ACTIVE_FILTERS_Theory_and_Design

Introduction 7
TABLE 1.3
0.5-dB Chebyshev filter
n b 0 b 1 b 2 b 3 b 4 b 5 b 6 b 7
1 2.863
2 1.516 1.426
3 0.716 1.535 1.253
4 0.379 1.025 1.717 1.197
5 0.179 0.753 1.310 1.937 1.172
6 0.095 0.432 1.172 1.590 2.172 1.159
7 0.045 0.282 0.756 1.648 1.869 2.413 1.151
8 0.024 0.153 0.574 1.149 2.184 2.149 2.657 1.146
consists of a number of second-order transfer functions multiplied together, possibly
with some first-order terms as well. We can think of the complex filter as being
made up of several second-order and first-order filters connected in series. The
transfer function thus takes the form:
K
Hs () =
( s2
+ a s+ a )( s2
+ a s+ a )… ( s2
+ a s+
a ) n0
11 10
21 20
n1
(1.11)
1.3.1 BUTTERWORTH FILTERS
The first, and probably best-known, filter is the Butterworth or maximally flat
response. It exhibits a nearly flat passband. The rolloff is 20 dB/decade or 6 dB/octave
for every pole. The general equation for a Butterworth filter’s amplitude response is
K
H( jω) =
/
n
s
+ ⎛ 2
⎡ ⎞ ⎤
⎢1
⎢ ⎝ ⎜ ω ⎠
⎟
⎥
1 ⎥
⎣ ⎦
12
(1.12)
TABLE 1.4
1-dB Chebyshev filter
n b 0 b 1 b 2 b 3 b 4 b 5 b 6 b 7
1 1.965
2 1.103 1.098
3 0.491 1.238 0.988
4 0.276 0.743 1.454 0.953
5 0.123 0.581 0.974 1.689 0.937
6 0.069 0.307 0.939 1.202 1.931 0.928
7 0.031 0.214 0.549 1.358 1.429 2.176 0.923
8 0.017 0.107 0.448 0.847 1.837 1.655 2.423 0.920