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This book develops the ideas behind and properties of wavelets and shows how they can be used as analytical tools for signal processing,numerical analysis, and mathematical modeling. We try to present this in a way that is accessible to the engineer, scientist, and appliedmathematician both as a theoretical approach and as a potentially practical method to solve problems. Although the roots of this subject goback some time, the modern interest and development have a history of only a few decades.

The early work was in the 1980's by Morlet, Grossmann, Meyer, Mallat, and others, but it was the paper by Ingrid Daubechies [link] in 1988 that caught the attention of the larger applied mathematics communities insignal processing, statistics, and numerical analysis. Much of the early work took place in France [link] , [link] and the USA [link] , [link] , [link] , [link] . As in many new disciplines, the first work was closely tied to a particular application or traditional theoreticalframework. Now we are seeing the theory abstracted from application and developed on its own and seeing it related to other parallel ideas. Ourown background and interests in signal processing certainly influence the presentation of this book.

The goal of most modern wavelet research is to create a set of basis functions (or general expansion functions) and transforms that will givean informative, efficient, and useful description of a function or signal and allow more effective and efficient processing. If the signal is represented as a function of time, wavelets provideefficient localization in both time and frequency or scale. Another central idea is that of multiresolution where the decomposition of a signal is in terms of the resolution of detail.

For the Fourier series, sinusoids are chosen as basis functions, then the properties of the resulting expansion are examined. For wavelet analysis,one poses the desired properties and then derives the resulting basis functions. An important property of the wavelet basis is providing amultiresolution analysis. For several reasons, it is often desired to have the basis functions orthonormal. Given these goals, you will seeaspects of correlation techniques, Fourier transforms, short-time Fourier transforms, discrete Fourier transforms, Wigner distributions, filterbanks, subband coding, and other signal expansion and processing methods in the results.

Wavelet-based analysis is an exciting new problem-solving tool for the mathematician, scientist, and engineer. It fits naturally withthe digital computer with its basis functions defined by summations not integrals or derivatives. Unlike most traditional expansion systems, thebasis functions of the wavelet analysis are not solutions of differential equations. In some areas, it is the first truly new tool we have had inmany years. Indeed, use of wavelets and wavelet transforms requires a new point of view and a new method of interpreting representations that weare still learning how to exploit.

Work by Donoho, Johnstone, Coifman, and others have added theoretical reasons for why wavelet analysis is so versatile and powerful,and have given generalizations that are still being worked on. They have shown that wavelet systems have some inherent generic advantages and arenear optimal for a wide class of problems [link] . They also show that adaptive means can create special wavelet systems forparticular signals and classes of signals.

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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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