THE WAVELET TUTORIAL PART II by ROBI POLIKAR Page 1 of 17 THE WAVELET TUTORIAL PART 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 http users rowan edu polikar WAVELETS WTpart2 html 11 10 2004 THE WAVELET TUTORIAL PART II by ROBI POLIKAR Page 2 of 17 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 http users rowan edu polikar WAVELETS WTpart2 html 11 10 2004 THE WAVELET TUTORIAL PART II by ROBI POLIKAR Page 3 of 17 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 X denotes the signal in frequency domain This convention is used to distinguish the two representations of the signal Equation 1 is called the Fourier transform of x t and equation 2 is called the inverse Fourier transform of X f which is x t For those of you who have been using the Fourier transform are already familiar with this Unfortunately many people use these equations without knowing the underlying principle Please take a closer look at equation 1 The signal x t is multiplied with an exponential term at some certain frequency f and then integrated over ALL TIMES The key words here are all times as will explained below Note that the exponential term in Eqn 1 can also be written as Cos 2 pi f t j Sin 2 pi f t 3 The above expression has a real part of cosine of frequency f and an imaginary part of sine of frequency f So what we are actually doing is multiplying the original signal with a complex expression which has sines and cosines of frequency f Then we integrate this product In other words we add all the points in this product If the result of this integration which is nothing but some sort of infinite summation is a large value then we say that the signal x t has a dominant spectral component at frequency f This means that a major portion of this signal is composed of frequency f If the integration result is a small value than this means that the signal does not have a major frequency component of f in it If this integration result is zero then the signal does not contain the frequency f at all It is of particular interest here to see how this integration works The signal is multiplied with the sinusoidal term of frequency f If the signal has a high amplitude component of frequency f then that component and the sinusoidal term will coincide and the product of them will give a relatively large value This shows that the signal x has a major frequency component of f However if the signal does not have a frequency component of f the product will yield zero which shows that the signal does not have a frequency component of f If the frequency f is not a major component of the signal x t then the product will give a relatively small value This shows that the frequency component f in the signal x has a small amplitude in other words it is not a major component of x Now note that the integration in the transformation equation Eqn 1 is over time The left hand side of 1 however is a function of frequency Therefore the integral in 1 is calculated for every value of f IMPORTANT The information provided by the integral corresponds to all time instances since the integration is from minus infinity to plus infinity over time It follows that no matter where in time the component with frequency f appears it will affect the result of the integration equally as well In other words whether the frequency component f appears at time t1 or t2 it will have the same effect on the integration This is why Fourier transform is not suitable if the signal has time varying frequency i e the signal is non stationary If only the signal has the frequency component f at all times for all f values then the result obtained by the Fourier transform makes sense http users rowan edu polikar WAVELETS WTpart2 html 11 10 2004 THE WAVELET TUTORIAL PART II by ROBI POLIKAR Page 4 of 17 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