Integral bounds for radar ambiguity functions and Wigner distributions

H. Lamouchi, Slim Omri

Maurice A. de Gosson

James Gareth Wood

Dries Sels, F. Brosens, Wim Magnus

Élliott H. Lieb

An upper bound is proved for the Lp norm of Woodward’s ambiguity function in radar signal analysis and of the Wigner distribution in quantum mechanics when p>2. A lower bound is proved for 1≤p<2. In addition, a lower bound is proved for the entropy. These bounds set limits to the sharpness of the peaking of the ambiguity function or Wigner distribution. The bounds are best possible and equality is achieved in the Lp bounds if and only if the functions f and g that enter the definition are both Gaussians.

Nuno Costa Dias, João Nuno Prata

Abstract We address a recent conjecture stated by Z. Van Herstraeten and N. J. Cerf. They claim that the Shannon entropy for positive Wigner functions is bounded below by a positive constant, which can be attained only by Gaussian pure states. We introduce an alternative definition of entropy for all absolutely integrable Wigner functions, which is the Shannon entropy for positive Wigner functions. Moreover, we are able to prove, in arbitrary dimension, that this entropy is indeed bounded below by a positive constant, which is not very distant from the constant suggested by Van Herstraeten and Cerf. We also prove an analogous result for another conjecture stated by the same authors for the Rényi entropy of positive Wigner functions. As a by-product we prove a new inequality for the radar-ambiguity function (and for the Wigner distribution) which is reminiscent of Lieb’s inequalities.