Statistical analysis of the covariance m 001

ARTICLE IN PRESS
Signal Processing 90 (2010) 1165–1175

Contents lists available at ScienceDirect

Signal Processing
journal homepage: www.elsevier.com/locate/sigpro

Statistical analysis of the covariance matrix MLE in
K-distributed clutter
Frédéric Pascal a,, Alexandre Renaux b
a
b

SONDRA/Supelec, 3 rue Joliot-Curie, F-91192 Gif-sur-Yvette Cedex, France
Laboratoire des Signaux et Systemes (L2S), Université Paris-Sud XI (UPS), CNRS, SUPELEC, Gif-Sur-Yvette, France

a r t i c l e in fo

abstract

Article history:
Received 22 January 2009
Received in revised form
13 July 2009
Accepted 26 September 2009
Available online 17 October 2009

In the context of radar detection, the clutter covariance matrix estimation is an
important point to design optimal detectors. While the Gaussian clutter case has been
extensively studied, the new advances in radar technology show that non-Gaussian
clutter models have to be considered. Among these models, the spherically invariant
random vector modelling is particularly interesting since it includes the K-distributed
clutter model, known to fit very well with experimental data. This is why recent results
in the literature focus on this distribution. More precisely, the maximum likelihood
estimator of a K-distributed clutter covariance matrix has already been derived. This
paper proposes a complete statistical performance analysis of this estimator through its
consistency and its unbiasedness at finite number of samples. Moreover, the closedform expression of the true Cramér–Rao bound is derived for the K-distribution
covariance matrix and the efficiency of the maximum likelihood estimator is
emphasized by simulations.
& 2009 Elsevier B.V. All rights reserved.

Keywords:
Covariance matrix estimation
Cramér–Rao bound
Maximum likelihood
K-distribution
Spherically invariant random vector

Notations
The notational convention adopted is as follows: italic
indicates a scalar quantity, as in A; lower case boldface
indicates a vector quantity, as in a; upper case boldface
indicates a matrix quantity, as in A. RefAg and I mfAg
are the real and the imaginary parts of A, respectively.
The complex conjugation, the matrix transpose operator,
and the conjugate transpose operator are indicated by , T ,
and H , respectively. The j th element of a vector a is
denoted aðjÞ . The n th row and m th column element of the
matrix A will be denoted by An;m. jAj and TrðAÞ are the
determinant and the trace of the matrix A, respectively.
 denotes the Kronecker product. J  J denotes any matrix
norm. The operator vec(A) stacks the columns of the
matrix A one under another into a single column vector.
The operator vech(A), where A is a symmetric matrix, does

the same things as vec(A) with the upper triangular
portion excluded. The operator veckðAÞ of a skewsymmetric matrix (i.e., AT ¼ A) does the same thing as
vechðAÞ by omitting the diagonal elements. The identity
matrix, with appropriate dimensions, is denoted I and the
zero matrix is denoted 0. E½ denotes the expectation
a:s:
operator. - stands for the almost sure convergence and
Pr
- stands for the convergence in probability. A zero-mean
complex circular Gaussian distribution with covariance
matrix A is denoted CN ð0; AÞ. A gamma distribution with
shape parameter k and scale parameter y is denoted
Gðk; yÞ. A complex m-variate K-distribution with parameters k, y, and covariance matrix A is denoted Km ðk; y; AÞ.
A central chi-square distribution with k degrees of freedom is denoted w2 ðkÞ. A uniform distribution with
boundaries a and b is denoted U ½a;b .
1. Introduction

 Corresponding author. Tel.: þ33 147405320; fax: þ33 147402199.

E-mail addresses: frederic.pascal@supelec.fr (F. Pascal),
alexandre.renaux@lss.supelec.fr (A. Renaux).

0165-1684/$ - see front matter & 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.sigpro.2009.09.029

The Gaussian assumption makes sense in many
applications, e.g., sources localization in passive sonar,

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F. Pascal, A. Renaux / Signal Processing 90 (2010) 1165–1175

radar detection where thermal noise and clutter are
generally modelled as Gaussian processes. In these
contexts, Gaussian models have been thoroughly investigated in the framework of statistical estimation and
detection theory (see, e.g., [1–3]). They have led to
attractive algorithms such as the stochastic maximum
likelihood method [4,5] or Bayesian estimators.
However, the assumption of Gaussian noise is not
always valid. For instance, due to the recent evolution of
radar technology, one can cite the area of space time
adaptive processing-high resolution (STAP-HR) where the
resolution is such that the central limit theorem cannot
be applied anymore since the number of backscatters is
too small. Equivalently, it is known that reflected signals
can be very impulsive when they are collected by a low
grazing angle radar [6,7]. This is why, in the last decades,
the radar community has been very interested in
problems dealing with non-Gaussian clutter modelling
(see, e.g., [8–11]).
One of the most general non-Gaussian noise model is
provided by spherically invariant random vectors (SIRV)
which are a compound processes [12–14]. More precisely,
an SIRV is the product of a Gaussian random vector (the
so-called speckle) with the square root of a non-negative
random scalar variable (the so-called texture). In other
words, a noise modelled as an SIRV is a non-homogeneous
Gaussian process with random power. Thus, these kind of
processes are fully characterized by the texture and the
unknown covariance matrix of the speckle. One of the
major challenging difficulties in SIRV modelling is to
estimate these two unknown quantities [15]. These
problems have been investigated in [16] for the texture
estimation while [17,18] have proposed different estimation procedures for the covariance matrix. Moreover, the
knowledge of these estimates accuracy is essential in
radar detection since the covariance matrix and the
texture are required to design the different detection
schemes.
In this context, this paper focuses on parameters
estimation performance where the clutter is modelled
by a K-distribution. A K-distribution is an SIRV, with a
gamma distributed texture depending on two real positive
parameters a and b. Consequently, a K-distribution
depends on a, b and on the covariance matrix M. This
model choice is justified by the fact that a lot of
operational data experimentations have shown the good
agreement between real data and the K-distribution
model (see [7,19–22] and references herein).
This K-distribution model has been extensively studied
in the literature. First, concerning the parameters estimation problem, [23,24] have estimated the gamma distribution parameters assuming that M is equal to the
identity matrix, [17] has proposed a recursive algorithm
for the covariance matrix M estimation assuming a and
b known and [25] has used a parameter-expanded
expectation-maximization (PX-EM) algorithm for the
covariance matrix M estimation and for a parameter n
assuming n ¼ a ¼ 1=b. Note also that estimation schemes
in K-distribution context can be found in [26,27] and
references herein. Second, concerning the statistical
performance of these estimators, it has been proved in

