小波变换教程(wavelet tutotial)

【实例简介】
这是90年代一位美国大学教授对小波的入门的介绍。面向小波新入门的人，看了以后一定知道了到底什么是小波？为英文版的，但不难明白，看了以后你对傅里叶变换也一定有新的认识，很有价值！！不妨看看
INDEX TO SERIES OF TUTORIALS TO WAVELET TRANSFORM BY ROBI POLIKAR 05/11/200604:39PM The Wavelet Tutorial is hosted by Rowan University, College of Engineering Web Servers ROWANUNIVERSITY College of Engineering R owan University The Wavelet Tutorial was originally developed and hosted( 1994-2000)at IOWA STATE UNIVERSITY OF SCIENCE AND TECHNOLOGY Last updated January 12, 2001 http://users.rowanedu/polikar/wavelets/wttutorial.html Page 3 of 3 THE WAVELET TUTORIAL PART i by ROBI POLlKAR 05/11/200603:36PM TTHE WAVELET TUTORIAL PART I ROBI POLIKAR FUNDAMENTAL CONCEPTS AN OVERVIEW OF THE WAVELET THEORY Second edition NEW!- Thanks to noel k. mamalet this tutorial is now available in french 之 14 4 1@ 12 AI RIght Reserved, R+bl POLIKAR, Ames, IA,1934 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 http://users.rowanedu/polikar/wavelets/wtpartl.html Page l of 15 THE WAVELET TUTORIAL PART i by ROBI POLlKAR 05/11/200603:36PM 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, duc to the lack of introductory lcvcl 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 thcorctical details are not ncccssarily ncccssary (!) In this tutorial I will try to give basic principles underlying the wavelet theory. The proofs of the theorems and related cquations will not be given in this tutorial duc 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**率****率**率****率*****率****率 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 arc number of transformations that can bc applicd, among which the Fourier transforms arc 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 relatcd 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, http://users.rowan.edu/polikar/wavelets/wtpartl.html age 2 of 15 THE WAVELET TUTORIAL PART i by ROBI POLlKAR 05/11/200603:36PM 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 clectric power we usc in our daily life in the us is 60 Hz (50 Hz elsewhere in the world). This mcans that if you try to plot the electric current, it will be a sine wave passing through the same point 50 times in I 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 1000 Figure 1. 1 3 Hz sign al sIgna 10口 Figure 1.3 50 Hz signa THE WAVELET TUTORIAL Al Rights Reserved, Robi Polikar, Ames, lowa 1996 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 http://users.rowan.edu/polikar/wavelets/wtpartl.html age 3 of 15 THE WAVELET TUTORIAL PART i by ROBI POLlKAR 05/11/200603:36PM 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 40 200 2口 80 100 re quency, HZ 400 20 50U Fre quency THE WAVELET TUTORIAL, Al Rights Reserved Robi Polikar, Ames, Iowa, 1996 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 cxactly a mirror imagc 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 Lct's give an cxample from biological signals. Suppose we arc looking at an ECG signal (Electro Cardio Graphy, 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 http://users.rowan.edu/polikar/wavelets/wtpartl.html age 4 of 15 THE WAVELET TUTORIAL PART i by ROBI POLlKAR 05/11/200603:36PM 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 decidc whether a pathological condition exists. A pathological condition can somctimes bc 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, 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 availablc at cnginccr'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. That is, no frequency information is available in the time-domain signal, and no time information is available in the Fourier transformed signal The natural question that comes to mind is that is it necessary to have both the time and the frequency information at the same time? As we will see soon, the answer depends on the particular application, and the nature of the signal in hand Recall that the ft gives the frequency information of the signal, which means that it tells us how much of each frequency exists in the signal, but it does not tell us when in time these frequency components exist This information is not required when the signal is so-called stationary Let's take a closer look at this stationarity concept more closely, since it is of paramount importance in signal analysis. Signals whose frequency content do not change in time are called stationary signals. In other words, the frequency content of stationary signals do not change in time. In this case, one does not need to know at what times frequency components exist, since all frequency components exist at all times !! For example the following signal x(t=cos(2 pi 10 t)+cos(2 pi 25*t)+cos(2*pi 50*t)+cos(2*pi 100*t) is a stationary signal, becausc it has frcqucncics of 10, 25, 50, and 100 Hz at any given timc instant. This signal is plotted below: http://users.rowanedu/polikar/wavelets/wtpartl.html Page 5 of 15 THE WAVELET TUTORIAL PART i by ROBI POLlKAR 05/11/200603:36PM 400 Time. ms gui And the following is its FT 400 20D 200 300 400 500 20 aD Frequency, Hz BO 120 Figure 1.6 The top plot in Figure 1. 6 is the(half of the symmetric) frequency spectrum of the signal in Figure 1.5. The bottom plot is the zoomed version of the top plot, showing only the range of frequencies that are of interest to us. Note the four spectral components corresponding to the frequencies 10, 25, 50 and 100 Hz Contrary to the signal in Figure 1.5, the following signal is not stationary. Figure 1. 7 plots a signal whose frequency constantly changes in time. This signal is known as the"chirp"signal. This is a non-stationary Signal http://users.rowan.edu/polikar/wavelets/wtpartl.html age 6 of 15 THE WAVELET TUTORIAL PART i by ROBI POLlKAR 05/11/200603:36PM 5 400 600 80 1000 Figure 1.7 Let's look at another example. Figure 1. 8 plots a signal with four different frequency components at four diffcrent timc intervals, hence a non-stationary signal. The interval o to 300 ms has a 100 Hz sinusoid, the interval 300 to 600 ms has a 50 Hz sinusoid, the interval 600 to 800 ms has a 25 Hz sinusoid, and finally the interval 800 to 1000 ms has a 10 Hz sinusoid Time. m: Figure 1. 8 And the following is its ft http://users.rowanedu/polikar/wavelets/wtpartl.html age 7 of 15 【实例截图】
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