Forward interval propagation through the Fourier discrete transform
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In this paper an algorithm for the forward interval propagation on the amplitude of the discrete Fourier transform (DFT) is presented. The algorithm yields best-possible bounds on the amplitude of the DFT for real and complex valued sequences. We show that computing the exact bounds for the amplitude of the DFT can be achieved with an exhaustive examination of all possible corners of the interval-shaped domain. However, because the number of corners increase exponentially (in base 2) 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 for the amplitude can be obtained propagating complex pairs only from the convex hull. 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 yield the true bounds.
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