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:

To satisfy condition (1), it is necessary to replace the original phase of each frame by the phase corresponding to smooth continuation from the previous frame (which is generally an interpolated frame). Altering the phase of a spectral frame changes its amplitude envelope in the time domain. Thus, it no longer looks like a windowed signal segment. Using the WOLA framework helps because the post-window guarantees a smooth cross-fade from frame to frame. Random amplitude-modulation distortion is generally heard as reverberation, also called phasiness [133].

When condition (2) is violated, the signal frame suffers dispersion in the time domain. For steady-state signals (filtered noise and/or steady tones), temporal dispersion should not be audible, while frames containing distinct pulses will generally become more ``smeared out'' in time.

It is not possible in general 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 (2), allowing the overlap-add cross-fade to take care of the phase discontinuity at the frame boundaries. For ``stationary'' frames, phase continuation, preserving condition (1), is more valuable.

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About the Author: 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.