Recent developments in applied mathematics and mechanics: theory, experiment and practice. Devoted to the 80th anniversary of academician N.N.Yanenko

Akademgorodok, Novosibirsk, Russia, June 24 - 29, 2001

Abstracts

Siberian participants

Time-frequency extrapolation of time series on the basis of minimization of semi-norms of wavelet decomposition

Slavskii V.V., Nikulin M.V.

Department of Mathematics,
Altai State University,
(Barnaul)

The classical wavelet-expansion is defined in space L²(R), that is works with signals of unlimited length. In applications the signal, as a rule, is final. There are various theoretical approaches to a solution of this problem [1], [2], [3]. Though formally these methods solve the given problem, actually it passes in other plane - the invariance of wavelet base concerning compilations is broken, in outcome the boundary effects remain, but already in other form.

Therefore in practice prefer to use classical wavelet expansion, and a signal consider unlimited in time, spreading it on all temporal axis ([4]). For example, the basis algorithm for DWT, used in a system Matlab, uses three different methods for prolongation of a signal: addition of zero, symmetrization, smooth addition. Thus there are distortions in the field of boundaries of a signal, which should be taken into account at removal of noise or at a determination of a wavelet-spectrum. The outcome can depend on a method of prolongation. Wavelet the expansion does not depend on a mode of prolongation of a signal only in one case, when the signal has length equal to a degree two and is used wavelet of the Haar.

There is a task: to continue signals so that prolongations rendered minimum influence on smoothings of signals. In work the new method of construction of prolongations of signals based on minimization of various Euclidean seminorms of wavelet-expansion is offered. The algorithm is realized in a system Matlab and is applicable for signals of arbitrary length and anyone wavelet.

1. A.Cohen, I.Daubechies and J.-C.Vial. Wavelets and fast wavelet transforms on the interval. ACHA 1(1994), 54-81.
2. A.Cohen, W.Dahmen, R.DeVore. Multiscale Decompositions on Bounded Domains, 1996.
3. T. Kilgore and J. Prestin Polynomial Wavelets on the Interval Constr. Approx. (1996) 12: 95-110
4. D. Zheng, B. F. Chao, Y. Zhou, N. Yu. Improvement of edge effect of the wavelet time-frequency spectrum: application to the length-of-day series Journal of Geodesy (2000) 74: 249-254

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Note. Abstracts are published in author's edition

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