This wavelet transform tutorial presents the basic principles underlying the wavelet theory in an accessible manner. Nice figures and explanations, very good examples.
Author's comment:
"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.
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.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).
Why do we need the frequency information?
Often times, the information that cannot be readily seen in the time-domain can be seen in the frequency domain.
Let's give an example from biological signals. Suppose we are looking at an ECG signal (ElectroCardioGraphy, graphical recording of heart's electrical activity). The typical shape of a healthy ECG signal is well known to cardiologists. Any significant deviation from that shape is usually considered to be a symptom of a pathological condition.
This pathological condition, however, may not always be quite obvious in the original time-domain signal. Cardiologists usually use the time-domain ECG signals which are recorded on strip-charts to analyze ECG signals. Recently, the new computerized ECG recorders/analyzers also utilize the frequency information to decide whether a pathological condition exists. A pathological condition can sometimes be diagnosed more easily when the frequency content of the signal is analyzed. Although FT is probably the most popular transform being used (especially in electrical engineering), it is not the only one. There are many other transforms that are used quite often by engineers and mathematicians. Hilbert transform, short-time Fourier transform (more about this later), Wigner distributions, the Radon Transform, and of course our featured transformation , the wavelet transform, constitute only a small portion of a huge list of transforms that are available at engineer's and mathematician's disposal. Every transformation technique has its own area of application, with advantages and disadvantages, and the wavelet transform (WT) is no exception. For a better understanding of the need for the WT let's look at the FT more closely. FT (as well as WT) is a reversible transform, that is, it allows to go back and forward between the raw and processed (transformed) signals. However, only either of them is available at any given time.
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http://users.rowan.edu/~polikar/WAVELETS/WTpart1.html
http://www.eso.org/sci/data-processing/software/esomidas//doc/user/98NOV/volb/node324.html
http://www.eso.org/sci/data-processing/software/esomidas//doc/user/98NOV/volb/node308.html
http://npg.dl.ac.uk/MIDAS/download/DataAcqApplications.html
Rights:
All Rights Reserved by Robi Polikar, at Rowan University. This tutorial is intended for educational purposes only. Unauthorized copying, duplicating and publishing is strictly prohibited.
http://users.rowan.edu/~polikar/WAVELETS/WTpart1.html
