Digital Signal Processing Reference
In-Depth Information
solutions and, as a consequence, it requires iterative numerical algorithms. This sec-
tion provides a survey of iterative techniques for kurtosis maximization proposed in
the literature. These include gradient-based algorithms (Sect.
9.4.1
) possibly includ-
ing some form of projection (Sect.
9.4.2
) or parametrization of the separation sys-
tem (Sect.
9.4.3
). Newton search is also considered, the popular FastICA algorithm
with cubic nonlinearity being arguably the most popular example (Sect.
9.4.4
). The
kurtosis-based FastICA can actually be recast as a gradient algorithm with constant
step size, which motivates the development of more elaborate algorithms with op-
timal selection of the step-size parameter (Sect.
9.4.5
). Our review concludes with
techniques based on reference signals leading to quadratic criteria that can be opti-
mized by algorithms with monotonic convergence (Sect.
9.4.6
).
The contrasts under study depend on the MISO filter
w
(n)
, but can also be con-
sidered as functions of the global MISO filter
g
(n)
w
(n)
M
(n)
. For ease of
notation, the corresponding filters will just be denoted in the sequel without refer-
ence to time index
n
. We focus on the maximization of the absolute kurtosis contrast
J
κ
(Eq. (
9.9
)), the treatment of
=
J
ε
(Eq. (
9.10
)) being totally analogous.
9.4.1 Gradient Search Algorithms
Since the seminal works establishing kurtosis as a deconvolution criterion [
28
,
58
,
66
], a wide variety of blind separation and equalization methods based on this con-
trast have been put forward using gradient optimization [
40
,
50
,
53
,
58
,
64
,
74
]. The
idea consists in taking small steps in the direction of the gradient:
w
+
=
w
+
μ
∇
J
κ
(
w
)
(9.13)
where
w
+
is the updated extracting vector and symbol
denotes the nabla, or gra-
dient vector, operator of first-order partial derivatives with entries
∇
[∇
J
κ
(
w
)
]
i
=
∂
J
κ
(
w
)/∂w
i
. From the first-order Taylor expansion of
J
κ
around
w
, and taking
into account update (
9.13
), we have:
J
κ
w
+
≈
J
κ
(
w
)
μ
∇
J
κ
(
w
)
+∇
J
κ
(
w
)
T
w
+
−
w
=
J
κ
(
w
)
2
.
+
J
κ
(
w
+
)
Hence, a finite sufficiently small positive value of
μ
guarantees
≥
J
κ
(
w
)
,
with equality if and only if
∇
J
κ
(
w
)
=
0
, i.e., when the algorithm has reached a
stationary point of
J
κ
. Likewise, a negative
μ
would allow the local minimization
of the contrast. In the instantaneous case, the absolute kurtosis gradient is given by
E
|
y
|
2
y
∗
x
−
E
y
∗
2
4sign
(
J
(
w
))
∇
J
κ
(
w
)
=
E
{
y
x
}
E
2
{|
|
2
}
y
.
4
y
2
2
)
E
y
∗
x
(
E
{|
y
|
}−|
E
{
}|
{
}
−
(9.14)
2
E
{|
y
|
}
This leads to the following algorithm:
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