On the condition number distribution of complex wishart matrices

Abstract

This paper investigates the distribution of the condition number of complex Wishart matrices. Two closely related measures are considered: the standard condition number (SCN) and the Demmel condition number (DCN), both of which have important applications in the context of multiple-input multiple-output (MIMO) communication systems, as well as in various branches of mathematics. We first present a novel generic framework for the SCN distribution which accounts for both central and non-central Wishart matrices of arbitrary dimension. This result is a simple unified expression which involves only a single scalar integral, and therefore allows for fast and efficient computation. For the case of dual Wishart matrices, we derive new exact polynomial expressions for both the SCN and DCN distributions. We also formulate a new closed-form expression for the tail SCN distribution which applies for correlated central Wishart matrices of arbitrary dimension and demonstrates an interesting connection to the maximum eigenvalue moments of Wishart matrices of smaller dimension. Based on our analytical results, we gain valuable insights into the statistical behavior of the channel conditioning for various MIMO fading scenarios, such as uncorrelated/semi-correlated Rayleigh fading and Ricean fading.

Full text

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 6, JUNE 20101705
On the Condition Number Distribution of
Complex Wishart Matrices
Michail Matthaiou, Member, IEEE, Matthew R. McKay, Member, IEEE,
Peter J. Smith, Senior Member, IEEE, and Josef A. Nossek, Fellow, IEEE
Abstract—This paper investigates the distribution of the con-
dition number of complex Wishart matrices. Two closely related
measures are considered: the standard condition number (SCN)
and the Demmel condition number (DCN), both of which have
important applications in the context of multiple-input multiple-
output (MIMO) communication systems, as well as in various
branches of mathematics. We first present a novel generic
framework for the SCN distribution which accounts for both
central and non-central Wishart matrices of arbitrary dimension.
This result is a simple unified expression which involves only a
single scalar integral, and therefore allows for fast and efficient
computation. For the case of dual Wishart matrices, we derive
new exact polynomial expressions for both the SCN and DCN
distributions. We also formulate a new closed-form expression
for the tail SCN distribution which applies for correlated central
Wishart matrices of arbitrary dimension and demonstrates an
interesting connection to the maximum eigenvalue moments of
Wishart matrices of smaller dimension.
Based on our analytical results, we gain valuable insights into
the statistical behavior of the channel conditioning for various
MIMO fading scenarios, such as uncorrelated/semi-correlated
Rayleigh fading and Ricean fading.
Index Terms—MIMO systems, complex Wishart matrices,
condition number, joint eigenvalue distribution.
I. INTRODUCTION
O
communications research and development, due to their huge
potential for delivering significant capacity gains compared
with conventional systems. The capacity and performance of
practical MIMO transmission schemes are often dictated by
the statistical eigenproperties of the instantaneous channel
correlation matrix1W = HH†or W = H†H, with H
denoting the matrix of random MIMO channel gains. The
VER the past decade, multiple-input multiple-output
(MIMO) systems have been at the forefront of wireless
Paper approved by N. Al-Dhahir, the Editor for Space-Time, OFDM and
Equalization of the IEEE Communications Society. Manuscript received June
10, 2009; revised October 21, 2009.
M. Matthaiou and J. A. Nossek are with the Institute for Circuit Theory
and Signal Processing, Technische Universität München (TUM), Arcistrasse
21, 80333, Munich, Germany (e-mail: {matthaiou, nossek}@nws.ei.tum.de).
M. R. McKay is with the Department of Electronic and Computer Engi-
neering, Hong Kong University of Science and Technology (HKUST), Clear
Water Bay, Kowloon, Hong Kong (e-mail: eemckay@ust.hk).
P. J. Smith is with the Department of Electrical and Computer Engineering,
University of Canterbury, Private Bag 4800, Christchurch, New Zealand (e-
mail: peter.smith@canterbury.ac.nz).
Part of this paper has been presented at the IEEE International Conference
on Communications (ICC), Cape Town, South Africa, May 2010.
