IEEE TRANSACTIONS ON INFORMATION THEORY, 2018

1

Capacity Scaling in MIMO Systems with General
Unitarily Invariant Random Matrices

arXiv:1306.2595v4 [] 5 Mar 2018

Burak Çakmak, Ralf R. Müller, Senior Member, IEEE, Bernard H. Fleury, Senior Member, IEEE

Abstract—We investigate the capacity scaling of MIMO systems with the system dimensions. To that end we quantify how
the mutual information varies when the number of antennas (at
either the receiver or transmitter side) is altered. For a system
comprising R receive and T transmit antennas with R > T ,
we find the following: By removing as many receive antennas as
needed to obtain a square system (provided the channel matrices
before and after the removal have full rank) the maximum
resulting loss of mutual information over all signal-to-noise ratios
(SNRs) depends only on R, T and the matrix of left-singular
vectors of the initial channel matrix, but not on its singular
values. In particular, if the latter
is Haar distributed the
P matrix
P
1
ergodic rate loss is given by Tt=1 R
r=T +1 r−t nats. Under the
same assumption, if T, R → ∞ with the ratio φ , T /R fixed, the
rate loss normalized by R converges almost surely to H(φ) bits
with H(·) denoting the binary entropy function. We also quantify
and study how the mutual information as a function of the system
dimensions deviates from the traditionally assumed linear growth
in the minimum of the system dimensions at high SNR.
Index Terms—multiple-input–multiple-output, mutual information, high SNR, multiplexing gain, unitary invariance, binary
entropy function, Haar random matrix, S-transform

I. I NTRODUCTION
HE capacity of a multiple-input–multiple-output (MIMO)
system with perfect channel state information at the
receiver can be expressed as [1]

T

min(T, R) log2 SNR + O(1)

(1)

whenever the channel matrix has full rank almost surely.
Here T and R denote the number of receive and transmit
antennas, respectively, and O(1) is a bounded function of the
signal-to-noise ratio (SNR) that does depend on T and R, in
general. The scaling term min(T, R) is often referred to as
the multiplexing gain. The explicit expression for the capacity
scaling when the number of transmit or receive antennas
varies, is difficult to calculate. Closed-form expressions can
be obtained only in few particular cases, e.g. for a channel
matrix of asymptotically large size with independent identically distributed (iid) zero-mean entries [2].
Burak Çakmak and Bernard H. Fleury were supported by the research
project VIRTUOSO funded by Intel Mobile Communications, Keysight,
Telenor, Aalborg University, and the Danish National Advanced Technology
Foundation. Ralf R. Müller was supported by the Alexander von Humboldt
Foundation.
Burak Çakmak is with the Department of Computer Science, Technical
University of Berlin, 10587 Berlin, Germany (e-mail: burak.cakmak@tuberlin.de).
Ralf R. Müller is with the Institute for Digital Communications, FriedrichAlexander Universität Erlangen-Nürnberg, 91058 Erlangen, Germany (e-mail:
mueller@lnt.de).
Bernard H. Fleury is with the Department of Electronic Systems, Aalborg
University, 9220 Aalborg, Denmark (e-mail: fleury@es.aau.dk).

In order to better understand capacity scaling in MIMO
channels with more complicated structures, such as correlation
at transmit and/or receive antennas, related works use either
implicit solutions, e.g. [3], or consider asymptotically high
SNR and express the capacity in terms of the multiplexing
gain, e.g. [4]. However, implicit solutions provide limited
intuitive insight into the capacity scaling and the multiplexing
gain is a crude measure of capacity.
In this article, we consider an affine approximation to the
mutual information at high SNR. In particular, we investigate
how mutual information varies when the numbers of antennas
(at either the receiver or transmitter side) is altered. Our affine
approximation to the mutual information leads to a generalization of the multiplexing gain which we call the multiplexing
rate. Such an approximation was formerly addressed in [1],
which was the baseline of many published works, e.g. [5]–[7].
We study the variation of the multiplexing rate when the
number of antennas either at the transmit or receive side varies.
More specifically, we formulate the reduction of the number of
antennas by means of a convenient linear projection operator.
This formulation allows us to asses the mutual information at
high SNR in insightful and explicit closed form. We consider
unitarily invariant matrix ensembles [8] which model a broad
class of MIMO channels [9]. Specifically, our sole restriction
is that the matrix of left (right) singular vectors of the initial
channel matrix, i.e. before the reduction, is Haar distributed.
Informally speaking, this implies that the channel matrix
involves some symmetry with respect to the antennas. An
individual antenna contributes in a “democratic fashion” to
the mutual information. There is no preferred antenna in the
system. In fact, such an invariance seems a natural property
for the mutual information to depend on T and R only, but
not on the specific antennas in the system.
Since the term O(1) in (1) is a bounded function of SNR,
the expression (1) has more than once led to misinterpretations
in the wireless communications community:
(i) when the number of antennas at either the transmit or
receive side varies, while the minimum of the system
dimensions (i.e. the numbers of transmit and receive
antennas) is kept fixed, the mutual information does not
vary at high SNR;
(ii) the mutual information scales linearly with the minimum
of the system dimensions at high SNR.
It is the goal of this paper to debunk these misinterpretations.
We summarize our main contributions as follows:
1) As regards misinterpretation (i) we find the following: For
a system comprising R receive and T transmit antennas

IEEE TRANSACTIONS ON INFORMATION THEORY, 2018

with R > T (T > R), let some of the receive (transmit)
antennas be removed from the system to obtain a system
with R̃ ≥ T receive (T̃ ≥ R transmit) antennas. Note
that min(T, R̃) = T (min(T̃ , R) = R). Then, the loss
of mutual information in the high SNR limit depends
only on R, T and R̃ (T̃ ) and the matrix of left (right)singular vectors of the initial R × T channel matrix,
but not on its singular values. Assuming the matrix
of left-(right-)singular vectors to bePHaarPdistributed,
T
R
1
the ergodic rate loss is given by
t=1
r=R̃+1 r−t
PR PT
1
( r=1 t=T̃ +1 t−r ) nats.
2) As regards misinterpretation (ii), we quantify how the
mutual information as a function of the number of antennas deviates from the approximate linear growth (versus
the minimum of the system dimensions) in the high SNR
limit. This deviation does depend on the singular values
of the channel matrix. We show that in the large system
limit the deviation is additive for compound unitarily
invariant channels and can be easily expressed in terms
of the S-transform (in free probability) of the limiting
eigenvalue distribution (LED) of the Gramian of the
channel matrix.
3) We show that the aforementioned results on the variation
of mutual information in the high SNR limit provide least
upper bounds on said variation over all SNRs. Thus, these
results have a universal character related to the SNR.
4) We derive novel formulations of the mutual information
and the multiplexing rate in terms of the S-transform
of the empirical eigenvalue distribution of the Gramian
of the channel matrix. These formulations establish a
fundamental relationship between the mutual information
and the multiplexing rate.
A. Related Work
The work presented in paper [5] is related to contribution 1).
Specifically, in [5, Section 3] the authors unveiled misinterpretation (i) for iid Gaussian unitarily invariant channel matrices.
We elucidate misinterpretation (i) by considering arbitrary
unitarily invariant matrices that need neither be Gaussian nor
iid. In particular, our results and/or statements do not require
any assumptions on the singular values of the channel matrix.
They solely depend on the singular vectors of the channel
matrix, e.g. see contribution 1). Our proof technique - which is
based on an algebraic manipulation of the projection operator
that we introduce - is different from any related work we are
aware of.
B. Organization
The paper is organized as follows. In Section II, we introduce the preliminary notations and definitions. In Section III,
we present the system model. In Section IV, we introduce new
formulations of the mutual information and the multiplexing
rate in terms of the S-transform. Section V and VI are
dedicated to lift misinterpretations (i) and (ii), respectively.
Conclusions are outlined in Section VII. The technical lemmas
and the proofs are located in the Appendix.

2

II. N OTATIONS & D EFINITIONS
N OTATION 1 We denote the binary entropy function as
(
(p − 1) log2 (1 − p) − p log2 p p ∈ (0, 1)
. (2)
H(p) ,
0
p ∈ {0, 1}
N OTATION 2 For an N × K matrix X, FK
X denotes the
empirical eigenvalue distribution function of X † X, i.e.
FK
X (x) =

1
| {λi ∈ L : λi ≤x} |
K

(3)

with L and |·| denoting the set of eigenvalues of X † X and the
cardinality of a set, respectively. Here, (·)† denotes conjugate
transposition. Moreover, for N, K → ∞ with φ = K/N fixed,
if FK
X converges weakly and almost surely to a LED function,
this limit is denoted by FX .
D EFINITION 1 A K-dimensional projector P β with β ≤ 1 is
a βK × K matrix with entries (P β )ij = δij , ∀i, j, where δij
denotes the Kronecker delta.
D EFINITION 2 For an N × K matrix X 6= 0, we define the
normalized rank of X † X as
K
αX
, 1 − FK
X (0)

(4)

and the distribution function of non-zero eigenvalues of X † X
as

1  K
αX − 1 u(x) + FK
(5)
F̃K
X (x)
X (x) , K
αX
with u(x) denoting the unit-step function.
The S-transform introduced by Voiculescu in the context of
free probability is defined as follows:
D EFINITION 3 [10] Let F be a probability distribution function with support in [0, ∞). Moreover, let α , 1 − F(0) 6= 0.
Define
Z
zx
Ψ(z) ,
dF(x), −∞ < z < 0.
(6)
1 − zx
Then, the S-transform of F is defined as
S(z) ,

z + 1 −1
Ψ (z),
z

−α < z < 0

(7)

where Ψ−1 denotes the composition inverse of Ψ.
N OTATION 3 For an N × K matrix X 6= 0, the S-transform
K
of FK
X is denoted by SX . For N, K → ∞ with φ = K/N
†
fixed, if X X has a LED function FX almost surely, the Stransform of FX is denoted by SX . Similarly, we define ΨK
X
and ΨX .
All large-system limits are assumed to hold in the almost
sure sense, unless explicitly stated otherwise. Where obvious,
limit operators indicating the large-system limit are omitted
for the sake of compactness and readability.

