SOME PRELIMINARIES

8.1 SOME PRELIMINARIES

We discuss two preliminary issues in this section. First, we consider the magnitude-squared response specifications, which are more typical of ana- log (and hence of IIR) filters. These specifications are given on the relative linear scale . Second, we study the properties of the magnitude-squared response.

8.1.1 RELATIVE LINEAR SCALE

Let H a (jΩ) be the frequency response of an analog filter. Then the lowpass filter specifications on the magnitude-squared response are given by

2 ≤ |H a (jΩ)| 2 ≤ 1,

0 ≤ |H a (jΩ)| 2 ≤

, Ω s ≤ |Ω|

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Chapter 8

IIR FILTER DESIGN

| H a 2 ( j Ω)| 1

FIGURE 8.1 Analog lowpass filter specifications

where ϵ is a passband ripple parameter, Ω p is the passband cutoff fre- quency 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 2

a (jΩ)| must satisfy

The parameters ϵ and A are related to parameters R p and A s , respec- tively, of the dB scale. These relations are given by

R p = −10 log 10

=⇒ ϵ = 10 R p /10 −1 (8.3)

(8.4) The ripples, δ 1 and δ 2 , of the absolute scale are related to ϵ and A by

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Some Preliminaries 389

8.1.2 PROPERTIES OF |H a (jΩ)| 2

Analog filter specifications (8.1), which are given in terms of the magnitude-squared response, contain no phase information. Now to eval-

uate the s-domain system function H a (s), consider

H a (jΩ) = H a (s)| s=jΩ

Then we have

2 |H ∗

a (jΩ)| =H a (jΩ)H a (jΩ) = H a (jΩ)H a (−jΩ) = H a (s)H a (−s)| s=jΩ

or

H a (s)H

a (−s) = |H a (jΩ)| # #

Ω=s/j

Therefore the poles and zeros of the magnitude-squared function are dis- tributed in a mirror-image symmetry with respect to the jΩ axis. Also for real filters, poles and zeros occur in complex conjugate pairs (or mirror- image symmetry with respect to the real axis). A typical pole-zero pat-

tern of H a (s)H a (−s) is shown in Figure 8.2. From this pattern we can construct H a (s), which is the system function of our analog filter. We want H a (s) to represent a causal and stable filter. Then all poles of H a (s) must lie within the left half-plane. Thus we assign all left-half poles of

H a (s)H a (−s) to H a (s). However, zeros of H a (s) can lie anywhere in the s-plane. Therefore they are not uniquely determined unless they all are on the jΩ axis. We will choose the zeros of H a (s)H a (−s) lying left to or on the jΩ axis as the zeros of Ha(s). The resulting filter is then called a minimum-phase filter.

jΩ

s-plane

FIGURE 8.2 Typical pole-zero pattern of H a (s)H a (−s)

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Chapter 8

IIR FILTER DESIGN