Fuzzy sets and fuzzy logic: theory and applications
Fuzzy sets and fuzzy logic: theory and applications
A first course in fuzzy logic
The scientist and engineer's guide to digital signal processing
The scientist and engineer's guide to digital signal processing
Introduction to Algorithms
Einschließungsmengen von Polynom-Nullstellen
Proceedings of the International Symposium on Interval Mathemantics
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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(n^2) steps, where n is the number of observations x(t"0),...,x(t"n"-"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 enclosure 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.