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Fig. 2.1
The instantaneous mixing and unmixing model for BSS-ICA
row of B. This can be seen as a projection pursuit density estimation problem to
find M directions such that the corresponding projections are the most mutually
independent. For the sake of simplicity, we will assume the square problem (the
same number of sources as mixtures, thus the order of A is MxM). Figure 2.1
shows a schema that illustrates the instantaneous mixing and unmixing model for
BSS-ICA.
Furthermore, the instantaneous linear model can be applied in the frequency
domain for the analysis of convolutive mixtures. Applying the Fourier transform to
both sides of Eq. ( 2.1 ), we obtain the following frequency expression
x ð x Þ¼ A ð x Þ s ð x Þ
ð 2 : 2 Þ
where x ð x Þ¼ FT fg; A ð x Þ¼ FT fg; and s ð x Þ¼ FT fg are the Fourier trans-
forms of the observation vector, mixing matrix, and source vector, respectively.
The time and frequency domain ICA models are equivalent, but the coefficients of
the transfer matrix may vary with x (see for instance [ 14 ] and the references
within). An attempt to generalize the BSS algorithms for MIMO signal processing
that exploits three signal properties nonwhitenes, nongaussianity, and nonsta-
tionarity in an information theoretic cost function has been recently formulated in
[ 15 , 16 ]. In some cases, the convolutive model can be solved as an ''instanta-
neous'' problem for selected frequencies. The frequency component permutation
problem is thus avoided. The frequencies to be analyzed are selected according to
the application; for instance, in a detection problem, the frequencies around the
working frequency of the excitation sensor are in the band of interest. We include
an example of this frequency ICA analysis applied in NDT in Chap. 5 .
It is well-known that A is identifiable, up to scaling and permutation of col-
umns, when s has at most one Gaussian component and Ais assumed to be non-
singular [ 17 ] The restriction in Gaussian components is explained by the central
limit theorem, considering that a linear mixture of independent random variables is
more Gaussian than the original variables. Thus, to specify B uniquely, we need to
put some scale and permutation constraints either on s or on B. Because of the ICA
indeterminacies the sources are usually assumed to be unit variance. Also, it is
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