IEEE TRANSACTIONS ON INFORMATION THEORY, 2018

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III. S YSTEM M ODEL

B. Unitary Invariance

Consider the MIMO system
y = Hx + n

(8)

where H ∈ CR×T , x ∈ CT ×1 , y ∈ CR×1 , n ∈ CR×1 are
respectively the channel matrix, the input vector, the output
vector, and the noise vector. The entries of x and n are
assumed to be independent (circularly symmetric) complex
Gaussian distributed with zero mean and variances σx2 and
σn2 , respectively. The transmit SNR is defined as
γ,

σx2
,
σn2

0 < γ < ∞.

(9)

The mutual information per transmit antenna of the communication link (8) is given by [11]
Z
I(γ; FTH ) , log2 (1 + γx) dFTH (x).
(10)
Similarly, I(γ; FR
) is the mutual information per receive
H†
antenna of (8).
A. Antenna Removal Via Projector

I(γ; FTV HU ) = I(γ; FTH )

In the sequel, we formulate the variation of mutual information when the number of antennas either at the transmit or
receive side of reference system (8) changes. This variation
is achieved by removing a certain fraction of antennas at the
corresponding side of the system. We formulate this removal
process via a multiplication of the channel matrix with a
rectangular projector matrix.
We distinguish between two cases: the removal of receive
antennas and the removal of transmit antennas. In the first case,
the system model resulting after removing a fraction 1 − β of
receive antennas in (8) reads
y β = P β (Hx + n)

(11)

= P β Hx + nβ .

(12)

The βR × R matrix P β is an R-dimensional projector which
removes a fraction 1 − β of receive antennas in reference
system (8) and nβ = P β n. The mutual information of the
MIMO system (12) is equal to
T I(γ; FTP β H ).

(13)

Similarly, removing a fraction 1 − β of transmit antennas in
(8) yields the R × βT system
ỹ = HP †β xβ + n.

(14)

Here, xβ is the vector obtained by removing from x the (1 −
β)T entries fed to the removed transmit antennas, i.e. xβ =
P β x with P β being a T -dimensional projector. The mutual
information of system (14) reads
βT I(γ; FβT

HP †β

For channel matrices that are unitarily invariant from right,
i.e. H and HU admit the same distribution for any unitary
matrix U independent of H, it does not matter which transmit
antennas are removed. Only their number counts. The same
applies to channel matrices that are unitarily invariant from
left for the removal of receive antennas. For channel matrices
that involve an asymmetry with respect to the antennas, i.e.
some antennas contribute more to the mutual information
than others, it must be specified which antennas are to be
removed and the mutual information will depend (typically in
a complicated manner) on the choice of the removed antennas.
In this paper, we restrict the considerations to cases where only
the number of removed antennas matters, since this leads to
explicit closed-form expressions.
For asymmetric channel matrices, one could obtain antennaindependent scaling laws if all antennas with equal contributions to mutual information are grouped together and all those
groups are decimated proportionally. Doing so would heavily
complicate the formulation of the antenna removal by means
of multiplication with projector matrices. However, we can
utilize the fact that for the channel in (8), mutual information
is invariant to multiplication with unitary matrices, i.e.

).

(15)

(16)

for all unitary matrices U and V . Since the channel matrix
U HV is bi-unitarily invariant for all random unitary matrices
U and V independent of H, and has the same mutual
information as H, we can assume without loss of generality
that H is unitarily invariant from left for receive and from
right for transmit antenna removal, respectively, and keep the
projector formulation of Section III-A as it is.
The multiplication with a random unitary matrix followed
by a fixed selection of antennas has statistically the same effect
as a random selection of antennas. It provides the symmetry required to make mutual information only depend on the number
of removed antennas and not on which antennas are removed.
Equivalence to the ergodic capacity variation: The ergodic
capacity of channel (8) is [12]
i
h
¯ FTH ) , max E I(γ; FT √ ) .
(17)
C(γ,
H Q
Q≥0
tr(Q)=T

Conceptually, we relax the iid assumption on the entries of x
in (8) and assume arbitrary correlation between these entries
described by the covariance matrix σx2 Q where Q is nonnegative definite with unit trace and σx2 , T1 E[x† x]. It is
shown in [12] that for channel matrices that are unitarily
invariant from right the ergodic capacity in (17) is attained
with Q = I, i.e.


¯ FT ) = E I(γ; FT ) .
C(γ,
(18)
H
H
In particular, the unitary invariance property of the channel
is not broken by removing some of the transmit or receive
antennas. For example, if H is invariant from right, then
HP †β is invariant from right too. In summary, for bi-unitarily
invariant channel matrices the variation of ergodic mutual
informations that results from removing some number of

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transmit or receive antennas does actually coincide with the
corresponding variation of ergodic capacities.
IV. M UTUAL I NFORMATION AND M ULTIPLEXING R ATE
The normalized mutual information in (10) can be decomposed as
Z
T
T
I(γ; FH ) = αH log2 (γx) dF̃TH (x)
{z
}
|
I0 (γ;FT
H)

+

T
αH

|

Z



1
log2 1 +
dF̃TH (x) .
xγ
{z
}

(19)

∆I(γ;FT
H)

We refer to the first term I0 (γ; FTH ) as the multiplexing
T
rate per transmit antenna. The factor αH
is the multiplexing
gain normalized by the number of transmit antennas. The
second term ∆I(γ; FTH ) is the difference between the mutual
information per transmit antenna and the multiplexing rate per
transmit antenna. To alleviate the terminology, in the sequel we
skip the explicit reference to the normalization by the number
of transmit (or receive, see later) antennas when we refer to
quantities such as those arising in (19). Whether the quantities
considered are absolute or normalized will be clear from the
context. We have
lim ∆I(γ; FTH ) = 0.

(20)

γ→∞

If H † H is invertible we have


1
(21)
I0 (γ; FTH ) = log2 det γH † H
T


1
∆I(γ; FTH ) = log2 det I + (γH † H)−1
(22)
T
with I denoting the identity matrix.
The affine approximation of the ergodic mutual information
at high SNR introduced in [1], see also [5, Eq. (9)] for a compact formulation of it, coincides with the ergodic formulation
of our definition of the multiplexing rate.
We next uncover a fundamental link between the mutual
information and the multiplexing rate. This result makes use
of the minimum-mean-square-error (MMSE) achieved by the
optimal receiver for (8) normalized by the number of transmit
antennas
Z
dFTH (x)
T
.
(23)
ηH (γ) ,
1 + γx
T
Clearly, ηH
(γ) is a strictly decreasing function of γ with range
T
(1 − αH , 1) [9].

T HEOREM 1 Define
Z
fH (x) , H(x)−

x

log2 STH (−z) dz,

T
0 ≤ x ≤ αH
. (24)

0

Then, we have
T
T
I(γ; FTH ) = fH (1 − ηH
) + (1 − ηH
) log2 γ
T
T
I0 (γ; FTH ) = fH (αH
) + αH
log2 γ.
T
T
For short we write ηH
for ηH
(γ) in (25).

(25)
(26)

P ROOF 1 See Appendix B.
Note that by definition the function fH (x) in (24) may
T
involve αH
via STH (z). We have the following implications of
Theorem 1: i) the mutual information can be directly expressed
as a function of the (normalized) MMSE; ii) for any expression
T
of the mutual information as a function of the MMSE ηH
the
T
multiplexing rate results immediately by substituting ηH for
T
1−αH
, e.g. see Examples 1 and 2; iii) the converse of ii) is not
always true: given an expression of the multiplexing rate as a
T
T
T
function of αH
, substituting αH
for 1 − ηH
does not always
yield the mutual information. An intermediate step is required
here to guarantee that the converse holds: the expression needs
T
first to be recast as a function of fH . Then substituting αH
T
for 1−ηH
in the latter function yields the mutual information.
If any probability distribution function with support in
[0, ∞), say F, is substituted for FTH in (19) the formulas (25)
and (26) remain valid provided I(γ; F) is finite and log(x) is
absolutely integrable over F̃, respectively1 . The absolute integrability condition holds if, and only if, I(γ; F) and ∆I(γ; F)
are finite, see (171)-(173). In the sequel we substitute FH for
FTH to calculate I(γ; FH ) and I0 (γ; FH ). In Appendix C, we
provide some sufficient conditions that guarantee the almost
sure convergence of I(γ; FTH ) and I0 (γ; FTH ) to I(γ; FH ) and
I0 (γ; FH ), respectively. We conclude that these asymptotic
convergence are reasonable assumptions in practice, for the
details see Appendix C.
It is well-known that the S-transform of the LED of the
product of asymptotically free matrices is the product of
the respective S-transforms of the LEDs of these matrices.
Therefore, for MIMO channel matrices that involve a compound structure, Theorem 1 provides a means to analytically
calculate the large-system limits of the mutual information
and multiplexing rate in terms of the large-system limits of
the MMSE and the multiplexing gain. We next address two
relevant random matrix ensembles that share this structure.
E XAMPLE 1 We consider the concatenation of vector-valued
fading channels described in [13]. Specifically, we assume that
the channel matrix H factorizes according to
H = X N X N −1 · · · X 2 X 1

(27)

where the entries of the Kn × Kn−1 matrix X n are iid with
zero mean and variance 1/Kn for n ∈ [1, N ]. Furthermore,
the ratios ρn , Kn /K0 n ∈ [1, N ] are fixed as Kn → ∞.
T
Moreover, let ηH denote the large-system limit of MMSE ηH
.
By invoking Theorem 1 we obtain an analytical expression of
the large-system limit of the mutual information in terms of
(the large-system limit of) the MMSE2 as
I(γ; FH ) = H(ηH ) + (1 − ηH )(log2 γ − N log2 e)
#
"N


X ρn
1 − ηH
1 − ηH
H
+ log2
.
+ (1 − ηH )
1 − ηH
ρn
ρn
n=1
(28)
F̃ is defined by substituting F̃T
H for F in (5).
explicit expression of the MMSE as a function of SNR is difficult
to obtain. However, ηH (γ) can be solved numerically from the fixed point
ηH (γ) QN
ηH (γ)+ρn −1
equation γ = 1−η
[13, Eq. (21)].
n=1
(γ)
ρ
1 Here
2 An

H

n

IEEE TRANSACTIONS ON INFORMATION THEORY, 2018

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Furthermore, as regards the multiplexing rate, we have
I0 (γ; FH ) =H(αH ) + αH (log2 γ − N log2 e)
#
"N
X ρn  αH 
αH
H
+ log2
(29)
+ αH
α
ρn
ρn
n=1 H
with αH = min(1, ρ1 , · · · , ρN ).

