WTpart1 - THE WAVELET TUTORIAL PART I by ROBI POLIKAR Page...

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T HE W AVELET T UTORIAL P ART I by ROBI POLIKAR FUNDAMENTAL CONCEPTS & AN OVERVIEW OF THE WAVELET THEORY Second Edition Welcome to this introductory tutorial on wavelet transforms. The wavelet transform is a relatively new concept (about 10 years old), but yet there are quite a few articles and books written on them. However, most of these books and articles are written by math people, for the other math people; still most of the math people don't know what the other math people are talking about (a math professor of mine made this confession). In other words, majority of the literature available on wavelet transforms are of little help, if any, to those who are new to this subject (this is my personal opinion). When I first started working on wavelet transforms I have struggled for many hours and days to figure out what was going on in this mysterious world of wavelet transforms, due to the lack of introductory level text(s) in this subject. Therefore, I have decided to write this tutorial for the ones who are new to Page 1 of 15 THE WAVELET TUTORIAL PART I by ROBI POLIKAR 11/10/2004 http://users.rowan.edu/~polikar/WAVELETS/WTpart1.html
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the this topic. I consider myself quite new to the subject too, and I have to confess that I have not figured out all the theoretical details yet. However, as far as the engineering applications are concerned, I think all the theoretical details are not necessarily necessary (!). In this tutorial I will try to give basic principles underlying the wavelet theory. The proofs of the theorems and related equations will not be given in this tutorial due to the simple assumption that the intended readers of this tutorial do not need them at this time. However, interested readers will be directed to related references for further and in-depth information. In this document I am assuming that you have no background knowledge, whatsoever. If you do have this background, please disregard the following information, since it may be trivial. Should you find any inconsistent, or incorrect information in the following tutorial, please feel free to contact me. I will appreciate any comments on this page. Robi POLIKAR ************************************************************************ TRANS... WHAT? First of all, why do we need a transform, or what is a transform anyway? Mathematical transformations are applied to signals to obtain a further information from that signal that is not readily available in the raw signal. In the following tutorial I will assume a time-domain signal as a raw signal, and a signal that has been "transformed" by any of the available mathematical transformations as a processed signal. There are number of transformations that can be applied, among which the Fourier transforms are probably by far the most popular. Most of the signals in practice, are TIME-DOMAIN signals in their raw format. That is, whatever that signal is measuring, is a function of time. In other words, when we plot the signal one of the axes is time (independent variable), and the other (dependent variable) is usually the amplitude. When we plot time- domain signals, we obtain a
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