Fractional Fourier transform in information processing, tomography of optical signal, and Green function of harmonic oscillator

A review of the properties of the fractional Fourier transform, which is used in information processing, is presented in connection with the symplectic tomography transform of optical signals. The relationship between the Green function of the quantum harmonic oscillator and the fractional Fourier transform is elucidated. An analysis of electromagnetic signals which uses an invertible map of analytic signal onto the tomographic probability distribution is made. The formal connection of the analysis with the tomography method of measuring quantum states is considered. The relation to other methods of time-frequency quasidistributions (for example, the Ville-Wigner quasidistribution) characterizing a signal is studied.