ROBUST ESTIMATION TECHNIQUES FOR COMPLEX-VALUED RANDOM VECTORS - Adaptive Signal Processing: Next Generation Solutions - page 98

B EXAMPLE 2.1
The complex normal (CN) distribution, labeled F k , is obtained with

g ( d ) ¼ exp ( d )

which gives c k , g ¼ p k as the value of the normalizing constant. At F k -distribution,
s C ¼ 1, so the parameters S and V coincide with the covariance matrix and
pseudo-covariance matrix of the distribution. Thus we write z CN k ( C , P ).

B EXAMPLE 2.2
The complex t-distribution with n degrees of freedom (0 , n , 1 ), labeled T k , n ,is
obtained with

g n ( d ) ¼ (1 þ 2 d=n ) (2 kþn ) = 2

which gives c k , g ¼ 2 k G ( 2 k þ 2 ) = [( pn ) k G ( 2 )] as the value of the normalizing constant.
The case n ¼ 1 is called the complex Cauchy distribution, and the limiting case
n !1 yields the CN distribution. We shall write z CT k , n ( S , V ). Note that
the T k , n -distribution possesses a finite covariance matrix for n . 2, in which
case s C ¼ n= ( n 2).

2.3.2 Circular Case

Definition 2 The subclass of CES distributions with V¼ 0 , labeled
F S ¼ CE k ( S , g ) for short, is called circular CES distributions.

Observe that V¼ 0 implies that D ( zjG ) ¼ z H S 1 z and jGj¼jSj

2 . Thus the pdf of
circular CES distribution takes the form familiar from the real case

f ( zjS ) ; f ( zjS ,0) ¼ c k , g jSj 1 g ( z H S 1 z ) :

Hence the regions of constant contours are ellipsoids in complex Euclidean k -space.
Clearly circular CES distributions belong to the class of circularly symmetric distri-
butions since f ( e ju zjS ) ¼ f ( zjS ) for all u [ R . For example, CN k ( C , 0), labeled
CN k ( C ) for short, is called the circular CN distribution (or, proper CN distribution
[38]), the pdf now taking the classical [24, 61] form

f ( zjC ) ¼ p k jCj 1 exp ( z H C 1 z ) :

See [33] for a detailed study of circular CES distributions.