WOLA Processing Steps

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

Extract the th windowed frame of data , $n=mR,\ldots,mR+N-1$ (assuming a length $M\leq N$ causal window and hop size ).
Take an FFT of the th frame translated to time zero,
${\tilde x}_m(n)=x_m(n+mR)$ , to produce the th spectral frame
${\tilde X}_m(\omega_k)$ , $k=0,\ldots,N-1$ .
Process ${\tilde X}_m(\omega_k)$ as desired to produce ${\tilde Y}_m(\omega_k)$ .
Inverse FFT ${\tilde Y}_m$ to produce ${\tilde y}_m(n)$ , $n=0,\ldots,N-1$ .
Apply a synthesis window to ${\tilde y}_m(n)$ to yield a weighted output frame ${\tilde y}^f_m(n) = {\tilde y}_m(n)f(n)$ , $n=0,\ldots,N-1$ .
Translate the th output frame to time as $y^f_m(n) = {\tilde y}^f_m(n-mR)$ and add to the accumulated output signal .

(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

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

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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