A Survey of Kurtosis Optimization Schemes for MISO Source Separation and Equalization - Advances in Heuristic Signal Processing and Applications - page 187

A2. The source processes s i (n) , i ∈{ 1 ,...,N } , are mutually statistically inde-

pendent.
These assumptions allow the separation of independent non-Gaussian sources using
kurtosis.
In all the following, we will focus on the MISO approach to BSS, which con-
sists in extracting one source after another. This is done by estimating one row of
W (n) , denoted by w (n) , in such a way that the extractor output y(n) w (n) x (n)

corresponds to one source up to a possible scalar filtering; this step is detailed in
Sect. 9.3.1 . If a full separation is to be accomplished, a deflation stage has to be
applied before searching for a new source, as will be described in Sect. 9.3.2 .

=

9.2.2 Instantaneous Mixtures

In the instantaneous mixture case, channel effects reduce to scale factors without
time delays, so that the MIMO channel is given by M (n) M δ n . As a result, the

LTI systems in Eqs. ( 9.2 )-( 9.4 ) reduce to constant matrices, and the respective con-
volution operations become matrix products:

=

x (n) = Ms (n), y (n) = Wx (n), G = WM .

Similarly, when dealing with MISO separation, the extracted source output reads
y(n)

= wx (n) , where w is a constant row vector corresponding to one row of ma-

trix W .
A similar matrix model holds when considering finite impulse response (FIR)
equalizers, a practical setting to deal with convolutive mixtures. If the separating
filter W (n) is represented by an order- R causal FIR MIMO filter, the summation
index in separation equation ( 9.3 ) extends from 0 to R . Hence, the convolution can
be expressed as the matrix product:

= W ( 0 ), W ( 1 ),..., W (R) x R + 1 (n) y (n)

where vector x R + 1 (n) is obtained by stacking (R + 1 ) consecutive delays of the

T . Thanks to
the equivalent matrix model enabled by the stacking device, results presented later
in the chapter for the instantaneous case can readily be extended to the convolutive
case with FIR equalizers. More details can be found, e.g., in [ 13 , 14 , 16 ].

x (n) T , x (n 1 ) T ,..., x (n R) T observed vector x (n) : x R + 1 (n) [ − − ]

9.3 Deflationary Source Separation

This section introduces the general methodology of the deflation-based BSS ap-
proach considered in this chapter. The approach iterates between the following two
fundamental steps: