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Phase Vocoder Sinusoidal Modeling

As mentioned in §G.7, the phase vocoder had become a standard analysis tool for additive synthesisG.8) by the late 1970s [186,187]. This section summarizes that usage.

In analysis for additive synthesis, we convert a time-domain signal $ x(t)$ into a collection of amplitude envelopes $ a_k(t)$ and frequency envelopes $ \omega_k+\Delta\omega_k(t)$ (or phase modulation envelopes $ \phi_k(t)=\int_t\Delta\omega_k(t)\,dt$ ), as graphed in Fig.G.12. 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 mean, of course, ``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 9).

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

$\displaystyle x_k(t)\eqsp a_k(t)\cos[ \omega_kt + \phi_k(t) ]. \protect$ (G.3)

In this expression, $ a_k(t)$ 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 resynthesize the signal using the classic oscillator summation, as shown in Fig.10.7 (ignoring the filtered noise in that figure).G.9

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

$\displaystyle \Delta \omega_k(t) \isdefs \frac{d}{dt} \phi_k(t)$ (G.4)

The analysis and synthesis signal models are summarized in Fig.G.9.

Figure G.9: Illustration of channel vocoder parameters in analysis (left) and synthesis (right).

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University