### Frequency-Domain COLA Constraints

Recall that for error-free OLA processing, we required the constant-overlap-add (COLA) window constraint:

(9.31) |

Thanks to the PSF, we may now express the COLA constraint in the frequency domain:

(9.32) |

In other terms,

**Notation:**

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.

#### Strong COLA

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

(9.34) |

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

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PSF Dual and Graphical Equalizers

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Poisson Summation Formula