Digital Signal Processing Reference
In-Depth Information
that is, the autocovariance of the component
s
i
. Thus the assumption
reads that the source components are to have different autocovariances
for given
τ
. In practice, if the eigenvalue decomposition is problematic,
a different choice of
τ
often resolves this problem. However, the AMUSE
algorithm is not applicable to sources with equal power spectra, meaning
sources for which such a
τ
does not exist.
Another solution is instead of using simple diagonalization to choose
more than one time lag and to do a simultaneous diagonalization of the
corresponding autocovariances. Such algorithms turn out to be quite
robust against noise, but of course also cannot overcome the problem of
equal source power spectra.
For this, other time-based ICA algorithms also use higher-order
moments in time, such as crosscumulants. A good overview of time-
based ICA/BSS algorithms is given in [123].
EXERCISES
1.
Define ICA and compare it with PCA.
2.
After having found an ICA separating matrix of a linear noisy
mixture
x
=
As
+
y
with white noise
y
, how can the sources be
estimated?
3.
How can maximization of non-Gaussianity find independent com-
ponents?
4.
Study the central limit theorem experimentally. Consider
T
i.i.d. samples
x
(
t
)
,t
=1
,...,T
of a uniform random variable,
and define
T
1
T
y
:=
x
(
t
)
.
t=1
Calculate 10
4
such realizations with corresponding
y
for
T
=
2
,
4
,
10
,
100 and compare these with a Gaussian with mean 0 and
variance var
x
by using histograms and kurtosis.
5.
In exercise 9 from chapter 3, calculate determine also an ICA of
the signals. Then compare the separated components with the
principal components, visually using scatter plots and numerically
by analyzing the mixing-separation-matrix products. For the ICA
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