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
![]() |
(9.45) |
We can say that
![$ R$](http://www.dsprelated.com/josimages_new/sasp2/img1170.png)
![$ w$](http://www.dsprelated.com/josimages_new/sasp2/img580.png)
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
by
![\begin{eqnarray*}
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}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img1534.png)
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