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402 Chapter 8 IIR FILTER DESIGN
• |H a (jΩ)| 2 approaches an ideal lowpass filter as N →∞.
• |H a (jΩ)| 2 is maximally flat at Ω = 0 since derivatives of all orders exist
and are equal to zero.
To determine the system function H a (s), we put (8.45) in the form of
(8.5) to obtain
H a (s)H a (−s) =|H a (jΩ)| 2∣ ∣ 1
(jΩ) 2N
∣Ω=s/j = ( ) 2N
=
s s
1+
2N +(jΩ c ) 2N
jΩ c
(8.46)
The roots of the denominator polynomial (or poles of H a (s)H a (−s)) from
(8.46) are given by
p k =(−1) 1
2N (jΩ) = Ωc e j π
2N (2k+N+1) , k =0, 1,...,2N − 1 (8.47)
An interpretation of (8.47) is that
• there are 2N poles of H a (s)H a (−s), which are equally distributed on
a circle of radius Ω c with angular spacing of π/N radians
• for N odd the poles are given by p k =Ω c e jkπ/N , k =0, 1,...,2N − 1
• for N even the poles are given by p k =Ω c e j( π
2N + kπ
N ) , k =0, 1,...,
2N − 1
• the poles are symmetrically located with respect to the jΩ axis
• apole never falls on the imaginary axis, and falls on the real axis only
if N is odd
As an example, poles of 3rd- and 4th-order Butterworth filters are shown
in Figure 8.13.
jΩ
jΩ
0
Ω c
k = 0
σ
0
Ω c
k = 0
σ
k = 2N − 1
k = 2N − 1
N = 3
N = 4
FIGURE 8.13
Pole plots for Butterworth filters
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