Digital Signal Processing Reference
In-Depth Information
bandpass, and bandstop filters along with equations for determining the
coefficients based on the parameters of the particular filter are given. The
frequency response in the passband of each filter is as defined in (7.8) and allows
us to determine causal coefficients directly. We will be assuming that the desired
passband magnitude response is 1, while the stopband response is 0. We will be
using
δ
p
and
δ
s
(or
a
pass
and
a
stop
) later to help determine the required length of
the filter.
In the first case, Figure 7.4 illustrates the lowpass filter specification with the
resulting derivation of the lowpass filter coefficients shown in (7.14) and (7.15).
The highpass, bandpass, and bandstop filter cases are portrayed respectively in
Figures 7.5 to 7.7 with the appropriate derivations in (7.16) to (7.21). In each
case, τ =
M
as determined from (7.9).
Example 7.1 Determining Ideal Coefficients for an FIR Filter
Problem:
Determine the ideal impulse response coefficients for a lowpass
filter of length 21 to satisfy the following specifications:
ω
pass
= 2
π⋅
3,000 rad/sec,
ω
stop
= 2
π⋅
4,000 rad/sec, and
f
s
= 20 kHz
Solution:
We first need to determine
Ω
c
, the cutoff frequency for the ideal
filter. This frequency can be set in the middle of the transition band and converted
to a digital frequency:
Ω
=
(
ω
+
ω
)
/(
2
⋅
f
)
=
1
.
0996
rad
/
sec
c
stop
pass
s
Using this value along with
τ
= 10 in (7.15), we can determine the following ideal
causal coefficients:
h
(
10
)
=
0
.
35000
h
(
=
h
(
11
)
=
0
.
28362
h
(
=
h
(
12
)
=
0
.
12876
h
(
7
)
=
h
(
13
)
=
−
0
.
01660
h
(
6
)
=
h
(
14
)
=
−
0
07568
h
(
=
h
(
15
)
=
−
0
04502
h
(
4
=
h
(
16
)
=
0
01639
h
(
=
h
(
17
)
=
0
04491
h
(
2
=
h
(
18
)
=
0
02339
h
(
=
h
(
19
)
=
−
0
.
01606
h
(
0
)
=
h
(
20
)
=
−
0
.
03183
