Fast and precise computations of discrete Fourier transforms using cyclotomic integers

Abstract

Applications of fast Fourier transforms (FFT) require high precision output. The usual method of fixed point computation of FFT's vectors of length 2l leads to an average loss of l/2 bits of precision. This phenomenon called computational noise is caused by inaccurate computations due to limited dynamic range. The calculation of FFT's with algebraic integers avoids computational noise. Complex numbers can be approximated accurately by cyclotomic integers, and when these integers are combined with Chinese remaindering strategies, the combination will give an algorithm to compute for bit precision of FFT length.