Digital Signal Processing Reference
In-Depth Information
E
|
|
,
E
|
|
.
i
0
=
a
i
1
=
2E
{
e
}
,
i
2
=
b
(9.35)
Hence, the derivative of
J
(
w
+
μ
g
)
with respect to
μ
is given by
P
(μ)Q(μ)
−
2
P(μ)Q
(μ)
Q
3
(μ)
p(μ)
Q
3
(μ)
.
J
(μ)
=
=
(9.36)
Combining Eqs. (
9.33
)-(
9.36
),
p(μ)
is given by the fourth-degree polynomial
(quartic)
4
a
n
μ
n
p(μ)
=
(9.37)
n
=
0
with
a
0
=−
2
h
0
i
1
+
h
1
i
0
,
a
1
=−
4
h
0
i
2
−
h
1
i
1
+
2
h
2
i
0
,
a
2
=−
3
h
1
i
2
+
3
h
3
i
0
,
3
=−
2
h
2
i
2
+
h
3
i
1
+
4
h
4
i
0
,
a
4
=−
h
3
i
2
+
2
h
4
i
1
.
The real parts of the roots of this polynomial are the step-size candidates. To deter-
mine the optimal step size
μ
opt
, the roots are plugged back into Eqs. (
9.31
)-(
9.33
)
to check which candidate maximizes
J
κ
(
w
+
μ
d
)
=|
J
(
w
+
μ
d
)
|
; if aiming at a
source with kurtosis sign
ε
, one should check
J
ε
(
w
+
μ
d
)
=
ε
J
(
w
+
μ
d
)
instead.
The extracting vector is then updated as
w
+
=
+
μ
opt
d
. Since the kurtosis is scale
invariant, the extracting vector can be normalized after updating, as in Eq. (
9.24
). It
should be remarked that this normalization is not forced by prewhitening—an op-
tional step when using kurtosis, as we saw before—but just performed by numerical
convenience (Sect.
9.4.2
). As a suitable search direction, one can use the gradient
of the full version of kurtosis, given by Eq. (
9.14
), which can be normalized for
increased numerical stability. This optimal step-size algorithm for iterative kurto-
sis maximization is referred to as
RobustICA
[
71
,
74
,
76
]. In the context of blind
single-channel equalization, the method had been suggested without details a few
years earlier under the name of
optimal step-size kurtosis maximization algorithm
(
OS-KMA
)[
69
]. A very similar optimization idea holds for other source separation
and equalization principles, both in blind and semi-blind operating modes, such as
the constant power [
68
] and the constant modulus criterion [
69
,
70
,
72
]
w
RobustICA algorithm for kurtosis optimization with optimal step size
Set an initial value
w
(
0
)
for the extracting vector.
•
•
For
k
=
1
,
2
,...,k
max
, do:
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