Phase Vocoder Sinusoidal Modeling

In this section, we describe a classic computer-music implementation of the phase vocoder (using the STFT) for purposes of measuring additive-synthesis parameters (amplitude, frequency, and occasionally phase versus frame time), as introduced by Moorer [175,176].

In analysis for additive synthesis,^H.6 we convert a time-domain signal into a collection of amplitude and frequency envelopes, as graphed in Fig.H.15. It is usually desired that these envelopes be slowly varying relative to the original signal. This leads to the assumption that we have at most one sinusoid in each filter-bank channel. (By ``sinusoid'' we really mean ``quasi sinusoid,'' since its amplitude and phase may be slowly time-varying.) The channel-filter frequency response is given by the FFT of the analysis window used (Chapter 8).

The signal in the $k^{th}$ subband (filter-bank channel) can be written

$\displaystyle x_k(t)\eqsp a_k(t)\cos[ \omega_kt + \phi_k(t) ]. \protect$

(H.1)

In this expression, is an amplitude modulation term, $\omega_k$ is a fixed channel center frequency, and $\phi_k(t)$ is a phase modulation (or, equivalently, the time-integral of a frequency modulation). Using these parameters, we can re-synthesize the signal using the classic oscillator summation, as shown in Fig.9.12 (ignoring the filtered noise in that figure).^H.7

Typically, the instantaneous phase modulation $\phi_k(t)$ is differentiated to obtain instantaneous frequency deviation:

$\displaystyle \Delta \omega_k(t) \isdefs \frac{d}{dt} \phi_k(t)$

The analysis and synthesis signal models are summarized in Fig.H.12.

**Figure H.12:** Illustration of channel vocoder parameters in analysis (left) and synthesis (right).
$\includegraphics[width=\twidth]{eps/pvchan}$

Subsections

Previous: Further Reading about FM Synthesis
Next: Computing Vocoder Parameters

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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.

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Spectral Audio Signal Processing
A History of Spectral Audio Signal Processing
Phase Vocoder Sinusoidal Modeling