The Phase Vocoder

Homer Dudley. The Vocoder. Bell Laboratories Record, December, 1939. Reprinted in IEEE Transactions on Acoustics, Speech and Signal Processing ASSP 29(3) 347-351(1981). 

James Andy Moorer. The Use of the Phase Vocoder in Computer Music Applications. Journal of the Audio Engineering Society 26 (1/2) 42—45(1978). 

Mark Dolson. The Phase Vocoder A Tutorial. Computer Music Journal 10 (4) 14-27(1986). 

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Time scaling. Because the phase-vocoder (the short-time Fourier transform) gives access to the implicit sinusoidal model parameters, the ideal time-scale operation described by Eq. (7.3) can be implemented in the same framework. Synthesis time-instants / are usually set at a regular interval / +1 - / = R. From the series of synthesis time-instants / analysis time-instants / are calculated according to the desired time warping function tua = T l(t ). The short-time Fourier transform of the time-scaled signal is then ... 

Early approaches to music analysis relied on a running Fourier transform to measure sine-wave amplitude and frequency trajectories. This technique evolved into a filter-bank-based processor and ultimately to signal analysis/synthesis referred to as the phase vocoder [Flanagan and Golden, 1966], This section describes the history of the phase vocoder, its principles, and limitations that motivate sinusoidal analysis/synthesis. Other formulations and refinements of the phase vocoder are given in chapter 7. 

The phase vocoder has been useful in a number of applications4. In time-scale modification, for example, the goal is to maintain the perceptual quality of the original signal while changing its apparent rate of articulation . In performing time-scale modification with the phase vocoder, the instantaneous frequency and amplitude of each channel are interpolated or decimated to a new time scale . In one scenario, the phase of each filter output in Equation (9.9) is first unwrapped, and the channel amplitude and unwrapped phase are then time scaled. With time-scale modification by a factor p, the modified filter output is given by... 

In spite of the many successes of the phase vocoder, numerous problems have limited its use. In the applications of time-scale modification and compression, for example, it is assumed that only one sine wave enters each bandpass filter within the filter bank. When more than one sine wave enters a bandpass filter, the meaning of the input sine-wave amplitude and phase envelope is lost. A particular sine wave also may not be adequately estimated when it falls between two adjacent filters of the filter bank. In addition, sine waves with rapidly-varying frequency due to large vibrato or fast pitch change are difficult to track. A result of using a fixed filter bank is that the frequency of... 

Since each harmonic is a multiple of the time-varying fundamental, higher harmonics vibrato have a larger bandwidth than lower harmonics. With rapid pitch vibrato, the temporal resolution required for frequency estimation increases with harmonic number. One approach to improve resolution time-warps the waveform inversely to pitch to remove vibrato [Ramalho, 1994]. This approach may also be useful in reducing channel cross-talk within the phase vocoder. 

Dolson, 1986] Dolson, M. (1986). The phase vocoder A tutorial. Computer Music Journal, 10(4) 14-27. 

The Phase Vocoder. The Phase Vocoder [Flanagan and Golden, 1966][Gordon and Strawn, 1985] is a common analysis technique because it provides an extremely flexible method of spectral modification. The phase vocoder models the signal as a bank of equally spaced bandpass filters with magnitude and phase outputs from each band. Portnoff s implementation of the Short Time Fourier Transform (STFT) provides a time-efficient implementation of the Phase Vocoder. The STFT requires a fast implementation of the Fast Fourier Transform (FFT), which typically involves bit addressed arithmetic. 

Pitch-scaling. There are several ways of using the phase-vocoder for pitch-scaling operations. 

Gordon and Strawn, 1985] Gordon, J. W. and Strawn, J. (1985). An Introduction to the Phase Vocoder. In Strawn, J., editor, Digital audio signal processing an anthology. William Kaufmann. 

The Phase Vocoder system implemented by the composer Trevor Wishart and his collaborators at Composer s Desktop Project (CDP) in England is a classic example of additive res)mthesis. In order to operate the Phase Vocoder the musician first needs to define a number of parameters, including the envelope for the window, the size of the window and the overlap factor. 

The size of the window defines the number of input samples to be analysed at a time. The larger the window, the greater the number of channels, but the lower the time resolution, and vice versa. This should be set large enough to capture four periods of the lowest frequency of interest. The sampling rate divided by the number of channels should be less than the lowest pitch in the input sound. This may be set in terms of the lowest frequency of interest or in terms of the number of channels required. Note that the term channel is the Phase Vocoder jargon for what we referred earlier to as harmonic detectors (Figure 3.11). 

Figure 3.11 The Phase Vocoder uses channels to act as harmonic detectors . The greater the number of channels, the better the frequency resolution. Note that (b) does not distinguish between the two uppermost partials that are perfectly distinct In (a)...

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