Further preprocessing

The success of ICA for a given data set may depende crucially on performing some application-dependent preprocessing steps. For example, if the data consists of time-signals, some band-pass filtering may be very useful. Note that if we filter linearly the observed signals x_i(t) to obtain new signals, say x_i^*(t), the ICA model still holds for ${\bf x}_i^*(t)$, with the same mixing matrix.

This can be seen as follows. Denote by ${\bf X}$ the matrix that contains the observations ${\bf x}(1), ... , {\bf x}(T)$ as its columns, and similarly for ${\bf S}$. Then the ICA model can be expressed as:

$${\bf X}={\bf A}{\bf S}$$ (34)

Now, time filtering of ${\bf X}$ corresponds to multiplying ${\bf X}$ from the right by a matrix, let us call it ${\bf M}$. This gives

$${\bf X}^*={\bf X}{\bf M}={\bf A}{\bf S}{\bf M}={\bf A}{\bf S}^*,$$ (35)

which shows that the ICA model remains still valid.