Random Beamforming over Correlated Fading Channels

Abstract

We study a multiple-input multiple-output (MIMO) multiple access channel
(MAC) from several multi-antenna transmitters to a multi-antenna receiver. The
fading channels between the transmitters and the receiver are modeled by random
matrices, composed of independent column vectors with zero mean and different
covariance matrices. Each transmitter is assumed to send multiple data streams
with a random precoding matrix extracted from a Haar-distributed matrix. For
this general channel model, we derive deterministic approximations of the
normalized mutual information, the normalized sum-rate with
minimum-mean-square-error (MMSE) detection and the
signal-to-interference-plus-noise-ratio (SINR) of the MMSE decoder, which
become arbitrarily tight as all system parameters grow infinitely large at the
same speed. In addition, we derive the asymptotically optimal power allocation
under individual or sum-power constraints. Our results allow us to tackle the
problem of optimal stream control in interference channels which would be
intractable in any finite setting. Numerical results corroborate our analysis
and verify its accuracy for realistic system dimensions. Moreover, the
techniques applied in this paper constitute a novel contribution to the field
of large random matrix theory and could be used to study even more involved
channel models.

Full text

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Random Beamforming over Correlated Fading
Channels
Jakob Hoydis, Romain Couillet, and M´ erouane Debbah
Abstract
We study a multiple-input multiple-output (MIMO) multiple access channel (MAC) from several
multi-antenna transmitters to a multi-antenna receiver. The fading channels between the transmitters and
the receiver are modeled by random matrices, composed of independent column vectors with zero mean
and different covariance matrices. Each transmitter is assumed to send multiple data streams with a
random precoding matrix extracted from a Haar-distributed matrix. For this general channel model, we
derive deterministic approximations of the normalized mutual information, the normalized sum-rate with
minimum-mean-square-error (MMSE) detection and the signal-to-interference-plus-noise-ratio (SINR) of
the MMSE decoder, which become arbitrarily tight as all system parameters grow infinitely large at the
same speed. In addition, we derive the asymptotically optimal power allocation under individual or sum-
power constraints. Our results allow us to tackle the problem of optimal stream control in interference
channels which would be intractable in any finite setting. Numerical results corroborate our analysis
and verify its accuracy for realistic system dimensions. Moreover, the techniques applied in this paper
constitute a novel contribution to the field of large random matrix theory and could be used to study
even more involved channel models.
J. Hoydis is with the Department of Telecommunications and the Alcatel-Lucent Chair on Flexible Radio, Sup´ elec, 91192
Gif-sur-Yvette, France (e-mail: jakob.hoydis@supelec.fr).
R. Couillet is with the EDF Chair on System Sciences and the Energy Challenge, Centrale Paris-Sup´ ele, 91192 Gif-sur-Yvette,
France (e-mail: romain.couillet@supelec.fr).
M. Debbah is with the Alcatel-Lucent Chair on Flexible Radio, Sup´ elec, 91192 Gif-sur-Yvette, France (e-mail: mer-
ouane.debbah@supelec.fr).
arXiv:1105.0569v1 [cs.IT] 3 May 2011

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I. INTRODUCTION
The gains of having multiple antennas at the transmitter and receiver in wireless fading point-to-
point channels are well established [1], [2]. It is known since the Telatar’s seminal paper [2] that when
channel state information (CSI) is available at the transmitter, the optimal transmission strategy is to
send independent data streams along the eigenmodes of the channel and to allocate power over these
eigenmodes according to the water-filling principle [3]. If no CSI is available at the transmitter but
the statistical properties of the channel are known, an optimal static power allocation which does not
depend on the actual channel realizations and maximizes the ergodic mutual information can be found.
Uniform power allocation is optimal when the channel entries are independent and identically distributed
(i.i.d.) Gaussian [2] or zero-mean symmetric [4]. It was shown in [5] that the optimality of uniform
power allocation also extends to the multiple-access channel. When not ergodic but outage capacity is
considered, it was conjectured [2] that allocating equal power to only a subset of the available transmit
antennas is optimal. This conjecture was proved for the Gaussian multiple-input single-output (MISO)
channel in [6].
In the presence of co-channel interference, much less is known about the optimal transmission strategies.
Recently, an exact expression of the ergodic mutual information of a Rayleigh fading multiple-input
multiple-output (MIMO) system in the presence of different MIMO interferers with arbitrary transmit
power levels was derived [7]. Since more involved channel models are intractable by exact analysis,
many works resort to asymptotic analyses where each transmitter and receiver is equipped with a large
number of antennas. The authors of [8] consider a doubly-correlated fading MIMO channel with correlated
interference and derive asymptotically tight approximations of the mutual information and its variance.
In [9], the asymptotic mutual information and its fluctuations are studied for arbitrary fading channels
with a variance profile and correlated noise. However, even in the asymptotic setting, the optimal transmit
strategies in interference channels are in general unknown.
An important question in MIMO systems with co-channel interference is whether a transmitter should
use all of its antennas to transmit independent data streams or whether it should restrict itself to a
smaller number of streams or antennas in order to reduce the interference to other receivers. In general,
this problem does not have a simple solution and the optimal number of antennas to be used (or streams
to be sent) depends on the strength of the interference and, thus, on the cross-channel gains between
the interferers. The authors of [10] pioneered this question, assuming that no CSI is available at the
transmitters while full CSI is available at the receivers. Their main finding is that when the interference

