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WOLA Processing Steps

The sequence of operations in a WOLA processor can be expressed as follows:

  1. Extract the $ m$ th windowed frame of data $ x_m(n)=x(n)w(n-mR)$ , $ n=mR,\ldots,mR+N-1$ (assuming a length $ M\leq N$ causal window $ w$ and hop size $ R$ ).

  2. Take an FFT of the $ m$ th frame translated to time zero,
    $ {\tilde x}_m(n)=x_m(n+mR)$ , to produce the $ m$ th spectral frame
    $ {\tilde X}_m(\omega_k)$ , $ k=0,\ldots,N-1$ .

  3. Process $ {\tilde X}_m(\omega_k)$ as desired to produce $ {\tilde Y}_m(\omega_k)$ .

  4. Inverse FFT $ {\tilde Y}_m$ to produce $ {\tilde y}_m(n)$ , $ n=0,\ldots,N-1$ .

  5. Apply a synthesis window $ f(n)$ to $ {\tilde y}_m(n)$ to yield a weighted output frame $ {\tilde y}^f_m(n) = {\tilde y}_m(n)f(n)$ , $ n=0,\ldots,N-1$ .

  6. Translate the $ m$ th output frame to time $ mR$ as $ y^f_m(n) =
{\tilde y}^f_m(n-mR)$ and add to the accumulated output signal $ y(n)$ .

(The overlap-add method discussed previously is obtained from the above procedure by deleting step 5.)

To obtain perfect reconstruction in the absence of spectral modifications, we require

\begin{eqnarray*}
x(n) &=& \sum_{m=-\infty}^{\infty} x(n) w(n-mR)f(n-mR) \\
&=& x(n) \sum_{m=-\infty}^{\infty} w(n-mR)f(n-mR),
\end{eqnarray*}

which is true if and only if

$\displaystyle \zbox {\sum_m w(n-mR)f(n-mR) = 1, \,\forall n\in\mathbb{Z}.}$ (9.44)



Subsections
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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA