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
We can say that -shifts of the window 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 by
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.
System Diagram of the Running-Sum Filter Bank