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Fast Convolution and Fast Fourier Transform under Interval and Fuzzy Uncertainty
, 2009
"... 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) ..."
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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(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 enclosure is too wide. In this paper, we show how to provide asymptotically accurate ranges for y(t) in time O(n · log(n)).
On Average Bit Complexity of Interval Arithmetic
 Bulletin of the European Association for Theoretical Computer Science (EATCS) 68
, 1999
"... In many practical situations, we know only the intervals which contain the actual (unknown) values of physical quantities. If we know the intervals x for a quantity x and y for another quantity y, then, for every arithmetic operation fi, the set of possible values of x fi y also forms an interval; t ..."
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In many practical situations, we know only the intervals which contain the actual (unknown) values of physical quantities. If we know the intervals x for a quantity x and y for another quantity y, then, for every arithmetic operation fi, the set of possible values of x fi y also forms an interval; the operations leading from x and y to this new interval are called interval arithmetic operations. For addition and subtraction, corresponding interval operations consist of two corresponding operations with real numbers, so there is no hope of making them faster. The best known algorithms for interval multiplication consists of 3 realnumber multiplications and several comparisons. We describe a new algorithm for which the average time is equivalent to using only 2 multiplications of real numbers. What is interval arithmetic. Many computer algorithms for processing real numbers have been designed to process measurement results. Measurements are never 100% precise; therefore, when after mea...
Interval Computations as an Important Part of Granular Computing: An Introduction
 in Handbook of Granular Computing, Chapter 1
, 2008
"... This chapter provides a general introduction to interval computations, especially to interval computations as an important part of granular computing. ..."
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This chapter provides a general introduction to interval computations, especially to interval computations as an important part of granular computing.
On the Efficiency of Interval Multiplication Algorithms
"... In this paper we present the theoretical base for some modifications in interval multiplication algorithms. A diversity of proposed implementation approaches is summarized along with a discussion on their costefficiency. It is shown that some improvements can be achieved by utilizing some properties ..."
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In this paper we present the theoretical base for some modifications in interval multiplication algorithms. A diversity of proposed implementation approaches is summarized along with a discussion on their costefficiency. It is shown that some improvements can be achieved by utilizing some properties of interval multiplication formulae and no special hardware support. Both conventional and extended interval multiplication operations are considered.