Normalizing this loss with the number of transmit antennas
yields
I(γ; FTH ) − I(γ; FTP β H ).
(34)
Assume that H and P β H have both full rank almost surely.
Then, we define the rate loss
χTH (R, βR) , lim I(γ; FTH ) − I(γ; FTP β H ), β ≥ φ. (35)
γ→∞

= I0 (γ; FTH ) − I0 (γ; FTP β H )

P ROOF 2 See Appendix D.

(36)

†

E XAMPLE 2 We consider a Jacobi matrix ensemble, see e.g.
[14], [15], which find application in the context of optical
MIMO communications [16], [17]. Accordingly, the channel
matrix factorizes as
H = P β2 U P †β1

(30)

where U is an N × N Haar unitary matrix. From Theorem 1
we obtain
I(γ; FH ) =H(ηH ) + (1 − ηH ) log2 γ


β1
H(β1 (1 − ηH )) β2
+ H
(1 − ηH )
−
β1
β1
β2
(31)
where ηH = ηH (γ) is given by
p
−(1 + κγ) + (1 + κγ)2 − 4β1 β2 γ(1 + γ)
ηH (γ) = 1 +
2β1 (1 + γ)
(32)
with κ , β1 + β2 . Moreover, we have
I0 (γ; FH ) =H(αH ) + αH log2 γ


H(β1 αH ) β2
β1
−
+ H
αH
β1
β1
β2

(33)

with αH = min(1, β2 /β1 ).
P ROOF 3 See Appendix E.
V. T HE U NIVERSAL R ATE L OSS
In Section 1 we underlined the following misinterpretation
of mutual information: when the number of antennas (at either
the transmit or receive side) varies, with the minimum of the
system dimensions kept fixed, the mutual information does not
vary at high SNR. It is the goal of this section to elucidate this
misinterpretation. To do so we need to distinguish between two
cases as to reference system (8): (i) T ≤ R; (ii) T ≥ R. In
the former (latter) case we consider the removal of receive
(transmit) antennas. In both cases the reduction of antennas is
constrained in a way that keeps the minimum of the numbers
of antennas at both sides fixed.
A. Case (i) – Removing receive antennas
We remove a fraction (1 − β) of receive antennas in system
(8) to obtain system (12). We constrain the reduction with
the condition β ≥ φ , T /R to ensure that min(T, βR) = T .
This reduction of the number of receive antennas causes a loss
in mutual information given by T I(γ; FTH ) − T I(γ; FTP β H ).

=

det H H
1
log2
.
T
det H † P †β P β H

(37)

T
T
The full-rank assumption implies αH
= αP
which is
βH
essential in the definition (35). Otherwise the difference in (35)
diverges as γ → ∞. Next, we present some general important
properties of the rate loss χTH (R, βR).
1) Universality related to SNR: Note that both quantities
in (34) increase with the SNR. It is shown in Appendix F that
their difference, i.e. (34), increases with the SNR too. Hence,
the rate loss χTH (R, βR) provides the least upper bound on
the mutual information loss over the entire SNR range.

R EMARK 1 Let H and P β H have both full rank almost
surely. Then, we have
χTH (R, βR) = sup{I(γ; FTH ) − I(γ; FTP β H )}.

(38)

γ

P ROOF 4 See Appendix F
2) Equivalence to capacity loss: Let us denote the capacity
of channel (12) as
C(γ; FTP β H ) , max I(γ; FTP β H √Q ).
Q≥0
tr(Q)=T

(39)

It turns out that (35) also holds when the mutual informations
in (35) are replaced by the respective capacities.
R EMARK 2 Let H and P β H have both full rank almost
surely. Then, we have
χTH (R, βR) = lim C(γ; FTH ) − C(γ; FTP β H ).
γ→∞

(40)

P ROOF 5 See Appendix G.
3) The invariance related to singular values: Though
χTH (R, βR) is defined through the distribution functions FTH
and FTP β H in (35), it actually depends solely on the matrix of
left singular vectors of H:
T HEOREM 2 Let H and P β H have both full rank almost
surely. Consider the spectral decomposition
H = LSR

(41)

where L is a R × R unitary matrix whose columns are the left
singular vectors of H, R is a T × T unitary matrix whose

IEEE TRANSACTIONS ON INFORMATION THEORY, 2018

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columns are the right singular vectors of H and the diagonal
entries of S are the singular values of H. Then, we have
χTH (R, βR) = −

1
log2 det P φ L† P †β P β LP †φ .
T

(42)

P ROOF 6 See Appendix H.
4) Statistical properties resulting from unitarily invariance:
Let H † H have full rank almost surely and be unitarily
invariant3 . Thereby, the matrix of left singular vectors of H,
i.e. L, is Haar, see [9, Lemma 2.6]. Thus, P φ L† P †β P β LP †φ
belongs to the Jacobi matrix ensemble, see Example 2. In other
words, the rate loss χTH (R, βR) becomes nothing but minus
the log det of the Jacobi matrix ensemble normalized by T .
We also refer the reader to [14] for a detailed study of the
determinant of the Jacobi matrix ensemble. In particular, from
[14, Proposition 2.4], the rate loss admits the explicit statistical
characterization
χTH (R, βR) ∼ −

T
1X
log2 ρt
T t=1

(43)

where {ρ1 , · · · , ρT } are independent random variables and
ρt ∼ Be ((βR + 1 − t), (1 − β)R). Here, X ∼ Y indicates
that random variables X and Y are identically distributed. For
a > 0 and b > 0, Be(a, b) denotes the Beta distribution with
density
Be(x; a, b) =

Γ(a + b) a−1
x
(1 − x)b−1 ,
Γ(a)Γ(b)

x>0

(44)

C OROLLARY 1 (U NIVERSAL R ATE L OSS ) Let H † H have
full rank almost surely and be unitarily invariant. Define4
R
T
1 X X
T ln 2 t=1
0

r=R +1

R EMARK 3 The function T χT (R, R0 ) (see (45)) satisfies the
symmetry property
0

T χT (R, T ) = T 0 χT (R, T 0 ) ,

T <R

(49)

where T 0 , R − T .

where Γ is the gamma function.

χT (R, R0 ) ,

Note that χT (R, T ) equals to the ergodic rate loss when we
remove as many antennas as needed to obtain a square system.
Furthermore, if R, T → ∞ with the ratio φ = T /R fixed, the
first and the second terms of (48) converge to respectively the
first and the second terms of (47).
We coin the limit (47) the binary entropy loss as it only
involves the binary entropy function evaluated at the aspect
ratios φ and β/φ of two channel matrices – the one before
and the one after the removal of the antennas. In particular,
for β = φ, i.e. we remove as many receive antennas as needed
to obtain a square system, the binary entropy loss has the
compact expression H(φ)/φ.
5) A symmetry property of the universal rate loss: We show
a symmetry property of the universal rate loss in the case when
the end system after (completion of the antenna removal) is
square, i.e. β = φ. Let us start with an illustrative example.
Consider two separate MIMO systems one of dimensions 3×2
and one of dimensions 3×1. Let the antenna removal processes
be 3 × 1 → 1 × 1 for the former system and 3 × 2 → 2 × 2
for the latter. Thus, in both cases two communication links
are removed from the reference systems. Let the channel
matrices of the reference systems fulfill the conditions stated
in Corollary 1 (i.e. full-rank and unitary invariance). Both
removal process lead to the same the binary entropy loss equal
to 3H(1/3) = 3H(2/3) = 2.75 bit.

1
,
r−t

T ≤ R0 ≤ R.
(45)

Then, we have
E[χTH (R, βR)] = χT (R, βR).

(46)

Moreover, if R, T → ∞ with φ = T /R fixed, we have almost
surely
 
H (φ) β
φ
T
χH (R, βR) →
− H
.
(47)
φ
φ
β
P ROOF 7 See Appendix I.
The name Universal Rate Loss refers to the fact that the
results in Corollary 1 solely refer to the number of transmit and
receive antennas before and after the variation. The ergodic
rate loss has the additive property
χT (R, R0 ) = χT (R, T ) − χT (R0 , T ) ,

T ≤ R0 ≤ R. (48)

3 Provided H † H is unitarily invariant, when H has almost surely full
rank, so does P β H too, see Appendix I.
4 The sum over an empty index set is by definition zero.

