Novel nearly tridiagonal commuting matrix and fractionalizations of generalized DFT matrix

Based on discrete Hermite-Gaussian-like functions, a discrete fractional Fourier transform (DFRFT), which provides sample approximations of the continuous fractional Fourier transform, was defined and investigated recently. In this paper, we propose a novel nearly tridiagonal matrix, which commutes with the generalized discrete Fourier transform (GDFT) matrix. It doesn´t has repeated eigenvalue. We can determine a unique orthonormal eigenvector set based on block diagonalization of new commuting matrix. The eigenvectors of the new nearly tridiagonal matrix are shown to be GDFT eigenvectors, which are more similar to the continuous Hermite-Gaussian functions than those developed before. The fractional transform of GDFT (GDFRFT) is defined through eigendecomposition. The new version of GDFRFT produce their transform outputs closer to the samples of the continuous fractional Fourier transform through numerical comparison.