Phase Distortion at Passband Edges

For many applications (such as lowpass, bandpass, or highpass filtering), the most phase dispersion occurs at the extreme edge of the passband (i.e., in the vicinity of cut-off frequencies). This phenomenon was clearly visible in the example of Fig.7.6.4. Only filters without feedback can have exactly linear phase (unless forward-backward filtering is feasible), and such filters generally need many more multiplies for a given specification on the amplitude response $ G(\omega)$ [68]. One should keep in mind that phase dispersion near a cut-off frequency (or any steep transition in the amplitude response) usually appears as ringing near that frequency in the time domain. (This can be heard in the upcoming matlab example of §11.6, Fig.11.1.)

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. (Elementary examples in this category are discussed in Appendix B.) Notable exceptions are echo and reverberation [86], in which delay characteristics are as important as magnitude characteristics.

When designing an ``instantaneous'' filtering operation, i.e., when not designing a ``delay effect'' such as an echo unit or reverberator, the amplitude response $ G(\omega)$ should be as smooth as possible as a function of frequency $ \omega$. Smoother amplitude responses correspond to shorter impulse responses (when the phase is zero, linear, or ``minimum phase'' as discussed in the next chapter). By keeping impulse-responses as short as possible, phase dispersion is minimized, and ideally inaudible. Linearizing the phase response with a delay equalizer (a type of allpass filter) does not eliminate ringing, but merely shifts it in time. A general rule of thumb is to keep the total impulse-response duration below the time-discrimination threshold of hearing in the context of the intended application.

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Definition of Minimum Phase Filters
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Forward-Backward Filtering