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The multiresolution decomposition seems to separate components of a signal in a way that is superior to most other methods for analysis, processing,or compression. Because of the ability of the discrete wavelet transform to decompose a signal at different independent scales and to do it in avery flexible way, Burke calls wavelets “The Mathematical Microscope" [link] , [link] . Because of this powerful and flexible decomposition, linear and nonlinear processing of signals in the wavelettransform domain offers new methods for signal detection, filtering, and compression [link] , [link] , [link] , [link] , [link] , [link] . It also can be used as the basis for robust numerical algorithms.

You will also see an interesting connection and equivalence to filter bank theory from digital signal processing [link] , [link] . Indeed, some of the results obtained with filter banks are the same as with discrete-timewavelets, and this has been developed in the signal processing community by Vetterli, Vaidyanathan, Smith and Barnwell, and others. Filter banks,as well as most algorithms for calculating wavelet transforms, are part of a still more general area of multirate and time-varying systems.

The presentation here will be as a tutorial or primer for people who know little or nothing about wavelets but do have a technical background. Itassumes a knowledge of Fourier series and transforms and of linear algebra and matrix theory. It also assumes a background equivalent to a B.S.degree in engineering, science, or applied mathematics. Some knowledge ofsignal processing is helpful but not essential. We develop the ideas in terms of one-dimensional signals [link] modeled as real or perhaps complex functions of time, but the ideas and methods have also proveneffective in image representation and processing [link] , [link] dealing with two, three, or even four or more dimensions. Vector spaces have proved to be a natural setting for developing both the theory andapplications of wavelets. Some background in that area is helpful but can be picked up as needed [link] . The study and understanding of wavelets is greatly assisted by using some sort of wavelet software system to work outexamples and run experiments. Matlab T M programs are included at the end of this book and on our web site (noted at the end of thepreface). Several other systems are mentioned in Chapter: Wavelet-Based Signal Processing and Applications .

There are several different approaches that one could take in presenting wavelet theory. We have chosen to start with the representation of asignal or function of continuous time in a series expansion, much as a Fourier series is used in a Fourier analysis. From this seriesrepresentation, we can move to the expansion of a function of a discrete variable (e.g., samples of a signal) and the theory of filter banks toefficiently calculate and interpret the expansion coefficients. This would be analogous to the discrete Fourier transform (DFT) and itsefficient implementation, the fast Fourier transform (FFT). We can also go from the series expansion to an integral transform called thecontinuous wavelet transform, which is analogous to the Fourier transform or Fourier integral. We feel starting with the series expansion gives thegreatest insight and provides ease in seeing both the similarities and differences with Fourier analysis.

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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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