Optimality property of the Gaussian window spectrogram

It is shown that for any signal x(t) the minimum of over ¿-88¿-88 [(t-tx)2 + (f-fx)2] Sx(w)(t, f) dt df over all normalized time-windows w(t) is achieved by the Gaussian window w(t) = 21/4 exp (-pt2). Here (tx, fx) is the center of gravity of the signal x(t), Sx(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 Sx(w) (t, f) around (tx, fX) in the time-frequency plane