The work of M. R. McKay was supported by the research grant DAG
S08/09.EG04.
Digital Object Identifier 10.1109/TCOMM.2010.06.090328
1The notation (⋅)†denotes the conjugate-transpose operation.
matrix H is typically modeled via a complex Gaussian dis-
tribution with mean and covariance structure defined by the
system configuration (eg. antenna spacing) and the nature
of the surrounding environment (eg. line-of-sight (LoS)). As
such, W is known to follow a complex Wishart distribution.
In recent years, the statistical properties of Wishart matrices
have been extensively studied and applied to a large number
of MIMO applications. For example, the unordered eigenvalue
distributions and determinant properties of Wishart matrices
have been derived in [1–16] and applied to explore the ergodic
capacity of the MIMO channel under different conditions,
whereas ordered/marginal eigenvalue distributions have been
derived in [17–26] and used to investigate the performance of
MIMO beamforming strategies. Many other statistical prop-
erties of Wishart matrices, such as those involving Gaussian
quadratic forms, have also been applied to MIMO analysis
in different contexts (eg. [27]). For a contemporary review of
random matrix theory and its application to wireless commu-
nications, see [28].
In this paper, we investigate the distribution of the condition
number of Wishart matrices, considering two closely related
measures: (i) the standard condition number (SCN), defined
as the ratio of the largest to the smallest eigenvalue, and
(ii) the Demmel condition number (DCN), which returns the
ratio of the matrix trace to the minimum eigenvalue [29].
Both of these condition numbers give a measure of the
relative conditioning (or rank-deficiency) of a matrix, and
the statistical properties of both have been shown to be
important for a variety of applications. For example, the SCN
has been used to indicate the degree of multipath activity
in MIMO communication channels [30,31], and has been
shown to have a close connection with the performance of
linear MIMO receivers in spatial multiplexing systems [32–
34]; further, it has been adopted to formulate novel spectrum
sensing algorithms in cognitive radio applications [35–37].
The distribution of the SCN also has many applications beyond
the realm of wireless communications; for example, it has
been employed in classical linear algebra for investigating
the sensitivity of matrix inversion problems to perturbation
errors and the convergence rate of iterative schemes [38]. The
distribution of the DCN, on the other hand, has been employed
in the context of MIMO to determine diversity-multiplexing
switching criteria for adaptive MIMO systems [39], and also
in the design of link adaptation protocols [40].
Despite the importance of the SCN and DCN of Wishart
matrices, little is still known about their statistical properties.
In particular, this issue was first considered in the pioneering
0090-6778/10$25.00 c ⃝ 2010 IEEE

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1706 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 6, JUNE 2010
work of Edelman [41] whose SCN analysis focused on 2×2
(i.e. dual) uncorrelated central Wishart matrices with arbitrary
degrees of freedom (DoF), and also on asymptotic distribu-
tions, valid for infinitely large dimensions. Some results on
the DCN were also presented, which gave asymptotic approx-
imations for the DCN distribution under certain conditions.
In [42–44], the authors proposed simple efficient bounds on
the high-tails of the SCN distributions, which again focused
primarily on uncorrelated central Wishart matrices. The SCN
probability density function (PDF) for the uncorrelated central
case was recently presented in tensor form in [45], using the
results of [46] for the ordered eigenvalues of Wishart matrices.
The SCN analysis of [41] was extended in [47] to allow for
matrices of arbitrary size and arbitrary correlation structures;
however, the final result for the SCN PDF was given as a
complicated expression involving an infinite series of zonal
polynomials, and is thus difficult to manage. Finally, the
exact SCN distributions for dual complex non-central Wishart
matrices with two DoF were recently presented in [48].
