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Discrete Convolutions via Mersenne Transrorms
December 1972 (vol. 21 no. 12)
pp. 1269-1273
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.
Index Terms:
Convolution, fast Fourier transforms, Fermat numbers, Mersenne numbers, number theoretic transform, transforms.
Citation:
C.M. Rader, "Discrete Convolutions via Mersenne Transrorms," IEEE Transactions on Computers, vol. 21, no. 12, pp. 1269-1273, Dec. 1972, doi:10.1109/T-C.1972.223497
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