We can note the following points regarding our single-sideband FIR filter design by means of direct Fourier intuition, frequency-sampling, and the window-method:

- 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 (
dB),
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
[270,218].

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.)

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

[Watch the Video] [Work some Exercises] [Examination]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University