# Windowing function squared?

At page 2 there's a "time-warped STFT's synthesis function" involving $v(n)=w^2(n)$ which is a windowing function. A windowing function squared.

What's the interpretation/meaning of windowing function squared?
What does it mean visually?

As said by @robert bristow-johnson, a squared window is another window. If you take a quite standard type, symmetric, unimodal, positive (with the related question Conditions for symmetric and unimodal windows in both time and frequency domains) and peaking at $1$, like a Hamming window, you can see the following behavior:

The squared one is below the original, due to the contractive effect of $t \to t^2$ in $[0,1]$ (in other words, $t^2\le t$), while keeping the same maximum location. So the window is better localized, and gets improved apodizing at the ends, which can be useful to reduce "blocking" artifacts at the edges of overlapped frames: if the original window ends at $0.1$, its squared version ends at $0.01$, a ten-fold reduction.

The other effect is that the new window is close to energy-normalized. Even if your "window" is zero-mean (like a wavelet), the energy of its square is never zero (unless the window is zero), so you can use it as a weighted average, of the shape (similar to equation 4): $$\frac{\sum a^2(k).x(k)}{\sum a^2(k)}\,.$$

it's just another window. usually this happens when a window is applied on the time-domain data going into the STFT (which is a DFT), some process happens to the frequency-domain data after the STFT, then there is an inverse DFT bringing it back to the time domain, and the window is applied again to that output time-domain data. if that process done to the data in the frequency domain happens to be the null process, then what comes out of the iDFT is exactly what went into the DFT. so then, in that case, the window is applied twice and that is the same as applying the square of the window once.

i have been having a little argument with someone on comp.dsp about this. there is a term i hadn't seen before ("Princen-Bradley condition") which i think means the same as a "complementary-power" window. if the square of a complementary-power window is applied to an STFT system above with a null process, the output will be exactly the input when 50% overlap is used. but if the square of a complementary window (such as the Hann) is used, then, with 50% overlap you do not get perfect reconstruction. so the square-root of a Hann window would be a good candidate for a complementary-power window. and you would have to apply that window twice to the time-domain data to get perfect reconstruction.

• I feel useful insights in your question, yet I cannot get the details from the words only ("some process happens to the frequency-domain data", and the link with the 50% overlap). Could you please please a reference? – Laurent Duval Mar 6 '16 at 10:18

Adding to Laurent's answer. I've plotted the magnitude of the FFTs of the two windows. I've padded the FFT to 1024 and the plot is shown in dB. Here's a regular plot:

Here's a zoomed version:

There effect of squaring the window is more obvious in the frequency domain. There is a loss in the coherent gain because more of the window coefficients are closer to zero. The main lobe widens because multiplication in time corresponds to convolution in frequency domain are this tends to spread the response. The sidelobes are lower and they decay faster - the rate of sidelobe decay is asymptotically proportional to the number of continuous derivatives of the time domain window - See Papoulis - "The Fourier Integral and its Applications".

It means that w(n) was applied both before the STFT and after the ISTFT.

w(n) is both an analysis window and a synthesis window. https://www.dsprelated.com/freebooks/sasp/Weighted_Overlap_Add.html

Typically w(n) will be chosen to be the square root of a COLA window, so that when squared and overlapped, it sums to unity and reproduces the original signal exactly with no ripple: https://www.dsprelated.com/freebooks/sasp/COLA_Examples.html