P ROOF 8 See Appendix J
0

Note that the expressions T χT (R, T ) and T 0 χT (R, T 0 ) corresponds to the ergodic rate losses for the antenna removal
processes R×T → T ×T and R×T 0 → T 0 ×T 0 , respectively.
In both cases T × (R − T ) communications links are removed
from the reference systems. In other words, for R being fixed
the ergodic rate loss T χT (R, T ) is a symmetric function of T
with respect to T = R/2 (see Figure 2).
Since χφR (R, βR) ≤ χφR (R, φR), the symmetry property
(49) implies that the maximum ergodic rate loss is attained
when φ = β = 1/2.
R EMARK 4 Let H † H have full rank almost surely and be
unitarily invariant. Then, for φ ≤ β ≤ 1 we have


1 1
,
= arg max E[χφR
(50)
H (R, βR)].
φ,β
2 2
B. Case (ii) – Removing transmit antennas
We remove a fraction (1 − β) of transmit antennas in (8)
to obtain system (14). We constrain the reduction of receive
antennas with β ≥ 1/φ (φ = T /R) to ensure min(βT, R) =
T . Reducing the number of transmit antennas results in a loss
of mutual information equal to T I(γ; FTH ) − βT I(γ; FβT † ).
HP β

IEEE TRANSACTIONS ON INFORMATION THEORY, 2018

7

Normalizing this loss with the number of transmit antennas of
the reference system gives
I(γ; FTH ) − βI(γ; FβT

HP †β

).

(51)

Let H and HP †β have both full rank almost surely. Then, we
define the large SNR limit
1
.
φ
(52)
Again the full rank assumption is important here. Otherwise
the difference (52) may diverge as γ → ∞.
βT
T
χ̃R
H (T, βT ) , lim I(γ; FH ) − βI(γ; F

HP †β

γ→∞

),

β≥

C OROLLARY 2 Let H and HP †β have both full rank almost
surely. Then, we have
1
(53)
log2 det P φ1 RP †β P β R† P †1
φ
T
where R is a T × T unitary matrix whose columns are the
right singular vectors of H, see (41).
χ̃R
H (T, βT ) = −

Note that the right-hand side in (53) is obtained by formally
replacing φ with φ−1 in the right-hand side of (47). This
follows from the identity
βI(γ; FβT

HP †β

)=

1
I(γ; FR
P β H † ).
φ

(54)

This substitution is valid for any result that refers to mutual
information, e.g. as in Corollary 1. However, it does not apply
in general to capacity related results, such as in Remark 2, due
to the placement of the projection operator on the transmitter
side.
C. The rate loss with antenna power profile
In this subsection we address the rate loss χTH for a channel
model that takes into consideration the power imbalance at the
transmitter and receiver sides:
H = ΛR H̃ΛT .

(55)

Here, the matrices ΛR ∈ CR×R and ΛT ∈ CT ×T are
diagonal, full-rank, and deterministic. The matrix ΛR (ΛT )
represents the power imbalance at receive (transmit) side.
We generalize Theorem 2 for the model (55) as (see
Appendix H)
†

χTH (R, βR)

det P φ L̃ Θ1 L̃P †φ
1
= log2
†
T
det P φ L̃ Θβ L̃P †φ

Moreover, let X β , P β X where X is a R × T matrix with
iid zero-mean complex Gaussian entries. Let D β be the βR ×
βR diagonal matrix whose diagonal entries are the non-zero
eigenvalues of Θβ , ΛR † P †β P β ΛR . Then, we have
"
#
det X †1 D 1 X 1
1
T
.
(57)
E[χH (R, βR)] = E log2
T
det X †β D β X β

(56)

where Θβ , ΛR † P †β P β ΛR for β ≤ 1 and L̃ is a R × R
unitary matrix whose columns are the left singular vectors of
H̃, see (41). Note that the rate loss does not depend on the
singular values of H̃. This property allows for obtaining a
convenient expression for the ergodic rate loss E[χTH (R, βR)]
†
when H̃ H̃ is unitarily invariant, i.e. L̃ in (56) is Haar
distributed.
C OROLLARY 3 Let H be defined as in (55). Furthermore, let
†
H̃ H̃ have full rank almost surely and be unitarily invariant.

P ROOF 9 See Appendix K
The expectation in (57) can be simply computed by using the
following result.
L EMMA 1 [5, Lemma 2] Let X be an n × m matrix with iid
zero-mean complex Gaussian entries such that n > m. Let D
be an n × n deterministic Hermitian positive-definite matrix
whose jth eigenvalue is denoted by λj . Moreover, let Ω be the
n × n Vandermonde matrix with (Ω)ij = λij−1 and Γ be the
(n − m) × (n − m) principal submatrix of Ω. Then, we have
m

E[ln det X † DX] =

det Γ X
det Ψi
det Ω i=1

(58)

where Ψi is m × m matrix whose entries are
n−m−1+l
(Ψi )k,l =νn−m+k λn−m+k

−

n−m
X

d−1 d−1
νq (Γ−1 )d,q
λn−m+k λqn−m−1+1 . (59)

d=1,q=1

In this expression, νq = ψ(l) + ln λq if l = i else νq = 1 with
ψ(·) denoting the digamma function.
D. Further discussions based on numerical results
As a warm up example, consider a 4 × 2 MIMO system
that is stripped off two of its four receive antennas. For fullrank channel matrices that are unitarily invariant from left
Theorem 1 gives the exact high SNR limit of the ergodic loss
equal to 4χ2 (4, 2) = 3.37 bit. The asymptotic loss (47) is
4H(2/4) = 4 bit. Note also that 4χ2 (4, 2) is the supremum
of the mutual information loss over all SNRs. This is depicted
in Figure 1 for a Gaussian channel.
We illustrate the universal rate loss and the tightness of
the approximation provided by the binary entropy loss, i.e.
RH(T /R), already for small system dimensions. To this end
we consider three different channel models that are unitarily
invariant from the left: (i) the channel matrix H = U Λ where
U ∈ CR×T is uniformly distributed over the manifold of
complex R×T matrices such that U † U = I and Λ ∈ RT ×T is
a positive diagonal matrix that represents the power imbalance
at the transmitter. This is a typical channel model in the
context of massive MIMO, i.e. in the regime of T  R.
Here we point out that Λ does not affect the rate loss.
Therefore for convenience we set Λ = I. (ii) the channel
matrix H = X with the entries of X being zero-mean iid
complex-valued Gaussian with finite variance; (iii) the channel
matrix H = X 2 DX 1 . Here X 1 ∈ CS×T and X 2 ∈ CR×S
represent the propagation channel from the transmit antennas

IEEE TRANSACTIONS ON INFORMATION THEORY, 2018

8

15

VI. D EVIATION FROM L INEAR G ROWTH
T E[I(.; FTH )]
T E[I0 (.; FTH )]
T E[I(.; FTP- H )]
T E[I0 (.; FTP- H )]

[bits]

10

3:37

5

0
-10

-5

0

5

10

15

20

.(dB)

Fig. 1: Ergodic mutual information (continuous lines) and
ergodic multiplexing rate (dashed lines) versus the SNR of a
zero-mean iid complex Gaussian MIMO channel with T = 2
transmit antennas and the number of receive antennas decreased from R = 4 (blue curves) to R = 2 (red curves).

4H(T =4)
H=U
H=X
H = X2 DX1

4

4E[@TH (4; T )]

3

2

In this section we clarify the second misinterpretation
underlined in Section 1. Specifically, we analyze the variation
of the multiplexing rate when either the number of receive or
the number transmit antennas varies while their maximum is
kept fixed.
For a channel matrix having orthogonal columns when
the number of transmit or receive antennas varies, the linear
growth of mutual information is obvious. However, for a
channel matrix with e.g. iid entries, a substantial crosstalk
arises due to the lack of orthogonality of its columns. The
effect of this crosstalk onto mutual information is non-linear
in the number of antennas.
The mutual information scales approximately linearly in the
minimum of the numbers of transmit and receive antennas.
For a tall rectangular channel matrix that becomes wider and
wider, the mutual information can only grow approximately
linearly until the matrix becomes square. The same holds for
a wide rectangular channel matrix growing taller and taller.
Therefore, we have to distinguish between two cases: (i) the
number of receive antennas is smaller than the number of
transmit antennas, i.e. a wide channel matrix, and (ii) the
converse of (i), i.e. a tall channel matrix. Since case (ii) can
be easily treated by replacing the channel matrix with its
conjugate transpose, we restrict our investigations to case (i).
The linear growth cannot continue once the channel matrix
has grown square. Thus, it makes sense to constrain the matrix
of reference system (8) to be square; i.e. we assume that the
channel matrix H in (8) is N × N i.e. N = R = T .
The exact mutual information of the (rectangular) system
(14) of size βN × N , β ≤ 1 is
N I(γ; FN
P β H ).

(60)

The mutual information (60) scales approximately linearly
with the number of receive antennas, if it is close to

1

βN I(γ; FN
H ).
0
0

1

2

3

4

T

Fig. 2: The maximal ergodic mutual information loss over
the SNR range: The entries of X ∈ C4×T , X 1 ∈ CS=4×T
and X 2 ∈ C4×S=4 are zero mean iid complex Gaussian. The
matrix U ∈ C4×T is uniformly distributed over the manifold
of complex 4 × T matrices. The S × S matrix D is positive
diagonal. Its diagonal entries are iid and uniformly distributed.

to the scatterers and from the scatterers to the receive antennas
respectively, while the diagonal entries in the diagonal matrix
D are the individual scattering coefficients of the scatterers.
This random matrix ensemble models the channel under the
assumption of propagation via one-bounce scattering only
[18]. To fulfill the full-rank condition we restrict to the case
S ≥ T . From Figure 2, we conclude that the binary entropy
loss yields an accurate approximation even for small system
dimensions.

(61)

Thus, in the high SNR limit, the deviation from the linear
growth normalized to N (the deviation from linear growth for
short) is given by
N
N
∆L(β; FN
H ) , lim I(γ; FP β H ) − βI(γ; FH )
γ→∞

N
= I0 (γ; FN
P β H ) − βI0 (γ; FH )

(62)
(63)

where H is assumed to have full rank almost surely. The fullT
T
rank assumption implies αH
= αP
which is necessary in
βH
the definition (62). Otherwise, (62) is divergent.
E XAMPLE 3 Let H be unitary. Then, we have
∆L(β; FN
H ) = 0.