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is weak, a transmitter should send independent streams with equal power from each of its antennas,
but when the interference is strong, all power should be put into a single stream which is transmitted
by a single antenna. In [11], it was shown that optimizing the number of transmitted data streams is
also helpful when CSI is available at the transmitter. Several later works [12], [13], [14] have studied
the same problem in the context of dense random ad hoc networks under different assumptions about
the availability of CSI at the transmitters and receivers and the corresponding transmit and reception
strategies. Surprisingly, the conclusions in all of these works are similar, confirming the optimality of
single-stream transmissions in dense, interference-limited networks.
The aforementioned references share the underlying assumption that the channel matrices/vectors are
composed of i.i.d. elements without any form of correlation. Thus, the problem of how many antennas
should be used for transmission and how many independent data streams should be sent are the same. With
transmit antenna correlation, however, it makes a difference which antennas are selected for transmission
and the question of the optimal number of antennas to be used becomes a combinatorial problem. To
circumvent this issue, random isometric precoding can be used to mitigate transmit correlation. The
remaining question is then how many orthogonal streams should be sent, using all available antennas.
This is the primal motivation of this paper, as our results allow to study the sum-rate of systems composed
of multiple transmitter-receiver pairs, each applying random isotropic beamforming. Random isotropic
beamforming [15] is a well studied technique in multi-user MIMO communication systems and unitary
precoders [16] are now proposed as limited feedback beamforming solutions in future wireless standards
[17], [18]. Nevertheless, only few related works relying on tools from large random matrix theory have
been published until today (e.g. [19]) and this paper might stimulate further research in this area.
The contributions of this paper are as follows. We consider a MIMO multiple access channel (MAC)
from several multi-antenna transmitters to a multi-antenna receiver. The transmitters are unaware of the
channel realizations and send an arbitrary number of independent data streams using isometric random
beamforming vectors. The receiver is assumed to be aware of all instantaneous channel realizations
and beamforming vectors. We assume a very general channel model where the channel matrices are
composed of independent, zero mean column vectors, each with a possibly different covariance matrix.
This channel model allows to treat many classes of well-known channel models, such as matrices with
a variance profile [20] as well as the Kronecker model [21]. Under these general assumptions, we
derive deterministic approximations of the normalized mutual information, the normalized sum-rate with
minimum-mean-square-error (MMSE) detection and the signal-to-interference-plus-noise-ratio (SINR) of
the MMSE decoder, which become arbitrarily tight as all system parameters grow infinitely large at

