Applying the fractional Fourier transform and the Wigner distribution on a signal in a cascade fashion is equivalent with a rotation of the time and frequency parameters of the Wigner distribution. This report presents a formula for all unitary operators that are related to energy preserving transformations on the parameters of the Wigner distribution by means of such a cascade of operators. Furthermore, such operators are used to solve certain type of energy localization problems via the Weyl correspondence.

Applications of group representations to physics (msc 20C35), Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) (msc 33D45), Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type (msc 42A38), Representations of groups, semigroups, etc. (msc 43A65), Signal theory (characterization, reconstruction, filtering, etc.) (msc 94A12)
CWI. Probability, Networks and Algorithms [PNA]

ter Morsche, H.G, & Oonincx, P.J. (1999). Integral representations of affine transformations in phase space with an application to energy localization problems. CWI. Probability, Networks and Algorithms [PNA]. CWI.