Issue No. 12 - December (1972 vol. 21)
C.M. Rader , Lincoln Laboratory, Massachusetts Institute of Technology
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.
Convolution, fast Fourier transforms, Fermat numbers, Mersenne numbers, number theoretic transform, transforms.
C. Rader, "Discrete Convolutions via Mersenne Transrorms," in IEEE Transactions on Computers, vol. 21, no. , pp. 1269-1273, 1972.