Phase Continuation

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Spectral Audio Signal Processing

Time Scale Modification

TSM by Resampling STFTs Across Time

TSM Examples

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

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

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

Reverberation is the sound created by many echoes blending together, such as you hear when you clap your hands together in an acoustically 'live' room. — Click for https://ccrma.stanford.edu/~jos/pasp/Artificial_Reverberation.html

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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 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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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

Time scaling refers to modifying the time-duration of a signal without modifying its audio spectral characteristics. It is often used to slow down music without changing the pitch of any of the instruments. It is not possible to do this exactly in general, but good approximate methods exist using, e.g., the phase vocoder or Synchronous OverLap-Add (SOLA) algorithms. — Click for https://ccrma.stanford.edu/~jos/sasp/Time_Scale_Modification.html

The Short Time Fourier Transform (STFT) computes the spectrum (DFT) of successive time frames of a signal. — Click for https://ccrma.stanford.edu/~jos/sasp/Short_Time_Fourier_Transform.html

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Time Scale Modification

TSM by Resampling STFTs Across Time

TSM Examples

Phase Continuation

When interpolating the STFT across time for TSM, it is straightforward to interpolate spectral magnitude, as we saw above. Interpolating spectral phase, on the other hand, is tricky, because there's no exact way to do it [220]. There are two conflicting desiderata when deciding how to continue phase from one frame to the next:

(1): Sinusoids should ``pick up where they left off'' in the previous frame.
(2): The relative phase from bin to bin should be preserved in each FFT.

To satisfy condition (1), it is necessary to replace the original phase of each frame by the phase corresponding to smooth continuation across time from the previous frame (which is generally an interpolated frame) for each FFT bin. Altering the phase of a spectral frame changes its amplitude envelope in the time domain. Thus, it may no longer looks like a windowed signal segment. Using the WOLA framework (§8.6) helps because the post-window guarantees a smooth cross-fade from frame to frame. Random amplitude-modulation distortion is generally heard as reverberation, also called phasiness [140].

When condition (2) is violated, the signal frame suffers dispersion in the time domain. For steady-state signals (filtered noise and/or steady tones), temporal dispersion should not be audible, while frames containing distinct pulses will generally become more ``smeared out'' in time.

It is not possible in general to satisfy both conditions (1) and (2) simultaneously, but either can be satisfied at the expense of the other. Generally speaking, ``transient frames'' should emphasize condition (2), allowing the WOLA overlap-add cross-fade to take care of the phase discontinuity at the frame boundaries. For stationary segments, phase continuation, preserving condition (1), is more valuable.

It is often sufficient to preserve relative phase across FFT bins (i.e., satisfy condition (2)) only along spectral peaks and their immediate vicinity [142,143,141,138,215,238].

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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

Phase Continuation

``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