Digital Signal Processing Reference
In-Depth Information
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
0
20
40
60
80
100
120
140
160
180
200
Figure 4.9
Kurtosis maximization, second example: absolute kurtosis versus angle. Again, we
plot the function
α →|
kurt((cos(
α
)sin(
α
))
z
)
|
with the super-Gaussian
z
from
figure 4.6.
some suitable abort condition is defined. Furthermore, there are various
ways of increasing the convergence speed of this type of algorithm.
In our case the gradient of
f
(
w
):=
kurt(
w
z
)
|
|
can be easily
calculated as
kurt(
w
z
)
(
w
)=
∂
|
|
kurt(
w
z
)
∇|
|
∂
w
=4 gn(ku
w
z
))
E
(
z
(
w
z
)
3
)
|
2
w
(4.3)
−
3
|
w
because by assumption Cov(
z
)=
I
,so
E
((
w
z
)
2
)=
w
E
(
zz
)
w
=
|
2
.
|
w
By definition of the kurtosis, for white
z
we therefore get
kurt(
w
z
)=
E
((
w
z
)
4
)
|
4
−
3
|
w
hence
∂
kurt(
w
z
)
∂w
i
=4
E
((
w
z
)
3
Z
i
)
|
2
w
i
−
12
|
w
so
∂
kurt(
w
z
)
∂
w
=4
E
((
w
z
)
3
z
)
2
w
.
−
3
|
w
|
On
S
1
, the second part of the gradient can be neglected and we get
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