Phase Vocoder Sinusoidal Modeling

International Standard Book Number — Click for https://mathworld.wolfram.com/ISBN.html

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Noise (in signal processing) is a random signal. In the language of statistical signal processing, a noise signal is typically modeled as 'stochastic process', which is in turn defined as a sequence of random variables. — Click for https://ccrma.stanford.edu/~jos/sasp/What_Noise.html

When displaced from its equilibrium state, a harmonic oscillator experiences a restoring force according to Hooke's law. — Click for http://ccrma.stanford.edu/~jos/filters/Mass_Spring_Oscillator_Analysis.html

Frequency Modulation (FM) is said to occur when one signal modulations the frequency of another signal, usually a sinusoid. The most typical case of FM is sinusoidal modulation of a sinusoid. Originally, FM was invented for high quality radio signal broadcast. More recently, FM synthesis was invented as the first widely used digital sound synthesis technique. — Click for https://ccrma.stanford.edu/~jos/mdft/Sinusoidal_Frequency_Modulation_FM.html

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The midpoint of the passband. — Click for http://en.wikipedia.org/wiki/Bandpass

Amplitude Modulation (AM) is said to occur when one signal modulations the amplitude of another signal. The most typical case of AM is multiplying by an offset sinusoid 1 + g sin(ω t+φ), where g is a real number between 0 and 1. — Click for https://ccrma.stanford.edu/~jos/mdft/Sinusoidal_Amplitude_Modulation_AM.html

A set of filters that decompose a signal into a set of components — Click for http://en.wikipedia.org/wiki/Filter_bank

Fast Fourier Transforms (FFT) are fast algorithms for computing the Discrete Fourier Transform (DFT) — Click for https://ccrma.stanford.edu/~jos/mdft/Fast_Fourier_Transform_FFT.html

The frequency response is defined for LTI filters as the Fourier transform of the filter output signal divided by the Fourier transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html

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A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

A sinusoid is any function of the form A sin(ω t+φ), where t is the independent variable, and A, ω, φ are fixed parameters of the sinusoid called the amplitude, (radian) frequency, and phase, respectively. Sinusoidal motion is produced by any 'pure' vibration, such as that of an ideal tuning fork or mass-spring system. — Click for https://ccrma.stanford.edu/~jos/mdft/Sinusoids.html

The amplitude envelope, or relatively slowly-changing outline of a sound waveform, makes for a useful first approximation of the instantaneous loudness of the sound. — Click for http://en.wikipedia.org/wiki/ADSR_envelope

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

A sinusoidal model for sound approximates each tonal component of the sound as a sum of slowly varying sinusoids. For tonal sounds such as from vibrating strings or wind instruments (including voiced speech), a sinusoidal model can provide a compact, high-fidelity representation. In addition to providing an intuitive, malleable representation for sound, sinusoidal models are also used in advanced audio compression. — Click for https://ccrma.stanford.edu/~jos/sasp/Sinusoidal_Modeling_Sound.html

Phase Vocoder Sinusoidal Modeling

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

In analysis for additive synthesis, we convert a time-domain signal into a collection of amplitude envelopes 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).

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

**Figure G.9:** Illustration of channel vocoder parameters in analysis (left) and synthesis (right).
$\includegraphics[width=\twidth]{eps/pvchan}$

Phase Vocoder Sinusoidal Modeling

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1. Copyright © 2022-02-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University