Digital Signal Processing Reference
In-Depth Information
Step 7
Determine the Kaiser window by substituting the values of
β
(obtained
in step 4) and
m
(obtained in step 6) into Eq. (15.18). Let the determined Kaiser
window be denoted by
w
kaiser
[
k
].
Step 8
The impulse response of the FIR filter is given by:
h
lp
[
k
]
=
h
ilp
[
k
]
w
kaiser
[
k
]
.
(15.22)
If the pass-band gain
H
lp
(0)
at
Ω
=
0, given by
h
lp
[
k
], is not equal to one,
we normalize
h
lp
[
k
] with
h
lp
[
k
].
Step 9
Confirm that the impulse response
h
lp
[
k
] satisfies the initial specifica-
tions by plotting the magnitude spectrum
H
lp
(
Ω
)
of the FIR filter obtained in
step 8.
Example 15.3 uses the above algorithm to design an FIR filter using the Kaiser
window.
Example 15.3
Using the Kaiser window, design the FIR filter specified in Example 15.2.
Solution
Following steps 1-3 of Example 15.2, we determine the following values for
the normalized cut-off frequency, impulse response of the ideal lowpass filter,
and minimum attenuation
A
:
=
0
.
4375;
h
lp
[
k
]
=
0
.
4375 sinc(0
.
4375(
k
−
m
));
A
=
50 dB
.
Ω
n
The value of
m
in the impulse response is set to (
N
−
1)
/
2.
Step 4 of Section 15.1.5 determines the value of
β
:
β
=
0
.
1102(
A
−
8
.
7)
=
4
.
5513
.
Step 5 computes the normalized transition bandwidth:
−
3
π
)
10
3
/
8
10
3
Ω
n
= ω
c
/
(0
.
5
ω
0
)
=
(4
π
0
.
5
2
π
=
0
.
1250
.
Using Step 6, the length of the Kaiser window is given by
A
−
7
.
95
2
.
285
π
Ω
n
50
−
7
.
95
2
.
285
π
0
.
125
N
≥
=
=
46
.
8619
,
which is rounded off to the closest higher odd number as 47.
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