DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

A Signal Processing Perspective 261
of its relatively good magnitude frequency response combined with a relatively good
phase response. Only a Bessel filter or a Gaussian filter of the same order would have
superior phase characteristics, compared with all other filter types in this book, and
yet the Butterworth magnitude frequency response is significantly superior to either
the Bessel or the Gaussian filter. Since the phase compensation filters are all-pass,
and either implemented in cascade with the Butterworth filter or combined with the
Butterworth transfer function prior to implementation (transfer functions multiplied),
the phase compensation has no effect on the magnitude frequency response.
Therefore, no magnitude frequency response figures are required, since they would
be identical to that given in Chapter 3: here, is also normalized to unity.
However, the phase response, phase delay, group delay, unit impulse response, and
unit step response are all effected by phase compensation.
The results of four compensation filters are illustrated in the figures that
follow. The four compensation filters are all-pass filters of order 1, 2, 3, and 4.
They were each empirically designed (trial and error), to achieve a group delay
response superior to that of the basic Butterworth filter (all-pass of order one), or of
the immediately lower order phase-compensated filter.
The 1st-order phase-compensation filter has the following transfer function:
where The 2nd-order phase-compensation filter has the following transfer
function:
where and The 3rd-order phase-compensation filter is as
follows:
where and as above, and The 4th-order phasecompensation
filter is as follows:
where and
In Figure 8.43 is shown the phase response of the uncompensated Butterworth
filter, and the phase response of the phase-compensated Butterworth filter using the
Section 8.7 Phase-Compensated Filters