- The pass-band ripple is much smaller than 0.1 dB, which is
``over designed'' and therefore wasting of taps.
- The stop-band response ``droops'' which ``wastes'' filter taps
when stop-band attenuation is the only stop-band specification. In
other words, the first stop-band ripple drives the spec (
while all higher-frequency ripples are over-designed. On the other
hand, a high-frequency ``roll-off'' of this nature is quite natural
in the frequency domain, and it corresponds to a ``smoother pulse''
in the time domain. Sometimes making the stop-band attenuation
uniform will cause small impulses at the beginning and end of
the impulse response in the time domain. (The pass-band and
stop-band ripple can ``add up'' under the inverse Fourier transform
integral.) Recall this impulsive endpoint phenomenon for the
Chebyshev window shown in Fig.3.33.
- The pass-band is degraded by early roll-off. The pass-band edge
is not exactly in the desired place.
- The filter length can be thousands of taps long without running
into numerical failure. Filters this long are actually needed for
sampling rate conversion
We can also note some observations regarding the optimal Chebyshev version designed by the Remez multiple exchange algorithm:
- The stop-band is ideal, equiripple.
- The transition bandwidth is close to half that of the
window method. (We already knew our chosen transition bandwidth was
not ``tight'', but our rule-of-thumb based on the Kaiser-window
main-lobe width predicted only about
% excess width.)
- The pass-band is ideal, though over-designed for static audio spectra.
- The computational design time is orders of magnitude larger
than that for window method.
- The design fails to converge for filters much longer than 256
taps. (Need to increase working precision or use a different
method to get longer optimal Chebyshev FIR filters.)
Comparison to Optimal Chebyshev FIR Filter