Digital Signal Processing Reference
In-Depth Information
The aforementioned M
ATLAB
code produces the following transfer functions
for the four filters:
s
6
+
1
.
125
10
5
s
4
+
4
.
219
10
9
s
2
+
5
.
273
10
13
s
6
+
430
.
2
s
5
+
2
.
05
10
5
s
4
+
4
.
221
10
7
s
3
+
7
.
688
10
9
s
2
+
6
.
049
10
11
s
+
5
.
273
10
13
;
Butterworth
H
(
s
)
=
0
.
7943
s
4
+
5
.
957
10
4
s
2
+
1
.
117
10
9
s
4
+
262
.
4
s
3
+
1
.
627
10
5
s
2
+
9
.
839
10
6
s
+
1
.
406
10
9
;
Type I Chebyshev
H
(
s
)
=
0
.
7943
s
4
+
8
.
015
10
4
s
2
+
1
.
406
10
9
s
4
+
304
.
5
s
3
+
1
.
265
10
5
s
2
+
1
.
142
10
6
s
+
1
.
406
10
9
;
Type II Chebyshev
H
(
s
)
=
0
.
7943
s
4
+
6
.
776
10
4
s
2
+
1
.
117
10
9
s
4
+
227
.
5
s
3
+
1
.
568
10
5
s
2
+
8
.
53
10
6
s
+
1
.
406
10
9
.
elliptic
H
(
s
)
=
7.5 Summary
Chapter 7 defines the CT filters as LTI systems used to transform the frequency
characteristics of the CT signals in a predefined manner. Based on the magnitude
spectrum
H
(
ω
)
, Section 7.1 classifies the frequency-selective filters into four
different categories.
(1) An ideal lowpass filter removes frequency components above the cut-off
frequency
ω
c
from the input signal, while retaining the lower frequency
components
ω ≤ ω
c
.
(2) An ideal highpass filter is the converse of the lowpass filter since it removes
frequency components below the cut-off frequency
ω
c
from the input signal,
while retaining the higher frequency components
ω ≤ ω
c
.
(3) An ideal bandpass filter retains a selected range of frequency components
between the lower cut-off frequency
ω
c1
and the upper cutoff frequency
ω
c2
of the filter. All other frequency components are eliminated from the
input signal.
(4) A bandstop filter is the converse of the bandpass filter, which rejects all
frequency components between the lower cut-off frequency
ω
c1
and the
upper cut-off frequency
ω
c2
of the filter. All other frequency components
are retained at the output of the bandstop filter.
The ideal frequency filters are not physically realizable. Section 7.2 introduces
practical implementations of the ideal filters obtained by introducing ripples
in the pass and stop bands. A transition band is also included to eliminate the
sharp transition between the pass and stop bands.
In Section 7.3, we considered the design of practical lowpass filters. We pre-
sented four implementations of practical filters: Butterworth, Type I Chebyshev,
Type II Chebyshev, and elliptic filters, for which the design algorithms were
covered. The Butterworth filters provide a maximally flat gain within the pass
band but have a higher-order
N
than the Chebyshev and elliptic filters designed
with the same specifications. By introducing ripples within the pass band,
Type I Chebyshev filters reduce the required order
N
of the designed filter.
Alternatively, Type II Chebyshev filters introduce ripples within the stop band
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