# ROBI POLIKAR WAVELET TUTORIAL PDF

The wavelet transformis a relatively new concept about 10 years old , but yet there are quite a fewarticles and books written on them. In other words, majority of theliterature available on wavelet transforms are of little help, if any, to thosewho are new to this subject this is my personal opinion. When I first started working on wavelet transforms I have struggled for manyhours and days to figure out what was going on in this mysterious world ofwavelet transforms, due to the lack of introductory level text s in thissubject. Therefore, I have decided to write this tutorial for the ones who arenew to the this topic.

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The wavelet transformis a relatively new concept about 10 years old , but yet there are quite a fewarticles and books written on them. In other words, majority of theliterature available on wavelet transforms are of little help, if any, to thosewho are new to this subject this is my personal opinion.

When I first started working on wavelet transforms I have struggled for manyhours and days to figure out what was going on in this mysterious world ofwavelet transforms, due to the lack of introductory level text s in thissubject. Therefore, I have decided to write this tutorial for the ones who arenew to the this topic. I consider myself quite new tothe subject too, and I have to confess that I have not figured out all thetheoretical details yet. However, as far as the engineering applications areconcerned, I think all the theoretical details are not necessarily necessary!

In this tutorial I will try to give basic principles underlying the wavelettheory. The proofs of the theorems and related equations will not be given inthis tutorial due to the simple assumption that the intended readers of thistutorial do not need them at this time. However, interested readers will bedirected 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 followinginformation, since it may be trivial. Should you find any inconsistent, or incorrectinformation in the following tutorial, please feel free to contact me. I willappreciate 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 readilyavailable in the raw signal.

In the following tutorial I will assume atime-domain signal as araw signal, and a signal that has been "transformed"by any of the available mathematical transformations as aprocessed signal. There are number of transformations that can beapplied, among which the Fourier transforms are probably by far the mostpopular.

That is, whatever that signal is measuring, is afunction of time. In other words, when we plot the signal one of the axes istime independent variable , and the other dependent variable is usually theamplitude. When we plot time-domain signals, we obtain atime-amplituderepresentation of the signal. This representation is not always the bestrepresentation of the signal for most signal processing related applications.

In many cases, the most distinguished information is hidden in the frequencycontent of the signal. The frequencyspectrum of a signal shows what frequencies exist in the signal. Intuitively, we all know that the frequency is something to do with thechange in rate of something. If something mathematical or physical variable, would be the technically correct term changes rapidly, we say that it is of high frequency, where as if this variabledoes not change rapidly, i. If this variable does not change at all, then we say it has zerofrequency, or no frequency.

For example the publication frequency of a dailynewspaper is higher than that of a monthly magazine it is published morefrequently. For example the electric power we use in our daily life inthe USis 60 Hz 50 Hz elsewhere in the world. This means that if you try to plot theelectric current, it will be a sine wave passing through the same point 50times in 1 second.

Now, look at the following figures. The first one is a sinewave at 3 Hz, the second one at 10 Hz, and the third one at 50 Hz. So how do we measure frequency, or how do we find the frequency content of asignal? If the FT of a signal intime domain is taken, the frequency-amplitude representation of that signal isobtained.

In other words, we now have a plot with one axis being the frequencyand the other being the amplitude. This plot tells us how much of each frequencyexists in our signal. The frequency axis starts from zero, and goes up to infinity. For everyfrequency, we have an amplitude value. For example, if we take the FT of theelectric 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. Noother signal, however, has a FT which is this simple.

For most practicalpurposes, signals contain more than one frequency component. The followingshows the FT of the 50 Hz signal: Figure 1. Note that two plots are givenin Figure 1. The bottom one plots only the first half of the top one. Due toreasons that are not crucial to know at this time, the frequency spectrum of areal 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 secondpart is usually not shown. In most of the following figures corresponding toFT, 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-domaincan be seen in the frequency domain.

The typical shape of a healthy ECG signal iswell known to cardiologists. Any significant deviation from that shape isusually considered to be a symptom of a pathological condition. This pathological condition, however, may not always be quite obvious in theoriginal time-domain signal. A pathologicalcondition can sometimes be diagnosed more easily when the frequency content ofthe signal is analyzed.