[28] that the recursive scheme proposed by [17] converges
and has a unique solution which is the maximum
likelihood (ML) estimator. Consequently, this estimator
has become very attractive. In order to evaluate the
ultimate performance in terms of mean square error, [23]
has derived the true Cramér–Rao Bound (CRB) for the
parameters of the gamma texture (namely, a and b)
assuming M equal to the identity matrix. Gini [29] has
derived the modified CRB on the one-lag correlation
coefficient of M where the parameters of the gamma
texture are assumed to be nuisance parameters. Concerning the covariance matrix M, a first approach for the true
CRB study, which is known to be tighter than the modified
one, has been proposed in [25] whatever the texture
distribution. However, note that, for the particular case of
a gamma distributed texture, the analysis of [25] involves
several numerical integrations and no useful information
concerning the structure of the Fisher information matrix
(FIM) is given. Finally, classical covariance matrix estimators are compared in [30] in the more general context of
SIRV.
The knowledge of an accurate covariance matrix
estimate is of the utmost interest in context of radar
detection since this matrix is always involved in the
detector expression [30]. Therefore, the goal of this
contribution is twofold. First, the covariance matrix ML
estimate statistical analysis is provided in terms of
consistency and bias. Second, the closed-form expression
of the true CRB for the covariance matrix M is given and is
analyzed. Finally, through a discussion and simulation
results, classical estimation procedures in Gaussian and
SIRV contexts are compared.
The paper is organized as follows. Section 2 presents
the problem formulation while Sections 3 and 4 contain
the main results of this paper: the ML estimate statistical
performance in terms of consistency and bias and the
derivation of the true CRB. Finally, Section 5 gives
simulations which validate theoretical results.
2. Problem formulation
In radar detection, the basic problem consists in
detecting if a known signal corrupted by an additive
clutter is present or not. In order to estimate the clutter
parameters before detection, it is generally assumed that
K signal-free independent measurements, traditionally
called the secondary data ck , k ¼ 1; . . . ; K are available.
As stated in the introduction, one considers a clutter
modelled thanks to a K-distribution denoted
ck Km ða; ð2=bÞ2 ; MÞ:

From the SIRV definition, ck can be written as
pffiffiffiffiffi
ck ¼ tk xk ;

ð1Þ
ð2Þ

where tk is gamma distributed with parameters a and
ð2=bÞ2 , i.e., tk Gða; ð2=bÞ2 Þ and, where xk is a complex
circular zero-mean m-dimensional Gaussian vector with
covariance matrix E½xk xH
k  ¼ M independent of tk . For
identifiability considerations, M is normalized according
to TrðMÞ ¼ m (see [17]). Note that the parameter a
represents the spikiness of the clutter. Indeed, when a is

ARTICLE IN PRESS
F. Pascal, A. Renaux / Signal Processing 90 (2010) 1165–1175

high the clutter tends to be Gaussian and, when a is small,
the tail of the clutter becomes heavy.
The probability density function (PDF) of a random
variable tk distributed according to Gða; ð2=bÞ2 Þ is given by
!
!a

b2

pðtk Þ ¼

4

tak 1
b2
exp  tk ;
GðaÞ
4

ð3Þ

where GðaÞ is the gamma function defined by
Z þ1
GðaÞ ¼
xa1 expðxÞ dx:
0

ð4Þ

From Eq. (2), the PDF of ck can be written
!
Z þ1
1
cH
ck
1
kM
pðtk Þ dtk ;
exp 
pðck ; M; a; bÞ ¼
tm
pm jMj
tk
0
k
ð5Þ
which is equal to
pðck ; M; a; bÞ ¼

aþm

b

1
ðcH
ck ÞðamÞ=2
kM
aþm1 m
2
jMjGð Þ

p

a

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Kma b cH
M1 ck ;
k

ð6Þ

where Kn ðÞ is the modified Bessel function of the second
kind of order n [31].
Gini et al. have derived the ML estimator as the
solution of the following equation [17]:
^ ML
M

K
1X
H
^ 1
¼
cm ðcH
k M ML ck Þck ck ;
K k¼1

ð7Þ

where the function cm ðqÞ is defined as
pffiffiffi
b Kam1 ðb qÞ
cm ðqÞ ¼ pffiffiffi
ð8Þ
pffiffiffi :
2 q Kam ðb qÞ
^ ML has to be normalized as
Note that the ML estimate M
^ ML Þ ¼ m. Finally, it has been shown in [28] that
M : TrðM
the solution to Eq. (7) exists and is unique for the
aforementioned normalization.

^ ML
3. Statistical analysis of M
^ ML
This section is devoted to the statistical analysis of M
in terms of consistency and bias.
3.1. Consistency
^ of M is said to be consistent if
An estimator M
Pr

^  MJ - 0;
JM
K-þ1

where K is the number of secondary data ck ’s used to
estimate M.
^ ML is a consistent
^ ML consistency). M
Theorem 3.1 (M
estimate of M.
^ ML will be denoted MðKÞ
^
Proof. In the sequel, M
to show
^ ML and the number K
explicitly the dependence between M
of xk0 s. Let us define the function fK;M such that
8
D!D;
>
>
<
K
1X
ð9Þ
fK;M :
1
A!
cm ðcH
ck Þck cH
>
kA
k;
>
K k¼1
:

1167

where cm ðÞ is defined by Eq. (8), where D ¼ fA 2
Mm ðCÞjAH ¼ A; A positive definite matrix} with Mm ðCÞ ¼
fm  m matrices with elements in Cg, and where C is the
^
set of complex scalar. As MðKÞ
is a fixed point of function
fK;M , it is the unique zero, which respects the constraint
^
TrðMðKÞÞ
¼ m, of the following function:
(
D!D;
gK :
A!gK ðAÞ ¼ A  fK;M ðAÞ:
^
To prove the consistency of MðKÞ,
Theorem 5.9 of
[32, p. 46] will be used. First, the strong law of large
numbers (SLLN) gives
a:s:

gK ðAÞ - gðAÞ;

8A 2 D;

K-þ1

where
gðAÞ ¼ A  E½cm ðcH A1 cÞccH 

8A 2 D;

ð10Þ

for cKm ða; ð2=bÞ2 ; MÞ.
Let us now apply the change of variable y ¼ A1=2 c. We
obtain
yKm ða; ð2=bÞ2 ; A1=2 MA1=2 Þ
and
gðAÞ ¼ A1=2 ðI  E½cm ðyH yÞyyH ÞA1=2

8A 2 D;
and

gK ðAÞ ¼ A1=2 I 

8A 2 D;

!
K
1X
1=2
H
cm ðyH
y
Þy
y
:
k k k k A
K k¼1

Let us verify the hypothesis of Theorem 5.9 of
[32, p. 46]. We have to prove that for every e40,
Pr

