Phase-Continued STFT TSM

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Spectral Audio Signal Processing

TSM Examples

Relative-Phase-Preserving STFT TSM

Spectrum analysis of sound is analogous to decomposing white light into its component colors by means of a prism — Click for https://ccrma.stanford.edu/~jos/mdft/Example_Applications_DFT.html

Click for http://en.wikipedia.org/wiki/Vocoder

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

Click for https://ccrma.stanford.edu/~jos/sasp/Modulation_Complex_Sinusoid.html#24349

Click for http://ccrma.stanford.edu/realsimple/SimpleStrings/Nonlinear_Overdrive.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

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Click for http://www.csounds.com/ezine/summer1999/beginners/

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

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Spectral Audio Signal Processing

TSM Examples

Relative-Phase-Preserving STFT TSM

Phase-Continued STFT TSM

Figure 10.18 shows the phase-continued-frames case in which relative phase is not preserved across FFT bins. As a result, the amplitude envelope is not preserved in the time domain within each frame. Figure 10.19 shows the spectrum of the same case, revealing significant distortion products at multiples of the frame rate due to the intra-frame amplitude-envelope distortion, which then ungracefully transitions to the next frame. Note that modulation sidebands corresponding to multiples of the frame rate are common in nonlinearly processed STFTs.

**Figure 10.18:** Phase-continued vocoder waveforms at 2X expansion.
$\includegraphics[width=\twidth]{eps/pv-ellis-wave}$

**Figure 10.19:** Phase-continued vocoder spectra at 2X expansion.
$\includegraphics[width=\twidth]{eps/pv-ellis-spec}$

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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-Continued STFT TSM

``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