Blind Source Separation - Unsupervised Signal Processing - page 190

algorithms [148]. Let us consider again the whitening transform in (6.12),
which leads us to

x ¯ = Tx

2 U T As

=

1
2 U T U

1
2 V T s

= −

= Vs (6.15)

where V is an orthogonal matrix. We can observe that the whitening process
reduces the problem to one in which the mixing matrix is orthogonal, and
hence, limits the search for the separating matrix to the group of orthogonal
matrices. In order to illustrate the effect of preprocessing, we present the
following example.

Example 6.1 (Two-Source Mixture)

Let us c on sider two independent sources uniformly distributed in the interval

− √ 3, √ 3 . The joint pdf of the sources is then given by

⎨
⎩

− √ 3 ≤ √ 3

1
12 ,

p x , y =

≤ x , y

0, otherwise

and is illustrated in Figure 6.2a .
Let the mixing matrix be

⎡
⎣

⎤
⎦

1 .5

A =

0.3 0.9

Then, the distribution of the observed data becomes as illustrated in
Figure 6.2b. Notice that the mixing matrix distorts and rotates the original source
distribution. Interestingly, after a prewhitening procedure, the distribution of
the transformed data, as shown in Figure 6.2c, resembles the original distribu-
tion, except for a rotation factor. So, the remaining step consists in determining
the orthogonal matrix corresponding to the rotation that will restore the source
distribution.

6.2.2 Criteria for Independent Component Analysis

As previously discussed, the very essence of ICA, as discussed in Section
6.2 , consists of determining a separating structure that provides estimates
as independent as possible. Different criteria were proposed in order to
implement the idea of ICA, and in the following we discuss some of them.