COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 60

where

' t ( W ) ¼ log p ( x ( t ) jW ) ¼ log p S ( Wx ) þ log j det Wj

and the density of the transformed random variables is written through the compu-
tation of the Jacobian as

p ( x ) ¼j det Wjp S ( Wx ) (1 : 48)

where W is defined in (1.11).

We use the notation that p S ( Wx ) W Q n¼ 1 p S n ( w n x ), where w n is the n th row of W ,
p S n ( u n ) W p S n ( u n r , u n i ) is the joint pdf of source n , n ¼ 1, ... , N , with u n ¼ u n r þ ju n i ,
and defined W ¼ A 2 1 , that is, we express the likelihood in terms of the inverse mixing
matrix, which provides a convenient change of parameter. Note that the time index in
x ( t ) has been omitted in the expressions for simplicity.
To

take advantage of Wirtinger calculus, we write each pdf as

N N N

p S n ( u r , u i ) ¼ g n ( u , u ) to define g ( u , u ): C C 7! R so that we can directly

evaluate

@ log g ( u , u )

@W ¼ @ log g ( u , u )

x H W c ( u , u ) x H

(1 : 49)

@u

where u ¼ Wx and we have defined the score function c ( u , u ) that is written directly
by using the result in Brandwood's theorem given by (1.5)

1
2

@ log p S ( u r , u i )

@u r þ j @ log p S ( u r , u i )

c ( u , u ) ¼

: (1 : 50)

@u i

When writing (1.49) and (1.50), we used a compact vector notation where each
element of the score function is given by

c n ( u , u ) ¼ @ log g n ( u n , u n )

1
2

@ log p S n ( u r , n , u i , n )
@u r , n

þ j @ log p S n ( u r , n , u i , n )
@u i , n

¼ :

@u n

(1 : 51)

To compute @ log j det Wj=@W , we first observe that @ log j det Wj¼ Trace( W 1
@W ) ¼ Trace( @WPP 1 W 1 ), and then choose

1
2

I jI
jII

P ¼

to write

@ log j det Wj¼ Trace( W 1 @W ) þ Trace(( W ) 1 @W )

¼hW H , @WiþhW T , @W i:

(1 : 52)