart with one-sided correlation. Using where the expectation is over the distribution of W while

this distribution, we then obtain new expressions for the

E[ x 2]

capacity. Other communication-theoretical quantities that are =g

SNR 1 2

nR E [ n ]

functionals of this distribution (e.g., the signal-to-interference

0-7803-8408-3/04/$20.00 2004 IEEE 515

Denoting by R, T and P the respective diagonal eigen- We shall consider channels where either R =I or T P=I.

value matrices of R, T and, and invoking the unitary Both cases basically mirror each other with the caveat that,

invariance of W and the fact that to achieve capacity the while R is determined exclusively by the receive correla-

eigenvectors of must coincide [13] with those of the transmit tions, T P involves both the transmit correlations and the

correlation, T, the capacity can be rewritten as resulting power allocation [14]. In either case, the eigenvalues

(diagonal entries) are required to be nonzero and distinct.

R W T PW

C (SNR) = E log2 det I + SNR

Nonetheless, as will be illustrated in the examples, zero

eigenvalues or eigenvalues with plural multiplicity can be

where P de nes the capacity-achieving power allocation [14].

accommodated through simple perturbation techniques.

If T =I, then capacity is achieved when the input is isotropic

The proofs of the various results are relegated to the

(P=I). If T =I, however, then in general P=I [14].

Appendix.

The capacity, as well as other quantities of interest, is

determined by the nonzero eigenvalues of H H, which are

III. C HARACTERIZATION OF THE U NORDERED

given by those of R W T PW if nT nR and by those

E IGENVALUES

of T PW R W if nT nR . For notational convenience

we therefore let m=min(nT, nR ) and n=max(nT, nR ) and In this section, we provide expressions for the marginal

distribution of the unordered eigenvalues of S. Let us start with

introduce the matrix

correlations at the end of the link with the fewest antennas.

S = m W n W

Theorem 1: Let n m with n =I and with m =I where

where W is (m n) with zero-mean IID Gaussian entries 1, . . ., m are its m distinct eigenvalues. The marginal density

while distribution of an unordered eigenvalue of S is

m = T P and n = R nT nR m m

n m+j 1 e / i D(i, j )

f = L (4)

m = R and n = T P nT nR

i=1 j =1

With that,

with

m i (S)

SNR

C (SNR) = E log2 1 + L= 1 1

m m

)! det( m )n (

m =1 (n k )

i=1 k 0 is a positive constant, 0 F 0 is the hyperge-

which corresponds to a d-wavelength antenna separation and a

ometric function of exponential type of two square matrix

broadside (truncated) Gaussian power azimuth spectrum with

arguments of different dimensions [15], and p (q ) with p q

2 root-mean-square spread. For d=2, the marginal density

is the complex multivariate Gamma function [16]

distribution of an arbitrary eigenvalue of S=H H, given by

(4), is depicted in Fig 1. Also shown is the histogram of 10000

p (q ) = p(p 1)/2 (q )!

random realizations obtained via Montecarlo, which closely

matches the analytical function.

Starting from (3), we set out to obtain the density distribution

Next, we turn our attention to the scenario where the

of the unordered eigenvalues of S, from which the capacity

correlation is at the end of the link with the most antennas.

in (2) can be then computed. As detailed in the Appendix,

Theorem 2: Let n>m with m =I and with n =I where

we leverage the technique pioneered in [17] to evaluate (2) by

1, . . ., n are its n distinct eigenvalues. The marginal density

means of the Mellin transform [18].

516

by 1, . . ., m the m distinct eigenvalues of the corresponding

0.8

correlation matrix, the capacity is given by

analytical

0.7

m m

D(i, j )

simulation

(Ri,j + Ti,j )

C (SNR) = (7)

i n+ m j

0.6

loge 2

0.6

j =1 i=1

with

0.5

( 1)k+1 ( k )!

0.4 1

0.4

Ri,j = ! loge SNR i + +

pdf

m u k

k (SNR i /m)

u=1 k=1

0.3

d=2 k 1 1

u=1 u

SNR i

loge + + k +

Ti,j = ( 1)

0.2

(k 1)!

( + k ) (SNR i /m)

receiver

k=1

transmitter

0.1

where L and D(i, j ) are as in Theorem 1 while =n m+j 1

and 0.5772 is the Euler-Mascheroni constant.

0 0

0 2 4 6 *-**-**-**-**-** 20

Proof: See Appendix.

0 20

(HH+) Before proceeding with the converse scenario, we proceed

to exemplify Theorem 3.

Fig. 1. Marginal density distribution of an unordered eigenvalue of S with Example 2: Consider the same scenario of Example 1. The

nT =4, nR =2, T =I and with R the diagonal eigenvalue matrix of R

capacity for various antenna separations, d, is shown in Fig. 2

in (5).

alongside corresponding Montecarlo simulations. (The case

d=10, in particular, results in the eigenvalues of R being

1.01 and 0.99, which can be seen as a perturbation of R =I.)

distribution of an unordered eigenvalue of S is

In every case, the agreement is excellent.

m m

Theorem 4: Consider a Rayleigh-faded channel with cor-

n m 1

j 1 D(i, j ) n m+i e / n m+i

f = L

relation at the end of link with the most antennas. Denoting

i=1 j =1

by 1, . . ., n the n distinct eigenvalues of the corresponding

n m n m

( 1 )k, n 1 +i n m 1 e / correlation matrix, the capacity is given by

m m

=1 k=1

D(i, j ) +i 1 (R +i,j + T +i,j )

loge 2

with i=1 j =1

det L= (6)

m 1

( 1 )k, +i +j 1 (R

k 1

( k ) +T

m !,j ),j

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