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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

The analysis filter $ h_k$ is typically complex bandpass (as in the STFT case). The integers $ R_k$ give the downsampling factor for the output of the $ k$ th channel filter: For critical sampling without aliasing, we set $ R_k= \pi/\hbox{Width}(H_k)$ . The impulse response of synthesis filter $ f_k$ can be regarded as the $ k$ 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