(64)

A. The large-system limit consideration
The deviation from linear growth (63) differs from the
quantity χTH defined in (35) only by the factor β scaling the
second term. Unlike χTH , ∆L does depend on the singular
values of channel matrix. This makes the analysis somehow

IEEE TRANSACTIONS ON INFORMATION THEORY, 2018

9

A SSUMPTION 1 The channel matrix H has full rank almost
surely. Furthermore, HH † is unitarily invariant, has a uniformly bounded spectral norm, and its empirical eigenvalue
distribution converges almost surely as N → ∞. Moreover,
∆I(1; FH ) is finite.
We carry out the analysis on the basis of the LED function
FP β H . Specifically, we consider
∆L(β; FH ) = I0 (γ; FP β H ) − βI0 (γ; FH ).

(65)

When we interpret the asymptotic results in the numerical
investigations we assume that

A

40
N E[I0 (.; FN
P- H )]
-N E[I0 (.; FN
H )]

[bits]

30
20
10
0
0

1

2

3

4

5

3

4

5

-N

B
3

N E["L(-; FN
H )]
N (- ! 1) log2 (1 ! -)

2

[bits]

intractable. On the other hand, it is well-known that asymptotic
results when the numbers of antennas grow large provide very
good approximations already for systems with a dozen (or
even less) of antennas in practice. Thus, we can resort to the
asymptotic regime in the number of antennas to study the
deviation from linear growth. To that end, in this section we
make use of the following underlying assumption:

1

0
0

1

2

-N

lim E[I0 (γ; FN
P β H )] = I0 (γ; FP β H ) ,

N →∞

β ≤ 1.

(66)

It is easy to show that the convergence (66) is a mild assumption for β < 1: as FH is assumed to have a compact support,
FP β H has a compact support too, see [19, Corollary 1.14].
Note that a compactly supported probability distribution can
be uniquely characterized by its moments. This fact allow us
to use the machinery provided
in Proposition 1 in Appendix C.
R
Specifically, supN E[ x−1 dF̃N
P β H (x)] < ∞ is sufficient for
(66) to hold. Indeed this is a reasonable condition for β < 1
since
Z
1
dF̃P β H (x) , 0 < β < 1
(67)
x

Fig. 3: Ergodic multiplexing rate and corresponding linear
growth (A) and (ergodic) deviation from linear growth (B) versus number of receive antennas βN . The entries of H ∈ C5×5
are iid Gaussian with zero mean and variance 1/5. The SNR
is γ = 20 dB.
L EMMA 2 Let H fulfill Assumption 1. Then, we have
Z 1
SH (−βz)
dz.
∆L(β; FH ) = −β
log2
SH (−z)
0

(69)

is strictly increasing with β, see Remark 5.

P ROOF 11 See Appendix M.

E XAMPLE 4 Let the entries of H be iid with zero mean and
variance σ 2 /N . Then, we have

Alternatively, we may bypass the need for using the Stransform by invoking the following result:

∆L(β; FH ) = (β − 1) log2 (1 − β)

R EMARK 5 Let H fulfill Assumption 1. Furthermore, let P t
be an N -dimensional projector with 0 < t < 1. Then, we have
Z
1
SH (−t) =
dF̃P t H (x) , 0 < t < 1.
(70)
x

(68)

where by convention 0 log2 0 = 0.
P ROOF 10 See Appendix L.
In other words, at high SNR the normalized mutual information of a MIMO system of sufficiently large dimensions with
zero-mean iid channel entries grows approximately linearly
with the minimum of the numbers of transmit and receive
antennas up to 1st order and the deviation from the linear
growth is close to (β − 1) log2 (1 − β). Figure 3 illustrates this
behavior.

The result in (70) also provides a convenient means to calculate the deviation from linear growth in the large-system
limit. The right-hand side of (70) is nothing but the asymptotic
inverse spectral mean of the channel matrix P t H.

B. The S-transform formulation

C. The universality related to the SNR range

The result in Example 4 can be obtained from previous
capacity results, e.g. [9, Eq. (2.63)]. We obtained it as a special
case of the following lemma.

Note that the difference I(γ; FP β H ) − βI(γ; FH ) converges to ∆L(β; FH ) as the SNR tends to infinity, see (63).
In Appendix O we show that this difference actually increases
with SNR unless FH is a Dirac distribution function. Thus, we

P ROOF 12 See Appendix M.

IEEE TRANSACTIONS ON INFORMATION THEORY, 2018

10

VII. C ONCLUSIONS

have the following universal characterization over the whole
SNR range.
R EMARK 6 Let H fulfill Assumption 1. Then, we have

∆L(β, FH ) = sup I(γ; FP β H ) − βI(γ; FH ) .
(71)
γ

P ROOF 13 See Appendix O.
D. The additive property
We now draw the attention to another important property of
the deviation from linear growth:
T HEOREM 3 Let X and Y be independent CN ×N random
matrices. Moreover, let X and Y fulfill Assumption 1. Then,
we have
∆L(β; FXY ) = ∆L(β; FX ) + ∆L(β; FY ).

(72)

P ROOF 14 See Appendix P.
E XAMPLE 5 Consider a random matrix defined as
H=

M
Y

Am

(73)

m=1

where the N × N matrices Am , m = 1, . . . , M , are independent, have iid entries with zero mean and variance σ 2 /N .
Then we have almost surely
∆L(β; FH ) = M ∆L(β; FA1 )

(74)

= M (β − 1) log2 (1 − β).

A variation of the number of antennas in a MIMO system
affects the mutual information at asymptotically large SNR
in following way: If the minimum number of antennas at
transmitter and receiver side stays unaltered, the change of
mutual information depends only on the system dimensions
and the matrix of left (or right) singular vectors of initial
channel matrix but not on its singular values. For channel
matrices that are unitarily invariant from left (or right) this
change of mutual information in the ergodic sense can be
expressed with a simple analytic function of the system
dimensions. Moreover, the large system limit of this expression
involves only the binary entropy functions of the aspect ratios
of two varying channel matrices – the one before and the one
after altering the number of antennas.
Mutual information grows only approximately linear with
the minimum of the system dimensions even at high SNR.
This deviation from that linear growth, i.e. the error of the
linear approximation, does depend on the singular values of
the channel matrix. It can be quantified and has the following
remarkable property in the large system limit: For certain
factorizable MIMO channel matrices, the deviation is the sum
of the deviations of the individual factors.
The results derived in this work for asymptoticly large SNR
are least upper bounds over the whole SNR range. This gives
them a universal character.
Finally, a fundamental relation between mutual information
and its affine approximation (the multiplexing rate) was unveiled. This relation can be conveniently described via the
S-transform of free probability.

(75)

As mentioned previously, the crosstalk due to non-orthogonal
columns in H affects the mutual information in a way that
is non-linear in the number of antennas. Thus, it causes the
deviation from linear growth. Let us be more precise here and
(inspired from [20, Eq. (1)]) introduce the concept of crosstalk
ratio:
PN P
†
2
i=1
j<i |hi hj |
.
(76)
CTH , lim
PN
†
2
N →∞
i=1 |hi hi |
Here hi denotes the ith column of H. For example, for an
unitary matrix H, we have CTH = 0. As a second example,
let the entries of H be iid complex Gaussian with zero mean
and variance 1/N . Then, from (196) we get
1
CTH = .
(77)
2
We next show that the crosstalk ratio has the same additive
property as the deviation from linear growth.

A PPENDIX A
P RELIMINARIES
L EMMA 3 Let p ∈ [0, 1]. Then, we have
Zp
log2

P ROOF 16 We first recast (79) into the equivalent identity
Zx
lim

log2

x→p

1−z
dz = H(p).
x−z

(80)

0

To prove (80), we first apply a variable substitution
Zx

1−t
log2
dt = x
x−t

0

Z1
log2

x−1 − z
dz
1−z

(81)

0

and decompose the right hand side of (81) as
Z1
x

P ROOF 15 See Appendix Q.

(79)

0

N ×N

R EMARK 7 Let X and Y be independent C
random
matrices. Moreover let X and Y fulfill Assumption 1. Then
we have
CTXY = CTX + CTY .
(78)

1−z
dz = H(p).
p−z

log2 (x
0

−1

Z1
− z) dz − x

log2 (1 − z) dz.
0

(82)

IEEE TRANSACTIONS ON INFORMATION THEORY, 2018

11

Define u , log2 (x−1 −z) and v = z. Applying the integration
by part rule, we obtain for the first integral:
Z1
log2 x

−1



− z dz

1
uv|0

=

0

Z1
−

v du

(83)

lim (Ψn )−1 (z) = Ψ−1 (z),

n→∞

0

= x−1 H(x) − log2 e.

Furthermore, Ψn (z) is a strictly increasing homeomorphism
of (−∞, 0) onto (−α, 0) [22]. This implies that (see e.g. [23,
Proposition 0.1])

(84)

−α < z < 0.

(93)

This completes the proof.

Using (84), we compute the second integral:
Z1

Z1
log2 (1 − z) dz = lim

x→1

0

0

L EMMA 6 Consider a random matrix X and a projector P β .
Assume that X † X and P †β P β are asymptotically free. Then,
log2 (x−1 − z) dz = − log2 e. (85) we have
SXP † (z) = SX (βz).
(94)
β

This completes the proof.
L EMMA 4 [21] Let A and A + B be invertible and B have
rank 1. Furthermore let g , tr(BA−1 ) 6= −1. Then, we have
(A + B)

−1

1
A−1 BA−1 .
= A−1 −
g+1

(86)

L EMMA 5 [22, Lemma 2 & Lemma 4] Let F be a probability
distribution function with support in [0, ∞) and S its Stransform. Moreover, let F be not a Dirac distribution function.
Then, S is strictly decreasing on (−α, 0) with α , 1 − F(0).
In particular, we have
Z
−1
lim− S(z) =
x dF(x)
(87)
z→0
Z
1
lim + S(z) =
dF(x)
(88)
x
z→−α

P ROOF 18 The S-transform of P β reads [9, Example 2.32]
SP β (z) =

z+1
.
z+β

(95)

By invoking the identity [9, Theorem 2.32] and the asymptotic
freeness between X † X and P †β P β , we obtain
SXP † (z)

=

β

=

z+1
SP (βz)SX (βz)
z + 1/β β
SX (βz).

