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448 Chapter 8 IIR FILTER DESIGN
TABLE 8.1
Comparison of three filters
Prototype Order N Stopband Att.
Butterworth 6 15
Chebyshev-I 4 25
Elliptic 3 27
8.5.1 COMPARISON OF THREE FILTERS
In our examples we designed the same digital filter using four different
prototype analog filters. Let us compare their performance. The specifications
were ω p =0.2π, R p =1dB, ω s =0.3π, and A s =15dB. This
comparison in terms of order N and the minimum stopband attenuations
is shown in Table 8.1.
Clearly, the elliptic prototype gives the best design. However, if we
compare their phase responses, then the elliptic design has the most nonlinear
phase response in the passband.
8.6 FREQUENCY-BAND TRANSFORMATIONS
In the preceding two sections we designed digital lowpass filters from
their corresponding analog filters. Certainly, we would like to design other
types offrequency-selective filters, such as highpass, bandpass, and bandstop.
This is accomplished by transforming the frequency axis (or band)
of a lowpass filter so that it behaves as another frequency-selective filter.
These transformations on the complex variable z are very similar
to bilinear transformations, and the design equations are algebraic. The
procedure to design a general frequency-selective filter is to first design
a digital prototype (of fixed bandwidth, say unit bandwidth) lowpass filter
and then to apply these algebraic transformations. In this section we
will describe the basic philosophy behind these mappings and illustrate
their mechanism through examples. MATLAB provides functions that
incorporate frequency-band transformation in the s-plane. We will first
demonstrate the use of the z-plane mapping and then illustrate the use
of MATLAB functions. Typical specifications for most commonly used
types offrequency-selective digital filters are shown in Figure 8.30.
Let H LP (Z)bethe given prototype lowpass digital filter, and let H(z)
be the desired frequency-selective digital filter. Note that we are using
two different frequency variables, Z and z, with H LP and H, respectively.
Define a mapping of the form
Z −1 = G(z −1 )
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