Strong COLA

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

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

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

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