Prime factor DFT algorithms for new small-N DFT modules

In the paper the construction of nesting and row-column prime factor discrete Fourier transform (DFT) algorithms using recently developed small-N DFT modules is investigated. For nesting algorithms it is shown how to construct algorithms requiring theoretically less than 2N real-valued multipliers, hence attaining probably the theoretical lower limit on the number of multiplications. If practical polynomial product algorithms are used, the DFT algorithms require less than 0(N log N) multiplications, but more than 0(N log N) additions. It is shown, however, that for N ¿ 216 the total numbers of arithmetical operations for these algorithms are similar to those for the best common factor fast Fourier transforms. The structures of nesting algorithms are composed of a greater number of stages than row-column ones, hence their realisations often require a greater number of nonarithmetical operations. The paper also contains some remarks on the construction of row-column prime factor DFT algorithms requiring fewer arithmetical operations than the best FFTs.