ðH1 Þ : supfJgK ðAÞ  gðAÞJg - 0;
K-þ1

A2D

ðH2 Þ :

inf

A:JAMJZe

fJgðAÞJg40 ¼ gðMÞ:

For every A 2 D, we have



1 X
K

H
H
H 
JgK ðAÞ  gðAÞJ ¼ JAJ
ðcm ðyH
y
Þy
y

E
½c
ðy
yÞyy

:
m
k k k k

K
k¼1

Since E½cm ðyH yÞyyH o þ 1, one can apply the SLLN to
H
the K i.i.d. variables cm ðyH
k yk Þyk yk , with same first order
moment. This ensures ðH1 Þ.
Moreover, the function cm ðcH A1 cÞ is strictly decreasing
w.r.t. A. Consequently, E½cm ðcH A1 cÞccH  too. This implies
that E½cm ðcH A1 cÞccH aA, except for A ¼ M. This ensures
ðH2 Þ.
Finally, Theorem 5.9 of [32, p. 46] concludes the proof
Pr
^ ML M. &
and M
K-þ1

3.2. Bias
This subsection provides an analysis of the bias B
^ ML Þ ¼ E½M
^ ML   M.
defined by BðM
^ ML ). M
^ ML is an unbiased
Theorem 3.2 (Unbiasedness of M
estimate of M at finite distance (i.e., at finite number K).

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F. Pascal, A. Renaux / Signal Processing 90 (2010) 1165–1175

^ ML will be denoted M
^
Proof. For the sake of simplicity, M
in this part. By applying the following change of variable,
yk ¼ M1=2 ck , to Eq. (7), one has

where pðc1 ; . . . ; cK ; hÞ is the likelihood function of the
observations ck , k ¼ 1; . . . ; K. Concerning our model, the
parameter vector is

K
X
^ 1 y ÞM1=2 y yH M1=2 ;
^ ¼1
cm ðyH
M
k
k k
kT
K k¼1

h ¼ ðvechT ðRefMgÞ veckT ðI mfMgÞÞT ;

^ 1=2 :
T^ ¼ M1=2 MM
Therefore,
K
1X
^ 1 y Þy yH :
cm ðyH
T^ ¼
k k k
kT
K k¼1

T^ is thus the unique estimate (see [28, Theorem III.1]) of
^ ¼ m. Its statistic is clearly
the identity matrix, with TrðTÞ
independent of M since the yk ’s are i.i.d. SIRVs with a
gamma distributed texture and identity matrix for the
Gaussian
covariance
matrix.
In
other
words,
yk Km ða; ð2=bÞ2 ; IÞ.
Moreover, for any unitary matrix U,
K
1X
^ H 1 zk Þzk zH ;
cm ðzH
k ðUTU Þ
k
K k¼1

where zk ¼ Uyk are also i.i.d. and distributed as
^ H has the same distribution as T.
^
Km ða; ð2=bÞ2 ; IÞ and UTU
Consequently,
^ ¼ UE½TU
^ H
E½T

for any unitary matrix U:

^ is different from 0, Lemma A.1 of [30] ensures
Since E½T
^ 1=2 ,
^
that E½T ¼ gI for g 2 R. Remind that T^ ¼ M1=2 MM
^ ¼ TrðMÞ ¼ m, one
^ ¼ gM. Moreover, since TrðMÞ
then E½M
has
^ ¼ TrðEðMÞÞ
^ ¼ gTrðMÞ ¼ gm;
m ¼ EðTrðMÞÞ

ð13Þ

With the parametrization of Eq. (13), the structure of
CRB becomes
!1
F1;1 F1;2
;
ð14Þ
CRBh ¼
F2;1 F2;2

where

^ H¼
UTU

m2  1:

ð11Þ

where the Fi;j ’s are the elements of the FIM given by
"
#
@2 ln pðc1 ; . . . ; cK ; hÞ
;
F1;1 ¼ E
T
@vechðRefMgÞ@vech ðRefMgÞ
mðm þ 1Þ mðm þ 1Þ

;
ð15Þ
2
2
"
#
@2 ln pðc1 ; . . . ; cK ; hÞ
;
F1;2 ¼ FT2;1 ¼ E
T
@vechðRefMgÞ@veck ðI mfMgÞ
mðm þ 1Þ mðm  1Þ

;
ð16Þ
2
2
"
#
@2 ln pðc1 ; . . . ; cK ; hÞ
;
F2;2 ¼ E
T
@veckðI mfMgÞ@veck ðI mfMgÞ
mðm  1Þ mðm  1Þ

:
ð17Þ
2
2
Since the ck ’s are i.i.d. random vectors, one have from
Eq. (6),
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K
Y
baþm ðcHk M1 ck ÞðamÞ=2
Kma ðb cH
pðc1 ; . . . ; cK ; hÞ ¼
M1 ck Þ:
k
aþm1 m
2
p jMjGðaÞ
k¼1

ð18Þ

Consequently, the log-likelihood function can be written as
ln pðc1 ; . . . ; cK jhÞ ¼ Kln

which implies that g ¼ 1.
^ is an unbiased estimate of M, for any
In conclusion, M
number K of secondary data. &

baþm
2aþm1 pm GðaÞ

K
X

þ

k¼1

!

 KlnðjMjÞ

1
lnððcH
ck ÞðamÞ=2 Kma ðb
kM

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M1 ck ÞÞ:
cH
k

4.1. Result

ð19Þ

3.3. Comments
Theorems 3.1 and 3.2 show the attractiveness of the
estimator (7) in terms of statistical properties, i.e.,
consistency and unbiasedness. Note also that this estimator is robust since the unbiasedness property is at finite
number of samples. In the next section, the Cramér–Rao
bound is derived for the observation model (2).
4. Cramér–Rao bound
In this Section, the Cramér–Rao bound w.r.t. M is
derived. The CRB gives the best variance that an unbiased
estimator can achieve. The proposed bound will be
compared to the mean square error of the previously
studied ML estimator (Eq. (7)) in the next section.
The CRB for a parameter vector h is given by
  2
1
@ ln pðc1 ; . . . ; cK ; hÞ
CRBh ¼ E
;
ð12Þ
@h @hT

The next subsections will show that the CRB w.r.t. h, in
this context, is given by
!
T
1
0
1 ðH FHÞ
;
ð20Þ
CRBh ¼
K
0
ðKT FKÞ1
where the matrices H and K are constant transformation
matrices filled with ones and zeros such that vecðAÞ ¼
HvechðAÞ and vecðAÞ ¼ KveckðAÞ, with A a skew-symmetric matrix, and where


2ða þ 1Þ jða; mÞ

F ¼ ðMT  MÞ1 
mþ1
8
ðMT  MÞ1=2 ðI þ vecðIÞvecT ðIÞÞðMT  MÞ1=2 ;

ð21Þ

where jða; mÞ is given by Eq. (F.3).
Remarks.

jða; mÞ is a constant which does not depend on b since
b2 tk0 Gða; 4Þ (see Eq. (F.3)).

