3 The cross-Wigner transform

Maurice A. de Gosson

Zhichao Zhang, Liang He, Dong Li +2 more

The metaplectic operator has shown to be a valid technique for generalizing the notion of cross Wigner distribution to achieve time-frequency superresolution. Inspired by the latest work (Cordero and Rodino, 2022), we revisit the notion of cross Wigner distribution in metaplectic transform domains by introducing two$2N\times 2N$symplectic matrices rather than integrating them into one$4N\times 4N$symplectic matrix. We style the derived general formulation as the cross metaplectic Wigner distribution and obtain its basic properties including time translation property, frequency modulation property, time translation and frequency modulation property, Moyal formula, complex conjugate symmetry, time reversal symmetry, and scaling property. We use it to define the so-called short-time metaplectic transform and clarify the equivalence between them. We establish the standard Heisenberg’s uncertainty principles for the cross metaplectic Wigner distribution, i.e., an attainable lower bound for two real-valued functions in cross metaplectic Wigner distribution domains and a sequence of attainable (unattainable) lower bounds for two complex-valued (one real-valued and the other complex-valued) functions in orthogonal, orthonormal, the minimum eigenvalue commutative and the maximum eigenvalue commutative cross metaplectic Wigner distribution domains. We further demonstrate the time-frequency superresolution superiority of the derived results over the conventional one through theoretical analyses and numerical experiments.

Gastón García‐Calderón, M. Moshińsky

Shows how for classical canonical transformations the authors can pass, with the help of Wigner distribution functions, from their representation U in the configurational Hilbert space to a kernel K in phase space. The latter is a much more transparent way of looking at representations of canonical transformations, as the classical limit is reached when h(cross) to 0 and the successive quantum corrections are related with the power of h(cross) 2n , n=1,2, et seq.

Maurice A. de Gosson