Fast convolution and Fast Fourier Transform under interval and fuzzy uncertainty

Convolution y(t) = ∫ a(t− s) ·x(s) ds is one of the main techniques in digital signal processing. A straightforward computation of the convolution y(t) requires O(n2) steps, where n is the number of observations x(t0), . . . , x(tn−1). It is well known that by using the Fast Fourier Transform (FFT) algorithm, we can compute convolution much faster, with computation time O(n · log(n)). In practice, we only know the signal x(t) and the function a(t) with uncertainty. Sometimes, we know them with interval uncertainty, i.e., we know intervals [x(t), x(t)] and [a(t), a(t)] that contain the actual (unknown) functions x(t) and a(t). In such situations, it is desirable, for every t, to compute the range [y(t), y(t)] of possible values of y(t). Of course, it is possible to use straightforward interval computations to compute this range, i.e., replace every computational step in FFT by the corresponding operations of interval arithmetic. However, the resulting enPreprint submitted to Elsevier Science 4 March 2009 closure is too wide. In this paper, we show how to provide asymptotically accurate ranges for y(t) in time O(n · log(n)). We also explain how to use these new algorithms to compute the convolution (and the Fourier transform) under fuzzy uncertainty.