Optimality property of the Gaussian window spectrogram

It is shown that for any signal x(t) the minimum of ∫_-∞^∞∫_-∞ ^∞ [(t-t_x)²+(f-f _x)²] S_x^(w)(t , f) dt df over all normalized time-windows w(t) is achieved by the Gaussian window w(t)=2^1/4 exp (-πt²). Here (t_x, f _x) is the center of gravity of the signal x(t ), S_x^(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 _x^(w) (t, f) around (t _x, f_X) in the time-frequency plane