Optimum smoothing of the Wigner--Ville distribution

A compromise is found between the different requirements that we would like to be fulfilled by a time frequency distribution, namely, positivity and obtention of a distribution close to the Dirac one for the unimodular signal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s(t) = \exp i\phi (t)</tex> (the fulfillment of the marginal conditions being of less interest in signal theory). Starting from the usual Wigner-Ville distribution, we define an optimum smoothing 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 σ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> and σ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ω</inf> such that σ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</inf> σ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ω</inf> = 1/2. Two possibilities appear: in the first one, we do not introduce any correlation between t and ω in the convolution kernel, and obtain a simple result. In the second one, extrapolating the frequency variation, and still using a Gaussian, we obtain a better result although the smoothing process becomes more complex. These results, to be physically meaningful, impose inequalities on the successive derivatives of φ which are equivalent to those used for the obtention of the classical limit for the corresponding quantum problem.