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S/CNAf 6J /MAG€ PROCESSING An Introduction to Wavelets Wavelets were developed independently by mathematicians, quantum physicists, electrical engineers, and geologists, but collaborations among these fields during the last decade have led to new and varied applications. What are wavelets, and why might they be useful to you? .: Amara Graps analyze according to scale. Indeed, some researchers feel that using wavelets means adopting a whole new mind-set or perspective in process- ing data. Wavelets are functions that satisfy certain mathematical requirements and are used in representing data or other functions. This idea is not new. Approximation using superposition of functions has existed since the early 18OOs, when Joseph Fourier discovered that he could superpose sines and cosines to represent other functions. However, in wavelet analy- sis, the scale that we use to look at data plays a special role. Wavelet algo- rithms process data at different scales or resolutions. If we look at a signal (or a function) through a large “window,” we would notice gross features. Similarly, if we look at a signal through a small “window,” we would no- tice small features. The result in wavelet analysis is to see both the forest and the trees, so to speak. This makes wavelets interesting and useful. For many decades scientists have wanted more appropriate functions than the sines and cosines, which are the basis of Fourier analysis, to approximate choppy signals.’ By their definition, these functions are nonlocal (and stretch out to infin- ity). They therefore do a very poor job in approximating sharp spikes. But with wavelet analysis, we can use approximating functions that are con- tained neatly in finite domains. Wavelets are wtll-suited for approximat- ing data with sharp discontinuities. The wavelet analysis procedure is to adopt a wavelet prototype function, called an analyzing wavelet or mother wavelet. Temporal analysis is per- formed with a contracted, high-frequency version of the prototype wavelet, while frequency analysis is performed with a dilated, low-frequency version of the same wavelet. Because the original signal or function can be repre- sented in terms of a wavelet expansion (using coefficients in a linear combi- An earlier version of this article will appear in the [email protected] Sczewtzfic d? Enginerring Applications of the :Ilizcintush 1991 Cuiference CD-ROiI.1, MacSciTech, Worcester, Mass., 1995. 50 1070-9924/95/$4.00 0 1995 IEEE IEEE COMPUTATIONAL SCIENCE & ENGINEERING Authorized licensed use limited to: ULAKBIM UASL - DOGU AKDENIZ UNIV. Downloaded on March 09,2010 at 10:45:51 EST from IEEE Xplore. Restrictions apply.
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nation of the wavelet functions), data operations can be performed using just the corresponding wavelet coefficients. And if you further choose the wavelets best adapted to your data, or trun- cate the coefficients below a threshold, your data are sparsely represented. This sparse coding makes wavelets an excellent tool in the field of data compression.
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This note was uploaded on 05/28/2010 for the course EE EE564 taught by Professor Runyiyu during the Spring '10 term at Eastern Mediterranean University.

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getPDF - S/CNAf 6J /MAG PROCESSING An Introduction to...

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