Elliptic discrete Fourier transforms of type II

• Authors: Artyom M. Grigoryan
• Venue: SMC'09 Proceedings of the 2009 IEEE international conference on Systems, Man and Cybernetics
• Year: 2009

This paper presents a novel concept of the N-point elliptic DFT of type II (EDFT-II), by considering and generalizing the N-point DFT in the real space R2N. In the definition of such Fourier transformation, the block-wise representation of the matrix of the DFT is reserved and the Givens transformations for multiplication by the twiddle coefficients are substituted by other basic transformations. The elliptic transformations are defined by different Nth roots of the identity matrix 2 × 2, whose groups of motion move the point (1, 0) around ellipses. The elliptic DFTs of type II are parameterized by two vector-parameters, exist for any order N, and differ from the class of elliptic DFT of type I whose basic transformations are defined by the elliptic matrix cos(Φ)I + sin(Φ)R, where R is such a matrix that R2 = -I and I is the identity matrix 2 × 2. Examples of application of the proposed N-block EDFT-II in signal and image processing are given.