This, of course, is only one simple example why frequency content might beuseful. Today Fourier transforms are used in many different areas including allbranches of engineering. Although FT is probably the most popular transform being used especially inelectrical engineering , it is not the only one.

There are many othertransforms that are used quite often by engineers and mathematicians. Everytransformation technique has its own area of application, with advantages anddisadvantages, and the wavelet transform WT is no exception. 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 giventime. That is, no frequency information is available in the time-domain signal,and no time information is available in the Fourier transformed signal. Thenatural question that comes to mind is that is it necessary to have both thetime 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 frequencyinformation of the signal, which means that it tells us how much of eachfrequency exists in the signal, but it does not tell us when in time thesefrequency components exist.

This information is not required when the signal isso-calledstationary. Signals whose frequency content do not change in time are calledstationarysignals In other words, the frequency content of stationary signals not change in time.

In this case, one does not need to knowat what timesfrequency components exist since all frequency components exist at all times!!!. This signal is plottedbelow: Figure 1. The bottom plot is the zoomed version of the topplot, showing only the range of frequencies that are of interest to us. Notethe 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. This is anon-stationary signal. The interval 0 to ms has a Hz sinusoid, the interval to ms has a 50 Hz sinusoid, the interval to ms has a 25 Hz sinusoid, andfinally the interval to ms has a 10 Hz sinusoid.

Note that the amplitudes of higher frequency components are higherthan those of the lower frequency ones. This is due to fact that higherfrequencies last longer ms each than the lower frequency components ms each. The exact value of the amplitudes are notimportant.

Other than those ripples, everything seems to be right. The FT has fourpeaks, corresponding to four frequencies with reasonable amplitudes Well, not exactly wrong, but not exactly right either Here is why: For the first signal, plotted in Figure 1. Answer: At all times! Remember that in stationary signals, all frequency componentsthat exist in the signal, exist throughout the entireduration of the signal. There is 10 Hz at all times, there is 50 Hz at alltimes, and there is Hz at all times.

Now, consider the same question for the non-stationary signal in Figure 1. At what times these frequency components occur? For the signal in Figure 1. Therefore, for these signals the frequency components donot appear at all times! Now, compare the Figures 1. The similarity between these two spectrum should be apparent.

Both of them show four spectralcomponents at exactly the same frequencies, i. Other than the ripples, and the difference in amplitude which can always benormalized , the two spectrums are almost identical, although the correspondingtime-domain signals are not even close to each other. Both of the signals involves the same frequency components, but the first onehas these frequencies at all times, the second one has these frequencies atdifferent intervals. So, how come the spectrums of two entirely differentsignals look very much alike?

Recall that the FT gives the spectral content ofthe signal, but it gives no information regardingwhere in time thosespectral components appear Therefore, FT is not a suitable technique for non-stationary signal, with oneexception: FT can be used for non-stationary signals, if we are only interested in whatspectral components exist in the signal, but not interested where these occur.

However, if this information is needed, i. For practical purposes it is difficult to make the separation, since thereare a lot of practical stationary signals, as well as non-stationary ones.

Almost all biological signals, for example, are non-stationary.

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## Wavelet Tutorial - Part 3

We basically need Wavelet Transform WT to analyze non-stationary signals, i. I have written that Fourier Transform FT is not suitable for non-stationary signals, and I have shown examples of it to make it more clear. For a quick recall, let me give the following example. Suppose we have two different signals. Also suppose that they both have the same spectral components, with one major difference. Say one of the signals have four frequency components at all times, and the other have the same four frequency components at different times. The FT of both of the signals would be the same, as shown in the example in part 1 of this tutorial.

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## Wavlet Tutorial - Part 2

Multiresolution Analysis and the Continuous Wavelet Transform Multiresolution Analysis Although the time and frequency resolution problems are results of a physical phenomenon the Heisenberg uncertainty principle and exist regardless of the transform used, it is possible to analyze any signal by using an alternative approach called the multiresolution analysis MRA. MRA, as implied by its name, analyzes the signal at different frequencies with different resolutions. Every spectral component is not resolved equally as was the case in the STFT. MRA is designed to give good time resolution and poor frequency resolution at high frequencies and good frequency resolution and poor time resolution at low frequencies. This approach makes sense especially when the signal at hand has high frequency components for short durations and low frequency components for long durations. Fortunately, the signals that are encountered in practical applications are often of this type.