Page 73 - Analog and Digital Filter Design

70 Analog and Digital Filter Design

Inverse Chebyshev design is that the Q factor of its components is lower than
in the Chebyshev design and therefore easier to achieve. The disadvantage
is that Inverse Chebyshev designs are more complex and require more
components.

The underlying method used to find the component values, which will be
described in the next chapter, is pole positions derived from Chebyshev
designs. The disadvantage of this is that the frequency response stopband is
normalized to w = 1, instead of the usual 3dB attenuation frequency. This
description is not very helpful to practicing engineers because the 3 dB point will
vary, depending upon the stopband attenuation and the filter order. Fortunately,
it is possible to correct this and produce pole and zero positions based on a
3dB cutoff. Passive filter component values can also be corrected to give a
3 dB cutoff frequency.

Inverse Chebyshev filters have a smooth passband with a gentle roll-off, a steep
skirt, and ripples in the stopband. Poles and zeroes will be explained in the
next chapter, but you may like to know that the “inverse” in Inverse Chebyshev
filters comes from the filter pole positions, which are the inverse of those for
Chebyshev filters. Pole and zero positions can be obtained using formulae,
and these can be used directly in the design of active filters. Formulae to find
the zero positions are given in the Appendix.

Inverse Chebyshev filters can achieve the same performance as Chebyshev
filters of the same order, however they are more complex. The smooth passband
with a gentle roll-off in the frequency domain transforms into the time domain
as a group delay that is flatter than Chebyshev designs. The other advantage
is that circuit elements require a lower Q factor; this makes them easier to
produce.

These filters have not been popular because there are no simple algorithms
to find passive filter component values. The exception to this is equations
for third-order filters, which were derived by John Rhodes, Professor at the
University of Leeds in the U.K., and these are presented in the Appendix.
Rhodes’s book, Theory of Electrical Filters (Wiley, 1976) is difficult to read, and
for Inverse Chebyshev filters Rhodes assumes a highpass prototype. Some con-
version is needed for a lowpass prototype and to give 3dB attenuation at the
passband edge at a frequency of w = 1 rads, but the results are given in this
chapter.