WTpart2 - THE WAVELET TUTORIAL PART II by ROBI POLIKAR Page...

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T HE W AVELET T UTORIAL P ART 2 by ROBI POLIKAR FUNDAMENTALS: THE FOURIER TRANSFORM AND THE SHORT TERM FOURIER TRANSFORM FUNDAMENTALS Let's have a short review of the first part. We basically need Wavelet Transform (WT) to analyze non-stationary signals, i.e., whose frequency response varies in time. 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 Page 1 of 17 THE WAVELET TUTORIAL PART II by ROBI POLIKAR 11/10/2004 http://users.rowan.edu/~polikar/WAVELETS/WTpart2.html
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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. Although the two signals are completely different, their (magnitude of) FT are the SAME ! . This, obviously tells us that we can not use the FT for non-stationary signals. But why does this happen? In other words, how come both of the signals have the same FT? HOW DOES FOURIER TRANSFORM WORK ANYWAY? An Important Milestone in Signal Processing: THE FOURIER TRANSFORM I will not go into the details of FT for two reasons: 1. It is too wide of a subject to discuss in this tutorial. 2. It is not our main concern anyway. However, I would like to mention a couple important points again for two reasons: 1. It is a necessary background to understand how WT works. 2. It has been by far the most important signal processing tool for many (and I mean many many) years. In 19th century (1822*, to be exact, but you do not need to know the exact time. Just trust me that it is far before than you can remember), the French mathematician J. Fourier, showed that any periodic function can be expressed as an infinite sum of periodic complex exponential functions. Many years after he had discovered this remarkable property of (periodic) functions, his ideas were generalized to first non-periodic functions, and then periodic or non-periodic discrete time signals. It is after this generalization that it became a very suitable tool for computer calculations. In 1965, a new algorithm called fast Fourier Transform (FFT) was developed and FT became even more popular. (* I thank Dr. Pedregal for the valuable information he has provided) Now let us take a look at how Fourier transform works: FT decomposes a signal to complex exponential functions of different frequencies. The way it does this, is defined by the following two equations: Page 2 of 17 THE WAVELET TUTORIAL PART II by ROBI POLIKAR 11/10/2004 http://users.rowan.edu/~polikar/WAVELETS/WTpart2.html
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Figure 2.1 In the above equation, t stands for time, f stands for frequency, and x denotes the signal at hand. Note that x denotes the signal in time domain and the
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This note was uploaded on 06/09/2011 for the course CAP 5015 taught by Professor Mukherjee during the Spring '11 term at University of Central Florida.

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WTpart2 - THE WAVELET TUTORIAL PART II by ROBI POLIKAR Page...

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