Fast and precise computations of discrete Fourier transforms using cyclotomic integers

Abstract

Many applications of fast fourier transforms (FFT's), such as computer- tomography, geophysical signal processing, high resolution imaging radars, and prediction filters, require high precision output. The usual method of fixed point computation of FFT's of vectors of length 2l leads to an average loss of l/2 bits of precision. This phenomenon, often referred to as computational noise, causes major problems for arithmetic units with limited precision which are often used for real time applications. Several researchers have noted that calculation of FFT's with algebraic integers avoids computational noise entirely, see, e.g., [3]. We will show that complex numbers can be approximated accurately by cyclotomic integers, and combine this idea with Chinese remaindering strategies in the cyclotomic integers to, roughly, give a O(b1+? L log (L)) algorithm to compute b-bit precision FFT's of length L. The first part of the paper will describe the FFT strategy, assuming good app..