Optimality property of the Gaussian window spectrogram

It is shown that for any signal x(t) the minimum of integral /sub - infinity //sup infinity / integral /sub - infinity //sup infinity / ((t-t/sub x/)/sup 2/+(f-f/sub x/)/sup 2/) S/sub x//sup (w)/(t, f) dt df over all normalized time-windows w(t) is achieved by the Gaussian window w(t)=2/sup 1/4/ exp (- pi t/sup 2/). Here (t/sub x/, f/sub x/) is the center of gravity of the signal x(t), S/sub x//sup (w)/ (t, f) is the spectrogram of x(t) due to the window w(t), and the double integral is a measure of the spread of S/sub x//sup (w)/ (t, f) around (t/sub x/, f/sub X/) in the time-frequency plane.<
>