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4.6 Specification of IIR Filters 129

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where C(e-> ) is the characteristic function. The magnitude of the characteristic
function should be small in the passband and large in the stopband. We define the
ripple factors:

The attenuation requirements can be rewritten in terms of the ripple factors:

Synthesis of the transfer function, also referred to as the approximation prob-
lem, involves finding a proper characteristic function, C(z), satisfying Equations
(4.17) and (4.18). Various approaches to solving this problem will be discussed in
the next sections.

4.6.1 Analog Filter Approximations
Many filter solutions, so-called filter approximations, have been developed to meet
different requirements, particularly for analog filters [4, 24]. The main work has
focused on approximations to lowpass filters, since highpass, bandpass, and stop-
band filters can be obtained from lowpass filters through frequency transforma-
tions [4, 15-18, 24, 27]. It is also possible to use these results to design digital
filters. The classical lowpass filter approximations, which can be designed by using
most standard filter design programs, are:

Butterworth The magnitude function is maximally flat at the origin and mono-
tonically decreasing in both the passband and the stopband. The variation of the
group delay in the passband is comparatively small. However, the overall group
delay is larger compared to the filter approximations we will discuss shortly. This
approximation requires a larger filter order than the filter approximations dis-
cussed shortly to meet a given magnitude specification.

Chebyshev I The magnitude function has equal ripple in the passband and
decreases monotonically in the stopband. The variation of the group delay is some-
what worse than for the Butterworth approximation. The overall group delay is
smaller than for Butterworth filters. A lower filter order is required compared to
the Butterworth approximation.

Chebyshev II (Inverse Chebyshev) The magnitude function is maximally
flat at the origin, decreases monotonically in the passband, and has equal ripple in
the stopband. The group delay has a variation similar to the Butterworth approxi-
mation, but much smaller overall group delay. The same filter order is required as
for the Chebyshev I approximation.

Cauer The magnitude function has equal ripple in both the passband and the
stopband, but the variation of the group delay is larger than that for the other
approximations. The overall group delay is the smallest of the four filter