Digital Signal Processing Reference
In-Depth Information
For types 1-4 FIR filters, the frequency response were shown in (5.22) to be
of the following form:
e
−
j
[
(N/
2
)ω
]
⎧
⎫
⎬
h
N
2
h
N
n
cos
(nω)
⎨
N/
2
H(e
jω
)
=
+
2
2
−
⎩
⎭
n
=
1
for type I
(5.47)
e
−
j
[
(N/
2
)ω
]
⎧
ω
⎫
h
N
n
cos
n
⎨
⎬
(N
+
1
)/
2
+
1
2
1
2
H(e
−
jω
)
=
2
−
−
⎩
⎭
n
=
1
for type II
(5.48)
e
−
j
[
(Nω
−
π)/
2]
⎧
⎫
⎬
⎭
h
N
n
sin
(nω)
⎨
⎩
N/
2
H(e
−
jω
)
=
2
2
−
n
=
1
for type III
(5.49)
e
−
j
[
(Nω
−
π)/
2]
⎨
⎩
ω
⎬
⎭
h
N
n
sin
n
(N
+
1
)/
2
+
1
2
1
2
H(e
−
jω
)
=
2
−
−
n
=
1
for type IV (5.50)
In general, Equations (5.47)-(5.50) are of the form
H(e
jω
)
e
−
j(Nω/
2
)
e
jβ
×
H
R
(ω)
,where
4
β
is either 0 or
π/
2 depending on the type of filter, and
H
R
(ω)
is a real function of
ω
, which can have positive or negative values. It is easy to
see that
H
R
(ω)
for type I filters can be reduced to the form (2
M
=
=
N
)
M
H
R
(ω)
=
0
a
[
k
]cos
(kω)
(5.51)
k
=
where
a
[0]
=
h
[
M
]
,
a
[
k
]
=
2
h
[
M
−
k
]
,
1
≤
k
≤
M
(5.52)
Consider
H
R
(ω)
for the type II filter shown in (5.48) and given below:
h
N
n
cos
n
ω
(N
+
1
)/
2
+
1
2
1
2
H
R
(ω)
=
2
−
−
n
=
1
This can be reduced to the form
b
[
k
]cos
k
ω
(
2
M
+
1
)/
2
1
2
H
R
(ω)
=
−
(5.53)
k
=
1
4
This is not the same parameter
β
that is used in Kaiser's window.
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