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Characteristics of Prototype Analog Filters 403
FIGURE 8.14 Pole plot for Example 8.1
A stable and causal filter H a (s) can now be specified by selecting
poles in the left half-plane, and H a (s) can be written in the form
H a (s) = ∏
Ω N c
LHP poles
(s − p k )
(8.48)
□ EXAMPLE 8.1 Given that |H a(jΩ)| 2 =
H a(s).
1
, determine the analog filter system function
1+64Ω6 Solution
From the given magnitude-squared response,
|H a(jΩ)| 2 1
=
1+64Ω = 1
6 ( Ω
1+
0.5
) 2(3)
Comparing this with expression (8.45), we obtain N =3and Ω c =0.5. The
poles of H a(s)H a(−s) are as shown in Figure 8.14.
Hence
H a(jΩ) =
=
Ω 3 c
(s − p 2)(s − p 3)(s − p 4)
1/8
(s +0.25 − j0.433)(s +0.5)(s +0.25 + j0.433)
=
0.125
(s +0.5)(s 2 +0.5s +0.25)
□
8.3.2 MATLAB IMPLEMENTATION
MATLAB provides a function called [z,p,k]=buttap(N) to design a normalized
(i.e., Ω c =1)Butterworth analog prototype filter of order N,
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