Digital Signal Processing Reference
In-Depth Information
As given in (1.57), we have
@
E
{
l
}
@Z
¼ E
{
c
(
u
)
u
H
}
I:
To derive the Newton update, we consider the diagonal and the off-diagonal elements
, and can write
@z
ii
@z
ii
of
E
{
@
2
l
} separately. We define
@z
ii
W
(
E
{
c
(
u
)
u
H
}
I
)
ii
(
E
{
c
(
u
)
u
H
}
I
)
ii
@
E
{
l
}
@z
ii
¼
and
2
E
{
l
}
@ z
ii
¼H
3
@ z
ii
:
@
Therefore the Newton rule for updating
@z
ii
can be written by solving
2
E
{
l
}
@
@z
ii
¼
@
E
{
l
}
@ z
ii
as in (1.35) to obtain
(
E
{
c
(
u
)
u
H
}
I
)
ii
(
E
{
c
(
u
)
u
H
}
I
)
ii
@z
ii
¼H
(1
:
58)
3
2
3
@z
ij
@z
ji
@z
ij
@z
ji
and the update for
@z
ii
is simply the conjugate of
@z
ii
.
For each off-diagonal element pair
@z
ij
, we write
@z
ij
W
4
5
. As in the updates
of the diagonal elements, we obtain
2
3
(
E
{
c
(
u
)
u
H
}
I
)
ij
(
E
{
c
(
u
)
u
H
}
I
)
ji
(
E
{
c
(
u
)
u
H
}
I
)
ij
(
E
{
c
(
u
)
u
H
}
I
)
ji
4
5
@
E
{
l
}
@z
ij
¼
H
1
H
2
H
2
H
1
2
E
{
l
}
@z
ij
¼
@
@z
ij
and obtain the Newton update rule for the parameters
@z
ij
as in the previous case
2
3
(
E
{
c
(
u
)
u
H
}
I
)
ij
(
E
{
c
(
u
)
u
H
}
I
)
ji
(
E
{
c
(
u
)
u
H
}
I
)
ij
(
E
{
c
(
u
)
u
H
}
I
)
ji
1
4
5
H
1
H
2
H
2
H
1
@z
ij
¼
(1
:
59)
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