COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 68

If a given source has a circular distribution, that is, p S n ( u ) ¼ g ( juj ), the corres-
ponding entry of the score function vector can be easily evaluated as

c n ( u ) ¼ @ log g (

uu p )
@u ¼

:

g 0 ( j u j )
g ( juj )

u
2 juj

Thus, the score function always has the same phase as its argument. This is the form of
the score function proposed in [9] where all sources are assumed to be circular.
If the real and imaginary parts of a given source are mutually independent, the score
function takes the form

1
2

@ log p S r ( u r )

@u r þ j @ log p S i ( u i )

c n ( u , u ) ¼

@u i

and suggests the need to use separate real-valued functions for processing the real and
imaginary arguments. For example, the score function proposed in [103] for complex
Infomax, c ( u ) ¼ tanh( u r ) þ j tanh( u i ), is shown to provide good performance for
independent and circular sources [3].
For density matching, approaches such as the Gram-Charlier and Edgeworth
expansions are proposed for the real case [19], and for the complex case, bivariate
expansions such as those given in [76] can be adopted. However, such expansions
usually perform well for unimodal distributions that are close to the Gaussian and
their estimators are very sensitive to outliers thus usually requiring large number of
samples. With the added dimensionality of the problem for the complex case, in
comparison to the real (univariate) case, such expansions become even less desirable
for complex density matching. Limitations of such expansions are discussed in detail
in [104] where an efficient procedure for least-mean-square estimation of the score
function is proposed for the real case.
Next, we discuss a number of possible density models and nonlinearity choices
for performing complex ICA and discuss their properties. Simple substitution of
u r ¼ ( uþ u ) / 2 and u i ¼ ( u - u ) / 2 j allows us to write a given pdf that is p ( u r , u i ):
R R 7! R in terms of a function f ( u , u ): C C 7! R . Since all smooth functions
that define a pdf can be shown to satisfy the real differentiability condition, they can be
used in the development of ICA algorithms and in their analyses using Wirtinger
calculus.

Generalized Gaussian Density Model A generalized Gaussian density of
order c of the form given in [26] can be written as a function C C 7! R as

f GG ( u , u ; s r , s i , r , c ) ¼ b exp( [ ga ( u , u )] c ) (1 : 69)