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SUMMARY OF FILTER CHARACTERISTICS
the Gibbs phenomenon, and is explained mathemati-
cally by the fact that the Fourier Series expansion of a
function fails to converge uniformly at discontinuities.
This point is significant because it shows that the
transient response of a filter is closely related to the
selectivity of the filter, and that phase linearization will
not eliminate overshoot and ringing.
5.2 Minimum Phase Filters
A minimum phase filter is defined as a filter whose
transfer function has all poles in the left plane and no
zeros in the right half plane of the complex frequency
s-plane. When a function is minimum phase, its phase
response is uniquely related to its amplitude response,
i.e., an amplitude specification uniquely defines the
phase characteristic of the filter. This attribute makes it
possible to categorize filter design parameters for a
limited selection of amplitude and phase character-
istics. These filters are the ones most often taken from
design tables ("cookbook filter design") and include
Butterworth, Chebyshev, Bessel, Transitional Bessell
Butterworth, Equal Ripple Phase, Elliptic Function,
and other less well-known designs. Each of these filters
provides unique characteristics which makes it a
valuable contribution to filter design. The "best" filter
from this group depends on the application: in some
cases a number of them may be satisfactory, while in
others none will suffice. The following sections describe
the basic characteristics of these filters from an applica-
tions viewpoint.
5.2.1 Butterworth Filters
Butterworth filters are also called "maximally flat
amplitude" filters because they are specifically designed
to have monotonically flat amplitude response, with a
nearly zero slope over the majority of the passband.
Outside the passband these filters exhibit good selectiv-
ity, with the amplitude rolloff approaching 6dB/octave-
bandwidth
per
pole
and
an
ultimate
rejection
approaching infinity as the frequency moves away from
the passband.
The pulse response of these filters is characterized by
high overshoot and significant ringing. The overshoot
increases monotonically with the number of poles, and
the duration of the ringing is inversely proportional to
the filter bandwidth (as it is in all minimum phase
filters.) The poor pulse response is caused by both the
high selectivity and the nonlinear phase characteristics
of the filters.
5-2
Typical applications of this type of filter include signal
rejection, frequency domain shaping of a spectrum, or
other amplitude response applications where the phase
distortion of the signal is not important (e.g., filtering a
fixed tuned linear oscillator.)
5.2.2 Chebyshev Filters
These filters are also called "equal ripple amplitude"
filters because the passband gain is defined within
specific limits, between which the gain varies in a
sinusoidal manner. The number of amplitUde ripples is
related to the number of poles in the transfer function.
The amount of ripple is determined by specifying the
transfer function with typical values of 0.01 to 0.5 dB.
The
attenuation
outside
the
passband
increases
monotonically towards infinity. For an equal number of
poles, the Chebyshev filter has a higher rate of rolloff in
the vicinity of the passband cutoff frequency than the
Butterworth filter. This rolloff eventually approaches a
6 dB/octave-bandwidth per pole rate.
The pulse response exhibits only slightly greater over-
shoot than the Butterworth, but condsiderably more
ringing. The amount of ringing is proportional to the
amplitUde ripple magnitude. The phase response also
has a ripple component roughly proportional to the
amplitUde ripple. One would properly suspect that this
filter introduces more distortion into a signal which
occupies the entire filter bandwidth. It can be shown
that over restricted portions of the passband, the com-
bined phase linearity and amplitude distortion effects
may be less for Chebyshev than for Butterworth and
Bessel filters.
The Chebyshev filter is usually used in applications
where a high selectivity, high ultimate rejection, and low
complexity are desired.
5.2.3 Elliptic Function Filters
The highest selectivity minimum phase filters are elliptic
function filters, characterized by equal ripple amplitude
in the passband and the stopband. The design
parameters are the amount of ripple in the passband,
the minimum attentuation in the stopband, and the
transition region width from passband to stopband.
These filters use zeros in the stopband to achieve excep-
tional selectivity, but have a finite ultimate attenuation
equal to the minimum attenuation. In some applications
this finite rejection can be a problem for interference
which is not sufficiently attenuated in the stop band.
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