ARTICLE IN PRESS
F. Pascal, A. Renaux / Signal Processing 90 (2010) 1165–1175

The first term of the right hand side of Eq. (21) is the

Gaussian FIM (i.e., when tk ¼ 1 8k). Indeed, using
Eqs. (24), (49) and (30),1 the Gaussian CRB, denoted
GCRBh , is straightforwardly obtained as
!
T
T
1
1
0
1 ðH ðM  MÞ HÞ
GCRBh ¼
:
K
0
ðKT ðMT  MÞ1 KÞ1
ð22Þ
By identification with Eq. (21), it means that


2ða þ 1Þ g
¼ 0:

lim
a-1
8
mþ1

Eq. (23) is given by
1
@2 lnððcH
ck ÞðamÞ=2 Kma ðb
kM
T

@vechðRefMgÞ@vech ðRefMgÞ
2

¼

@ f ðzÞ

2

ð28Þ

T

2

T

þ ib @veck ðImfMgÞKT ðMT  MÞ1 vecðck cH


ð29Þ

and
T

 M1 ÞH@vechðRefMgÞ
2

T

þ 2b @veck ðI mfMgÞKT ððMT ck cTk MT Þ
 M1 ÞK@veckðI mfMgÞ

ð30Þ

and (details are given in Appendix C)
pffiffiffi
@f ðzÞ
1 Kmaþ1 ð zÞ
pffiffiffi Þ
d1 ðzÞ ¼
¼  pffiffiffi
@z
2 z Kma ð z

4.2.1. Analysis of F1;1
With Eq. (19) one has

ð31Þ

and

@2 ln pðc1 ; . . . ; cK ; hÞ
T

@vechðRefMgÞ@vech ðRefMgÞ
@ lnðjMjÞ

T

@vechðRefMgÞ@vech ðRefMgÞ

@2 f ðzÞ
@z2
pffiffiffi 
pffiffiffi 

1
Kmaþ1 ð zÞ 2
K
ð zÞ
pffiffiffi Þ pffiffiffi  ma1 pffiffiffi Þ
:
¼

4z
Kma ð z
Kma ð z
z

d2 ðzÞ ¼

2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

H 1 c ÞÞ
K @2 lnððcH M1 c ÞðamÞ=2 K
X
ma ðb ck M
k
k
k
:
T
@vechðRefMgÞ@vech ðRefMgÞ
k¼1

ð23Þ

The first part of the right hand side of Eq. (23) is given

ð32Þ

Consequently, the structure of Eq. (28) becomes
2

T

@2 f ðzÞ ¼ 2b d1 ðzÞ@vech ðRefMgÞHT P1 H@vechðRefMgÞ
2

T

þ 2b d1 ðzÞ@veck ðI mfMgÞKT P1 K@veckðI mfMgÞ

by

4

T

4

T

þ b d2 ðzÞ@vech ðRefMgÞHT P2 H@vechðRefMgÞ

@2 lnðjMjÞ

þ b d2 ðzÞ@veck ðI mfMgÞKT P2 K@veckðI mfMgÞ;

T

@vechðRefMgÞ@vech ðRefMgÞ
ð24Þ

thanks to Appendix A, Eq. (A.3), and thanks to the fact that
T

@veck ðI mfMgÞ
¼0
@vechðRefMgÞ

ð27Þ

@z ¼ b @vech ðRefMgÞHT ðMT  MÞ1 vecðck cH


To make the reading easier, only the outline of the CRB
derivation is given below. All the details are reported into
the different appendices.

¼ KHT ðMT  MÞ1 H

:

Note that


@ @f ðzÞ
@2 f ðzÞ
@f ðzÞ 2
@2 f ðzÞ ¼
@z @z þ
@z @z ¼
@ z;
@z
@z
@z
@z2

2

K

T

@vechðRefMgÞ@vech ðRefMgÞ

@2 z ¼ 2b @vech ðRefMgÞHT ððMT ck cTk MT Þ

4.2. Outline of the proof

þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cH
M1 ck ÞÞ
k

with (details are given in Appendix B)

Consequently, due to the structure of I þ vecðIÞvecT ðIÞ,
the FIM for K-distributed observations is given by the
Gaussian FIM minus a sparse matrix depending on a, M
and m.

¼ K

1169

ð25Þ

ð33Þ

where
P1 ¼ ðMT ck cTk MT Þ  M1 ;

ð34Þ

T
1
H
H
P2 ¼ ðMT  MÞ1 vecðck cH
k Þvec ðck ck ÞðM  MÞ :

ð35Þ

Therefore, Eq. (27) is given by

and

@2 f ðzÞ

T

@vech ðRefMgÞ
¼ I:
@vechðRefMgÞ

ð26Þ

Through the remain of the paper, let us set z ¼
pffiffiffi
b2 cHk M1 ck and f ðzÞ ¼ lnððz=b2 ÞðamÞ=2 Kma ð zÞÞ. The
second term (inside the sum) of the right-hand side of

T

@vechðRefMgÞ@vech ðRefMgÞ
2

4

¼ 2b d1 ðzÞHT P1 H þ b d2 ðzÞHT P2 H

ð36Þ

due to Eqs. (25) and (26).
Using Eqs. (23), (24), and (36), F1;1 is given by
2

F1;1 ¼ KHT ððMT  MÞ1 þ 2b E½d1 ðzÞP1 
4

þ b E½d2 ðzÞP2 ÞH;
2

1
1
With b ¼ 1, z ¼ cH
ck which is the term to be derived w.r.t. h in
kM
the exponential term of the multivariate Gaussian distribution.

ð37Þ

where d1 ðzÞ and d2 ðzÞ are defined by Eqs. (31) and (32),
respectively.

