Fast Digital Convolutions using Bit-Shifts

Details Fast Digital Convolutions using Bit-Shifts Fast Digital Convolutions using Bit-Shifts

2010-05-10

arxiv.org/abs/1005.1497v1

Computer Science

Numerical Analysis

4 pages, 2 figures, submitted to IEEE Signal Processing Letters

Scientific paper

An exact, one-to-one transform is presented that not only allows digital circular convolutions, but is free from multiplications and quantisation errors for transform lengths of arbitrary powers of two. The transform is analogous to the Discrete Fourier Transform, with the canonical harmonics replaced by a set of cyclic integers computed using only bit-shifts and additions modulo a prime number. The prime number may be selected to occupy contemporary word sizes or to be very large for cryptographic or data hiding applications. The transform is an extension of the Rader Transforms via Carmichael's Theorem. These properties allow for exact convolutions that are impervious to numerical overflow and to utilise Fast Fourier Transform algorithms.

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