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388 Chapter 8 IIR FILTER DESIGN
|H a (jΩ)| 2 Ω p Ω s
1
1
1 +
1
A 2
0
0
Ω
FIGURE 8.1
Analog lowpass filter specifications
where ɛ is a passband ripple parameter, Ω p is the passband cutoff frequency
in rad/sec, A is a stopband attenuation parameter, and Ω s is the
stopband cutoff in rad/sec. These specifications are shown in Figure 8.1,
from which we observe that |H a (jΩ)| 2 must satisfy
|H a (jΩ p )| 2 = 1
1+ɛ 2 at Ω = Ω p
|H a (jΩ s )| 2 = 1 (8.2)
A 2 at Ω = Ω s
The parameters ɛ and A are related to parameters R p and A s , respectively,
of the dB scale. These relations are given by
R p = −10 log 10
1
1+ɛ 2 =⇒ ɛ = √
10 Rp/10 − 1 (8.3)
and
1
A s = −10 log 10
A 2 =⇒ A =10As/20 (8.4)
The ripples, δ 1 and δ 2 ,ofthe absolute scale are related to ɛ and A by
√
1 − δ 1 1
=
1+δ 1 1+ɛ 2 =⇒ ɛ = 2√ δ 1
1 − δ 1
and
δ 2
1+δ 1
= 1 A =⇒ A = 1+δ 1
δ 2
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