Conclusions

In the window-method filter design:

• The passband ripple is much smaller than 0.1 dB, which is ``over designed'' and therefore wasting of taps.

• The stopband response ``droops'' which ``wastes'' filter taps when stopband attenuation is the only stopband specification. In other words, the first stopband 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 stopband attenuation uniform will cause small impulses at the beginning and end of the impulse response in the time domain. (The passband and stopband ripple can ``add up'' under the inverse Fourier transform integral.)

• The passband is degraded by early roll-off. The passband 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 [247,201].

In the optimal Remez-exchange filter:

• The stopband is ideal, equiripple.

• The transition bandwidth is close to half that of the window method.

• The pass-band is ideal, though over-designed for static audio spectra.

• The expected time-domain ``ears'' due to passband ripple are quite small.

• 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 to get longer filters.)

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Order a Hardcopy of Spectral Audio Signal Processing

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.