In this paper, we first present a novel generic framework
for the exact SCN distribution for different classes of complex
Wishart matrices. This key result is a unified expression which
applies for both central and non-central Wishart matrices of
arbitrary dimension (for the central case, allowing for arbitrary
correlation structure), and involves only a single scalar integral
which allows for fast and efficient computation. We note that
the analysis of non-central Wishart matrices is based on both
an exact infinite series representation as well as on a tractable
approximation which extensively simplifies the mathematical
formulations. For the specific case of dual uncorrelated and
correlated central Wishart matrices, we also derive exact
polynomial expressions for the SCN and DCN distributions.
Moreover, for correlated central Wishart matrices of arbitrary
dimension, we propose a new closed-form expression for
the SCN tail distribution. This result generalizes recent work
from [44] to account for arbitrary correlation structures, and
reveals an interesting connection between the tail behavior of
the SCN distribution and the maximum eigenvalue moments
of a Wishart matrix of smaller dimension. Based on our
analytical results, we gain valuable insights into the statistical
behavior of the channel conditioning for various MIMO fading
scenarios, such as uncorrelated/semi-correlated Rayleigh and
Ricean fading.
The rest of the paper is organized as follows. In Section II,
three common MIMO Gaussian fading models are introduced,
along with their connections to different classes of complex
Wishart distributions. The corresponding joint ordered eigen-
value distributions are also addressed. In Section III, our
new results on the SCN distribution are presented, whereas
Section IV deals with the DCN. In Section V, the theoretical
results are validated via Monte-Carlo simulations, while Sec-
tion VI concludes the paper and summarizes the key findings.
Notation: We use upper and lower case boldface for ma-
trices and vectors, respectively. The 푛 × 푛 identity matrix is
expressed as I푛, while the (푖,푗)-th entry of a matrix A is
denoted as 푎푖,푗. The symbol (⋅)푇denotes the transpose, tr(⋅)
yields the matrix trace, etr(⋅) is shorthand for exp(tr(⋅)),
and ∣ ⋅ ∣ represents the determinant. The symbol
d∼ denotes
“distributed as”, while ∼ will be used for asymptotic notation.
II. MIMO CHANNELS MODELS AND THEIR EIGENVALUE
DISTRIBUTIONS
Consider a MIMO system equipped with 푁푡transmit and
푁푟receive antennas. The wireless channel can be effectively
characterized by the matrix H ∈ ℂ푁푟×푁푡, whose entries rep-
resent the complex responses between each antenna pair. For
most performancemeasures of interest, such as the ergodic and
outage capacities [1,2,4,5,20,25], the diversity-multiplexing
trade-off [49], and symbol error rate of practical transmission
schemes [18,19,21,26,27], the effect of the channel is re-
flected via the statistical properties of the instantaneous MIMO
correlation matrix
{
The channel matrix H is typically modeled as complex
Gaussian. In this paper, we consider three common classes of
channels, for which W ∈ ℂ푠×푠follows a Wishart distribution
with 푡 DoF, where 푠 ≜ min(푁푡,푁푟) and 푡 ≜ max(푁푡,푁푟).
Note that, in all cases, the channel is normalized such that
퐸 [tr(W)] = 푁푟푁푡. Hereafter, we wiil use H푤to represent
an 푁푟×푁푡matrix whose entries are i.i.d.
variables.
Definition 1 (Uncorrelated Rayleigh Fading): The uncor-
related Rayleigh model is valid when the antenna spacings
and/or the angular spreads are high enough to induce indepen-
dent fading, and there is no LoS path between the transmitter
and receiver. Under these conditions,
W ≜
HH†,
H†H,
if 푁푟≤ 푁푡
if 푁푟> 푁푡.
(1)
d∼ 풞풩(0,1) random
H = H푤
(2)
and W is uncorrelated central Wishart with 푡 DoF, commonly
denoted as W
Definition 2 (Semi-Correlated Rayleigh Fading): In prac-
tice, the MIMO spatial subchannels are often correlated due to
the limited angular spreads or restrictions on the array sizes.
The effect of spatial correlation can be reflected as follows
{
H푤Σ1/2
d∼ 풞풲푠(푡,I푠) [50].
