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Downsampling with Anti-Aliasing

Figure 9.19: Processing in one filter-bank analysis channel.
\includegraphics[width=3in]{eps/FBSonechan}

In OLA, the hop size $ R$ is governed by the COLA constraint

$\displaystyle \sum_{m=-\infty}^\infty w(n+mR) = \hbox{constant}$ (10.26)

In FBS, $ R$ is the downsampling factor in each of the filter-bank channels, and thus the window $ w$ serves as the anti-aliasing filter (see Fig.9.19). We see that to avoid aliasing, $ W(\omega)$ must be bandlimited to $ (-\pi/R, \pi/R)$ , as illustrated schematically in Fig.9.20.

Figure 9.20: Schematic illustration of a window transform that suppresses all aliasing.
\includegraphics[width=3in]{eps/IdealWindowFD}



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``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
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