Interval propagation through the discrete Fourier transform

We present an algorithm for the forward propagation of intervals through the discrete Fourier transform. The algorithm yields best-possible bounds when computing the amplitude of the Fourier transform for real and complex valued sequences. We show that computing the exact bounds of the amplitude can be achieved with an exhaustive examination of all possible corners of the interval domain. However, because the number of corners increases exponentially with the number of intervals, such method is infeasible for large interval signals. We provide an algorithm that does not need such an exhaustive search, and show that the best possible bounds can be obtained propagating complex pairs only from the convex hull of endpoints at each term of the Fourier series. Because the convex hull is always tightly inscribed in the respective rigorous bounding box resulting from interval arithmetic, we conclude that the obtained bounds are guaranteed to enclose the true values.