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Some Preliminaries 387
8.1 SOME PRELIMINARIES
The first approach is used in MATLAB to design IIR filters. A
straightforward use of these MATLAB functions does not provide any
insight into the design methodology. Therefore we will study the second
approach because it involves the frequency-band transformation in the
digital domain. Hence in this IIR filter design technique we will follow
the following steps:
• Design analog lowpass filters.
• Study and apply filter transformations to obtain digital lowpass filters.
• Study and apply frequency-band transformations to obtain other digital
filters from digital lowpass filters.
The main problem with these approaches is that we have no control
over the phase characteristics of the IIR filter. Hence IIR filter designs
will be treated as magnitude-only designs. More sophisticated techniques,
which can simultaneously approximate both the magnitude and the phase
responses, require advanced optimization tools and hence will not be covered
in this book.
We begin with a discussion on the analog filter specifications and the
properties of the magnitude-squared response used in specifying analog
filters. Next, before we delve into basic techniques for general IIR filters,
we consider the design of special types of digital filters—for example,
resonators, notch filters, comb filters, etc. This is followed by a brief description
of the characteristics of three widely used analog filters: namely.
Butterworth, Chebyshev, and elliptic filters. Finally, we will study transformations
to convert these prototype analog filters into different frequencyselective
digital filters and conclude this chapter with several IIR filter
designs using MATLAB.
We discuss two preliminary issues in this section. First, we consider the
magnitude-squared response specifications, which are more typical of analog
(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
1
1+ɛ 2 ≤|H a(jΩ)| 2 ≤ 1, |Ω| ≤Ω p
0 ≤|H a (jΩ)| 2 ≤ 1 (8.1)
A 2 , Ω s ≤|Ω|
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