Digital Signal Processing Reference
In-Depth Information
Example 7.3
Determine the cut-off frequency for the lowpass filter specified in Example 7.2.
Solution
Based on the magnitude spectrum, we note that the maximum gain of the filter
is given by 1 or 0 dB. At the cut off frequency
ω
c
,
H
(
ω
c
)
=
0
.
7071
1
=
0
.
7071
,
which implies that
5
.
018
10
3
(j
ω
c
)
4
+
2
.
682
10
14
(j
ω
c
)
2
−
1
.
026
10
4
(j
ω
c
)
+
3
.
196
10
24
(j
ω
c
)
5
+
9
.
863
10
4
(j
ω
c
)
4
+
2
.
107
10
10
(j
ω
c
)
3
+
1
.
376
10
15
(j
ω
c
)
2
+
1
.
026
10
20
(j
ω
c
)
+
3
.
196
10
24
=
0
.
7071
.
The above equality can be solved for
ω
c
using numerical techniques in
M
ATLAB
. The value of the cut-off frequency is given by
ω
c
=
3
.
462
π
10
4
radians/s. Note that the cut-off frequency lies within the transitional
band in between the pass and stop bands of the lowpass filter as derived in
Example 7.2.
7.3 Design of CT lo
wpass filters
To begin our discussion of the design of CT filters, we consider a prototype or
normalized lowpass filter, defined as a lowpass filter, with a cut-off frequency
of
ω
c
=
1 radians/s. The remaining specifications for the pass and stop bands
of the normalized lowpass filter are assumed to be given by
pass band (0
≤ω≤ω
p
radians
/
s)
1
− δ
p
≤
H
(
ω
)
≤
1
+ δ
p
;
(7.12)
stop band (
ω >ω
s
radians
/
s)
H
(
ω
)
≤δ
s
,
(7.13)
with
ω
p
≤ ω
c
≤ ω
s
. Using the transfer function of the normalized lowpass filter,
it is straightforward to implement any of the more complicated CT filters.
Section 7.4 considers the frequency transformations used to convert a lowpass
filter into another category of frequency-selective filters.
There are several specialized implementations such as Butterworth, Type I
Chebyshev, Type II Chebysev, and elliptic filters, which may be used to design
a normalized lowpass filter. Figure 7.5 shows representative characteristics of
these implementations, where we observe that the Butterworth filter (Fig. 7.5(a))
has a monotonic transfer function such that the gain decreases monotonically
from its maximum value of unity at
ω =
0 along the positive frequency axis.
The magnitude spectrum of the Butterworth filter has negligible ripples within
the pass and stop bands, but has a relatively lower fall off leading to a wide
transitional band. By allowing some ripples in either the pass or stop band,
the Type I and Type II Chebyshev filters incorporate a sharper fall off. The
Type I Chebyshev filter constitutes ripples within the pass band, while the
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