Convolution using a conjugate symmetry property for the generalized discrete Fourier transform

Often, signals which lie in a ring S are convolved using a generalized discrete Fourier transform (DFT) over an extension ring R in order to allow longer sequence lengths. In this paper, a conjugate symmetry property which generalizes the well known property of the complex DFT for real data is presented for this situation. This property is used to obtain a technique for computing the DFT of μ sequences with values in a ring S using a single DFT in an extension ring R of degree μ over S. From this result, a method to compute the convolution of length μn S-sequences using a length n DFT in R is derived. Example of the application to the complex DFT and to a number theoretic transform are presented to illustrate the theory.