Digital Signal Processing Reference
In-Depth Information
It is clear that the nonlinearly distorted signal does not present a Gauss-
sian distribution anymore. Thus, considering the PNL model, in order
to obtain the separating system, first the nonlinearities
g
i
(
·
)
should be
adjusted so that the signals after each nonlinearity
g
i
(
·
)
present a Gaussian
distribution, which can be accomplished by
F
−
1
Gauss
g
i
(
·
)
=
(
F
x
i
(
·
))
(6.117)
where
F
x
i
(
x
i
)
denotes the cumulative distribution function of
x
i
F
−
1
Gauss
(
·
)
the inverse of a cumulative distribution function of a normalized
Gaussian, i.e., the quantile function
Once the observations have been “Gaussianized,” they should represent
a linear mixture of the sources, and hence, the sources could be recovered by
using any ICA method developed for linear mixtures.
The results of this approach heavily depend on the validity of the Gaus-
sianity assumption, which is sounder when a large number of sources is
present in the mixture. However, in situations in which this assumption is
not representative, there will be a considerable distortion in the transformed
data, and ICA methods will not be efficient.
6.7 Concluding Remarks
In this chapter, we studied the instigating problem of BSS. As a natural
progression on the theme of blind equalization, BSS widens the scope of
our study in terms of theoretical tools as well as of potential of practical
applications.
We first presented the problem in its general form before reaching the
linear and instantaneous mixture model, which was a reference in the ensu-
ing discussion. The starting point of this discussion was ICA, a widespread
BSS tool with major theoretical and historical importance and a significant
degree of applicability.
In order to discuss some fundamental criteria to perform ICA, we intro-
duced important concepts and definitions. BSS makes use of important
notions, issues from information theory like mutual information and negen-
tropy, the definitions of which have been stated. From there, we enounced
the Infomax principle. We discussed on about the role of high-order statis-
tics in BSS and described the procedure of whitening preprocessing and the
notion of nonlinear decorrelation.