(96)
(97)

R EMARK 8 Let H = P β2 U P †β1 with U an N -dimensional
Haar unitary. Then, we have almost surely
SH (z) =

1 + β1 z
.
β2 + β1 z

(98)

where we use the convention 1/0 = ∞ in (88) when F(0) > 0.
T HEOREM 4 [22, Proposition 1] Let F be a probability
distribution function
R with support in (0, ∞) and S its Stransform. Then | log x| dF(x) is finite if, and only if,
R1
| log S(−z)| dz is finite. If either of these integrals is finite,
0
Z1

Z
log(x) dF(x) = −

log S(−z) dz.

(89)

0

T HEOREM 5 For n ∈ N+ , {1, 2, ...} let Fn be probability
distribution functions on [0, ∞). Furthermore let 1 − Fn (0) =
α > 0, ∀n ∈ N+ . Moreover let Sn denote the S-transform of
Fn . Then if Fn converges weakly to a probability distribution
function F as n → ∞, we have
lim Sn (z) = S(z),

n→∞

−α < z < 0

(90)

P ROOF 19 By invoking Lemma 6 to (95) we obtain (98).
A PPENDIX B
P ROOF OF T HEOREM 1
A. Proof of (25)
By definition, I(γ; FTH ) < ∞. Then, from identity [22,
Eq. (5)] we write
Z 1
I(γ; FTH ) = −
log2 (s)∂ΨT√γH (−s) ds
(99)
0
dΨT
H (x)

where ∂ΨTH (ω) , dx
.
x=ω
At this stage we point out two identities:
T
ΨT√γH (−1) + 1 = ΨTH (−γ) + 1 = ηH
(γ)

where S is the S-transform of F.

lim

x→0−

P ROOF 17 Let us consider the function, (see (6))
Z
zx
dFn (x), −∞ < z < 0.
Ψn (z) ,
1 − zx

(91)

zx
For z ∈ (−∞, 0), z → 1−zx
is bounded and continuous.
Hence, the weak convergence of Fn implies that
n

lim Ψ (z) = Ψ(z),

n→∞

−∞ < z < 0.

(92)

ΨT√

γH (x)

+ 1 = 1.

(100)
(101)

Now we apply the variable substitution z , ΨT√γH (−s) + 1
in the integral in (99). Notice that with this substitution the
upper and lower limits of this integral read (100) and (101),
respectively. As a result (99) is recast in the form
I(γ; FTH ) =

Z
1

T
ηH



<−1>
log2 −ΨT√γH (z − 1) dz

(102)

IEEE TRANSACTIONS ON INFORMATION THEORY, 2018

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<−1>

with ΨTH
denoting the inverse of ΨTH . Then, by the
definition of the S-transform, see (7), we obtain
T
T
Z ηH
Z ηH
1−z
dz +
log2 ST√γH (z − 1) dz
I(γ; FTH ) =
log2
z
1
1
(103)
T
Z ηH
T
= H(ηH
)+
(104)
log2 ST√γH (z − 1) dz
1
T
= H(ηH
)−

T
1−ηH

Z

log2 ST√γH (−z) dz.

(105)

0

Finally, we obtain (25) by using the scaling property of the
S-transform [24, Lemma 4.2].
B. Proof of (26)
Let S̃TH be the S-transform of F̃TH . By using [9, Theorem
2.32] we write
z+1
T
STH (αH
z), −1 < z < 0.
(106)
S̃TH (z) =
T
z + 1/αH
Note that F̃TH is an empirical distribution function. Thus,
log2 (x) is absolutely integrable over it. We use Theorem 4
and Lemma 3 to complete the proof:
I0 (γ; FTH )

=

T
αH

log2 γ −

T
αH

Z1

log2 S̃TH (−x) dx

(107)

we have almost surely
lim I0 (γ; FTH ) = I0 (γ; FH ).

T →∞

Condition (111) is reasonable in practice. Otherwise the
power amplification per dimension of the MIMO system
explodes as its dimensions grow to infinity. One can show that
for rectangular and unitarily invariant channel matrices, the
condition (113) is reasonable too due to the strict decreasing
property of the function of β in (67). However, it might
not hold when the channel matrix is square. As an example,
consider a channel matrix H whose entries are iid with zero
mean and variance σ 2 /T . Then, condition (113) holds if
φ 6= 1, but is violated if φ = 1. Indeed the latter case turns
out critical for the “log det” convergence of the zero-mean
iid matrix ensemble, e.g. see [9], [25]. Nevertheless, both [26,
Proposition 2.2] and numerical evidence lead us to conjecture
that (114) holds when φ = 1 as well. Thus, we conclude that
the asymptotic convergence of the multiplexing rate, i.e. (114),
is a mild assumption in practice.
Proof of Proposition 1
For the sake of readability of the proof, whenever we use the
limit operator indicating that T tends to infinity, we implicitly
assume that the ratio φ = T /R is fixed.
For convenience we define
Y , I + γH † H.

0
T
T
= αH
log2 γ − αH

Z1
log2
0

1−x
T
ST (−αH
x) dx
T −x H
1/αH

T
= αH
log2 γ −

Z

log2

T
αH
−x T
S (−x) dx
1−x H

I(γ; FTH )

(109)

T

T
T
= αH
log2 γ + H(αH
)−

log2 STH (−x) dx.

(110)

0

A PPENDIX C
O N THE CONVERGENCE OF M UTUAL INFORMATION AND
M ULTIPLEXING R ATE
In this section we provide some sufficient conditions that
guarantee the convergence of the mutual information (10) and
multiplexing rate (see (19)) in the large system limit.
P ROPOSITION 1 As R, T → ∞ with the ratio φ , T /R fixed
let H † H have a LED FH . Furthermore, let
Z
sup x dFTH (x) < ∞ a.s..
(111)
T

Then we have almost surely
lim I(γ; FTH ) = I(γ; FH ).

T →∞

Moreover if in addition
Z
1
dF̃TH (x) < ∞ a.s.
sup
x
T

Z0
=−

log2 ST√Y (z) dz.

(116)

−1

0

ZαH

(115)

By Theorem 4 we have

(108)
αT
H

(114)

(112)

(113)

The function ST√Y is strictly decreasing on (−1, 0) if, and only
if, FTH is not a Dirac distribution function, see Lemma 5. If
FTH is a Dirac distribution function then ST√Y is a constant
function. Without loss of generality, we can assume that FTH is
not a Dirac distribution function. Then, by invoking Lemma 5
again we have
−1

1
1
tr(Y )
< ST√Y (z) < tr(Y −1 ), −1 < z < 0.
T
T
(117)
For convenience we define the random variable
Z
T
M , sup x dFTH (x) s.t. φ = .
(118)
R
T
Since the upper bound in (117) is smaller than one we have
| log2 ST√Y (z)| = − log2 ST√Y (z),
1
< log2 tr(Y )
T
≤ log2 (1 + γM ).

−1 < z < 0

(119)
(120)
(121)

Because of (121), we can apply Lebesgue’s dominated convergence theorem [27, Theorem 10.21]:
lim I(γ; FTH ) = −

Z0

T →∞

−1

lim log2 ST√Y (z) dz.

T →∞

(122)

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By invoking Theorem 5 we complete the proof of (111):
lim log2 ST√Y (z) = log2 lim ST√Y (z)

(123)

=

(124)

T →∞

T →∞
log2 S√Y (z).

I0 (γ, FTH )

where the result (131) follows from the identity (84). We
obtain (31) from (25) with (131) inserted in (24). Moreover,
by the definition of the S-transform we have
β1 (1 − z)Ψ2H (z) + (1 − (β1 + β2 )z)ΨH (z) − β2 z = 0. (132)

To prove (114), we use the same arguments for
as for I(γ, FTH ). In particular, by invoking Lemma 5 again
we can write
Z
−1
Z
1
x dF̃TH (x)
< S̃TH (z) <
dF̃TH (x), −1 < z < 0
x
(125)
with S̃TH denoting the S-transform of F̃TH . Unlike (117), the
right-most integral is not bounded in general, so we need the
additional assumption (113). This completes the proof.

Note that 1 + ΨH (−γ) = ηH (γ). Thus, (132) has two
solutions for ηH (γ). Only one fulfills the properties of ηH (γ)
in [9, pp. 41]. Specifically, from the property ηH (γ) → 1 as
γ → 0 we concludeR that (32) is this solution. Finally it is also
1
easy to show that R 0 | log2 S̃H (−z)| dz is finite in this case.
This implies that | log(x)|dF̃H (x) is finite too. Thus, the
multiplexing rate is obtained by replacing the term (1 − ηH )
in (31) with αH , which leads to (33).

A PPENDIX D
P ROOF OF E XAMPLE 1

A PPENDIX F
P ROOF OF R EMARK 1

With a convenient re-parameterization of [13, Eq. (19)] we
write
N
Y
ρn
.
(126)
SH (z) =
z
+
ρn
n=1

We can write the integral terms in (127) as
Z 1−ηH
z
log2 (1 −
) dz =
ρn
0
Z 1
1 − ηH
ρn
log2
+
− z) dz
log2 (
ρn
1 − ηH
0

d{I(γ; FTH ) − I(γ; FTP β H )}
dγ

=

T
T
ηP
(γ) − ηH
(γ)
βH

γ ln 2

(127)

where the equality holds when β = 1. To prove (134) it is
sufficient to consider the removal of a single receive antenna,
i.e. β = (R − 1)/R. It is immediate that
H † H = H † P †β P β H + h†R hR

(128)

for n ∈ [1, N ]. By invoking the result in (84) we obtain (28).
From the linearity
R 1 property of the Lebesgue integral, it is
easy Rto show that 0 | log2 S̃H (−z)dz| is finite, which implies
that | log(x)|dF̃H (x) is finite too, due to Theorem 4. Thus,
the multiplexing rate is obtained by replacing the term (1−ηH )
in (28) with αH (due to Theorem 1). This leads to (29).
Finally, we note that if αH < 1 the S-transform SH (z)
diverges as z → (−αH ), see Lemma 5. Thus, from (126)
the unique solution of αH is αH = min(1, ρ1 , ρ2 , . . . , ρN ).

(135)

with hR ∈ C1×T representing the Rth row of H. Then (134)
follows directly from Lemma 4 in Appendix A.
A PPENDIX G
P ROOF OF R EMARK 2
We decompose the capacity expression in (39) as
C(γ, FTP β H ) = I0 (γ; FTP β H √Q∗ ) + ∆I(γ; FTP β H √Q∗ )
(136)
with Q? denoting the capacity achieving covariance matrix.
We define
C0 (γ, FTP β H ) , max I0 (γ; FTP β H √Q ).

Recall (98):
1 + β1 z
.
β2 + β1 z

(137)

Q≥0
tr(Q)=T

A PPENDIX E
P ROOF OF E XAMPLE 2
SH (z) =

. (133)

Hence, to prove Remark 1 we simply need to show that
n
o
tr (I + γH † P †β P β H)−1 − (I + γH † H)−1 ≥ 0 (134)

From Theorem 1 we have
I(γ; FH ) = H(ηH ) + (1 − ηH ) log2 γ
N Z 1−ηH
X
z
) dz.
+
log2 (1 −
ρ
n
n=1 0

We first point out the relationship [9]

(129)

Moreover, notice that αH = 1 − FH (0) = min(1, β2 /β1 ). For
convenience let a , 1 − ηH (γ) < αH . Then, we have
Z a
Z 1
1 − β1 at
log2
log2 SH (−z) dz = a
dt
(130)
β2 − β1 at
0

0
β1
H(β1 a) β2
=
− H
a
(131)
β1
β1
β2

In particular, by the definitions in (39) and (137) we have
C0 (γ, FTP β H ) ≥ I0 (γ; FTP β H √Q∗ ). Hence, we have
lim C0 (γ; FTP β H ) − C(γ; FTP β H ) ≥ 0.

(138)

γ→∞

Since H † P †β P β H has almost surely full rank, we have
T
= α√
and thereby
Q

T
√
αP
βH Q

T
T
√
I0 (γ; FTP β H √Q ) = α√
Q log2 γ+α Q

Z

T

log2 x dF̃P β H √Q (x).
(139)

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For a sufficiently large SNR a full-rank matrix Q maximizes
(139). Therefore, to prove the result we can assume without
loss of generality that Q has full rank. Doing so, we have
I0 (γ; FTP β H √Q )

1
= log2 γ + log2 det H † P †β P β H
T
1
+ log2 det Q.
(140)
T

Due to the constraint tr(Q) = T , the identity operator
maximizes (140). Hence, from (138) we have
lim I0 (γ; FTP β H ) − C(γ; FTP β H ) ≥ 0.

γ→∞

(141)

On the other hand we have
I0 (γ; FTP β H ) < I(γ; FTP β H ) ≤ C(γ; FTP β H ).

(142)

Thus (141) must be zero. This completes the proof.

A PPENDIX I
P ROOF OF C OROLLARY 1
We first show that provided H † H is unitarily invariant,
when H † H has full rank almost surely, so does H † P †β P β H
too for φ ≤ β: From (147) we have
det H † P †β P β H = det Σ2 det P φ L† P †β P β LP †φ

(149)

where Σ is a T ×T diagonal matrix whose diagonal entries are
the positive singular values of H. By the unitary invariance
assumption, P φ L† P †β P β LP †φ is a Jacobi matrix ensemble
with a positive determinant for φ ≤ β [14]. Thereby, (149) is
positive.
Given x ∼ Be(a, b) we have E[ln x] = ψ(a) − ψ(a + b)
where ψ(·) denotes the digamma function. For natural arguments, the digamma function can be expressed as
n−1
X

1
.
l

A PPENDIX H
P ROOF OF T HEOREM 2

ψ(n) = ψ(1) +

We prove (56) which is a generalization of Theorem 2. We
make use of (55) to write

Hence, from (43) we can write the ergodic rate loss as

l=1

(150)

T

†

det H † P †β P β H = det Λ†T ΛT det H̃ Θβ H̃

(143)

E[χTH (R, βR)] = −

where Θβ , ΛR † P †β P β ΛR for β ≤ 1. Hence, from (37) the
rate loss reads as
det H̃ Θ1 H̃
1
log2
.
†
T
det H̃ Θβ H̃

T

1 X
[ψ(R + 1 − t) − ψ(βR + 1 − t)]
T ln 2 t=1
(152)
"
#
βR−t
T
R−t
X 1
1 X X1
−
(153)
=
T ln 2 t=1 r=1 r
r
r=1

(144)

To simplify this expression, we consider the singular value
decomposition of H̃
†

H̃ = L̃[Σ|0] R̃

H̃ =L̃P †φ ΣR̃.

(146)

For notational compactness, let us define Z β
,
†
†
P φ L̃ Θβ L̃P φ and A , ΣR̃. Thereby, we can write
†

H̃ Θβ H̃ = A† Z β A. Note that A† Z β A and Z β AA† have
the same eigenvalues. Thus, we have
†

det H̃ Θβ H̃ = det Σ2 det Z β .

T

=

(145)

where L̃ and R̃ are respectively R × R and T × T unitary
matrices, Σ is a T × T positive diagonal matrix and 0 is a
(R − T ) × T zero matrix. Remark that we can actually write
(145) as

(147)

(151)

=

†

χTH (R, βR) =

1 X
E[ln ρt ]
T ln 2 t=1

1 X
T ln 2 t=1

R−t
X
r=βR−t+1

1
.
r

(154)

This completes the derivation of (46).
As regards to derivation of (47), we first note the almost sure
convergence of the limit [14, Theorem 3.6 and Eq. (4.23)]
1
log2 det P φ L† P †β P β LP †φ =
T →∞ T

Z

lim

log2 (x) dFP β U P † (x).
φ

(155)
Using (33) we express this limit in terms of binary entropy
function:
 
Z
1
β
φ
log2 (x) dFP β U P † (x) = − H(φ) + H
. (156)
φ
φ
φ
β
This completes the derivation of (47).

We complete the derivation of (56) by plugging (147) in (144):
χTH (R, βR) =

det Z 1
1
log2
.
T
det Z β

(148)

Note also that Z 1 = I for ΛR = I. This completes the proof
of Theorem 2.

A PPENDIX J
P ROOF OF R EMARK 3
It is sufficient to prove the result for (R − T ) <PT . For the
p
sake of notational compactness, we define hp , l=1 1l and

IEEE TRANSACTIONS ON INFORMATION THEORY, 2018

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g(R, T ) , ln(2)T χT (R, T ). Then, from (153) we write
g(R, T ) =

=

T
X

hR−t −

t=1
R−T
X

hT −t

(157)

t=1
T
X

hR−t +

t=1

−

T
X

A PPENDIX L
S OLUTION OF E XAMPLE 4

hR−t

t=(R−T )+1

2T
−R
X

T
X

hT −t −

hT −t .

(158)

t=2T −R+1

t=1

Notice that
T
X

hR−t =

hT −t

(159)
A PPENDIX M
P ROOF OF L EMMA 2

t=1

t=(R−T )+1
T
X

2T
−R
X

Note that we do not assume that H has Gaussian entries.
However it is well known that for any distribution of the entries
of H, the distribution function FN
P β H converges weakly and
almost surely to the Marc̆enko-Pastur law. In other words,
we get the same asymtotic results regardless of whether we
restrict the entries of H to Gaussian or not. Thus, without
loss of generality we can assume that the entries of H are
Gaussian, so that HH † is unitarily invariant. Doing so we
have SH (z) = (1 + z)−1 [9]. Then, we immediately obtain
(68) from (33).

hT −t =

t=2T −R+1

R−T
X

h(R−T )−t .

(160)

t=1

We have αP β H = β. Thus
I0 (γ; FP β H ) = βI0 (γ; F̃P β H ) = βI0 (γ; FH † P † ).

Thereby, we get

(168)

β

g(R, T ) =

R−T
X

hR−t −

R−T
X

t=1

Furthermore, with Lemma 6 we have
h(R−T )−t

(161)
SH † P † (z) = SH † (βz) = SH (βz).

t=1

= g(R, R − T ).

(162)

In the sequel we first show that

This completes the proof.

(170)
where S̃P β H is the S-transform of F̃P β H . To do so, it is
R1
sufficient to show that |log2 SH (−z)| dz < ∞. Since FH
0

(163)

has a compact support, I(γ; FH ) is finite. Now we show that
log x is absolutely integrable over FH if, and only if, I(1; FH )
and ∆I(1; FH ) are finite [22]:

where U is a R × R unitary matrix whose columns are the
left singular vectors of the Gaussian random matrix X. Since
X † X is unitarily invariant U is Haar distributed. The matrix
of the left singular vectors of H̃, i.e. L̃, is Haar distributed too
†
as H̃ H̃ is unitarily invariant. Thereby, from (56) and (163)
we have
(164)

Thereby, we have

which completes the proof.

Z1
|log2 (x)| dFH (x) =

0

 
1
dFH (x)
log2
x

0

Z∞
+

log2 (x) dFH (x).

