Digital Signal Processing Reference
In-Depth Information
Even though this is not apparent, the Infomax approach to source sepa-
ration is closely related to the maximum likelihood method [55,56]. In [56],
Cardoso shows that, in the context of BSS, the log likelihood function is
expressed by
E
N
log
det
log
p
s
i
w
i
x
j
J
ML
(
W
)
∝
+
(
W
)
(6.43)
i
=
1
Comparing (6.42) and (6.43), it is clear that both approaches have very
similar cost functions, the only difference being the nonlinearities. In fact,
the cost functions are exactly the same when
f
i
(
·
)
equals the cumulative dis-
tribution function (CDF) of the
i
th source. If that is the case, the pdf of
f
i
y
i
will be uniform in the interval [0, 1] when
y
i
is equal to
s
i
or to some other
source with the same pdf [56].
If the likelihood function is rewritten in terms of the Kullback-Leibler
divergence, i.e.,
D
p
y
y
J
ML
(
W
)
=−
p
s
(
s
)
(6.44)
it becomes clear that the maximum likelihood approach is, in a certain sense,
a pdf matching criterion: matrix
W
should be chosen in order that the distri-
bution of the estimates be “as close as possible” to the distribution of the
sources. In the Infomax approach, however, the true pdfs of the sources
are replaced by the derivatives of nonlinear functions
f
i
(
·
)
. Nonetheless, it
is important to remark that even if there is no perfect match between these
functions and the pdf of the sources, it can still be possible to separate the
signals [56].
6.3 Algorithms for Independent Component Analysis
The ICA criteria exposed in
Section 6.2
indicate theoretical solutions to
the BSS problem. However, the effectiveness of such solutions depends
on finding feasible algorithms to implement them in practical scenarios.
This section discusses some classical algorithms, starting from Hérault and
Jutten's seminal approach.
6.3.1 Hérault and Jutten's Approach
The proposal of Hérault and Jutten [143, 144] is considered to be the first
algorithm capable of extracting signals from linear mixtures. The method is
inspired in elements of neural networks and based on the structure presented
