DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

A Signal Processing Perspective 265
response shifts farther to the right (increased delay) as the order of the phasecompensation
filter is increased. Note that the time at which the impulse response
peak occurs correlates well with the low-frequency delay as indicated in either Figure
8.44 or 8.45. It is interesting to note that the impulse response increasingly exhibits
even symmetry about the time of the response peak as the order of the phasecompensation
filter is increased. Compare the unit impulse response of the
Butterworth filter compensated by the 4th-order all-pass with that shown in Figure 2.3
and expressed in (2.2) for the ideal lowpass filter. The peak value for (2.2) with
is 0.3183, whereas the peak value in Figure 8.46 is 0.3399. From (2.2), the
impulse response first zero-crossings would be at the time of the peak (0 for (2.2))
whereas the first zero-crossings in Figure 8.46 are 11.9 and 17.8,
which are 14.9 (the time of the peak) -3 and +2.9. Therefore, the unit impulse
response of the 10th-order Butterworth filter compensated by the 4th-order all-pass is
beginning to approximate the ideal unit impulse response, with a delay, rather well.
In Figure 8.47 is shown the unit step response of the uncompensated
Butterworth filter, and the unit step response of the phase-compensated Butterworth
filter using the 1st-order, 2nd-order, 3rd-order and 4th-order all-pass filters. Compare
the unit step response of the Butterworth filter compensated by the 4th-order all-pass
with that shown in Figure 2.4 for the ideal lowpass filter. The unit step response of
the 10th-order Butterworth filter compensated by the 4th-order all-pass is beginning
to approximate the ideal unit step response, with a delay, rather well.
Section 8.7 Phase-Compensated Filters