The generalized discrete Fourier transform in rings of algebraic integers

Abstract

Conditions are presented for a transform of the DFT structure, defined in a ring of residues of a ring of algebraic integers, to map cyclic convolution isomorphically into a pointwise product. The conditions are used to verify that a number of potentially useful transforms (which require no general multiplications) satisfy this property. In particular, transforms defined in residue rings of the Gaussian integers, the Eisenstein integers, and a biquadratic domain are studied.