小波变换.pdf

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title

preface

part I

part 2

part III

part IV

acknowledments

INDEX TO SERIES OF TUTORIALS TO WAVELET TRANSFORM BY ROBI POLIKAR THE ENGINEER'S ULTIMATE GUIDE TO WAVELET ANALYSIS THE WAVELET TUTORIAL by ROBI POLIKAR Also visit Rowan’s Signal Processing and Pattern Recognition Laboratory pages Two new tutorials: Pattern Recognition & Ensemble Based Systems in Decision Making PREFACE OVERVIEW: WHY WAVELET TRANSFORM? PART I: NEW! – Thanks to Noël K. MAMALET, this tutorial is now available in French PART II: FUNDAMENTALS: THE FOURIER TRANSFORM AND THE SHORT TERM FOURIER TRANSFORM, RESOLUTION PROBLEMS PART III: MULTIRESOLUTION ANALYSIS: THE CONTINUOUS WAVELET TRANSFORM PART IV: MULTIRESOLUTION ANALYSIS: THE DISCRETE WAVELET TRANSFORM ACKNOWLEDGMENTS http://users.rowan.edu/~polikar/wavelets/wttutorial.html[2010-11-24 14:08:17]

INDEX TO SERIES OF TUTORIALS TO WAVELET TRANSFORM BY ROBI POLIKAR Please note: Due to large number of e-mails I receive, I am not able to reply to all of them. I will therefore use the following criteria in answering the questions: 1. The answer to the question does not already appear in the tutorial; 2. I actually know the answer to the question asked. If you do not receive a reply from me, then the answer is already in the tutorial, or I simply do not know the answer. My apologies for the inconvenience this may cause. I appreciate your understanding. For questions, comments or suggestions, please send an e-mail to ROBI POLIKAR MAINPAGE Thank you for visiting THE WAVELET TUTORIAL. Including your current access, this page has been visited The Wavelet Tutorial is hosted by Rowan University, College of Engineering Web Servers times since March 07,1999. The Wavelet Tutorial was originally developed and hosted (1994-2000) at Last major update: January 12, 2001. http://users.rowan.edu/~polikar/wavelets/wttutorial.html[2010-11-24 14:08:17]

WAVELET TUTORIAL INTRODUCTION - ROBI POLIKAR WELCOME Welcome to The Wavelet Tutorial ! It was end of October 1994. I had recently studied fundamentals of wavelet transform as my graduation project at my undergraduate institution, and I was planning to use this technique in analyzing signals of biological origin for my Master's degree thesis. My major professor suggested that I should work on EEG signals, since they are less studied compared to many other biological signals. This, of course, would require a database of EEG signals. I was in desperate need of finding a database of EEG signals when I decided to use this project for my Master's degree thesis, Multiresolution Wavelet Analysis of Event Related Potentials for the Detection of Alzheimer's Disease . The hospitals were refusing to cooperate in sharing their files, stating that all patient files were confidential. I have then decided to search the Internet hoping to find people who may have a database that might be of any use to me. I told them that I would use the wavelet transform to analyze EEG signals, and asked them if they had such data to share with me. The majority of the mails I have received from them were of the type: Sorry! We do not have any EEG data , but what is this wavelet business anyway? If you can provide some information, we may be able to direct you ... So I replied and tried to explain what I was after. It didn't take me too long to realize that I was writing a 4-6 pages of information on wavelet transform all over again every time someone asked for more information. Furthermore, since most of those people were from the medical community and had little or no background in signal processing, I had to start from the definition of a transform. Trying to explain a relatively new signal processing technique backed with a highly complex mathematical theory, starting from the definition of the transform was no easy task. One thing I suffered while I was learning the basics of the wavelet transform is the fact that the majority of the articles and books (if not all of them) are written by math people, for the math people, in a language which even most of the math people themselves cannot understand what is going on. I remember that I got frustrated with all those equations, trying to figure out how and where to use them. I was so frustrated at that time that I decided to write my own book in some day. When I received so many mails about the wavelet transform, I thought that writing a tutorial could be a starting point for my future dream of writing my own book of wavelet trasforms. I knew that I had to put it in simple words to make it understandable to those people. This is how this tutorial was first created. In the first version of the tutorial, there were absolutely no equations, and it simply consisted of basic concepts what wavelet transform is all about. I received an unexpected number of replies from many people around all the world who were pleasantly surprised in how simple words wavelet transform can be explained . They asked me to give more information, going into a little more detail. I have then decided to write a complete tutorial covering everything from Fourier transforms to short time Fourier transform and wavelet transforms. Part I of this tutorial presents an overview of the basic concepts that are of importance in understanding the wavelet theory. This part is strictly for those who have no background in signal processing, somehow heard that some wavelet thing or other is the way to go. This part summarizes the concept of transforming, and talks about when and why Fourier transform, by far the most often used transform in signal processing, might not be a suitable technique to use. Part II introduces the Short Term Fourier Transform (STFT), which has been used to obtain time-frequency representations of non-stationary signals. I think it is important to fully understand STFT, since wavelet transform was developed as an alternative to the STFT, to overcome some problems that are inherent to it. By the end of this part, the reader should be comfortable why and when wavelet transform needs to be used. Part III introduces the continuous wavelet transform (CWT), explaining how the problems inherent to the STFT are http://users.rowan.edu/~polikar/WAVELETS/WTpreface.html[2010-11-24 14:11:34]

