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Spectral Audio Signal Processing
    The Filter Bank Summation (FBS) Interpretation of the Short Time Fourier Transform (STFT)
       Downsampled STFT Filter Banks
          Downsampled STFT Filter Bank

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Downsampled STFT Filter Bank

So far we have considered only (the ``sliding'' DFT) in our filter-bank interpretation of the STFT. For we obtain a downsampled version of $X_m(\omega_k)$ :

$\begin{eqnarray*} X_{mR}(\omega_k) &=& \sum_{n=-\infty}^\infty [x(n)e^{-j\omega_... ...Delta}{=}}\hbox{\sc Flip}(w)) \\ &=& (x_k \ast {\tilde w})(mR) \end{eqnarray*}$

Let us define the downsampled time index as $\tilde{m} \mathrel{\stackrel{\Delta}{=}}mR$ so that

$\displaystyle X_{\tilde{m}}(\omega_k) = \sum_{n=-\infty}^\infty [x(n)e^{-j\omeg... ...}-n) \mathrel{\stackrel{\Delta}{=}}\left(x_k \ast {\tilde w}\right)(\tilde{m})$

i.e., $X_{\tilde{m}}$ is simply evaluated at every $R^{th}$ sample, as shown in Fig.9.17.

$\begin{psfrags} % latex2html id marker 26131\psfrag{w}{{\Large $\protect\hbox{... ...]{eps/fbs2} \caption{Downsampled STFT filter bank.} \end{figure} \end{psfrags}$

Note that this can be considered an implementation of a phase vocoder filter bank [195].

Subsections

Filter Bank Reconstruction

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written by 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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