Fractionalisation of an odd time odd frequency DFT matrix based on the eigenvectors of a novel nearly tridiagonal commuting matrix

The discrete equivalent of Hermite-Gaussian functions (HGFs) plays a critical role in the definition of a discrete fractional Fourier transform (DFRFT). The discrete equivalents are typically calculated through the eigendecomposition of a commutator matrix. In this study, the authors mainly focus on the fractionalisation of an odd time odd frequency discrete Fourier transform (O-ODFT) matrix. First, the authors propose a novel nearly tridiagonal matrix, which commutes with the O-ODFT matrix. It does not have multiple eigenvalues. The authors can determine a unique orthonormal eigenvector set based on block diagonalisation of a new commuting matrix. The eigenvectors of the new nearly tridiagonal matrix are shown to be O-ODFT eigenvectors, which are similar to the continuous HGFs. Then, the result of the eigendecomposition of the transform matrix is used in order to define the fractionalisation of O-ODFT (O-ODFRFT). The definition is exactly unitary, index additive and reduces to the O-ODFT for unit order. Finally, numerical examples are illustrated to demonstrate that the proposed O-ODFRFT is approximated to the continuous fractional Fourier transform.