Some properties The STFT preserves frequency shifts and time shifts up to a modulation: $$\begin{eqnarray*} y(t) = x(t)\ e^{j2\pi \nu_0 t} &\ \Rightarrow\ & F_y(t,\nu;h)=... ...\Rightarrow\ & F_y(t,\nu;h)=F_x(t-t_0,\nu;h)\ e^{j2\pi t_0 \nu} \end{eqnarray*}$$ Generalizing what has been said previously, the signal $x(t)$ can be reconstructed from its STFT with a synthesis window $g(t)$ different from the analysis window $h(t)$: $$x(t)=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} F_x(u,\xi;h)\ g(t-u)\ e^{j2\pi t \xi}\ du\ d\xi$$ providing that the windows $g$ and $h$ validate the constraint $$\int_{-\infty}^{+\infty} g(t)\ h^*(t)\ dt =1.$$
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