Two Classes of Elliptic Discrete Fourier Transforms: Properties and Examples

Abstract

This paper analyzes the block structure of the matrix of the N-point discrete Fourier transform (DFT) in the real space R 2N . Each block of this matrix corresponds to the Givens transformation, or elementary rotation describing the multiplications by twiddle coefficients. Such rotations around the circle can be substituted by other kinds of rotations, for instance rotations around ellipses, while reserving the block-wise representation of the matrix and main properties of the DFT. To show that, we present two classes of the elliptic discrete Fourier transforms (EDFT), that are defined by different types of the Nth roots of the identity matrix 2×2, whose groups of motion move points around different ellipses. These two classes (the N-block EDFT of types I and II) are parameterized and exist for any order N. Properties and examples of application of the proposed elliptic EDFTs in signal and image processing are given.