Generalized Commuting Matrices and Their Eigenvectors for DFTs, Offset DFTs, and Other Periodic Operations
It is well known that some matrices (such as Dickinson-Steiglitz's matrix) can commute with the discrete Fourier transform (DFT) and that one can use them to derive the complete and orthogonal DFT eigenvector set. Recently, Candan found the general form of the DFT commuting matrix. In this paper, we further extend the previous work and find the general form of the commuting matrix for any periodic, quasi-periodic, and offset quasi-periodic operations. Using the general commuting matrix, we can derive the complete and orthogonal eigenvector sets for offset DFTs, DCTs of types 1, 4, 5, and 8, DSTs of types 1, 4, 5, and 8, discrete Hartley transforms of types 1 and 4, the Walsh transform, and the projection operation (the operation that maps a whole vector space into a subspace) successfully. Moreover, several novel ways of finding DFT eigenfunctions are also proposed. Furthermore, we also extend our theories to the continuous case, i.e., if a continuous transform is periodic, quasi-periodic, or offset quasi-periodic (such as the FT and some cyclic operations in optics), we can find the general form of the commuting operation and then find the complete and orthogonal eigenfunctions set for the continuous transform.