Generalized STFT

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Spectral Audio Signal Processing

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The impulse response of a system is its output signal in response to the impulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

Click for https://ccrma.stanford.edu/~jos/mdft/Alias_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

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

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

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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Spectral Audio Signal Processing

Geometric Signal Theory

Dyadic Filter Bank Design

Generalized STFT

A generalized STFT may be defined by [287]

$\displaystyle x_k(n)$	$\displaystyle =$	$\displaystyle (x\ast h_k)(nR_k) \eqsp \sum_{m=-\infty}^{\infty}x(m) \underbrace{h_k(nR_k - m)}_{\hbox{analysis filter}}$
$\displaystyle x(n)$	$\displaystyle =$	$\displaystyle \sum_k (x_k\ast f_k)(n) \eqsp \sum_{k=0}^{N-1}\sum_{m=-\infty}^{\infty}x_k(m) \underbrace{f_k(n-m R_k)}_{\hbox{synthesis filter}}$

This filter bank and its reconstruction are diagrammed in Fig.11.35.

**Figure 11.35:** Generalized STFT
$\includegraphics[width=\twidth]{eps/GenSTFT}$

The analysis filter is typically complex bandpass (as in the STFT case). The integers give the downsampling factor for the output of the th channel filter: For critical sampling without aliasing, we set $R_k= \pi/\hbox{Width}(H_k)$ . The impulse response of synthesis filter can be regarded as the th basis signal in the reconstruction. If the $\{f_k\}$ are orthonormal, then we have $f_k(n) = h_k^\ast(-n)$ . More generally, $\{h_k,f_k\}$ form a biorthogonal basis.

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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

Generalized STFT

``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