Overview

COURSE OUTLINE: The word 'wavelet' refers to a little wave. Wavelets are functions designed to be considerably localized in both time and frequency domains. There are many practical situations in which one needs to analyze the signal simultaneously in both the time and frequency domains, for example, in audio processing, image enhancement, analysis and processing, geophysics and in biomedical engineering. Such analysis requires the engineer and researcher to deal with such functions, that have an inherent ability to localize as much as possible in the two domains simultaneously. This poses a fundamental challenge because such a simultaneous localization is ultimately restricted by the uncertainty principle for signal processing. Wavelet transforms have recently gained popularity in those fields where Fourier analysis has been traditionally used because of the property which enables them to capture local signal behavior. The whole idea of wavelets manifests itself differently in many different disciplines, although the basic principles remain the same. The aim of the course is to introduce the idea of wavelets. Haar wavelets has been introduced as an important tool in the analysis of signal at various level of resolution. Keeping this goal in mind, idea of representing a general finite energy signal by a piecewise constant representation is developed. The concept of Ladder of subspaces, in particular the notion of 'approximation' and 'Incremental' subspaces is introduced. Connection between wavelet analysis and multirate digital systems have been emphasized, which brings us to the need of establishing equivalence of sequences and finite energy signals and this goal is achieved by the application of basic ideas from linear algebra. Towards the end, the relation between wavelets and multirate filter banks, from the point of view of implementation is explained.

Syllabus

Introduction.
Haar wavelet.
Origin of wavelets.
Dyadic wavelet.
Dilates and translates of Haar wavelet.
L2 norm of a function.
Piecewise constant representation of a function.
Ladder of subspaces.
Scaling function of Haar wavelet.
Demonstration: Piecewise constant approximation of functions.
Vector representation of sequences.
Properties of norm.
Parsevals theorem.
Equivalence of functions & sequences.
Angle between Functions & their Decomposition.
Additional Information on Direct-Sum.
Introduction to filter banks.
Haar Analysis filter bank in Z-domain.
Haar Synthesis filter bank in Z-domain.
Moving from Z-domain to frequency domain.
Frequency Response of Haar Analysis Low pass Filter bank.
Frequency Response of Haar Analysis High pass Filter bank.
Disqualification of Ideal Filter bank.
Ideal Two-band Filter bank.
Realizable Two-band Filter bank.
Demonstration: DWT of images.
Relating Fourier transform of scaling function to filter bank.
Fourier transform of scaling function.
Construction of scaling and wavelet functions from filter bank.
Demonstration: Constructing scaling and wavelet functions..
Conclusive Remarks and Future Prospects.