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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} $2\pi/R$.}$


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

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University