H =
Σ1/2
푠 H푤,
if 푁푟≤ 푁푡
if 푁푟> 푁푡
푠 ,
(3)
where Σ푠 ∈ ℂ푠×푠> 0 denotes the spatial correlation
matrix. Note that here we have assumed that correlation occurs
only at the side with minimum number of antennas2, in
which case W is semi-correlated central Wishart, expressed
as W
Definition 3 (Uncorrelated Ricean Fading): This scenario
is relevant when there is a direct LoS path between the
transmitter and receiver. Then, the channel matrix consists of
a deterministic component, HL, and a Rayleigh-distributed
random component, H푤, which accounts for the scattered
signals, or
√
d∼ 풞풲푠(푡,Σ푠).
H =
퐾푟
퐾푟+ 1HL+
√
1
퐾푟+ 1H푤
(4)
2Correlation at the side with the maximum number of antennas can also
be handled, however the notation is more cumbersome. Thus, we choose to
omit the explicit presentation of this scenario throughout the paper.

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MATTHAIOU et al.: ON THE CONDITION NUMBER DISTRIBUTION OF COMPLEX WISHART MATRICES1707
where 퐾푟 stands for the Ricean 퐾-factor, representing the
ratio of the power of the deterministic component to the power
of the fading component. Clearly, H has a non-zero mean,
given by 퐸 {H} = M =
W is uncorrelated non-central Wishart, usually expressed as
W
scaling factor, and
{
is the non-centrality matrix. For the sake of consistency with
previous studies, we consider a scaled version of W, that is
S = 휀−2W
W have, by definition, the same SCN.
Explicit solutions can sometimes be obtained for the distri-
butional properties of non-central Wishart matrices. However,
they are usually more complicated than the corresponding
expressions for central Wishart matrices. In this light, the fol-
lowing well-known result, which approximates a non-central
Wishart distribution with a central Wishart distribution of
modified correlation structure, will be particularly useful.
Lemma 1 ([51]): Acomplex
Wishart matrix W
풞풲푠(푡,ˆΣ푠), with the effective
first-order moments and second-order moments differing
by (1/푡)Ω, as the complex non-central Wishart matrix
W
√퐾푟/(퐾푟+ 1)HL. In this case,
(푡,휀2I푠,Ω), where 휀 = 1/√퐾푟+ 1 is a power
휀−2MM†,
if 푁푟≤ 푁푡
휀−2M†M,
if 푁푟> 푁푡
d∼ 풞풲푠
Ω =
(5)
d∼ 풞풲푠(푡,I푠,Ω) [18,19,50]. Note that S and
semi-correlatedcentral
d∼
correlation matrix beingˆΣ푠= 휀2I푠+ (1/푡)Ω, has the same
d∼ 풞풲푠(푡,휀2I푠,Ω).
A. Joint Ordered Eigenvalue Distributions
The instantaneous MIMO correlation matrix W (or S) is
Hermitian and positive definite, hence its eigenvalues are real
and strictly positive. Let us denote these eigenvalues by the
vector 흀 ≜ [휆1,휆2,...,휆푠]푇, with 휆1≥ 휆2... ≥ 휆푠> 0. For
each of the three channel models introduced in Definitions 1–
3, the corresponding joint ordered eigenvalue distributions
admit the following generic form [8,22,23,52]
푓(흀) = 퐾∣Φ(흀)∣ × ∣Ψ(흀)∣
푠∏
ℓ=1
휉(휆ℓ).