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the same speed. The expressions are given as functions of a set of deterministic quantities which can
be computed by a standard fixed-point algorithm which is proved to converge. Moreover, we derive
the optimal power allocation under individual or sum-power constraints which can be computed by an
iterative water-filling algorithm. We then apply these results to find the optimal number of independent
streams to be transmitted in a 2×2 interference channel. Although the use of deterministic approximations
in this context requires an exhaustive search over all possible stream-configurations, it is computationally
much less expensive than Monte Carlo simulations. Extensions to more than two transmit-receive pairs
and possible different objective functions, e.g. weighted sum-rate or sum-rate with MMSE decoding, are
straightforward. Our numerical results show that the deterministic approximations are very tight for even
small system dimensions. We further show that with random beamforming, it is optimal to (i) send as
many independent data streams as transmit antennas and (ii) allocate power uniformly over the transmitted
streams. For the interference channel, we find that at low SNR, it is optimal to use all streams while at
high SNR, stream-control, i.e., transmitting less than the maximal number of streams, is beneficial. Apart
from these practical applications, our work also constitutes a novel contribution to the field of random
matrix theory as will be highlighted in Section III.
The remainder of this paper is organized as follows: In Section II, we present a detailed description of
the system model and make several definitions of frequently used quantities. Section III summarizes recent
results on large random matrix theory involving Haar distributed matrices, which will be extended in this
paper. We present our main results in Section IV and show numerical results for practical applications
such as optimal stream control for the MIMO interference channel in Section V. The paper is concluded
in Section VI. Related results, lemmas and the proofs of all theorems are provided in the appendix.
Notations: Boldface lower and upper case symbols represent vectors and matrices, respectively. IN
is the size-N identity matrix and diag(x1,...,xN) is a diagonal matrix with elements xi. The trace,
transpose and Hermitian transpose operators are denoted by tr(·), (·)Tand (·)H, respectively. The spectral
norm of a matrix A is denoted by ?A?, and, for two matrices A and B, the notation A ? B means
that A − B is positive-definite. The notations ⇒ and
respectively. We use CN (m,R) to denote the circular symmetric complex Gaussian distribution with
mean m and covariance matrix R. We denote by R+the set [0,∞) and by C+the set {z ∈ C,Im[z] > 0}.
Denote by C the set of continuous functions from X ⊂ C to Y ⊂ C and by S the class of functions
f analytic over C \ R+, such that, for z ∈ C+, f ∈ C+, zf ∈ C+and limy→∞−iyf(iy) < ∞. Such
functions are known to be Stieltjes transforms of finite measures supported by R+(see e.g. [22]).
a.s.
− → denote weak and almost sure convergence,

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II. SYSTEM MODEL
Consider the following discrete-time MIMO channel with output vector y ∈ CN:
y =
K
?
k=1
HkWkP
1
2
kxk+ n
(1)
where, for k ∈ {1,...,K},
(i) Hk∈ CN×Nkis a random channel matrix whose jth column vector hkj∈ CNis modeled as
1
2
kjukj,
hkj= R
j = 1,...,Nk
(2)
where Rkj∈ CN×Nare Hermitian nonnegative definite matrices and the vectors ukj∈ CNhave
independent and identically distributed (i.i.d.) elements with zero mean, variance 1/N and finite
moment of order 4 + ?, for some common ? > 0,
(ii) Wk∈ CNk×nkis a complex precoding matrix which contains nk≤ Nkorthonormal columns of an
Nk× NkHaar-distributed random unitary matrix,
(iii) Pk= diag(pk1,...,pknk) ∈ Rnk×nk
(iv) xk∼ CN(0,Ink) is the transmit vector of the kth transmitter,
(v) n ∼ CN(0,ρIN) is a noise vector.
+
is a nonnegative diagonal matrix,
Remark 1: The statistical model (2) generalizes several well-know channel models of interest (see
[23], [24] for examples). It comprises in particular the Kronecker channel model with transmit and receive
correlation matrices [25], [21], where the matrices Hkare given as
Hk= R
1
2
kUkT
1
2
k
(3)
where Uk∈ CN×Nkis a random matrix whose elements are independent CN(0,1/N) and Rk∈ CN×N
and Tk∈ CNk×Nkare covariance matrices. Since both Ukand Wkare unitarily invariant, we can assume
without loss of generality for the statistical properties of y that Tk= diag(tk1,...,tkNk). Defining the
matrices Rkj= tkjRkfor j = 1,...,Nk, we fall back to the channel model in (2).
Remark 2: Whenever the distribution of Hkis invariant by multiplications by unitary matrices from
the right side (e.g. for (3) with Tk= INk), our channel model boils down to
y =
K
?
k=1
HkP
1
2
kxk+ n
which has been studied in [23] for the general case (2) and in [21] for the Kronecker model (3). The same
holds for nk= Nkfor all k with uniform power allocation, i.e., Pk= Ink, since WkPkWH
k= INk.