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JCSS
2010

Fast convolution and Fast Fourier Transform under interval and fuzzy uncertainty

13 years 2 months ago
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 Else...
Guoqing Liu, Vladik Kreinovich
Added 28 Jan 2011
Updated 28 Jan 2011
Type Journal
Year 2010
Where JCSS
Authors Guoqing Liu, Vladik Kreinovich
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