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SUMMARY OF FILTER CHARACTERISTICS
5.3 Non-Minimum Phase And Allpass
Networks
An allpass filter can be defined as one whose transfer
function has all its zeros in the right half plane and all its
poles in the left half plane as images of the zeros.
Allpass functions have unit amplitude for all real fre-
quencies and, for all positive frequencies, a negative
phase. Any non-minimum phase function can be written
as the product of a minimum phase function and an
allpass network.
The use of an allpass network with a minimum phase
network (such as those described above) enables the
filter designer to realize simultaneously both the
amplitude and phase characteristics desired (at least on
paper). This allpass filter is being used as a phase
equalizer. In general, it takes one zero-pole allpass net-
work for each pole of a minimum phase filter to achieve
a significant improvement in phase linearity (assuming a
high selectivity filter such as a Chebyshev). This rule of
thumb hints at the substantial complexity that can result
from the phase equalization of multi-pole minimum
phase network.
Direct synthesis techniques can be used to design net-
works which meet both amplitude and phase constraints
simultaneously. The situation is nevertheless unfavor-
able because substantial design effort and complexity
are involved.
5.4 Review of Analog Filter
Characteristics
5.4.1 Effect of Pole/Zero Locations on Filter
Parameters
Relative comparisons of filter types can be made for
filters having the same number of elements (poles) and
the same 3 dB bandwidth. The relative pole-zero loca-
tions for variol;ls 4-pole lowpass filters are illustrated in
Figure 5-2.
For filters of the same type having a different number of
poles, the pole locations will fall on the same contours
as the 4-pole case and the filter will have the same
general characteristics.
It can be seen that filters optimized for a particular
amplitude characteristic have pole locations inside the
circle passing through the cutoff frequency; similarly,
5-4
)w
•
0 laB CHEBYSHEV
)(
BUTTERWORTH
•
±
'h'EQUAL RIPPLE PHASE
T
BESSEL
ALL ZEROS AT
w
=
~
I
\
'f'
•
\
.
~"~
Figure 5-2. Pole-Zero Locations Of 4-Pole
Lowpass Prototype Filters
filters optimized for linear-phase properties have pole
locations outside this circle. The amplitude, phase, and
time delay characteristics corresponding to common
prototype filters are illustrated in Figure 5-3. The cor-
respondence of good amplitude characteristics and pole
locations near the jw axis,
or of good phase
characteristics and pole locations away from the jw axis,
is observable. Signal distortion results if changes in
amplitude or phase occur at frequencies within the
signal spectrum. A measure of the degree of distortion
can be obtained by observing the step and impulse
responses of the system, as illustrated in Figure 5-3 for
the 4-pole low-pass prototype filters. The degree of
signal distortion can be determined qualitatively by
observing the rise time, overshoot, and settling time of
the filter step response. Figure 5-4 indicates that approx-
imately the same percentage overshoot of the step
response is achieved for the Butterworth and Chebyshev
filters; the deviation from linearity of the phase
response across the 3 dB bandwidth is also approxi-
mately equal. Figure 5-3, however, shows that the time-
delay characteristic differs by as much as 2:
1.
Phase
linearity, therefore, provides a better measure of signal
distortion than the time-delay variation across the pass-
band. This is illustrated by noting the step response and
time-delay
characteri~tic
of a linear-phase filter that has
fine-grain variations. Because deviations from linearity
are
infinitesimal,
the
step
response
is virtually
unchanged. However, the time delay can have extremely
large variations simply by increasing the slope of the
fine-grain phase variations.
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