COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 67

source densities, the mutual information is given by

!

I ( W ) ¼ D kp ( u ) k Y ¼ X

N N

p S n ( u n ) H ( u n ) H ( u )

n¼ 1 n¼ 1

¼ X

N

H ( u n ) H ( x ) log j det Wj (1 : 68)

n¼ 1

where in the last line, we have again used the complex-to-real transformation for the
source density given in (1.48). Since H ( x ) is constant, using the mean ergodic theorem
for the estimation of entropy, it is easy to see that minimization of mutual information
is equivalent to ML, and when the weight matrix is constrained to be unitary, to the
MN criterion.

1.6.4 Density Matching

For all three approaches for achieving ICA, the ML, MN, and mutual information
minimization discussed in Sections 1.6.1-1.6.3, the nonlinearity used in the algorithm
is expected to be matched as much as possible to the density for each estimated source.
Also, the desirable large sample properties of the ML estimator assume their optimal
values when the score function is matched to the source pdf, for example, the asymp-
totic covariance matrix of the ML estimator is minimum when the score function is
chosen to match the source pdfs [89]. A similar result is given for the maximization
of negentropy in [52]. A number of source density adaptation schemes have
been proposed for performing ICA in the real-valued case, in particular for ML-
based ICA (see e.g. , [24, 59, 66, 112, 120]) and more recently for the complex case
[84, 85] for maximization of negentropy.
The most common approach for density adaptation has been the use of a flexible
parametric model and to estimate the parameters—or a number of key parameters—
of the model along with the estimation of the demixing matrix. In [89], a true ML
ICA scheme has been differentiated as one that estimates both the source pdfs and
the demixing matrix W , and the common form of ML ICA where the nonlinearity
is fixed and only the demixing matrix is estimated is referred to as quasi-maximum
likelihood. Given the richer structure of possible distributions in the two-dimensional
space compared to the real-valued, that is, single dimensional case, the pdf estimation
problem becomes more challenging for complex-valued ICA. In the real-valued case,
a robust nonlinearity such as the sigmoid nonlinearity provides satisfactory perform-
ance for most applications [29, 54] and the performance can be improved by matching
the nonlinearity to the sub- or super-Gaussian nature of the sources [66]. In the com-
plex case, the circular / noncircular nature of the sources is another important factor
affecting the performance [3, 84]. Also, obviously the unimodal versus multimodal
structure of the density requires special care in both the real and the complex
case. Hence, in general, it is important to take a priori information into account
when performing source matching.