Many applications of fast Fourier transforms (FFT's), such as computer tomo
graphy, geophysical signal processing, high-resolution imaging radars, and
prediction filters, require high-precision output. An error analysis reveal
s that the usual method of fixed-point computation of FFT's of vectors of l
ength 2(l) leads to an average loss of l/2 bits of precision. This phenomen
on, often referred to as computational noise, causes major problems for ari
thmetic units with limited precision which are often used for real-time app
lications, Several researchers have noted that calculation of FFT's with al
gebraic integers avoids computational noise entirely, see, e,g,, [1], We wi
ll combine a new algorithm for approximating complex numbers by cyclotomic
integers with Chinese remaindering strategies to give an efficient algorith
m to compute b-bit precision FFT's of length L, More precisely, we will app
roximate complex numbers by cyclotomic integers in Z[e(2 pi i/2 pi)] whose
coefficients, when expressed as polynomials in e(2 pi i/2n), are bounded in
absolute value by some integer M. For fixed n our algorithm runs in time O
(log(M)), and produces an approximation with worst case error of O(1/M2n-2-
1). We will prove that this algorithm has optimal worst case error by provi
ng a corresponding lower bound on the worst case error of any approximation
algorithm for this task. The main tool for designing the algorithms is the
use of the cyclotomic units, a subgroup of finite index in the unit group
of the cyclotomic held,
First implementations of our algorithms indicate that they are fast enough
to be used for the design of low-cost high-speed/high-precision FFT chips.