ARTICLE IN PRESS
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F. Pascal, A. Renaux / Signal Processing 90 (2010) 1165–1175


ðMT  MÞ1=2 ðI þ vecðIÞvecT ðIÞÞðMT  MÞ1=2 H;

The two expectation operators involved in the previous
equation can be detailed as follows:
E½d1 ðzÞP1  ¼ 12ðMT CMT Þ  M1

ð38Þ

and
E½d2 ðzÞP2  ¼ 14ðMT  MÞ1 ðW þ N  !ÞðMT  MÞ1 ;

ð39Þ

where
8
pffiffiffi


1 Kmaþ1 ð zÞ  T
>
>
>
p
ffiffi

p
ffiffi

;
c
C
¼
E
c
>
>
z Kma ð zÞ k k
>
>
>


>
>
1
>
H
H
H
>
W
¼
E
c
Þvec
ðc
c
Þ
;
vecðc
>
k
k
<
k
k
z
pffiffiffi


>
2 K
ð zÞ
>
>
N ¼ E 3=2 maþ1pffiffiffi vecðck cHk ÞvecH ðck cHk Þ ;
>
>
z
>
Kma ð zÞ
>
>
pffiffiffi
pffiffiffi

>
>
>
1 Kma1 ð zÞKmaþ1 ð zÞ
>
H
H
>
p
ffiffi

vecðck cH
!
¼
E
:
k Þvec ðck ck Þ:
2
z
Km
a ð zÞ

4.2.2. Analysis of F2;2
The analysis of F2;2 is similar to the one used for F1;1.
Indeed, one has to calculate
@2 ln pðc1 ; . . . ; cK ; hÞ
T

@veckðI mfMgÞ@veck ðI mfMgÞ
2

¼ K
þ



2

b2

MT :

ð41Þ

Concerning W, one has
"
#
H
H
vecðck cH
1
k Þvec ðck ck Þ
;
W¼ 2E
cH
M1 ck
b
k

T

ð44Þ

where the expectation is taken under a complex
K-distribution Km ða; ð2=bÞ2 ; MÞ.

H
H
The closed-form expression of E vecðck cH
k Þvec ðck ck Þ=
1
H
ck M ck  under a complex K-distribution is given in
Appendix E. One finds

a

b4 m þ 1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cH
M1 ck ÞÞ
k

@veckðI mfMgÞ@veck ðI mfMgÞ

k¼1

:

ð48Þ

Using Eq. (A.3), one has
K

@2 lnðjMjÞ
T

@veckðI mfMgÞ@veck ðI mfMgÞ

¼ KKT ðMT  MÞ1 K
@veckðI mfMgÞ

ð42Þ

where the expectation is taken under a complex
K-distribution Km ða  1; ð2=bÞ2 ; MÞ.
Concerning !, one has
"
#
pffiffiffi
pffiffiffi
H
H
1
K
ð zÞKmaþ1 ð zÞ vecðck cH
k Þvec ðck ck Þ
ffiffiffi
;
! ¼ 2 E ma1 2 p
Kma ð zÞ
cH
M1 ck
b
k

8

T

ð49Þ

due to Eq. (25) and due to

where the expectation is taken under a complex
K-distribution Km ða; ð2=bÞ2 ; MÞ.
Concerning N, one has
"
#
H
H
vecðck cH
1
k Þvec ðck ck Þ
;
ð43Þ

E
2
cH
M1 ck
ða  1Þb
k



@ lnðjMjÞ

@veckðI mfMgÞ@veck ðI mfMgÞ

K @2 lnððcH M1 c ÞðamÞ=2 K
X
ma ðb
k
k

ð40Þ

After some calculus detailed in Appendix D, one finds

ð47Þ

where jða; mÞ is given by Eq. (F.3).

ðMT=2  M1=2 ÞðI þ vecðIÞvecT ðIÞÞðMT=2  M1=2 Þ

ð45Þ

T

@veck ðI mfMgÞ

¼ I:

ð50Þ

By using the same notation as for the derivation of F1;1
and by using Eq. (33), one obtains for the second term on
the right hand side of Eq. (48) as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
@2 lnððcH
ck ÞðamÞ=2 Kma ðb cH
M1 ck ÞÞ
kM
k
T

@veckðI mfMgÞ@veck ðI mfMgÞ
2

4

¼ 2b d1 ðzÞKT P1 K þ b d2 ðzÞKT P2 K;

ð51Þ

where P1 , P2 , d1 ðzÞ and d2 ðzÞ are defined by Eqs. (34), (35),
(31), and (32), respectively. Therefore, the structure of F2;2
is the same as the structure of F1;1 except that one
replaces the matrix H by the matrix K.
4.2.3. Analysis of F1;2 ¼ FT2;1
Due to the structure of Eq. (A.3) and Eq. (33), and since
T
the derivation is w.r.t. @vechðRefMgÞ and @veck ðI mfMgÞ, it
is clear that
F1;2 ¼ FT2;1 ¼ 0

ð52Þ

by using Eq. (25).
This concludes the proof of the CRB derivation.
5. Simulation results

and



8

1
ðMT=2  M1=2 ÞðI þ vecðIÞvecT ðIÞÞðMT=2  M1=2 Þ:
b4 m þ 1

ð46Þ
The structure of ! is analyzed in Appendix F.
Consequently, Eq. (37) is reduced to



2ða þ 1Þ jða; mÞ
F1;1 ¼ KHT ðMT  MÞ1 

mþ1
8

In this section, some simulations are provided in order
to illustrate the proposed previous results in terms of
consistency, bias, and variance analysis (throughout the
CRB). While no mathematical proof is given in other
sections, we show the efficiency of the MLE meaning that
the CRB is achieved by the variance of the MLE.
The results presented are obtained for complex
K-distributed clutter with covariance matrix M randomly
chosen.

ARTICLE IN PRESS
F. Pascal, A. Renaux / Signal Processing 90 (2010) 1165–1175

All the results are presented different values of
pffiffiffi
parameter a and b ¼ 2 a following the scenario of [17].
The size of each vector ck is m ¼ 3. Remember that the
parameter a represents the spikiness of the clutter (when
a is high the clutter tends to be Gaussian and, when a is
small, the tail of the clutter becomes heavy). The norm
used for consistency and bias is the L2 norm.

1171

5.1. Consistency
Fig. 1 presents results of MLE consistency for 1000
Monte Carlo runs per each value of K. For that purpose, a
^ KÞ ¼ JM
^  MJ versus the number K of ck ’s is
plot of DðM;
presented for each estimate. It can be noticed that the
^ KÞ tends to 0 when K tends to 1 for
above criterion DðM;

^ KÞ versus the number of secondary data for different values of a.
Fig. 1. DðM;

^ KÞ versus the number of secondary data for different values of a.
Fig. 2. CðM;

ARTICLE IN PRESS
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F. Pascal, A. Renaux / Signal Processing 90 (2010) 1165–1175

0.09

Empirical MSE α =1
CRB α = 1
Empirical MSE α = 0.1
CRB α = 0.1
Gaussian CRB

0.08
0.07

MSE

0.06
0.05
0.04
0.03
0.02
0.01
100

200

300

400
500
600
700
Number of secondary data

800

900

1000

Fig. 3. GCRB, CRB and empirical variance of the MLE for different values of a.

each estimate. Moreover, note that the parameter a has
very few influence on the convergence speed which
highlights the robustness of the MLE.
5.2. Bias
Fig. 2 shows the bias of each estimate for the different
values of a. The number of Monte Carlo runs is given in
the legend of the figure.
For that purpose, a plot of the
___
^ KÞ ¼ JM
^  MJ versus
the number K of ck ’s is
criterion CðM;
___
^ is defined as the empirical
presented for each estimate. M
^
mean of the quantities MðiÞ
obtained from I Monte Carlo
runs. For each iteration i, a new set of K secondary data ck
^
is generated to compute MðiÞ.
It can be noticed that,
as enlightened by the previous theoretical analysis, the
^ tends to 0 whatever the value of K. Furthermore,
bias of M
one sees again the weak influence of the parameter a on
the unbiasedness of the MLE.

