Decimation-in-frequency split-radix algorithm for computing new Mersenne number transform

Abstract

The new Mersenne number transform (NMNT) has proved to be an important number theoretic transform (NTT) for error-free calculation of convolutions and correlations. Its main feature is that for a suitable Mersenne prime number (p), the allowed power-of-two transform lengths can be very large. In this paper, a new decimation-in-frequency split-radix algorithm for efficient computation of the 1-D NMNT is developed by deriving a general radix-based algorithm in finite fields modulo Mersenne numbers and applying the principles of the split-radix algorithm to the NMNT. The proposed algorithm possesses the in-place computation property, leading to substantial reductions in the number of multiplications and additions. The validity of the proposed algorithm is verified through examples involving numerical calculations of the forward and inverse transforms and digital filtering applications, using both the 1-D NMNT and the developed algorithm.