Fast convolution using generalized Fermat/Mersenne number transforms

Abstract

Applications of fast convolution using Fermat and Mersenne number transforms to digital filtering are greatly limited by the short transform sequence-lengths of these transforms. The generalized modulo numbers M generated by the following equation: M=p/sup q/t/+or-(p-1), which include the Fermat and Mersenne numbers, are proposed: p, q, and r, are integers and p is always prime. By using the generalized modulus numbers in computing fast convolution, the transform sequence lengths can be much greater than those obtained by either Fermat numbers or Mersenne numbers. The removal of the sequence length constraint by using the generalized modulus numbers and m-valued logic arithmetic implementation makes the fast convolution more practically useful. >