(at finite number of samples) have been proved. In order
to analyze the variance of the estimator, the Cramér–Rao
lower bound is derived. The Fisher information matrix in
this case is simply the Fisher information matrix of the
Gaussian case plus a term depending on the tail of the
K-distribution. Simulation results have been proposed to
illustrate these theoretical analyses. These results have
shown the efficiency of the estimator and the weak
influence of the spikiness parameter in terms of consistency and bias.
Appendix A. Derivation of @2 lnfjMjg
To find this term and several other, we will use the
following results [33]:
@TrðXÞ ¼ Trð@XÞ;

ðA:1aÞ

@vecðXÞ ¼ vecð@XÞ;

ðA:1bÞ

5.3. CRB and MSE

@A1 ¼ A1 @AA1 ;

ðA:1cÞ

The CRB and empirical variance of the MLE for 10 000
Monte Carlo runs are plotted in Fig. 3. For comparison, we
also plot the Gaussian CRB (i.e., when tk ¼ 1 8k) given by
Eq. (22). Although it is not mathematically proved in this
paper, one observes, in Fig. 3, the efficiency of the MLE
even for impulsive noise (a small).

@jAj ¼ jAj TrðA1 @AÞ;

ðA:1dÞ

6. Conclusion
In this paper, a statistical analysis of the maximum
likelihood estimator of the covariance matrix of a complex
multivariate K-distributed process has been proposed.
More particularly, the consistency and the unbiasedness

@ðA

BÞ ¼ @ðAÞ

BþA

@ðBÞ

where

¼  or 

ðA:1eÞ

@lnðjMjÞ ¼ TrðM1 @MÞ;

ðA:1fÞ

@ð@MÞ ¼ 0;

ðA:1gÞ

TrðABÞ ¼ TrðBAÞ

ðA:1hÞ

M1  M1 ¼ ðM  MÞ1 ;

ðA:1iÞ

TrðAH BÞ ¼ vecH ðAÞvecðBÞ;

ðA:1jÞ

ARTICLE IN PRESS
F. Pascal, A. Renaux / Signal Processing 90 (2010) 1165–1175

vecðABCÞ ¼ ðCT  AÞvecðBÞ:
@lnðjMjÞ ¼ TrðM

2

1
@MM1 ck Þ
@z ¼ b TrðcH
kM

from ðA:1fÞ

ð@MÞÞ

Concerning @2 z, one has

ðA:1kÞ

By using these properties, one has
1

2
2

from ðA:1aÞðA:1eÞðA:1gÞðA:1cÞ

¼ vecH ðM1 ð@MÞM1 Þvecð@MÞ

¼ b

2

¼ 2b Trð@MM

from ðA:1kÞ

ðM  MÞ

1

 M1 Þ@vecðMÞ

1
ck cH
Þ
kM

from ðA:1cÞðA:1hÞ

from ðA:1jÞðA:1kÞðA:1bÞ

2

T

@2 z ¼ 2b @vech ðRefMgÞHT ððMT ck cTk MT ÞM1 ÞH@vechðRefMgÞ

@vecðRefMg þ iI mfMgÞ

2

T

þ2b @veck ðI mfMgÞKT ððMT ck cTk MT Þ
M1 ÞK@veckðI mfMgÞ:

@vecH ðiI mfMgÞðMT  MÞ1 @vecðRefMgÞ
@vecH ðRefMgÞðMT  MÞ1 @vecðiI mfMgÞ

@vecH ðiI mfMgÞðMT  MÞ1 @vecðiI mfMgÞ:

ðA:2Þ

Since vecH ðiI mfMgÞ ¼ ivecT ðI mfMgÞ and vecH ðRefMgÞ
¼ vecT ðRefMgÞ, @2 lnðjMjÞ is reduced to
@2 lnðjMjÞ ¼ @vecT ðRefMgÞðMT  MÞ1 @vecðRefMgÞ

 @vecT ðI mfMgÞðMT  MÞ1 @vecðI mfMgÞ

¼ @vech ðRefMgÞH

from ðA:1eÞðA:1gÞ

By letting M ¼ RefMg þ iI mfMg, one obtains

¼ @vecH ðRefMgÞðMT  MÞ1 @vecðRefMgÞ

T

@MM

1

¼ 2b @vecH ðMÞððMT ck cTk MT Þ

By letting M ¼ RefMg þ iI mfMg in Eq. (A.3), one has
@ lnðjMjÞ ¼ @vec ðRefMg þ iI mfMgÞ

1

2

¼ @vecH ðMÞðMT  MÞ1 @vecðMÞ
from ðA:1bÞðA:1iÞ:
H

from ðA:1aÞ

1
TrðcH
Þ@MM1 ck
k @ðM

1
þ cH
@M@ðM1 Þck Þ
kM

from ðA:1jÞ

¼ vecH ð@MÞðMT  M1 ÞH vecð@MÞ

T

ðB:2Þ

1
@2 z ¼ b TrðcH
@MM1 Þck Þ
k @ðM

@2 lnðjMjÞ ¼ TrðM1 ð@MÞM1 @MÞ

2

1173

T

ðMT  MÞ1 H@vechðRefMgÞ
T

@veck ðI mfMgÞKT ðMT  MÞ1 K@veckðI mfMgÞ;
ðA:3Þ

where the matrix H and K are constant transformation
matrices filled with ones and zeros such that vecðAÞ ¼
HvechðAÞ and vecðAÞ ¼ KveckðAÞ.
Appendix B. Derivation of @z and @2 z

ðB:3Þ

Appendix C. Derivation of @f ðzÞ=@z and @2 f ðzÞ=@z2
Concerning the first derivative of f ðzÞ, one has
pffiffiffi
@f ðzÞ
@
2
¼ lnððz=b ÞðamÞ=2 Kma ð zÞÞ
@z
@z
pffiffiffi
am
1
@Kma ð zÞ
pffiffiffi
¼
þ
:
2z
@z
Kma ð zÞ

