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Wavelets trong Electromagnetics và mô hình thiết bị P2

Intuitive Introduction to Wavelets 2.1 TECHNICAL HISTORY AND BACKGROUND The first questions from those curious about wavelets are: What is a wavelet? Who invented wavelets? What can one gain by using wavelets? 2.1.1 Historical Development Wavelets are sometimes referred to as the twentieth-century Fourier analysis. Wavelets exploit the multiresolution analysis just like microscopes do in microbiology. The genesis of wavelets began in 1910 when A. Haar proposed the staircase approximation to approximate a function, using the piecewise constants now called the Haar wavelets [1] | Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright 2003 John Wiley Sons Inc. ISBN 0-471-41901-X CHAPTER TWO Intuitive Introduction to Wavelets 2.1 TECHNICAL HISTORY AND BACKGROUND The first questions from those curious about wavelets are What is a wavelet Who invented wavelets What can one gain by using wavelets 2.1.1 Historical Development Wavelets are sometimes referred to as the twentieth-century Fourier analysis. Wavelets exploit the multiresolution analysis just like microscopes do in microbiology. The genesis of wavelets began in 1910 when A. Haar proposed the staircase approximation to approximate a function using the piecewise constants now called the Haar wavelets 1 . Afterward many mathematicians physicists and engineers made contributions to the development of wavelets Paley-Littlewood proposed dyadic frequency grouping in 1938 2 . Shannon derived sampling theory in 1948 3 . Calderon employed atomic decomposition of distributions in parabolic Hp spaces in 1977 4 . Stromberg improved the Haar systems in 1981 5 . Grossman and Morlet decomposed the Hardy functions into square integrable wavelets for seismic signal analysis in 1984 6 . Meyer constructed orthogonal basis in L2 with dilation and translation of a smooth function in 1986 7 . 15 16 INTUITIVE INTRODUCTION TO WAVELETS Mallat introduced the multiresolution analysis MRA in 1988 and unified the individual constructions of wavelets by Stromberg Battle-Lemarie and Meyer 8 . Daubechies first constructed compactly supported orthogonal wavelet systems in 1987 9 . 2.1.2 When Do Wavelets Work Most of the data representing physical problems that we are modeling are not totally random but have a certain correlation structure. The correlation is local in time spatial domain and frequency spectral domain . We should approximate these data sets with building blocks that possess both time and frequency localization. Such building blocks will be able to reveal the intrinsic correlation .