  We elucidate why an interval algorithm that computes the exact bounds on the
amplitude and phase of the discrete Fourier transform can run in polynomial
time. We address this question from a formal perspective to provide the
mathematical foundations underpinning such an algorithm. We show that the
procedure set out by the algorithm fully addresses the dependency problem of
interval arithmetic, making it usable in a variety of applications involving
the discrete Fourier transform. For example when analysing signals with poor
precision, signals with missing data, and for automatic error propagation and
verified computations.
