An orthonormal class of exact and simple DFT eigenvectors with a high degree of symmetry

Abstract

The paper presents a novel orthonormal class of eigenvectors of the discrete Fourier transform (DFT) whose order N is factored as N=rM/sup 2/. The DFT eigenvectors have the form e=E/spl alpha/, where /spl alpha/ are eigenvectors of some /spl lscr/ /spl times//spl lscr/ matrices, given by, or related to, the DFT matrix of order r, with /spl lscr/ = r, 2r, or 4r, and the matrix E expands /spl alpha/ to the full DFT size N=rM/sup 2/. In particular, when N is an arbitrarily large power of 2, r may be 1 or 2. The resulting eigenvectors are expressed exactly with simple exponential expressions, have a considerable number of elements constrained to 0, and show a high degree of symmetry. The derivation of such a class is based on a partition of the N-dimensional linear space into subspaces of very small dimension (r, 2r or 4r).