Following Spectral Peaks

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Spectral Audio Signal Processing

Additive Synthesis Analysis

Sinusoidal Peak Finding

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Transients are short intervals during which the signal evolves quickly in some nontrivial or relatively unpredictable way (Bello, 2005). In computer music, the short attack portion of a sound is often treated as a transient signal. — Click for http://en.wikipedia.org/wiki/Transient_%28acoustics%29

To downsample a signal by the factor N means to form a new signal consisting of every Nth sample of the original signal. — Click for https://ccrma.stanford.edu/~jos/mdft/Downsampling_Operator.html

Sampling is the process of converting a continuous-time signal into a discrete-time signal. — Click for http://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

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

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The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.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

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

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

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Spectral Audio Signal Processing

Additive Synthesis Analysis

Sinusoidal Peak Finding

Following Spectral Peaks

In the analysis phase, sinusoidal peaks are measured over time in a sequence of FFTs, and these peaks are grouped into ``tracks'' across time. A detailed discussion of various options for this can be found in [246,174,271,84,248,223,10,146], and a particular case is detailed in Appendix H.

The end result of the analysis pass is a collection of amplitude and frequency envelopes for each spectral peak versus time. If the time advance from one FFT to the next is fixed (5ms is a typical choice for speech analysis), then we obtain uniformly sampled amplitude and frequency trajectories as the result of the analysis. The sampling rate of these amplitude and frequency envelopes is equal to the frame rate of the analysis. (If the time advance between FFTs is $\Delta t=5$ ms, then the frame rate is defined as $1/\Delta t = 200$ Hz.) For resynthesis using inverse FFTs, these data may be used unmodified. For resynthesis using a bank of sinusoidal oscillators, on the other hand, we must somehow interpolate the envelopes to create envelopes at the signal sampling rate (typically kHz or higher).

It is typical in computer music to linearly interpolate the amplitude and frequency trajectories from one frame to the next [271].^11.10 Let's call the piecewise-linear upsampled envelopes ${\hat A}_k(n)$ and $\hat{F}_k(n)$ , defined now for all at the normal signal sampling rate. For steady-state tonal sounds, the phase may be discarded at this stage and redefined as the integral of the instantaneous frequency when needed:

$\displaystyle \hat{\Theta }_k(n) \isdefs \hat{\Theta }_k(n-1) + 2\pi T \hat{F}_k(n). \protect$

(11.19)

When phase must be matched in a given frame, such as when it is known to contain a transient event, the frequency can instead move quadratically across the frame to provide cubic phase interpolation [174], or a second linear breakpoint can be introduced somewhere in the frame for the frequency trajectory (in which case the area under the triangle formed by the second breakpoint equals the added phase at the end of the segment).

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``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

Following Spectral Peaks

``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