Independent Component Analysis and Blind Source Separation - Biomedical Signal Analysis: Contemporary Methods and Applications - page 137

butwerequiretheautocovariance C τ ( t ) to be diagonal for all t and
τ . This second-order assumption holds for time signals which we would
typically call “independent”. Furthermore, note that we do not need the
source distributions to be non-Gaussian.
In terms of algorithm, we will now use simple second-order statistics
in the time domain instead of the higher-order statistics used before.
Without loss of generality, we can again assume E ( x ( t )) = 0 and

A ∈ O ( n ). Then

C τ ( t ):= E ( x ( t ) x ( t

τ ) ) .

−

Time decorrelation

Let the offset τ ∈ N be arbitrary, often τ =1.Definethe symmetrized

autocovariance

2 C τ +( C τ )
Using the usual properties of the covariance together with linearity, we
get

1

C τ

:=

C τ

= A C τ A .

(4.7)

By assumption C τ is diagonal, so equation 4.7 is an eigenvalue decom-
position of C τ . If we further assume that C τ has n different eigenvalues,
then the above decomposition is uniquely determined by C τ except for
orthogonal transformation in each eigenspace and permutation; since the
eigenspaces are one-dimensionalm this means A is uniquely determined
by equation 4.7 except for equivalence. Using this additional assump-
tion, we have therefore shown the usual separability result, and we get
an algorithm:
Algorithm: ( AMUSE )Let x ( t ) be whitened and assume that for
agiven τ the matrix

C τ

has n different eigenvalues. Calculate an

eigenvalue decomposition

C τ

= W DW

with D diagonal and W ∈ O ( n ). Then W is the separation matrix and

W ∼ A .

Note that by equation 4.7, C τ

and C τ havethesameeigenvalues.

Because C τ is diagonal, the eigenvalues are given by

E ( s i ( t ) s i ( t − τ ))