Digital Signal Processing Reference
In-Depth Information
Thus, one may say that
y
1
will be “less” Gaussian if its distribution is equal
to that of one of the sources, i.e., when
q
presents only one nonzero element.
Therefore, to obtain a vector
w
1
that maximizes the non-Gaussianity of
w
1
x
should also lead to source recovery.
The classical way of quantifying the gaussianity of a distribution is to use
the
kurtosis
, defined in Section 2.3.3. Most distributions present a nonzero
kurtosis value, the Gaussian being one of the few exceptions. In fact, it is
usual to classify pdfs according to their kurtosis value: if
K
(
x
)>
0, it is said
that
x
has a super-Gaussian distribution; if
K
0,
x
has a sub-Gaussian
distribution. Hence, one criterion that expresses the maximization of the non-
Gaussian character of a signal is given by
(
x
)<
ma
w
K
y
i
(6.35)
Another possibility of quantifying the non-Gaussianity is by means of the
concept of negentropy [148], defined as follows:
DEFINITION 6.5
(Negentropy)
The negentropy of a random variable
Y
is
defined as
J
Negentropy
(
Y
)
=
H
(
Y
Gauss
)
−
H
(
Y
)
(6.36)
where
Y
Gauss
represents a Gaussian variable with the same mean and
variance of
Y
.
The notion of negentropy is based on the fact that a Gaussian random
variable will present the largest entropy over all other distributions with
the same mean and variance [230]. Therefore, the negentropy will always
assume nonnegative values, being zero only if
y
is Gaussian. In this sense,
the negentropy quantifies the degree of proximity between the pdf of
y
and
that of a Gaussian variable.
One interesting point regarding the non-Gaussianity maximization is that
it can be used to estimate the sources individually. Due to this feature, tech-
niques based on this approach are usually associated within the framework
of blind source extraction (BSE) [79,185]. In BSE, we are not interested in all
sources, but only in a subset of the signals contained in the mixture. If the
number of signals to be extracted is the same as the total number of sources,
we reach again with the BSS problem.
The procedure for the extraction of more than one source can be carried
out with two distinct strategies, both exploring the fact that the extract-
ing vectors
w
i
, obtained considering whitened data, will necessarily be
orthogonal.
