Digital Signal Processing Reference
In-Depth Information
Thus, according to this theorem, it would not be possible to obtain
independent random variables from a linear mixture of non-Gaussian
sources. Consequently, if we consider that the coefficients
a
i
and
b
i
are the
parameters of the overall input-output mapping
WA
, it is clear that the esti-
mate will only be independent, assuming that none of them has a Gaussian
distribution, if the sources are no longer mixed.
Therefore, source separation can be achieved via ICA, which can be
formally defined as follows.
T
con-
DEFINITION 6.1
(ICA) The ICA of a random vector
x
=[
x
1
x
2
...
x
M
]
sists of determining a matrix
W
such that the elements of
y
Wx
be as
statistically independent as possible, in the sense of optimizing a cost func-
tion that expresses, direct or indirectly, the notion of independence between
signals.
∗
=
Therefore, source recovery relies on two important concepts: those of sta-
tistical independence and of non-Gaussianity. Nonetheless, it is important to
note that the solution found using ICA will recover the sources up to scale
and permutation ambiguities. That is because if a vector
s
is composed of
independent random variables, so will be a vector that is just a permutation
of
s
, or even a scaled version thereof. In other words,
y
=
Ps
(6.7)
will also present independent components,
where
is a diagonal matrix
P
is a permutation matrix, will also present independent components
Hence, the conditions under which the sources can be recovered using
ICA may be summarized in the following theorem [74].
THEOREM 6.2 (Separability)
The system presented in (6.3) is separable by ICA, i.e., it is possible to obtain
W
such that
y
Wx
correspond to the sources up to scale and permutation
ambiguities, if and only if the mixing matrix
A
is full rank and there is, at
most, one Gaussian source.
=
The first condition regarding the rank of
A
is self-evident, since we are
looking for a separating matrix that should, in some sense, invert the mixing
∗
In the context of source separation, the cost function is also termed a
contrast function
[74].
