Digital Signal Processing Reference
In-Depth Information
To minimize
E(k, D)
, we differentiate equation (7.100) with respect to
g
k
and
β
D
which are functions of
k
and
D
respectively. Thus,
L
−
1
∂E(k, D)
∂g
k
=
[
˜
s(n)
−
g
k
V
k
(n)
−
β
D
Z
D
(n)
]
×
(
−
2
)V
k
(n)
=
0
(7.101)
n
=
0
and similarly,
−
L
1
∂E(k, D)
∂β
D
=
[
˜
s(n)
−
g
k
V
k
(n)
−
β
D
Z
D
(n)
]
×
(
−
2
)Z
D
(n)
=
0
(7.102)
n
=
0
Rearranging the above equations we get
−
−
−
L
1
L
1
L
1
V
k
(n)
0
˜
s(n)V
k
(n)
=
g
k
+
β
D
Z
D
(n)V
k
(n)
(7.103)
n
=
n
=
0
n
=
0
L
−
1
L
−
1
L
−
1
Z
2
D
(n)
0
˜
s(n)Z
D
(n)
=
g
k
V
k
(n)Z
D
(n)
+
β
D
(7.104)
=
=
=
n
n
0
n
0
In matrix form,
G
1
G
2
G
2
G
3
−
1
K
1
K
2
g
k
β
D
=
(7.105)
where,
L
−
1
V
k
(n)
G
1
=
(7.106)
n
=
0
L
−
1
G
2
=
Z
D
(n)V
k
(n)
(7.107)
n
=
0
L
−
1
Z
2
D
(n)
G
3
=
(7.108)
n
=
0
L
−
1
=
0
˜
K
1
s(n)V
k
(n)
(7.109)
n
=
−
L
1
K
2
=
0
˜
s(n)Z
D
(n)
(7.110)
n
=
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