Digital Signal Processing Reference
In-Depth Information
becomes particularly interesting when used for MISO processing. Its most attrac-
tive feature in this context—which may probably explain the interest it has aroused
over more than two decades—lies in the absence of spurious local extrema under
ideal model conditions. As a result,
global
convergence to right source extraction
solutions is guaranteed even by means of local optimization algorithms. This highly
desirable property was first proven in [
58
] in the context of blind single-channel
equalization, and later in [
26
] for BSS in real-valued instantaneous mixtures after
prewhitening. The proof was finally extended to the more general convolutive com-
plex mixture case in [
59
,
64
]; see also [
27
] for the instantaneous scenario. The good
convergence properties of later algorithms such as the multiuser kurtosis optimiza-
tion of [
50
] are actually inherited from the decoupling of the MIMO criterion into
a set of MISO extraction criteria through a deflation approach. As reviewed in the
chapter, a good number of cost-effective iterative algorithms are available for kurto-
sis maximization.
Besides its mathematical tractability and computational convenience, kurtosis
proves more robust to finite sample effects than related criteria such as the fourth-
order moment or the fourth-order cumulant [
5
,
6
]. This interesting property is espe-
cially useful when processing short data records.
9.1.3 Historical Overview
Originally proposed by Wiggins [
66
] and Donoho [
28
] for single-channel decon-
volution in the context of seismic exploration, the use of kurtosis for interference
cancelation and signal recovery quickly spread to other application domains such as
digital communications, biomedical signal processing, image denoising, and speech
enhancement, as well as more involved signal models. Its application in digital
channel equalization dates back to the work of Shalvi and Weinstein [
58
], who
proved its validity as a blind deconvolution criterion for the non-Gaussian distri-
butions typically encountered in communications and proposed gradient algorithms
for kurtosis maximization based on spectral prewhitening. Extensions to multi-
channel models soon followed. The criterion was proposed for BSS by Delfosse-
Loubaton [
26
] and Papadias [
50
] using second-order sphering or prewhitening, and
by Tugnait [
64
] even without prewhitening. Connections with the popular constant
modulus criterion for blind equalization, which had been developed a few years ear-
lier in [
33
,
57
,
63
], were realized by Comon [
20
,
21
] and Regalia [
53
]; see also [
77
].
In its original definition, Hyvärinen's popular FastICA algorithm for BSS based on
independent component analysis also relied on the kurtosis criterion [
36
]; the algo-
rithm was independently developed by Moreau in [
40
]. More recent developments
include monotonically convergent algorithms optimizing quadratic contrasts based
on reference signals [
10
,
12
,
13
,
16
] as well as parameter-free iterative algorithms
with algebraic optimal step-size selection [
74
].
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