Signal denoising using wavelet-based methods  (Page 11/9)

Signal Denoising using Wavelets-based Methods Ioannis Kougioumtzoglou, Isaac Hernandez-Fajardo, Georgios Evangelatos

and Xin Ming.

George R. Brown School of Engineering, Rice University

Houston, TX - USA

Introduction

The basic idea which lies behind wavelets is the representation of an arbitrary function as a combination of simpler functions, generated as scaled and dilated versions of a particular oscillatory “mother” function.

Late Jean Morlet, a geophysical engineer, introduced the term “wavelet” while attempting to analyze signals related to seismic data. The mathematical formulation of the wavelet transform and its inverse was rigorouslyestablished by Grossman and Morlet citep( ). Since then, ideas from diverse scientific fields have resulted in developing wavelets into a powerful analysis tool.

The term wavelet is often used to denote a signal located in time with a concentrated amount of energy citep( ). This “mother” wavelet is used to generate a set of “daughter”functions through the operations of scaling and dilation applied to the mother wavelet. This set forms an orthogonal basis that allows, using inner products,to decompose any given signal much like in the case of Fourier analysis. Wavelets, however, are superior to Fourier analysis for time information is not lost when moving to the frequency domain. This property makes them suitable forapplications from diverse fields where the frequency content of a signal as well as the energy's temporal location is valuable.

The wavelets application of interest for this work is their use for data analysis, specifically for signals denoising. Denoising stands for the process of removing noise, i.e unwanted information, present in an unknown signal.The use of wavelets for noise removal was first introduced by Donoho and Johnstone citep( ). The shrinkage methods for noise removal, first introduced by Donoho citep( ), have led to a variety of approaches to signal denoising.