Prime number DFT computation via parallel circular convolvers

Abstract

A great deal of work has been carried out in recent years into the construction of computationally efficient small discrete Fourier transform (DFT) algorithms. Most small-DFT algorithms exploit the equivalence of prime number DFT computation with that of circular convolution, as well as Winograd's complexity theory results relating to the optimal computation of small circular convolutions, to achieve reduced-complexity solutions. The paper extends these results to the case of medium/large prime number DFT computation by means of the Agarwal—Cooley technique, whereby a multidimensional index mapping, combined with Winograd's results, converts the associated one-dimensional circular convolution into a multidimensional nested circular convolution. The resulting computational structure is then expressed in the form of an input addition phase, an output addition phase and, in between, a number of independent circular convolutions, which in hardware can be implemented in parallel, via both word-level and bit-level arithmetic techniques, to provide high-throughput solutions to the original prime number DFT computation.