# WTpart3 - THE WAVELET TUTORIAL PART III by ROBI POLIKAR...

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T HE W AVELET T UTORIAL P ART III MULTIRESOLUTION ANALYSIS & THE CONTINUOUS WAVELET TRANSFORM by Robi Polikar 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 Page 1 of 28 THE WAVELET TUTORIAL PART III by ROBI POLIKAR 11/10/2004 http://users.rowan.edu/~polikar/WAVELETS/WTpart3.html

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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. For example, the following shows a signal of this type. It has a relatively low frequency component throughout the entire signal and relatively high frequency components for a short duration somewhere around the middle. THE CONTINUOUS WAVELET TRANSFORM The continuous wavelet transform was developed as an alternative approach to the short time Fourier transform to overcome the resolution problem. The wavelet analysis is done in a similar way to the STFT analysis, in the sense that the signal is multiplied with a function, {\it the wavelet}, similar to the window function in the STFT, and the transform is computed separately for different segments of the time-domain signal. However, there are two main differences between the STFT and the CWT: 1. The Fourier transforms of the windowed signals are not taken, and therefore single peak will be seen corresponding to a sinusoid, i.e., negative frequencies are not computed. 2. The width of the window is changed as the transform is computed for every single spectral component, which is probably the most significant characteristic of the wavelet transform. The continuous wavelet transform is defined as follows Page 2 of 28 THE WAVELET TUTORIAL PART III by ROBI POLIKAR 11/10/2004 http://users.rowan.edu/~polikar/WAVELETS/WTpart3.html
Equation 3.1 As seen in the above equation , the transformed signal is a function of two variables, tau and s , the translation and scale parameters, respectively. psi(t) is the transforming function, and it is called the mother wavelet . The term mother wavelet gets its name due to two important properties of the wavelet analysis as explained below: The term wavelet means a small wave . The smallness refers to the condition that this (window) function is of finite length ( compactly supported ). The wave refers to the condition that this function is oscillatory . The term

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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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WTpart3 - THE WAVELET TUTORIAL PART III by ROBI POLIKAR...

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