Digital Signal Processing Reference
In-Depth Information
where
=
d
[0]
2
d
[1]
1
d
[1]
−
2
d
[
k
−
1]
−
d
[
k
]
,
1
2
M
−
1
d
[
k
]
=
2
≤
k
≤
2
d
2
M
=
d
2
M
+
1
2
−
1
2
c
[
.
],
d
[
.
]intermsof
a
[
.
],
b
[
.
],
c
[
.
],
d
[
.
] since they are linearly related. We express Equations (5.51), (5.54),
(5.57), and (5.60) in a common form, (5.61), in order to develop a common
algorithm that obtains a minimax approximation for all four types of filters
a
[
.
],
b
[
.
],
Note that we can express the coefficients
H
R
(ω)
=
Q(ω)P (ω)
(5.61)
where
⎨
⎩
1 for type I
cos
2
for type II
sin
(ω)
for type III
sin
2
for type IV
Q(ω)
=
(5.62)
and
K
P(ω)
=
α
[
k
]cos
(kω)
(5.63)
k
=
0
where
⎧
⎨
a
[
k
]
for type I
b
[
k
]
for type II
α
[
k
]
=
(5.64)
⎩
c
[
k
]
for type III
d
[
k
]
for type IV
and
⎧
⎨
⎩
M
for type I
2
M
−
1
for type II
2
K
=
(5.65)
M
−
1
for type III
2
M
−
1
2
for type IV
We define a weighted error function
W(e
jω
)
H
R
(ω)
H
d
(e
jω
)
=
W(e
jω
)
Q(ω)P (ω)
H
d
(e
jω
)
J(ω)
=
−
−
W(e
jω
)Q(ω)
P(ω)
H
d
(e
jω
)
Q(e
jω
)
=
−
(5.66)
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