WAVELET TUTORIAL INTRODUCTION - ROBI POLIKAR solved. This part gives an introduction to the mathematical backbone of the wavelet transform. Also given in this part are a couple examples that actually show how WT of a signal look like, something I could not find in any of the articles or books I have read on WT. Part IV talks about the discrete wavelet transform, a very effective and fast technique to compute the WT of a signal. Finally, a bibliography is included for those who need more than what is given in this tutorial. I would like to note that I am not an expert on wavelet transform, but just a user of this method. It is therefore, possible that I might have missed some important points, or even might have given false information. Should you find any incomplete, inconsistent, or incorrect information please feel free to inform me. I will appreciate any comments on this tutorial. This is absolutely necessary to make this tutorial complete and accurate. I will be most grateful to those sending their opinions and comments. I will be throughly happy, if I can be of any service to anyone who would like to learn wavelet transform with this tutorial. Robi POLIKAR 06/06/1995, 329 Durham Computation Center, Iowa State University Ames, IOWA, 50011 Return To Index Page http://users.rowan.edu/~polikar/WAVELETS/WTpreface.html[2010-11-24 14:11:34]

THE WAVELET TUTORIAL PART I by ROBI POLIKAR THE WAVELET TUTORIAL PART I by ROBI POLIKAR FUNDAMENTAL CONCEPTS & AN OVERVIEW OF THE WAVELET THEORY NEW! – Thanks to Noël K. MAMALET, this tutorial is now available in Second Edition French 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. http://users.rowan.edu/~polikar/WAVELETS/WTpart1.html[2010-11-24 14:12:07]

THE WAVELET TUTORIAL PART I by ROBI POLIKAR 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, or no frequency. For example the publication frequency of a daily newspaper is higher than that of a monthly magazine (it is published more frequently). The frequency is measured in cycles/second, or with a more common name, in "Hertz". For example the electric power we use in our daily life in the US is 60 Hz (50 Hz elsewhere in the world). This means that if you try to plot the electric current, it will be a sine wave passing through the same point 50 times in 1 second. Now, look at the following figures. The first one is a sine wave at 3 Hz, the second one at 10 Hz, and the third one at 50 Hz. Compare them. http://users.rowan.edu/~polikar/WAVELETS/WTpart1.html[2010-11-24 14:12:07]

THE WAVELET TUTORIAL PART I by ROBI POLIKAR So how do we measure frequency, or how do we find the frequency content of a signal? The answer is FOURIER TRANSFORM (FT). If the FT of a signal in time domain is taken, the frequency-amplitude representation of that signal is obtained. In other words, we now have a plot with one axis being the frequency and the other being the amplitude. This plot tells us how much of each frequency exists in our signal. The frequency axis starts from zero, and goes up to infinity. For every frequency, we have an amplitude value. For example, if we take the FT of the electric current that we use in our houses, we will have one spike at 50 Hz, and nothing elsewhere, since that signal has only 50 Hz frequency component. No other signal, however, has a FT which is this simple. For most practical purposes, signals contain more than one frequency component. The following shows the FT of the 50 Hz signal: http://users.rowan.edu/~polikar/WAVELETS/WTpart1.html[2010-11-24 14:12:07]

THE WAVELET TUTORIAL PART I by ROBI POLIKAR Figure 1.4 The FT of the 50 Hz signal given in Figure 1.3 One word of caution is in order at this point. Note that two plots are given in Figure 1.4. The bottom one plots only the first half of the top one. Due to reasons that are not crucial to know at this time, the frequency spectrum of a real valued signal is always symmetric. The top plot illustrates this point. However, since the symmetric part is exactly a mirror image of the first part, it provides no additional information, and therefore, this symmetric second part is usually not shown. In most of the following figures corresponding to FT, I will only show the first half of this symmetric spectrum. 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. This, of course, is only one simple example why frequency content might be useful. Today Fourier transforms are used in many different areas including all branches of engineering. 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, http://users.rowan.edu/~polikar/WAVELETS/WTpart1.html[2010-11-24 14:12:07]

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