(6)
Table I on the next page lists the associated parameters for
each of the channel scenarios considered. Note that here we
adopt similar notation as in [23]. In Table I, V1(흀) is a
Vandermonde matrix with (푖,푗)-th entry 휆푖−1
흈 ≜ [휎1,휎2,...,휎푠]푇and 흎 ≜ [휔1,휔2,...,휔푠]푇contain the
non-zero ordered eigenvalues (assumed distinct) of Σ푠and Ω,
respectively, i.e. 휎1 > 휎2... > 휎푠 > 0 and 휔1 > 휔2... >
휔푠 > 0; also, F(흀,흎) and E(흀,흈) have (푖,푗)-th entries
0ℱ1(푡 − 푠 + 1;휆푗휔푖) and exp(−휆푗/휎푖) respectively, with
0ℱ1(⋅;⋅) denoting the standard generalized hypergeometric
function [53, Eq. (9.14.1)]. Note that in all cases, the (푖,푗)-th
element of Φ(흀) and Ψ(흀) has the same general form:
푗
; the vectors
{Φ(흀)}푖,푗= 휙푖(휆푗),
with 휙푖(⋅) and 휓푖(⋅) defined according to the specific channel
scenario, as described above. The generic eigenvalue distribu-
tion (6) will be particularly important for deriving the SCN
distribution in the following.
{Ψ(흀)}푖,푗= 휓푖(휆푗)
(7)
III. STATISTICS OF THE STANDARD CONDITION NUMBER
The SCN, defined as
푧 ≜휆1
휆푠,푧 ≥ 1
(8)
is a metric which determines the invertibility of a matrix. A
condition number close to one indicates a well-conditioned
full-rank matrix with almost equal eigenvalues, whereas a very
high condition number implies a near rank-deficient matrix.
As was previously stated, the SCN arises in various areas in
MIMO communications, as well as other mathematical fields
such as numerical analysis and linear algebra. In general, the
statistical properties of the SCN of MIMO communication
channels have not been well investigated. In this section, we
present new expressions for the distribution of the SCN, which
embraces each of the channel models introduced in Section II.
A. General Framework for the SCN Distribution of Wishart
Matrices
The following theorem establishes a new general framework
for the cumulative distribution function (CDF) of the SCN
of different classes of Wishart matrices, which applies for
arbitrary matrix dimensions and arbitrary DoF. This constitutes
a key contribution of the paper.
Theorem 1: The CDF, 퐹푧(푥) = Pr(푧 ≤ 푥), with 푥 ≥
1, of the SCN of an uncorrelated central Wishart matrix,
W
W
matrix, S
∫∞
d∼ 풞풲푠(푡,I푠), semi-correlated central Wishart matrix,
d∼ 풞풲푠(푡,Σ푠), and uncorrelated non-central Wishart
d∼ 풞풲푠(푡,I푠,Ω), can be represented as3
푠
∑
퐹푧(푥)=퐾
ℓ=1
0
?????
[∫푥휆푠
휆푠
휙푖(푢)휓푗(푢)휉(푢)푑푢, 푖 ∕= ℓ
휙푖(휆푠)휓푗(휆푠)휉(휆푠),푖 = ℓ
]?????푑휆푠
(9)
with 휙푖(⋅) and 휓푖(⋅) defined as in (7), while 퐾 is the
normalization constant given in Table I.
Proof: See Appendix A.
It is important to note that all integrals inside the determi-
nant admit a closed-form solution, for all cases of interest.
Thus, only a single integration is required, whose numerical
evaluation is more robust and efficient compared to conven-
tional Monte-Carlo simulations. The closed-form solutions for
the definite integrals in (9) are summarized in Table II, where
we have made use of [53, Eq. (3.381.1)] and [18, Eq. (4)].
In these expressions, 훾(푎,푥) denotes the lower incomplete
gamma function [53, Eq. (8.350.1)], and 푄푝,푞(푎,푏) is the
Nuttall 푄-function defined as [54]
∫∞
with 퐼푞(⋅) denoting the 푞-th order modified Bessel function of
the first kind.
We now take a deeper look at the numerical evaluation of
the non-central case. We first recall that since the sum of
the indices of the Nuttall-푄 function in Table II is always
odd, a closed-form representation is given in [55, Eq. (13)]
푄푝,푞(푎,푏) =
푏
푥푝exp
(
−푥2+ 푎2
2
)
퐼푞(푎푥)푑푥
(10)
3Here we adopt the compact notation for the determinant of a matrix,
written in terms of the (푖,푗)-th element.

