Digital Signal Processing Reference
In-Depth Information
9.1.4 Chapter Outline
This chapter summarizes the basic concepts behind the use of kurtosis as an MISO
source separation and equalization criterion, and reviews some practical numeri-
cal algorithms proposed in the literature for kurtosis optimization. Our focus is on
the deflationary separation of statistically independent sources in linear mixtures,
which are assumed noiseless for the sake of simplicity. Although adaptive (online,
recursive, sample-by-sample) algorithms have also been devised, our interest lies
primarily in batch (offline, windowed, block) algorithms that reuse a whole set of
observed signal samples at each iteration. As stated in [
1
], batch processing leads to
statistically more efficient implementations, from which adaptive versions can often
be obtained with simple changes.
After presenting the BSS signal model and assumptions in Sect.
9.2
, the gen-
eral deflation procedure based on the kurtosis criterion is introduced in Sect.
9.3
.
Section
9.4
, the core of the chapter, reviews a number of iterative algorithms for
kurtosis contrast maximization. A few experimental results illustrating the perfor-
mance of such methods are reported in Sect.
9.5
. Finally, Sect.
9.6
summarizes the
main results of the chapter and points out some possible avenues of further research.
9.1.5 Mathematical Notations
Before beginning the exposition, defining some mathematical notations will be use-
ful. Throughout the chapter, unless otherwise stated, signals can be complex- or
real-valued. The letter
n
stands for a generic integer,
n
. Lightface (
x
,
X
), bold-
face lowercase (
x
), and boldface uppercase (
X
) characters denote scalar, vector, and
matrix quantities, respectively. The transpose and Hermitian (conjugate-transpose)
operators are denoted by superscripts
(
·
)
T
and
(
·
)
H
. The cumulant of a set of random
variables is represented by Cum
∈ Z
{·}
. In particular, the fourth-order marginal cumulant
of a zero-mean random variable
y
is given by:
Cum
y,y
∗
,y,y
∗
=
E
|
y
|
4
−
2E
|
y
|
2
2
−
E
y
2
2
C
{
y
}
(9.1)
where E
{·}
represents the mathematical expectation. Finally, symbols
and
·
stand
for, respectively, the convolution operator and the scalar product.
9.2 Blind Source Separation: Model and Assumptions
In this section, we introduce the different mixing models considered in this chapter.
The assumptions used to perform BSS are also given.
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