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 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 time-amplitude representation of the signal. This representation is not always
the best representation of the signal for most signal processing related applications. In many cases, the most
distinguished information is hidden in the frequency content of the signal. The frequency SPECTRUM of
a signal is basically the frequency components (spectral components) of that signal. The frequency spectrum
of a signal shows what frequencies exist in the signal.
Intuitively, we all know that the frequency is something to do with the change in rate of something. If
something ( a mathematical or physical variable, would be the technically correct term) changes rapidly, we
say that it is of high frequency, where as if this variable does not change rapidly, i.e., it changes smoothly,
we say that it is of low frequency. If this variable does not change at all, then we say it has zero frequency,
