A Survey of Kurtosis Optimization Schemes for MISO Source Separation and Equalization - Advances in Heuristic Signal Processing and Applications - page 188

Fig. 9.1 Schematic diagram of the signal model considered in this chapter

Extraction aims at estimating one source from the observed mixture by using a

MISO filter maximizing a suitable contrast function, as detailed in
Sect. 9.3.1 .

Deflation aims at canceling out the contribution of the source estimated in the

previous step, so that the number of sources contributing to the mixture
decreases by one. Section 9.3.2 explains how to perform this step.

Using the deflated mixture obtained in the second step, the algorithm goes back to
the first step to search for another source, and so forth. The procedure is repeated
until all sources have been estimated (full separation) or the source of interest has
been recovered.

9.3.1 Source Extraction with MISO Contrast Functions

The MISO source extraction problem consists in estimating one row of the separat-
ing filter W (n) in Eq. ( 9.3 ). The entries of this row vector, called vector equalizer
and denoted w (n) , represent a bank of Q scalar LTI filters with impulse responses
{

Q
i

w i (n) }

1 . The output of the separation procedure is the scalar signal =

Q

y(n) = w (n) x (n) = w i (n)x i (n). (9.6)

i = 1

Defining the global LTI vector filter g (n) w (n) M (n) , with impulse response

= p ∈Z g (n) w (p) M (n − p) , we then have

N

y(n) = g (n) s (n) = g i (n)s i (n). (9.7)

i = 1

Vector g (n) is a row of the global matrix defined in Eq. ( 9.4 ). These notations are
summedupinFig. 9.1 . According to the ambiguities described in Sect. 9.2.1 ,a
successful extraction restores one of the source components s i (n),i ∈{

,
possibly up to scalar filtering: y(n) = d ii (n)s i (n) . The global filter g (n) thus con-
tains a single nonzero entry, g i (n) = d ii (n) , g j (n) =

1 ,...,N }

0, j = i , and is actually a row