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1708IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 6, JUNE 2010
TABLE I
JOINT EIGENVALUE PDF OF COMPLEX WISHART MATRICES: PARAMETERS AND NORMALIZATION CONSTANTS
Φ(흀)Ψ(흀)휉(휆ℓ)퐾
Uncorrelated central (UC)
V1(흀)V1(흀)휆푡−푠
ℓ
푒−휆ℓ
퐾푢푐=
⎡
⎣
푠∏
푠∏
푖=1
(푠 − 푖)!
푠∏
푗=1
(푡 − 푗)!
⎤
⎦
−1
Semi-correlated central (CC)
V1(흀)E(흀,흈)휆푡−푠
ℓ
퐾푐푐=
푖=1
1
휎푡
푖(푡 − 푖)!
푠∏
푖<푗
휎푖휎푗
휎푗− 휎푖
Uncorrelated non-central (UN)
V1(흀)F(흀,흎)휆푡−푠
ℓ
푒−휆ℓ
퐾푢푛= (푡 − 푠)!−푠etr(−흎)∣V1(흎)∣−1
in terms of Marcum 푄-functions and a finite weighted sum
of Bessel functions. Hence, a direct evaluation of (9) involves
either Nuttall-푄 or Marcum-푄 functions both of which are
somehow difficult to manipulate. In order to simplify the
procedure, a second alternative is to apply the central/non-
central approximation of Lemma 1 on (4), so that the resulting
SCN CDF expression does not involve the Nuttall-푄 function.
While this approximation simplifies the non-central analysis,
its accuracy deteriorates at high Ricean 퐾-factors.
The third alternative, which still leads to an exact solution,
is to expand the involved hypergeometric function in the
integrand of (9), according to [53, Eq. (9.14.1)]
0ℱ1(푚;푥) =
∞
∑
푘=0
푧푘(푚 − 1)!
푘!(푚 + 푘 − 1)!
(11)
so that the corresponding definite integral inside the determi-
nant for the non-central case becomes
∑
ℐ푖,푗=
∞
푘=0
(
(푡 − 푠)!휔푘
푘!(푡 − 푠 + 푘)!
푗
)(
훾(푡 − 푠 + 푘 + 푖,푥휆푠)
− 훾(푡 − 푠 + 푘 + 푖,휆푠)
)
. (12)
This expression converges quickly for most practical MIMO
configurations and 퐾-factors, e.g. 푡 ≤ 20 and 퐾푟≤ 10 dB,
and thus can be truncated to a finite number of terms while still
yielding good accuracy. This accuracy will be demonstrated
through simulations in Section V.
B. Exact Expressions for the SCN Distribution of Dual
Wishart Matrices
We now focus on the case of dual Wishart matrices with
arbitrary DoF, i.e. 푠 = 2 and 푡 ≥ 2. This scenario is particu-
larly important for MIMO systems, where size/implementation
cost limitations of the mobile terminals typically restrict the
numbers of antennas to be small (e.g. hand-held devices). In
the following, we present a closed-form polynomial expres-
sions for the CDF and PDF of the SCN for the case of dual
uncorrelated central Wishart matrices. We then consider the
more general case of semi-correlated central Wishart.