(171)

Thus, we have
Z1

 
1
dFH (x) < ∞ ⇐⇒ ∆I(1; FH ) < ∞, (172)
log2
x

0

(165)

where U β is a R × R unitary matrix. Since X ∼ U β X, we
have
X † Θβ X ∼ X † P †β D β P β X.
(166)

#
"
1
det X † D 1 X
T
E[χH (R, βR)] = E log2
T
det X † P †β D β P β X

Z∞

1

Note that the rank of Θβ is βR. Thus, we can consider the
eigenvalue decomposition
Θβ = U †β P †β D β P β U β

0

0

Following the same line of argumentation as used to obtain
(148) we get

1
det X † Θ1 X
log2
.
T
det X † Θβ X

|log2 SH (−βz)| dz < ∞

log2 S̃P β H (−z) dz =

A PPENDIX K
P ROOF OF C OROLLARY 3

χTH (R, βR) ∼

Z1

Z1

det P φ U † Θ1 U P †φ
det X † Θ1 X
=
log
log2
2
det X † Θβ X
det P φ U † Θβ U P †φ

(169)

β

Z∞
log2 (x) dFH (x) < ∞ ⇐⇒ I(1; FH ) < ∞.
1

with ⇐⇒ implying ‘’if, and only if”. Hence (171) is finite.
R1
Due to Theorem 4 this implies that |log2 SH (−z)| dz is
0

(167)

(173)

finite too.
By invoking Theorem 4, (168) and (169) we obtain
Z 1
I0 (γ; FP β H ) = β log2 γ − β
log2 SH (−βz) dz. (174)
0

IEEE TRANSACTIONS ON INFORMATION THEORY, 2018

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Due to (170), it follows from the linearity property of the
Lebesgue integral that
∆L(β; FH ) = I0 (γ; FP β H ) − βI0 (γ; FH )
Z 1
SH (−βz)
dz.
= −β
log2
SH (−z)
0

Then, by using (178) and (182) we obtain
ηH † P † = lim S√Y β (z)

(175)

z→−1+

β

(176)

(187)

= lim + S√Y 1 (βz)

(188)

=

(189)

z→−1
S√Y 1 (−β)

,

0<β<1

This completes the proof.
which is strictly increasing with β, see Lemma 5. This
completes the proof.

A PPENDIX N
P ROOF OF R EMARK 5
Invoking Lemma 6 we can write
SH (−t) = SH † (−t) = lim SH † P † (z).
z→−1+

t

A PPENDIX P
P ROOF OF T HEOREM 3

(177)

Since H has almost surely full rank, αHP † = 1, so that
t
F̃P t H = FH † P † . Then from Lemma 5 we have
t
Z
1
lim + SH † P † (z) =
dFH † P † (x).
(178)
t
t
x
z→−1

The matrices XX † , Y Y † and P †β P β are asymptotically
free [28]. Then, from Lemma 2 and the linearity property of
the Lebesgue integral we have
Z1
∆L(β; FXY ) = −β

This completes the proof.

log2

SX (−βz)SY (−βz)
dz (190)
SX (−z)SY (−z)

0

= ∆L(β; FX ) + ∆L(β; FY ).

(191)

A PPENDIX O
P ROOF OF R EMARK 6
A PPENDIX Q
P ROOF OF R EMARK 7

For the sake of notational simplicity we introduce
Y β , I + γP β HH † P †β

(179)

= P β (I + γHH † )P †β

(180)

= P β Y 1 P †β .

(181)

It follows that Y 1 is unitarily invariant since HH † is.
Furthermore, since HH † has a compactly support LED so
does Y 1 . Thus Y 1 is asymptotically free of P †β P β [28]. Then,
with Lemma 6 we have in the limit N → ∞
S√Y β (z) = S√Y 1 (βz).

For an N × N matrix A, we define
φ(A) , lim

N →∞

1
tr(A)
N

whenever the limit exists. Since XX † and Y Y † are asymptotically free, we have (see [29, Eq. (120)])
φ(X † Y † Y X) = φ(X † X)φ(Y † Y )

(182)

φ((X † Y † Y X)2 ) = φ(X † X)2 φ((Y † Y )2 )

Here we note that S√Y β (z) is strictly decreasing on (−1, 0)
if, and only if, F√Y β is not a Dirac distribution function, see
Lemma 5.
We recall the following property of ηH (γ) see (23) [9]:

+ φ(Y † Y )2 φ((X † X)2 )

1 − ηP β H − β(1 − ηH )
d{I(γ; FP β H ) − βI(γ; FH )}
=
,
dγ
γ ln 2
(183)
where for convenience ηH is short for ηH (γ). Hence, in order
to prove the remark it is sufficient to show that
(1 − β) + βηH − ηP β H ≥ 0

− φ(X † X)2 φ(Y † Y )2 .

lim (A† A)ii → φ(A† A), ∀i.

N →∞

(185)

Thus, the right-hand side of (184) is equal to β(ηH −ηH † P † ).
β
Therefore we are left with proving ηH ≥ ηH † P † . Firstly,
β
remark that
Z
1
dF√Y β (x).
(186)
ηH † P † =
β
x

(194)

(195)

Inserting (195) in the definition of the crosstalk ratio in (76),
we get for A ∈ {X, Y , XY } that

(184)

β

(193)

Furthermore, from [30, Theorem 2.1] for A ∈ {X, Y , XY }
we have almost surely

CTA =

where the equality holds when β = 1. Furthermore, by using
[9, Lemma 2.26] we have
ηP β H = (1 − β) + βηH † P † .

(192)

φ((A† A)2 ) 1
− .
2
2φ(A† A)2

(196)

We complete the proof by plugging (193) and (194) in (196)
for A = XY :
CTXY =

φ(X † X)2 φ((Y † Y )2 ) + φ(Y † Y )2 φ((X † X)2 )
−1
2φ(X † X)2 φ(Y † Y )2
(197)

= CTX + CTY .

(198)

IEEE TRANSACTIONS ON INFORMATION THEORY, 2018

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17

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Burak Çakmak was born in Istanbul, Turkey, 1986. He received the B.Eng.
degree from Uludağ University, Turkey in 2009, M.Sc. degree from Norwegian
University of Science and Technology, Norway in 2012 and Ph.D degree from
Aalborg University, Denmark in 2017. Dr. Çakmak is a postdoctoral researcher
at the Department of Computer Science, Technical University of Berlin,
Germany. His research interests include random matrix theory, communication
theory, statistical physics of disorder systems, machine learning and Bayesian
inference.

Ralf R. Müller (S’96–M’03–SM’05) was born in Schwabach, Germany,
1970. He received the Dipl.-Ing. and Dr.-Ing. degree with distinction from
Friedrich-Alexander-Universität (FAU) Erlangen-Nürnberg in 1996 and 1999,
respectively. From 2000 to 2004, he directed a research group at The
Telecommunications Research Center Vienna in Austria and taught as an
adjunct professor at TU Wien. In 2005, he was appointed full professor at
the Department of Electronics and Telecommunications at the Norwegian
University of Science and Technology in Trondheim, Norway. In 2013, he
joined the Institute for Digital Communications at FAU Erlangen-Nürnberg
in Erlangen, Germany. He held visiting appointments at Princeton University,
US, Institute Eurcom, France, University of Melbourne, Australia, University
of Oulu, Finland, National University of Singapore, Babes-Bolyai University,
Cluj-Napoca, Romania, Kyoto University, Japan, FAU Erlangen-Nürnberg,
Germany, and TU München, Germany. Prof. Müller received the Leonard G.
Abraham Prize (jointly with Sergio Verdú) for the paper “Design and analysis
of low-complexity interference mitigation on vector channels” from the IEEE
Communications Society. He was presented awards for his dissertation “Power
and bandwidth efficiency of multiuser systems with random spreading”
by the Vodafone Foundation for Mobile Communications and the German
Information Technology Society (ITG). Moreover, he received the ITG award
for the paper “A random matrix model for communication via antenna
arrays” as well as the Philipp-Reis Award (jointly with Robert Fischer). Prof.
Müller served as an associate editor for the IEEE TRANSACTIONS ON
INFORMATION THEORY from 2003 to 2006 and as an executive editor
for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS from
2014 to 2016.

Bernard H. Fleury (M’97–SM’99) received the Diplomas in Electrical
Engineering and in Mathematics in 1978 and 1990 respectively and the
Ph.D. Degree in Electrical Engineering in 1990 from the Swiss Federal
Institute of Technology Zurich (ETHZ), Switzerland. Since 1997, he has been
with the Department of Electronic Systems, Aalborg University, Denmark,
as a Professor of Communication Theory. From 2000 till 2014 he was
Head of Section, first of the Digital Signal Processing Section and later of
the Navigation and Communications Section. From 2006 to 2009, he was
partly affiliated as a Key Researcher with the Telecommunications Research
Center Vienna (ftw.), Austria. During 19781985 and 19921996, he was a
Teaching Assistant and a Senior Research Associate, respectively, with the
Communication Technology Laboratory, ETHZ. Between 1988 and 1992, he
was a Research Assistant with the Statistical Seminar at ETHZ. Prof. Fleurys
research interests cover numerous aspects within communication theory, signal
processing, and machine learning, mainly for wireless communication systems
and networks. His current scientific activities include stochastic modeling
and estimation of the radio channel, especially for large systems (operating
in large bandwidths, equipped with large antenna arrays, etc.) deployed in
harsh conditions (e.g. in highly time-varying environments); iterative messagepassing processing (with focus on the design of efficient feasible architectures
for wireless receivers); localization techniques in wireless terrestrial systems;
and radar signal processing. Prof. Fleury has authored and coauthored more
than 150 publications and is co-inventor of 6 filed or published patents in
these areas. He has developed, with his staff, a high-resolution method for the
estimation of radio channel parameters that has found a wide application and
has inspired similar estimation techniques both in academia and in industry.

