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Choice of WOLA Window

The synthesis (output) window in weighted overlap-add is typically chosen to be the same as the analysis (input) window, in which case the COLA constraint becomes

$\displaystyle \zbox {\sum_m w^2(n-mR) = \hbox{constant}\,\forall n\in{\bf Z}.}$ (9.45)

We can say that $ R$ -shifts of the window $ w$ in the time domain are power complementary, whereas for OLA they were amplitude complementary.

A trivial way to construct useful windows for WOLA is to take the square root of any good OLA window. This works for all non-negative OLA windows (which covers essentially all windows in Chapter 3 other than Portnoff windows). For example, the ``root-Hann window'' can be defined for odd $ M$ by

w(n) &=& w_R(n) \sqrt{\frac{1}{2} + \frac{1}{2} \cos( 2\pi n/M) } \\
&=& w_R(n) \cos(\pi n/M),
\; n= -\frac{M-1}{2},\ldots,\frac{M-1}{2}

Notice that the root-Hann window is the same thing as the ``MLT Sine Window'' described in §3.2.6. We can similarly define the ``root-Hamming'', ``root-Blackman'', and so on, all of which give perfect reconstruction in the weighted overlap-add context.

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System Diagram of the Running-Sum Filter Bank
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Strong COLA