Downsampling with Anti-Aliasing

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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 is governed by the COLA constraint

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

In FBS, is the downsampling factor in each of the filterbank channels, and thus the window 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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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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Spectral Audio Signal Processing
    The Filter Bank Summation (FBS) Interpretation of the Short Time Fourier Transform (STFT)
       Downsampled STFT Filter Banks
          Downsampling with Anti-Aliasing