OPTIMUM SMOOTHING OF THE WIGNER-VILLE DISTRIBUTION

A compromise is found between the different requirements to be fulfilled by a time frequency distribution, namely, positivity and obtention of a distribution close to the Dirac one for the unimodular signal s(t) equals exp PHI (t). Starting from the usual Wigner-Ville distribution, optimum smoothing is defined by minimizing the width of the different functions approximating the desired Dirac distribution. The smoothing is obtained by a convolution through a double Gaussian of width sigma //t and sigma // omega such that sigma //t sigma // omega equals 1/2. Two possibilities appear. These results, to be physically meaningful, impose inequalities on the successive derivatives of PHI which are equivalent to those used for obtaining the classical limit for the corresponding quantum problem.