ðC:1Þ

Since @Kn ðyÞ=@y ¼ Knþ1 ðyÞ þ ðn=yÞKn ðyÞ [34]. It follows
that
pffiffiffi
pffiffiffi
pffiffiffi
@Kma ð zÞ
1
ma
Kma ð zÞ:
ðC:2Þ
¼  pffiffiffi Kmaþ1 ð zÞ þ
2z
@z
2 z
Plugging Eq. (C.2) in Eq. (C.1), one obtains
pffiffiffi
@f ðzÞ
1 Kmaþ1 ð zÞ
pffiffiffi :
¼  pffiffiffi
@z
2 z Kma ð zÞ

ðC:3Þ

Concerning the second derivative of f ðzÞ, one has
pffiffiffi 

@ f ðzÞ
1 @
1 K
ð zÞ
pffiffiffi maþ1 pffiffiffi Þ
¼
2
2 @z
@z
z Kma ð z
!


pffiffiffi
1 Kmaþ1 ð zÞ
1 @ Kmaþ1 ðyÞ


pffiffiffi 
: ðC:4Þ
¼ 3=2
4z @y Kma ðyÞ
y¼pffiffiz
4z
Kma ð zÞ
2

By using the properties from Eq. (A.1a) to (A.1k), one
has for @z
2

2

1
1
ck Þ ¼ b @TrðcH
ck Þ
@z ¼ b @ðcH
kM
kM
2

1
¼ b TrðcH
Þck Þ
k @ðM
2

from ðA:1aÞ

1
TrðcH
@MM1 ck Þ
kM

from ðA:1cÞ

1
¼ b Trð@MM1 ck cH
Þ
kM

from ðA:1hÞ

¼ b

2
2

1
¼ b vecH ð@MÞvecðM1 ck cH
Þ
kM
2

Since @Kn ðyÞ=@y ¼ Kn1 ðyÞ  ðn=yÞKn ðyÞ [34]. It follows
that


@ Kmaþ1 ðyÞ


@y Kma ðyÞ
y¼pffiffiz

from ðA:1jÞ

pffiffiffi
pffiffiffi
pffiffiffi
maþ1
pffiffiffi
Kmaþ1 ð zÞÞKma ð zÞ
ðKma ð zÞ þ
z
pffiffiffi
¼
2
Km
a ð zÞ
pffiffiffi
pffiffiffi
pffiffiffi
ma
Kmaþ1 ð zÞðKma1 ð zÞ þ pffiffiffi Kma ð zÞÞ
z
pffiffiffi
:
þ
2
Km
a ð zÞ

H

¼ b @vec ðMÞ

ðMT  MÞ1 vecðck cH


from ðA:1bÞðA:1kÞðA:1iÞ:

By letting M ¼ RefMg þ iI mfMg, one obtains
2

@z ¼ b @vecT ðRefMgÞðMT  MÞ1 vecðck cH

2

þ ib @vecT ðI mfMgÞðMT  MÞ1 vecðck cH

2

T

¼ b @vech ðRefMgÞHT ðMT  MÞ1 vecðck cH

2

T

þ ib @veck ðI mfMgÞKT ðMT  MÞ1 vecðck cH
k Þ:

ðB:1Þ

ðC:5Þ

Plugging Eq. (C.5) in Eq. (C.4), one obtains
pffiffiffi
pffiffiffi
pffiffiffi
@2 f ðzÞ
1
1 Kmaþ1 ð zÞ 1 Kmaþ1 ð zÞKma1 ð zÞ
pffiffiffi 
pffiffiffi
¼
:
þ 3=2
2
2
4z 2z
@z
Kma ð zÞ 4z
Kma ð zÞ

ðC:6Þ

ARTICLE IN PRESS
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F. Pascal, A. Renaux / Signal Processing 90 (2010) 1165–1175

Appendix D. Derivation of matrix C

since
xH
k xk ¼

The matrix C is given by


pffiffiffi

1 Kmaþ1 ð zÞ  T
Gða  1Þ T
pffiffiffi ck ck ¼
E ½ck cH
k ;
2GðaÞ
z Kma ð zÞ

C ¼ E pffiffiffi

¼

ðD:1Þ

Kmaþ1 ðb

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M1 ck Þ;
cH
k

ðD:2Þ

which is a complex K-distribution Km ða  1; ð2=bÞ2 ; MÞ.
Then


¼

1
1
ET ½ck cH
ET ½tk xk xH
k ¼
k
2ða  1Þ
2ða  1Þ
1
E½tk ET ½xk xH
k ;
2ða  1Þ

ðD:3Þ
2

where E½tk  ¼ ða  1Þð2=bÞ since tk follows a Gamma
T
distribution Gða  1; ð2=bÞ2 Þ and ET ½xk xH
k  ¼ M since xk is
a complex normal random vector (independent of tk ) with
zero mean and covariance matrix M. Consequently,
C ¼ ð2=b2 ÞMT .

E

M1 ck
cH
k

"

E

¼ ðM

1=2

M

Þ

#
H
H
vecðyk yH
k Þvec ðyk yk Þ
ðMT=2  M1=2 Þ;
yH
y
k k

ðE:1Þ

ðE:2Þ

where E½tk  ¼ að2=bÞ2 and where
"
#
H
H
vecðxk xH
k Þvec ðxk xk Þ
R¼E
:
ðE:3Þ
xH
x
k k
qffiffiffiffiffiffi
Let us set xðjÞ
¼ r2j expðiyj Þ for j ¼ 1; . . . ; m. Note that
k

r2j w2 ð2Þ is independent of yj U ½0;2p . Consequently, the

elements of the matrix R can be rewritten as
2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

Rk;l ¼ E4

r2p r2q r2p0 r2q0
Pm

j¼1

r2j

5E½expðiðyp  yp0 þ yq0  yq ÞÞ

r2p r2p0 expðiðyp  yp0 ÞÞ:

ðE:5Þ

(1) p ¼ p0 ¼ q ¼ q0 , i.e., l ¼ k ¼ p þ mðp  1Þ,
(2) p ¼ p0 , q ¼ q0 and paq, i.e., l ¼ k ¼ p þ mðq  1Þ,
(3) p ¼ q, p0 ¼ q0 and pap0 , i.e., l ¼ p0 þ mðp0  1Þ and
k ¼ p þ mðp  1Þ.
Consequently, the non-zero elements of Rk;l are given by
P
(1) Rpþmðp1Þ;pþmðp1Þ ¼ E½ðr2p Þ2 = m
r2  ¼ 4=ðm þ 1Þ,
Pj¼1 j2
r  ¼ 2=ðm þ 1Þ,
(2) Rpþmðq1Þ;pþmðq1Þ ¼ E½r2p r2p0 = m
Pj¼1 j 2
(3) Rpþmðp1Þ;p0 þmðp0 1Þ ¼ E½r2p r2q = m
j¼1 rj  ¼ 2=ðm þ 1Þ,

