Hermite-Gaussian-like eigenvectors of the DFT matrix generated by the eigenanalysis of an almost tridiagonal matrix

The development of the discrete fractional Fourier transform (DFRFT) necessitates having orthonormal eigenvectors for the DFT matrix, F. The objective of having the DFRFT approximate its continuous counterpart can be met if the eigenvectors of F approximate samples of the Hermite-Gaussian functions. Orthonormal Hermite-Gaussian-like eigenvectors for F are rigorously derived by a detailed analysis of an almost tridiagonal matrix, S, which commutes with F. By an appropriate similarity transformation, S is reduced to a 2×2 block diagonal form and the elements of the two exactly tridiagonal matrices forming the two diagonal blocks are explicitly derived in terms of the elements of matrix S.