Independent Component Analysis and Blind Source Separation - Biomedical Signal Analysis: Contemporary Methods and Applications - page 109

Definition

→ R

n

Definition 4.3 BSS: Let s :Ω be an independent random

μ R

n

−→ R

m

vector, and let : be a measurable mapping. An ICA

of x := μ ( s ) is called a BSS of ( s , μ ). Given a full-rank matrix A ∈

Mat( n ), called a mixing matrix , a linear ICA of x := As is called

a linear BSS of ( s , A ).

× m ; R

Again, we speak of square BSS if m = n . In the linear case this
means that the mixing matrix A is invertible: A Gl( n ).

If m>n , the model above is called overdetermined or undercomplete .
In the case m<n (i.e. in the case of less mixtures than sources) we speak
of underdetermined or overcomplete BSS .
Given an independent random vector s :Ω

∈

n and an invertible

→ R

matrix A Gl( n )

such that BAs is independent (i.e. the set of all square linear BSSs of
As ).

∈ Gl( n ), denote BSS ( s , A ) all invertible matrices B ∈

Properties

In the following we will mostly deal only with the linear case. So the goal
of BSS - one of the main applications of ICA - is to find the unknown
mixing matrix A , given only the observations/mixtures x . Using theorem
4.2, we see that in the linear case this is indeed possible, except for the
usual indeterminacies scaling and permutation.

n
be an independent random vector with existing covariance having at
most one Gaussian component, and let A

Theorem 4.2 Indeterminacies of linear BSS: Let s :Ω → R

∈ Gl( n ). If W is a BSS of

( s , A ), then W −1 ∼ A .

Proof This follows directly from theorem 4.2 because both A −1 and
W are ICAs of x := As .

So in this case BSS ( s , A )=Π( n ) A −1 ,whereΠ( n ) denotes the group
of products of n × n scaling and permutation matrices.