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Phase Distortion at Passband Edges

For many applications (such as lowpass, bandpass, or highpass filtering), the frequency region where most of the phase dispersion occurs is at the extreme edge of the desired range, or passband (i.e., in the vicinity of cut-off frequencies). Only filters without feedback can have exactly linear phase (unless forward-backwards filtering is feasible), and such filters generally need many more multiplies for a given specification on the amplitude response $ G(\omega)$ [65]. One should realize that phase dispersion at a cutoff frequency usually appears as ringing near that frequency in the time domain. (This can be heard in the matlab example of §12.3, Fig.12.1.) To be conservative, the amplitude response $ G(\omega)$ should be a smooth function of $ \omega$ so that the phase dispersion is inaudible whether or not the filter is linear phase. Linearizing the phase with a delay equalizer (a type of allpass filter) does not eliminate ringing, but merely shifts it in time. A good rule of thumb is to keep the total impulse-response duration below the time-discrimination threshold of hearing.

For musical purposes, $ G(\omega)$, or the effect that a filter has on the magnitude spectrum of the input signal, is usually of primary interest. This is true for all ``instantaneous'' filtering operations such as tone controls, graphical equalizers, parametric equalizers, formant filter banks, shelving filters, and the like. Notable exceptions are echo and reverberation, in which delay characteristics are at least as important.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (August 2006 Edition).
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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