PSF and Weighted Overlap Add

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PSF and Weighted Overlap Add

Using ``square-root windows'' $\sqrt{w}$ in the WOLA context, the valid hop sizes are identical to those for in the OLA case. More generally, given any window for use in a WOLA system, it is of interest to determine the hop sizes which yield perfect reconstruction.

Recall that, by the Poisson Summation Formula (PSF),

$\displaystyle \zbox {\underbrace{\sum_m w(n-mR)}_{\hbox{\sc Alias}_R(w)} = \und... ...e}_{\frac{2\pi}{R}}(W)\right]}} \quad \omega_k \isdef \frac{2\pi k}{R} \protect$

For WOLA, this is easily modified to become

$\displaystyle \zbox {\underbrace{\sum_m w(n-mR)f(n-mR)}_{\hbox{\sc Alias}_R(w\c... ...le}_{\frac{2\pi}{R}}(W\ast F)\right]}} \quad \omega_k \isdef \frac{2\pi k}{R}$

where is the analysis window and is the synthesis window.

When , this becomes

$\displaystyle \underbrace{\sum_m w^2(n-mR)}_{\hbox{\sc Alias}_R(w^2)} = \under... ...ple}_{\frac{2\pi}{R}}(W\ast W)\right]}, \quad \omega_k \isdef \frac{2\pi k}{R}$

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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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Chapters

Spectral Audio Signal Processing
    Overlap-Add (OLA) STFT Processing
       Dual of Constant Overlap-Add
          PSF and Weighted Overlap Add