On the effects of windowing on the discretization of the fractional Fourier transform

The eigenvalue degeneracy problem inherent in the discrete Fourier transform (DFT) matrix operator and the development of a full basis of orthogonal eigenvectors have been addressed via a commuting matrix, devoid of the aforementioned eigenvalue degeneracy problem, that also serves as a discrete version of the Gauss-Hermite (G-H) differential operator. This G-H operator is however, is not bandlimited, and existing discretization efforts run into distortion problems that manifest as deviation from the ideal linear eigenvalue spectrum, aliasing in the eigenvectors, and as a non-invertible peak to parameter mapping associated with the discretization restricting its ability to uniquely represent multicomponent chirp signals. Existing approaches do not account for the effects of windowing on discretization. In this paper, we focus on distortion issues associated with the discretization of the G-H operator and their sources. We specifically analyze the discrete version of the G-H operator based on quantum mechanics in finite dimensions (QMFD), by computing its underlying peak to parameter mapping and its invertibility to subsequently present a representation of the operator with improved mapping invertibility via use of suitable windowing of the eigenvalue spectrum.