COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 59

In this section based on the work in [6], we concentrate on the complex ICA
approaches that use nonlinear functions without imposing any limitations on the
type of source distribution and demonstrate how Wirtinger calculus can be used for
efficient derivation of algorithms and for working with probabilistic characterizations,
which is important in the development of density matching mechanisms that play a
key role in this class of ICA algorithms. We present the two main approaches for per-
forming ICA: maximum likelihood (ML) and maximization of non-Gaussianity
(MN). We discuss their relationship to each other and to other closely related
ICA approaches, and in particular note the importance of source density matching
for both approaches. We present extensions of source density matching mechanisms
for the complex case, and note a few key points for special classes of sources,
such as Gaussian sources, and those that are strictly second-order circular. We
present examples that clearly demonstrate the performance equivalence of ML- and
MN-based ICA algorithms when exact source matching is used for both cases.
In the development, we consider the traditional ICA problem such that

x ¼ As

NN , that is, the number of sources and observations are
equal and all variables are complex valued.
The sources s i where s ¼ [ s 1 , ... , s N ] T are assumed to be statistically independent
and the source estimates u i where u ¼ [ u 1 , ... , u N ] T , are given by u ¼ Wx . If the
mixtures are whitened and sources are assumed to have unit variance, WA approxi-
mates a permutation matrix when the ICA problem is solved, where we assume that
the mixing matrix is full rank. For the complex case, an additional component of
the scaling ambiguity is the phase of the sources since all variables are assumed to
be complex valued. In the case of perfect separation, the permutation matrix will
have one nonzero element. Separability in the complex case is guaranteed as long
as the mixing matrix A is of full column rank and there are no two complex
Gaussian sources with the same circularity coefficient [33], where the circularity
coefficients are defined as the singular values of the pseudo-covariance matrix of
the source random vector. This is similar to the real-valued case where second-
order algorithms that exploit the correlation structure in the mixtures use joint diago-
nalization of two covariance matrices [106].

N and A [ C

where x , s [ C

1.6.1 Complex Maximum Likelihood

As in the case of numerous estimation problems, maximum likelihood theory provides
a natural formulation for the ICA problem. For T independent samples x ( t ) [ C

N ,we

can write the log-likelihood function as

L ( W ) ¼ X

T
t¼ 1 ' t ( W ),