H
H
H 1
and the matrix E½vecðck cH
ck  can be
k Þvec ðck ck Þ=ck M
written as
"
#
 2
H
H
vecðck cH
2a
2
k Þvec ðck ck Þ
¼
E
mþ1 b
cH M1 ck
k

ðMT=2  M1=2 Þ

ðI þ vecðIÞvecT ðIÞÞðMT=2  M1=2 Þ:

1
H
ck  where ck Km ða  1; ð2=bÞ2 ; MÞ is
vecH ðck cH
k Þ=ck M
given by
"
#
 
H
H
vecðck cH
2ða  1Þ 2 2
k Þvec ðck ck Þ
E
¼
mþ1 b
cH
M1 ck
k

ðMT=2  M1=2 Þ

ðI þ vecðIÞvecT ðIÞÞðMT=2  M1=2 Þ:
ðE:7Þ

Appendix F. Analysis of !

pffiffiffiffiffi
where yk Km ða; ð2=bÞ2 ; IÞ. Since yk ¼ tk xk , where
2
tk Gða; ð2=bÞ Þ is independent of xk CN ð0; IÞ, one has
"
#
H
H
vecðck cH
k Þvec ðck ck Þ
E
¼ ðMT=2  M1=2 ÞE½tk R
1
cH
M
c
k
k
ðMT=2  M1=2 Þ;

qffiffiffiffiffiffiffiffiffiffiffiffi

In the same way, the expression of E½vecðck cH


In this appendix, we derive the expression of E½vecðck cH

1
2
H
Þ=c
M
c

where
c
K
ð
a
;
ð2=
b
Þ
;
MÞ.
The
case
vecH ðck cH
m
k
k
k
k
where ck Km ða  1; ð2=bÞ2 ; MÞ will be, of course, straightforward. Let us set the following change of variable:
ck ¼ M1=2 yk . One obtains from Eq. (A.1k)
T=2

r2j and ½vecðxk xHk Þn

ðE:6Þ

Appendix E. Derivation of
H
H
H 1
E½vecðck cH
k Þvec ðck ck Þ=ck M ck 

"
#
H
H
vecðck cH
k Þvec ðck ck Þ

j¼1

Note that E½expðiðyp  yp0 þ yq0  yq ÞÞa0 if and only if

where the last expectation is taken under the distribution

baþm1 jMj1
pðck Þ ¼ aþm2
ðcH M1 ck Þðam1Þ=2
2
pm Gða  1Þ k

m
X

Let us set the following change of variable: ck ¼
pffiffiffiffiffi
M1=2 tk xk , where tk Gða; ð2=bÞ2 Þ is independent of
xk CN ð0; IÞ, one has



~ ðMT=2  M1=2 Þ;
ðMT=2  M1=2 Þ!

where
2

!~ ¼ E6
4

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3

tk Kma1 ðb tk xHk xk ÞKmaþ1 ðb tk xHk xk Þ vecðxk xH ÞvecH ðxk xH Þ7
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H
2
Km
a ðb tk xk xk Þ

Let us set

xðjÞ
k

k

xH
x
k k

k

5:

ðF:1Þ
qffiffiffiffiffiffi
2
2
2
¼ rj expðiyj Þ for j ¼ 1; . . . ; m. rj w ð2Þ is

independent of yj U ½0;2p . Consequently, due to Eq. (E.5),
~ can be rewritten as
the elements of the matrix !
U~

ðE:4Þ

1

b2

k;l

2

6
¼ E4tk0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi3
P
Pm 2ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
Kma1 b tk0 m
r2p r2q r2p0 r2q0 7
j¼1 rj Kmaþ1 b tk0
j¼1 rj
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pm 2 5
Pm 2ffi
2
j¼1 rj
Km
a b tk0
j¼1 rj

E½expðiðyp  yp0 þ yq0  yq ÞÞ:

ðF:2Þ

ARTICLE IN PRESS
F. Pascal, A. Renaux / Signal Processing 90 (2010) 1165–1175

As before, E½expðiðyp  yp0 þ yq0  yq ÞÞa0 if and only if
(1) p ¼ p0 ¼ q ¼ q0 , i.e., l ¼ k ¼ p þ mðp  1Þ,
(2) p ¼ p0 , q ¼ q0 and paq, i.e., l ¼ k ¼ p þ mðq  1Þ,
(3) p ¼ q, p0 ¼ q0 and pap0 , i.e., l ¼ p0 þ mðp0  1Þ and
k ¼ p þ mðp  1Þ.
Consequently, the non-zero elements of U~ k;l are given by
~
(1) U pþmðp1Þ;pþmðp1Þ
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P

~
(2) U pþmðq1Þ;pþmðq1Þ
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3
ðr2p Þ2 7
Pm 2 5;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P

~
0 þmðp0 1Þ
(3) U pþmðp1Þ;p
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2p r2p0 7
Pm 2 5;
j¼1 rj

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P

r2p r2q 7
Pm 2 5:
j¼1 rj

6
¼ E4tk0
6
¼ E4tk0
6
¼ E4tk0

Note

Kma1 b

Kma1 b

6
¼ b E4tk0

j¼1

r2j Kmaþ1 b tk0

m
j¼1

r2j

tk0

Pm

j¼1

r2j Kmaþ1 b tk0

m
j¼1

r2j

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pm 2ffi
2
Km
a b tk0
j¼1 rj

Kma1 b

tk0

Pm

j¼1

r2j Kmaþ1 b tk0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pm 2ffi
2
Km
a b tk0
j¼1 rj

that

2

Pm

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pm 2ffi
2
Km
a b tk0
j¼1 rj

1 ~
2 U pþmðp1Þ;pþmðp1Þ

jða; mÞ 2

tk0

m
j¼1

r2j

j¼1

rj

3

3

U~ pþmðq1Þ;pþmðq1Þ ¼ U~ pþmðp1Þ;p0 þmðp0 1Þ ¼
8p 8p0 8q 8q0 . Consequently, only

3
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
Pm 2ffi
2
2 2
Kma1 b tk0 m
j¼1 rj Kmaþ1 ðb tk0
j¼1 rj Þ ðrp Þ
7
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pm 2 5
Pm 2ffi
2
j¼1 rj
Km
a ðb tk0
j¼1 rj Þ

ðF:3Þ

has to be computed and the matrix ! can be written as



jða; mÞ
4

2b

ðMT=2  M1=2 ÞðI þ vecðIÞvecT ðIÞÞ

ðMT=2  M1=2 Þ:

Note that
b2 tk0 Gða; 4Þ.

ðF:4Þ

jða; mÞ is independent of b since,

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