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PSF and Weighted Overlap Add

Using ``square-root windows'' $ \sqrt{w}$ in the WOLA context, the valid hop sizes $ R$ are identical to those for $ w$ in the OLA case. More generally, given any window $ w(n)$ 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),

$\displaystyle \zbox {\underbrace{\sum_m w(n-mR)}_{\hbox{\sc Alias}_R(w)} = \underbrace{\frac{1}{R}\sum_{k=0}^{R-1} W(\omega_k)e^{j\omega_k n}}_{\hbox{\sc DFT}_R^{-1} \left[\hbox{\sc Sample}_{\frac{2\pi}{R}}(W)\right]}} \quad \omega_k \isdef \frac{2\pi k}{R} \protect$ (9.39)

For WOLA, this is easily modified to become

$\displaystyle \zbox {\underbrace{\sum_m w(n-mR)f(n-mR)}_{\hbox{\sc Alias}_R(w\cdot f)} = \underbrace{\frac{1}{R}\sum_{k=0}^{R-1} (W\ast F)(\omega_k)e^{j\omega_k n}}_{\hbox{\sc DFT}_R^{-1} \left[\hbox{\sc Sample}_{\frac{2\pi}{R}}(W\ast F)\right]}} \quad \omega_k \isdef \frac{2\pi k}{R}$ (9.40)

where $ w(n)$ is the analysis window and $ f(n)$ is the synthesis window.

When $ w=f$ , this becomes

$\displaystyle \underbrace{\sum_m w^2(n-mR)}_{\hbox{\sc Alias}_R(w^2)} = \underbrace{\frac{1}{R}\sum_{k=0}^{R-1} (W\ast W)(\omega_k)e^{j\omega_k n}}_{\hbox{\sc DFT}_R^{-1} \left[\hbox{\sc Sample}_{\frac{2\pi}{R}}(W\ast W)\right]}, \quad \omega_k \isdef \frac{2\pi k}{R}$ (9.41)

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2019-07-26 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University