Introduction to complex random vectors and processes - Statistical Signal Processing of Complex-Valued Data - page 47

Complex elliptical distributions have a number of desirable properties.

The mean of x (if it exists) is independent of the choice of pdf generator g .

The augmented covariance matrix of x (if it exists) is proportional to the augmented
generating matrix H xx . The proportionality factor depends on g , and is most eas-
ily determined using the characteristic function of x . Therefore, if the second-order
moments exist, the pdf ( 2.76 ) is the pdf of a complex improper elliptical random
vector, and the pdf ( 2.78 )for H xx = 0 is the pdf of a complex proper elliptical random

vector with R xx =

0 . This is due to Result 2.8 , to be discussed in Section 2.5 .

C m

−→

All marginal distributions are also elliptical. If y : contains components of

C n , m

x : −→ < n , then y is elliptical with the same g as x , and y contains the

corresponding components of x . The augmented generating matrix of y , H yy ,isthe

sub-matrix of H xx that corresponds to the components that y extracts from x .

Let y = Mx + b , where M is a given 2 m × 2 n augmented matrix and b is a given

2 m × 1 augmented vector. Then y is elliptical with the same g as x and y = M x + b

and H yy = MH xx M H .

C m are jointly elliptically distributed with pdf generator
g , then the conditional distribution of x given y is also elliptical with

C n and y :

If x : −→ −→

H xy H − 1 x | y = x +

yy ( y

− y ) , (2.83)

which is analogous to the Gaussian case ( 2.60 ), and augmented conditional generating
matrix

H xy H − 1

yy H xy ,

H xx | y = H xx − (2.84)

which is analogous to the Gaussian case ( 2.61 ). However, the conditional distribution
will in general have a different pdf generator than x and y .

2.4 Sufficient statistics and ML estimators for covariances:

complex Wishart distribution

Let's draw a sequence of M independent and identically distributed (i.i.d.) samples
{

i

x i }

1 from a complex multivariate Gaussian distribution with mean

x and augmented

=

covariance matrix R xx . We assemble these samples in a matrix X = [ x 1 , x 2 ,..., x M ],

and let X x M ] denote the augmented sample matrix. Using the expression

for the Gaussian pdf in Result 2.4 , the joint pdf of the samples

= [ x 1 , x 2 ,...,

i

{ x i }

1 is =

= π − Mn (det R xx ) − M / 2 exp

xx ( x m − x )

M

1
2

( x m − x ) H R − 1

p ( X ) − (2.85)

m = 1

= π − Mn (det R xx ) − M / 2 exp

xx S xx )

M
2

tr( R − 1

− . (2.86)