Strong COLA

An overly strong (but sufficient) condition is to require that the window transform $ W(\omega)$ be bandlimited consistent with downsampling by $ R$ :

$\displaystyle W(\omega) = 0, \quad \vert\omega\vert \geq \pi/R
\quad\quad\hbox{(sufficient for COLA)}
$

This condition is sufficient, but not necessary, for perfect OLA reconstruction. Strong COLA implies weak COLA, but it cannot be achieved exactly by finite-duration window functions.

When either of the strong or weak COLA conditions are met, we have

$\displaystyle \zbox {\sum_m w(n-mR) = \frac{1}{R} W(0)}$ (9.34)

That is, the overlap-add of the window $ w$ at hop-size $ R$ is equal numerically to the dc gain of the window divided by $ R$ .


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Choice of WOLA Window
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Example 2: Time Domain Aliasing