Discrete Signal Windowing
Depending on the window function, you may be able to use a DFT-even version of the window function. &quot;DFT-even&quot; means that the periodic extension of the window function is symmetrical. In MATLAB and Octave you can get such a window like this (the first line in the source code):
a = hanning(10, &quot;periodic&quot;);
b = fftshift(a);
c = a + b;
plot(a, &quot;x&quot;, b, &quot;o&quot;, c, &quot;+&quot;);

Figure 1. DFT-even Hann window (blue x), shifted DFT-even Hann window (red o) and their sum (orange +).
The window function and the shifted window function have in some sense a 50 % overlap, and they sum to a constant, which I think was the desired quality that the 50 % overlap was though to facilitate.
Not a problem in the above, but if the window function is such that it does not end at a zero value, and the start and the end values are not equal, then one possibility would be to move one half of the value of the last sample to just before the first sample.