Frequency-Domain COLA Constraints

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

$$\sum_m w(n-mR) = \hbox{constant}$$ (9.31)

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

$$\zbox {w\in\hbox{\sc Cola}(R) \;\Leftrightarrow\; W(\omega_k) = 0, \quad \vert k\vert = 1,2, \dots, R-1}$$ (9.32)

In other terms,

$\parbox{0.8\textwidth}{A window $w$\ gives constant overlap-add at hop-size $R$\ if and only if the window transform $W$\ is \emph{zero at all harmonics of the frame rate} $2\pi/R$.}$

Notation:

$$\zbox {w \in \hbox{\sc Cola}(R) \quad \Leftrightarrow \quad W \in \hbox{\sc Nyquist}(2\pi/R)} \protect$$ (9.33)

The ``Nyquist($\Omega_R$ )'' property for a function $W$ simply means that $W$ is zero at all nonzero multiples of $\Omega_R$ (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.