This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Connexions module: m13890 1 m19  WavletBased Signal Analysis * C. Sidney Burrus This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License † Abstract A multiresolution formulation of signal decomposition give a signal expansion in terms of basis func tions called wavelets. 1 WaveletBased Signal Analysis There are wavelet systems and transforms analogous to the DFT, Fourier series, discretetime Fourier trans form, and the Fourier integral. We will start with the discrete wavelet transform (DWT) which is analogous to the Fourier series and probably should be called the wavelet series [2]. Wavelet analysis can be a form of timefrequency analysis which locates energy or events in time and frequency (or scale) simultaneously. It is somewhat similar to what is called a shorttime Fourier transform or a Gabor transform or a windowed Fourier transform. The history of wavelets and wavelet based signal processing is fairly recent. Its roots in signal expansion go back to early geophysical and image processing methods and in DSP to lter bank theory and subband coding. The current high interest probably started in the late 1980's with the work of Mallat, Daubechies, and others. Since then, the amount of research, publication, and application has exploded. Two excellent descriptions of the history of wavelet research and development are by Hubbard [4] and by Daubechies [ ? ] and a projection into the future by Sweldens [ ? ] and Burrus [1]. 1.1 The Basic Wavelet Theory The ideas and foundations of the basic dyadic, multiresolution wavelet systems are now pretty well developed, understood, and available [2][3][8][7]. The rst basic requirement is that a set of expansion functions (usually a basis) are generated from a single mother function by translation and scaling. For the discrete wavelet expansion system, this is φ j,k ( t ) = φ ( 2 j t k ) (1) where j,k are integer indices for the series expansion of the form f ( t ) = X j,k c j,k φ j,k ( t ) . (2) * Version 1.1: Sep 17, 2006 12:46 pm GMT5 † http://creativecommons.org/licenses/by/2.0/ http://cnx.org/content/m13890/1.1/ Connexions module: m13890 2 The coe cients c j,k are called the discrete wavelet transform of the signal f ( t ) . This use of translation and scale to create an expansion system is the foundation of all socalled rst generation wavelets [ ? ]....
View
Full Document
 Spring '10
 Baltazar
 Fourier Series, Wavelet, Discrete wavelet transform

Click to edit the document details