Page 266 - Analog and Digital Filter Design

Phase-Shift Networks (All-Pass Filters) 243

The most important plot in Figure 9.15 is the group delay. This falls steadily
over the passband of the filter, then peaks up just before the cutoff frequency.
Over most of the passband the group delay is between 630ps and 640~s.

Equalization of Chebyshev Filters

The Buttenvorth filter was reasonably simple to equalize because its group delay
had a smooth curve. Chebyshev filters are more difficult to equalize because the
peak group delay has greater amplitude; also, the group delay does not rise
smoothly. but has ripple.

The group delay of an equalizer can be set to have a high peak value, but this
causes the range of frequencies over which the delay occurs to become veiy
short. Generally there is a reciprocal relationship between the value of the peak
group delay and the steepness of the group delay versus frequency curve.
Butterworth filters had a low percentage group delay variation across the pass-
band and many could be equalized effectively by a first-order equalizer.

Chebyshev filters have a high percentage group delay variation across the pass-
band. The significance of this is that the equalizing sections must also produce
a high percentage group delay variation at other frequencies to compensate for
where the filter’s group delay is short. Equalizers that produce these high delay
variations do so for a limited range of frequencies. Unless there are several
equalizing sections, each compensating for different frequencies, the group delay
curve will have significant ripple.

Chebyshev Group Delay

A computer program subroutine given by Rorabaugh in Digital Filter Designer’s
Hmdbooks provides the amplitude and phase response of Chebyshev filters. This
subroutine is reproduced, with the kind permission of McGraw-Hill, as Listing
9.1. The subroutine, chebyshevFreqResponse( ), was used within a program of
my own to produce tables of phase versus frequency. The result was then nunier-
ically differentiated to find the group delay at each frequency. The tables of
group delay versus frequency were then used by one of my MATHCAD appli-
cations to produce a graph.