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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Multirate Filter Banks
       Wavelet Filter Banks
          Geometric Signal Theory
             Generalized STFT

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Generalized STFT

A generalized STFT may be defined by [266]

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

This filter bank and its reconstruction are diagrammed in Fig.10.36.

Figure 10.36: Generalized STFT
\includegraphics[width=\twidth]{eps/GenSTFT}

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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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