Theorem 2: The CDF of the SCN of W
given by
d∼ 풞풲2(푡,I2) is
퐹푧(푥) = 퐾푢푐(푝1(푥) − 푝1(1)),푥 ≥ 1
(13)
TABLE II
CLOSED-FORM EXPRESSIONS FOR THE DEFINITE INTEGRAL IN (9)
ℐ푖,푗=∫푥휆푠
휆푠
휙푖(푢)휓푗(푢)휉(푢)푑푢
UC
CC (훾(푡 − 푠 + 푖,푥휆푠/휎푗) − 훾(푡 − 푠 + 푖),휆푠/휎푗))휎푡−푠+푖
(푡 − 푠)!푒휔푗휔(푠−푡)/2
×
훾(푡 − 푠 + 푖 + 푗 − 1,푥휆푠) − 훾(푡 − 푠 + 푖 + 푗 − 1,휆푠)
푗
UN
푗
2(푠−푡−2푖+2)/2
(√2휔푗,√2휆푠
(√2휔푗,√2푥휆푠
(
푄푡−푠+2푖−1,푡−푠
)
−푄푡−푠+2푖−1,푡−푠
))
where
푝1(푦) = Δ1(푡,푡−1,푦)−2Δ1(푡−1,푡,푦)+Δ1(푡−2,푡+1,푦)
and
(
Δ1(푚,푛,푦) = (푛 − 1)!푚! −
푛−1
∑
푘=0
(푚 + 푘)!푦푘
푘!(푦 + 1)푚+푘+1
)
.
Proof: We start by particularizing (9) to the uncorrelated
central Wishart scenario using Table I. Then, by substituting
푠 = 2 and expanding the determinant gives
{∫∞
∫푥휆2
The result follows by integrating using [53, Eq. (3.351.1)] and
[53, Eq. (3.351.3)], and simplifying.
Corollary 1: The PDF of the SCN of W
given by
(
+Δ2(푡 − 2,푡 + 1,푧)
where
퐹푧(푥) = 퐾푢푐
0
푒−휆2휆푡−2
2
(
∫푥휆2
휆2
2
∫푥휆2
푢푡푒−푢푑푢
휆2
푢푡−2푒−푢푑푢
)
−2휆2
휆2
푢푡−1푒−푢푑푢 +
휆2
푑휆2
}
.
(14)
d∼ 풞풲2(푡,I2) is
푓(푧) = 퐾푢푐
Δ2(푡,푡 − 1,푧) − 2Δ2(푡 − 1,푡,푧)
)
,푧 ≥ 1
(15)
Δ2(푚,푛,푦) = −(푛 − 1)!
푛−1
∑
푘=0
(푚 + 푘)!푦푘−1(푘 − (푚 + 1)푦)
푘!(푦 + 1)푚+푘+2
.
Proof: Obtained by differentiating (13).
Note that an alternative expression for the SCN PDF of
uncorrelated central Wishart matrices was also originally given

Page 5

MATTHAIOU et al.: ON THE CONDITION NUMBER DISTRIBUTION OF COMPLEX WISHART MATRICES1709
by Edelman in [41, Eq. (7.2)].4We have, however,included the
result (15), derived using different methods and in the context
of our new general framework, for the sake of completeness.
In the following, we derive exact closed-form expressions for
the CDF and PDF of the SCN for dual semi-correlated central
Wishart matrices. To the best of our knowledge, these results
are new.
Theorem 3: The CDF of the SCN of W
given as
d∼ 풞풲2(푡,Σ2) is
퐹푧(푥) = 퐾푐푐(푝2(푥) − 푝2(1)),
where 푝2(푦) is defined in (17) while
푧 ≥ 1
(16)
Δ3(푚,푛,휇,휈,푦)
= (푛 − 1)!
(
푚!
휇푛휈푚+1−
푛−1
∑
푘=0
(푚 + 푘)!푦푘
푘!휇푛−푘(휇푦 + 휈)푚+푘+1
)
.
Proof: We start by particularizing (9) to the semi-
correlated central Wishart scenario using Table I. Then, sub-
stituting 푠 = 2 and expanding the determinant gives (18).
Integrating using [53, Eq. (3.351.1)] and [53, Eq. (3.351.3)],
we readily obtain (16).
Corollary 2: The PDF of the SCN of W
given in (19) at the bottom of the page while
d∼ 풞풲2(푡,Σ2) is
Δ4(푚,푛,휇,휈,푦)
= −(푛 − 1)!
