Wavelet theory and its application

Wavelet Theory and Applications
A literature study R.J.E. Merry DCT 2005.53

Prof. Dr. Ir. M. Steinbuch Dr. Ir. M.J.G. van de Molengraft

Eindhoven University of Technology Department of Mechanical Engineering Control Systems Technology Group

Eindhoven, June 7, 2005

Summary
Many systems are monitored and evaluated for their behavior using time signals. Additional information aboutthe properties of a time signal can be obtained by representing the time signal by a series of coefficients, based on an analysis function. One example of a signal transformation is the transformation from the time domain to the frequency domain. The oldest and probably best known method for this is the Fourier transform developed in 1807 by Joseph Fourier. An alternative method with some attractiveproperties is the wavelet transform, first mentioned by Alfred Haar in 1909. Since then a lot of research into wavelets and the wavelet transform is performed. This report gives an overview of the main wavelet theory. In order to understand the wavelet transform better, the Fourier transform is explained in more detail. This report should be considered as an introduction into wavelet theory and itsapplications. The wavelet applications mentioned include numerical analysis, signal analysis, control applications and the analysis and adjustment of audio signals. The Fourier transform is only able to retrieve the global frequency content of a signal, the time information is lost. This is overcome by the short time Fourier transform (STFT) which calculates the Fourier transform of a windowedpart of the signal and shifts the window over the signal. The short time Fourier transform gives the time-frequency content of a signal with a constant frequency and time resolution due to the fixed window length. This is often not the most desired resolution. For low frequencies often a good frequency resolution is required over a good time resolution. For high frequencies, the time resolution ismore important. A multi-resolution analysis becomes possible by using wavelet analysis. The continuous wavelet transform is calculated analogous to the Fourier transform, by the convolution between the signal and analysis function. However the trigonometric analysis functions are replaced by a wavelet function. A wavelet is a short oscillating function which contains both the analysis function andthe window. Time information is obtained by shifting the wavelet over the signal. The frequencies are changed by contraction and dilatation of the wavelet function. The continuous wavelet transform retrieves the time-frequency content information with an improved resolution compared to the STFT. The discrete wavelet transform (DWT) uses filter banks to perform the wavelet analysis. The discretewavelet transform decomposes the signal into wavelet coefficients from which the original signal can be reconstructed again. The wavelet coefficients represent the signal in various frequency bands. The coefficients can be processed in several ways, giving the DWT attractive properties over linear filtering.

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Samenvatting
Systemen worden vaak beoordeeld op hun gedrag en prestatiesdoor gebruik te maken van tijdsignalen. Extra informatie over de eigenschappen van de tijdsignalen kan verkregen worden door het tijdsignaal weer te geven met behulp van co¨ffici¨nten, die berekend worden door middel van e e vergelijkingssignalen. Een voorbeeld hiervan is de transformatie van een tijdsignaal naar het frequentiedomein. De oudste en meest bekende methode om een signaal tetransformeren naar het frequentiedomein is de Fourier transformatie, ontwikkeld in 1807 door Joseph Fourier. Een relatief nieuwe methode met aantrekkelijke eigenschappen is de wavelet transformatie die voor het eerst vermeld werd in 1909 door Alfred Haar. Vanaf die tijd is veel onderzoek uitgevoerd naar zowel de wavelet functies, als ook de wavelet transformatie zelf. Om de wavelet transformatie beter te...