Digital Signal Processing Reference
In-Depth Information
6.4 Other Approaches for Blind Source Separation
Independently of the algorithm we use to perform BSS, all methods based
on ICA rely on two hypotheses: that the sources are mutually independent,
and that at most one of them is Gaussian. However, in some applications,
one may have access to more a priori information about the mixing system
and the sources.
In this section, we will discuss other approaches that are based on distinct
assumptions about the sources and/or the mixing process. Under these addi-
tional hypotheses, it is possible to devise specific methods that may present
advantages over ICA.
6.4.1 Exploring the Correlation Structure of the Sources
A class of simple but effective algorithms for source separation can be
derived if the power spectral densities of the sources are mutually distinct,
i.e., if their correlation structure obeys
E
s
i
k
s
i
k
l
=
E
s
j
k
s
j
k
l
−
−
(6.86)
for
i
0.
Differently from the ICA approach, these algorithms explore the time
information of the observed signals in order to recover the sources. An inter-
esting point of this new approach is that, in this case, source separation
can be carried out using only second-order statistics. In order to under-
stand this fact, let us consider that data has been prewhitened. Then, the
autocorrelation matrix of the observations for a given delay
l
is given by
=
j
and some
l
=
E
x
l
=
x
T
U
T
R
¯
x
(
n
)
¯
(
n
−
l
)
=
UR
s
(
l
)
(6.87)
¯
where
E
R
s
l
=
s
T
¯
s
(
n
)
¯
(
n
−
l
)
(6.88)
denotes the autocorrelation matrix of the sources for a delay
l
.
If the sources are mutually independent (or, at least, uncorrelated),
R
s
l
is diagonal, which means that (6.87) represents the eigendecomposition of
R
x
l
reveals the rotation matrix that
leads to source separation. This idea is the very essence of the AMUSE
algorithm [286] and depends on the correct choice of the delay
l
for which
x
l
. Thus, the decomposition of
R
¯
¯
R
s
l
presents distinct eigenvalues. An extension of this method would be to
