Frequency-Domain COLA Constraints

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

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

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

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

In other terms,
$\textstyle \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:

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

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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