Abstract :
A transform analogous to the discrete Fourier transform is defined in the ring of integers with a multiplication and addition modulo a Mersenne number. The arithmetic necessary to perform the transform requires only additions and circular shifts of the bits in a word. The inverse transform is similar. It is shown that the product of the transforms of two sequences is congruent to the transform of their circular convolution. Therefore, a method of computing circular convolutions without quantization error and with only very few multiplications is revealed.
Keywords :
Convolution, fast Fourier transforms, Fermat numbers, Mersenne numbers, number theoretic transform, transforms.; Algebra; Arithmetic; Discrete Fourier transforms; Discrete transforms; Equations; Fast Fourier transforms; Fourier transforms; Modules (abstract algebra); Page description languages; Quantization; Convolution, fast Fourier transforms, Fermat numbers, Mersenne numbers, number theoretic transform, transforms.;
