PSF and Weighted Overlap Add
Using ``square-root windows'' 
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),
![$\displaystyle \zbox {\underbrace{\sum_m w(n-mR)}_{\hbox{\sc Alias}_R(w)} = \underbrace{\frac{1}{R}\sum_{k=0}^{R-1} W(\omega_k)e^{j\omega_k n}}_{\hbox{\sc DFT}_R^{-1} \left[\hbox{\sc Sample}_{\frac{2\pi}{R}}(W)\right]}} \quad \omega_k \isdef \frac{2\pi k}{R} \protect$](http://www.dsprelated.com/josimages_new/sasp2/img1466.png) |
(9.39) |
For WOLA, this is easily modified to become
![$\displaystyle \zbox {\underbrace{\sum_m w(n-mR)f(n-mR)}_{\hbox{\sc Alias}_R(w\cdot f)} = \underbrace{\frac{1}{R}\sum_{k=0}^{R-1} (W\ast F)(\omega_k)e^{j\omega_k n}}_{\hbox{\sc DFT}_R^{-1} \left[\hbox{\sc Sample}_{\frac{2\pi}{R}}(W\ast F)\right]}} \quad \omega_k \isdef \frac{2\pi k}{R}$](http://www.dsprelated.com/josimages_new/sasp2/img1482.png) |
(9.40) |
where
is the analysis window and 
is the synthesis window.
When 
, this becomes
![$\displaystyle \underbrace{\sum_m w^2(n-mR)}_{\hbox{\sc Alias}_R(w^2)} = \underbrace{\frac{1}{R}\sum_{k=0}^{R-1} (W\ast W)(\omega_k)e^{j\omega_k n}}_{\hbox{\sc DFT}_R^{-1} \left[\hbox{\sc Sample}_{\frac{2\pi}{R}}(W\ast W)\right]}, \quad \omega_k \isdef \frac{2\pi k}{R}$](http://www.dsprelated.com/josimages_new/sasp2/img1485.png) |
(9.41) |
Next Section:
Example COLA Windows for WOLA
Previous Section:
PSF Dual and Graphical Equalizers
