Phase Continuation in a Time-Scaling Vocoder

There are two conflicting desiderata when deciding how to continue the phase from one frame to the next:

• When condition (1) is violated, spectral components are phase modulated at the rate of one ``pulse'' per frame. This spreads their energy elsewhere in the spectrum. The use of a synthesis window helps, but the overlap of sinusoidal components at altered phases will still cause some amplitude error in the cross-over region between adjacent frames, where the amount of error depends on frequency and the amount of phase error.

• When condition (2) is violated, the signal frame suffers dispersion in the time domain. Equivalently, the amplitude envelope is modified for each spectral region, so that the final overlap-add exhibits more general amplitude modulation. We will illustrate this in some examples below.

• Random amplitude modulation tends to be perceived as reverberation distortion. The apparent ``room size'' increases as the frame size is increased.

• It is theoretically impossible to satisfy both conditions (1) and (2) simultaneously, but either can be satisfied at the expense of the other. Generally speaking, ``transient frames'' should emphasize condition (1), allowing the overlap-add cross-fade to take care of the phase discontinuity at the frame boundaries. For ``stationary'' frames, the amplitude envelope is relatively flat so that relative phase is usually not perceivable; in this case, it is best to optimize for condition (2).

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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.