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),

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

For WOLA, this is easily modified to become

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

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

When $w=f$ , this becomes

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