Introduction to complex random vectors and processes - Statistical Signal Processing of Complex-Valued Data - page 50

Thus, we may define the characteristic function of x as

E exp j

2 s H x

E exp[j Re( s H x )] .

ψ ( s ) = = (2.99)

2.5.1 Characteristic functions of Gaussian and elliptical distributions

The characteristic function of a real elliptical random vector z (cf. Fang et al . (1990 )) is

s u

R zz s u
s v

exp(j s u

z )

s T
v

s T
v

ψ ( s u , s v ) = φ , (2.100)

where is a scalar function that determines the distribution. From this we obtain the

following result, which was first published by Ollila and Koivunen (2004 ).

φ

Result 2.7. The characteristic function of a complex elliptical random vector x is

exp j

φ 4 s H H xx s

2 s H

ψ ( s ) = x

2 s H H xx s +

H xx s ∗ )

exp[j Re( s H

1
2

Re( s H

= x )] φ . (2.101)

R xx , H xx = R xx , and

If x is complex Gaussian, then H xx = φ ( t ) = exp( − t / 2) .

Both expressions in ( 2.101 ) - in terms of augmented vectors and in terms of un-
augmented vectors - are useful. It is easy to see the simplification that occurs when
H xx =

0 or R xx =

0 , as when x is a proper Gaussian random vector.
The characteristic function has a number of useful properties. It exists even when
the augmented covariance matrix R xx (or the covariance matrix R xx ) is singular, a
property not shared by the pdf. The characteristic function also allows us to state a
simple connection between R xx and H xx for elliptical random vectors (consult Fang
et al . (1990 ) for the expression in the real case).

Result 2.8. Let x be a complex elliptical random vector with characteristic function
( 2.101 ). If the second-order moments of x exist, then

d
d t φ

R xx = c H xx with c =− 2 ( t ) | t = 0 . (2.102)

c H xx and R xx = c H xx . In particular, we see that x is proper

It follows that R xx =

if and only if H xx = 0 . For Gaussian x , φ ( t ) = exp( − t / 2) and therefore c = 1, as

expected.

2.5.2 Higher-order moments

First- and second-order moments suffice for the characterization of many probability dis-
tributions. They also suffice for the solution of mean-squared-error estimation problems
and Gaussian detection problems. Nevertheless, higher-order moments and cumulants
carry important information that can be exploited when the underlying probability distri-
bution is unknown. For a complex random variable x , there are N + 1 different N th-order