ROBI POLIKAR. FUNDAMENTAL CONCEPTS. &. AN OVERVIEW OF THE WAVELET THEORY. Welcome to this introductory tutorial on wavelet transforms. Wavelet Tutorial - Part 1. by Robi Polikar. Fundamental Concepts and an Overview of the Wavelet Theory. Welcome to this introductory tutorial on wavelet . Although the discretized continuous wavelet transform enables the computation of the continuous wavelet transform by computers, it is not a true discrete.
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In the following sections I will present the wavelet transform and develop a scheme an interesting tutorial aimed at engineers by Robi Polikar from Iowa State. Robi Polikar. Most images and graphs are taken aracer.mobi~polikar/ WAVELETS/aracer.mobi Fourier Transform is by far the most popular. Wavelet Transform (WT) is a relatively new concept as a whole, even it though it incorporates some of .  RobiPolikar, The Wavelet Tutorial.
This is why Fourier transform is not suitable if the signal has time varying frequency, i. If only the signal has the frequency component "f" at all times for all "t" values , then the result obtained by the Fourier transform makes sense.
Note that the Fourier transform tells whether a certain frequency component exists or not. This information is independent of where in time this component appears. It is therefore very important to know whether a signal is stationary or not, prior to processing it with the FT.
The example given in part one should now be clear. The frequency axis has been cut here, but theoretically it extends to infinity for continuous Fourier transform CFT. Actually, here we calculate the discrete Fourier transform DFT , in which case the frequency axis goes up to at least twice the sampling frequency of the signal, and the transformed signal is symmetrical.
However, this is not that important at this time. Figure 2. However, these components occur at different times. Please look carefully and note the major four peaks corresponding to 5, 10, 20, and 50 Hz. I could have made this figure look very similar to the previous one, but I did not do that on purpose.
The reason of the noise like thing in between peaks show that, those frequencies also exist in the signal. Especially note how time domain signal changes at around time ms With some suitable filtering techniques, the noise like part of the frequency domain signal can be cleaned, but this has not nothing to do with our subject now. If you need further information please send me an e-mail.
By this time you should have understood the basic concepts of Fourier transform, when we can use it and we can not. As you can see from the above example, FT cannot distinguish the two signals very well. To FT, both signals are the same, because they constitute of the same frequency components. Therefore, FT is not a suitable tool for analyzing non-stationary signals, i. Please keep this very important property in mind.
Unfortunately, many people using the FT do not think of this. They assume that the signal they have is stationary where it is not in many practical cases. Of course if you are not interested in at what times these frequency components occur, but only interested in what frequency components exist, then FT can be a suitable tool to use.
So, now that we know that we can not use well, we can, but we shouldn't FT for non-stationary signals, what are we going to do? Remember that, I have mentioned that wavelet transform is only about a decade old. You may wonder if researchers noticed this non-stationarity business only ten years ago or not. Obviously not. Apparently they must have done something about it before they figured out the wavelet transform?
They have come up with Let's look at the problem in hand little more closer. What was wrong with FT? It did not work for non-stationary signals. Let's think this: Can we assume that , some portion of a non-stationary signal is stationary? The answer is yes. Just look at the third figure above. The signal is stationary every time unit intervals.
You may ask the following question? What if the part that we can consider to be stationary is very small? Well, if it is too small, it is too small. There is nothing we can do about that, and actually, there is nothing wrong with that either. We have to play this game with the physicists' rules. In STFT, the signal is divided into small enough segments, where these segments portions of the signal can be assumed to be stationary. For this purpose, a window function "w" is chosen. The width of this window must be equal to the segment of the signal where its stationarity is valid.
This window function is first located to the very beginning of the signal. Let's suppose that the width of the window is "T" s. The window function and the signal are then multiplied. Then this product is assumed to be just another signal, whose FT is to be taken. In other words, FT of this product is taken, just as taking the FT of any signal.
The next step, would be shifting this window for some t1 seconds to a new location, multiplying with the signal, and taking the FT of the product. This procedure is followed, until the end of the signal is reached by shifting the window with "t1" seconds intervals. As you can see from the equation, the STFT of the signal is nothing but the FT of the signal multiplied by a window function. I will correct this soon.
I have just noticed that I have mistyped it. These will correspond to three different FTs at three different times.
Therefore, we will obtain a true time-frequency representation TFR of the signal. Probably the best way of understanding this would be looking at an example. First of all, since our transform is a function of both time and frequency unlike FT, which is a function of frequency only , the transform would be two dimensional three, if you count the amplitude too. Let's take a non-stationary signal, such as the following one: Figure 2. The interval 0 to ms is a simple sinusoid of Hz, and the other ms intervals are sinusoids of Hz, Hz, and 50 Hz, respectively.
Apparently, this is a non-stationary signal. The "x" and "y" axes are time and frequency, respectively. Please, ignore the numbers on the axes, since they are normalized in some respect, which is not of any interest to us at this time. Just examine the shape of the time-frequency representation.
First of all, note that the graph is symmetric with respect to midline of the frequency axis. The symmetric part is said to be associated with negative frequencies, an odd concept which is difficult to comprehend, fortunately, it is not important; it suffices to know that STFT and FT are symmetric.
What is important, are the four peaks; note that there are four peaks corresponding to four different frequency components.
Also note that, unlike FT, these four peaks are located at different time intervals along the time axis. Remember that the original signal had four spectral components located at different times.
Now we have a true time-frequency representation of the signal. We not only know what frequency components are present in the signal, but we also know where they are located in time. It is grrrreeeaaatttttt!!!! Well, not really! The implicit problem of the STFT is not obvious in the above example.
Of course, an example that would work nicely was chosen on purpose to demonstrate the concept. This principle originally applied to the momentum and location of moving particles, can be applied to time-frequency information of a signal. Simply, this principle states that one cannot know the exact time-frequency representation of a signal, i.
The problem with the STFT has something to do with the width of the window function that is used. 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. 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. 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 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. 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.
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. So how do we measure frequency, or how do we find the frequency content of a signal?
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: Figure 1.
Note that two plots are given in Figure 1. 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.
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.
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, 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.
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!!! This signal is plotted below: Figure 1.
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 Hz. Contrary to the signal in Figure 1.
Figure 1. This signal is known as the "chirp" signal.