Reversible implementation of a disrete linear transformation

Discrete linear transformations form important steps in processing information. Many such transformations are injective and therefore are prime candidates for a physically reversible implementation into hardware. We present here the first steps towards a reversible digital implementation of two different integer transformations on four inputs: The Haar wavelet and the H.264 transform

Fast Digital Convolutions using Bit-Shifts

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.Comment: 4 pages, 2 figures, submitted to IEEE Signal Processing Letter

A discrete Fourier transform for virtual memory machines

An algebraic theory of the Discrete Fourier Transform is developed in great detail. Examination of the details of the theory leads to a computationally efficient fast Fourier transform for the use on computers with virtual memory. Such an algorithm is of great use on modern desktop machines. A FORTRAN coded version of the algorithm is given for the case when the sequence of numbers to be transformed is a power of two

Faster truncated integer multiplication

We present new algorithms for computing the low n bits or the high n bits of the product of two n-bit integers. We show that these problems may be solved in asymptotically 75% of the time required to compute the full 2n-bit product, assuming that the underlying integer multiplication algorithm relies on computing cyclic convolutions of real sequences.Comment: 28 page