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),

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

For WOLA, this is easily modified to become

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

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

When $w=f$ , this becomes

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