Thanks to the PSF, we may now express the COLA constraint in the frequency domain:
In other terms,
The ``Nyquist( )'' property for a function simply means that is zero at all nonzero multiples of (all harmonics of the frame rate here).
We may also refer to (8.33) as the ``weak COLA constraint'' in the frequency domain. It gives necessary and sufficient conditions for perfect reconstruction in overlap-add FFT processors. However, when the short-time spectrum is being modified, these conditions no longer apply, and a stronger COLA constraint is preferable.
An overly strong (but sufficient) condition is to require that the window transform be bandlimited consistent with downsampling by :
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
That is, the overlap-add of the window at hop-size is equal numerically to the dc gain of the window divided by .
PSF Dual and Graphical Equalizers
Poisson Summation Formula