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

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

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

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.