푛−1
∑
푘=0
(푚 + 푘)!푦푘−1(푘휈 − 휇(푚 + 1)푦)
푘!휇푛−푘(휇푦 + 휈)푚+푘+2
.
Proof: Obtained by differentiating (16).
In contrast to the dual central uncorrelated and semi-
correlated Wishart scenarios considered above, the case of
dual uncorrelated non-central Wishart matrices does not lend
itself to a tractable representation. This is due, primarily, to
the cross-products of hypergeometric and Nuttall-푄 functions
that arise upon the determinant expansion in (9). However,
an infinite series representation can be obtained as in [48],
4Reference [41] adopts a slightly different definition of the SCN, given by
the square root of (8).
while a closed-form approximation for the SCN CDF/PDF of
dual complex non-central Wishart matrices is directly obtained
from Theorem 3 and Corollary 2 respectively, upon application
of Lemma 1 (i.e. by invoking the central/non-central Wishart
approximation).
C. High-Tail Distribution for the SCN of Wishart Matrices
We now investigate the probability of experiencing an
“extremely large” condition number. This is an important
issue for various practical applications. For example, in the
MIMO context, it predicts the probability of experiencing an
extremely poor channel due to fading. Moreover, since the
performance of MIMO linear receivers is intimately related
to the channel condition number, the asymptotic analysis
provides insights into the likelihood of such receivers failing
in practice [32]. The asymptotic analysis is also important
for characterizing the performance and complexity of adaptive
MIMO detectors [34] or lattice-aided MIMO receivers [56],
and has direct implications for the design of detection thresh-
olds for spectrum sensing algorithms (see for instance [35–
37]).
The high-tail distribution of uncorrelated central Wishart
matrices was recently established in [44, Theorem 3.2], which
revealed an interesting relationship in terms of the maximum
eigenvalue moments of a smaller Wishart matrix. Using a
similar technique to that of [44], we herein generalize that
result to the case of semi-correlated central Wishart matrices,
and demonstrate a similar interesting relationship in terms of
maximum eigenvalue moments.
We first require the following lemma:
Lemma 2: Let 푓(휆1,휆푠) be the joint density of the maxi-
mum and minimum eigenvalues of W
for fixed 휆1,
d∼ 풞풲푠(푡,Σ푠). Then,
푓(휆1,휆푠) ∼ 푓asy(휆1,휆푠)
= (푡 − 푠 + 1)휆푡−푠
푠
푠
∑
ℓ=1
풦ℓ푔푠−1,푡
(휆1,휎[ℓ]
),(휆푠→ 0)
(20)
푝2(푦) = Δ3(푡 − 1,푡 − 1,1/휎1,1/휎2,푦) − Δ3(푡 − 2,푡,1/휎1,1/휎2,푦)
− Δ3(푡 − 1,푡 − 1,1/휎2,1/휎1,푦) + Δ3(푡 − 2,푡,1/휎2,1/휎1,푦).
(17)
퐹푧(푥) = 퐾푐푐
{∫∞
∫∞
0
푒−휆2/휎2휆푡−2
2
(
휆2
∫푥휆2
∫푥휆2
휆2
푢푡−2푒−푢/휎1푑푢 −
∫푥휆2
∫푥휆2
휆2
푢푡−1푒−푢/휎1푑푢
)
푑휆2
}
−
0
푒−휆2/휎1휆푡−2
2
(
휆2
휆2
푢푡−2푒−푢/휎2푑푢 −
휆2
푢푡−1푒−푢/휎2푑푢
)
푑휆2
.
(18)
푓(푧) = 퐾푐푐
(
Δ4(푡 − 1,푡 − 1,1/휎1,1/휎2,푧) − Δ4(푡 − 2,푡,1/휎1,1/휎2,푧)
−Δ4(푡 − 1,푡 − 1,1/휎2,1/휎1,푧) + Δ4(푡 − 2,푡,1/휎2,1/휎1,푧)
)
,푧 ≥ 1
(19)