Cross-ambiguity and Wigner Functions

Élliott H. Lieb

David K. Ferry, Xavier Oriols, Josef Weinbub

Mykola Yaremenko

The connection between cross-ambiguity and cross-Wigner functions is studied. The correspondence between the class of symbols and the Weyl operators is established; we obtain the necessary and sufficient conditions on the symbol under which the Weyl operator belongs to the trace class. The theory of Kaster-Loupias-Miracle is developed and the exact conditions on the Weyl operator to be positively defined in the Hilbert space is established.

David K. Ferry, Mihail Nedjalkov

M. Horibe, Akiyoshi Takami, T. Hashimoto +1 more

For the Wigner function of a system in N-dimensional Hilbert space, we propose the condition, which ensures that the Wigner function has correct marginal distributions along tilted lines. Under this condition we get the Wigner function without ambiguity if N is odd. If N is even, the Wigner function does not exist.

Martin J. Bastiaans

The description of a signal by means of a local frequency spec-trum resembles such things as the score in music, the phase space in mechanics, and the ray concept in geometrical optics. Two types of local frequency spectra are presented: the Wigner distribution function and the sliding-window spectrum, the latter having the form of a cross-ambiguity function. The Wigner distribution function in particular can provide a link between Fourier optics and geometrical optics; many properties of the Wigner distribution function, and the way in which it prop-agates through linear systems, can be interpreted in geometric-optical terms. The Wigner distribution function is linearly related to other signal representations like Woodward's ambiguity function, Rihaczek's complex energy density function, and Mark's physical spectrum. An advantage of the Wigner distribution function and its related signal representations is that they can be applied not only to deterministic signals but to stochastic signals as well, leading to such things as Walther's generalized radiance and Sudarshan's Wolf tensor. On the other hand, the sliding-window spectrum has the advantage that a sampling theorem can be formulated for it: the sliding-window spectrum is completely determined by its values at the points of a certain space-frequency lattice, which is exactly the lattice suggested by Gabor in 1946. The sliding-window spectrum thus leads naturally to Gabor's expansion of a signal into a discrete set of properly shifted and modulated versions of an elementary signal, which is again another space-frequency signal representation, and which is related to the degrees of freedom of the signal.