We are given two linear mixtures of two source signals which we know to be independent of each other, i.e. observing the value of one signal does not give any information about the value of the other. The BSS problem is then to determine the source signals given only the mixtures.

Putting this into mathematical notation, we model the problem by

x=As

A first step in many ICA algorithms is to whiten (sphere) the data.
This means that we remove any correlations in the data, i.e. the signals
are forced to be uncorrelated. Again putting the words in mathematical
terms, we seek a linear transformation **V** such that when **y**
= **Vx** we now have *E*{**yy'**} = **I**. This is easily
accomplished by setting
**V** = **C**^{-1/2},
where **C** = *E*{**xx'**} is the correlation matrix of the data,
since then we have *E*{**yy'**}
= *E*{**Vxx'V'**} =
**C**^{-1/2}**CC**^{-1/2}
= **I**.

The figure below shows the signals **y** and the joint density p(**y**) after such
an operation.

After sphering, the separated signals can be found by an orthogonal
transformation of the whitened signals **y** (this is simply a rotation
of the joint density). The appropriate rotation is sought by maximizing the
non-normality of the marginal densities (shown on the edges of the density
plot). This is because of the fact that a linear mixture of independent
random variables is necessarily more Gaussian than the original variables.
(This is the same phenomenon as is stated by the central limit theorem.)
This implies that in ICA we must restrict ourselves to at most one Gaussian source
signal.

There are many algorithms for performing ICA, but the most efficient
to date is the
**FastICA**.
(fixed-point) algorithm which was developed by us.
The plot below shows the result after one step of the FastICA algorithm.

The rotation continues...

...and continues...

...until it starts to converge...

Convergence! The source signals (components of **s**) in this example
were a sinusoid and impulsive noise, as can be seen in the left part of
the plot below. The right plot shows the joint density which can be
seen to be the product of the marginal densities, i.e.
p(**s**)=p(s_{1})p(s_{2}).
This is of course the definition of independence.