Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


Downsampled STFT Filter Bank

So far we have considered only $ R=1$ (the ``sliding'' DFT) in our filter-bank interpretation of the STFT. For $ R>1$ 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_kn}]\tilde{w}(mR-n)
\hspace{1.2cm} (\tilde{w} \mathrel{\stackrel{\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

$\displaystyle X_{\tilde{m}}(\omega_k) = \sum_{n=-\infty}^\infty [x(n)e^{-j\omega_kn}]\tilde{w}(\tilde{m}-n) \mathrel{\stackrel{\Delta}{=}}\left(x_k \ast {\tilde w}\right)(\tilde{m})$ (10.25)

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


\begin{psfrags}
% latex2html id marker 25754\psfrag{w}{{\Large $\protect\hbox{\sc Flip}(w)$\ }}\psfrag{x(n)}{\Large $x(n)$\ }\psfrag{Xm}{\Large $X_m$\ }\psfrag{Xmt}{\Large $X_{\tilde{m}}$\ }\psfrag{X0}{\Large $X_{\tilde{m}}(\omega_0)$\ }\psfrag{X1}{\Large $X_{\tilde{m}}(\omega_1)$\ }\psfrag{XNm1}{\Large $X_{\tilde{m}}(\omega_{N-1})$\ }\psfrag{ejw0}{\Large $e^{-j\omega_0n}$\ }\psfrag{ejw1}{\Large $e^{-j\omega_1n}$\ }\psfrag{ejwNm1}{\Large $e^{-j\omega_{N-1}n}$\ }\psfrag{dR}{\Large $\downarrow R$\ }\begin{figure}[htbp]
\includegraphics[width=\twidth]{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 [212]. (See §G.5 for an introduction to the vocoder.)



Subsections
Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA