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.

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. 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, relation between wavelets and multirate filter banks, from the point of view of implementation is explained.

Week .No

Topic

1

Introduction

Origin of Wavelets

Haar Wavelet

Dyadic Wavelet

Dilates and Translates of Haar Wavelets

L_{2}norm of a function

2

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

Parseval's Theorem

3

Equivalence of functions & sequences

Angle between Functions & their Decomposition

Additional Information on Direct-Sum

Introduction to Filter Bank

Haar Analysis Filter Bank in Z-domain

Haar Synthesis Filter Bank in Z-domain

4

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

Ideal Two-band Filter bank

Disqualification of Ideal Filter bank

Realizable Two-band Filter bank

Demonstration: DWT of images

5

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

Exposure to Signals and systems, Some basic Engineering Mathematics

Michael W. Frazier, "An Introduction to Wavelets Through Linear Algebra"?, Springer, 1999.

Stephane Mallat, "A Wavelet Tour Of Signal Processing", Academic Press, Elsevier, 1998, 1999, Second Edition.

http://nptel.ac.in/courses/117101001/?: The lecture series on Wavelets and Multirate Digital Signal Processing created by Prof. Vikram M. Gadre in NPTEL.

Barbara Burke Hubbard, "The World according to Wavelets - A Story of a Mathematical Technique in the making", Second edition, Universities Press (Private) India Limited 2003.

P.P. Vaidyanathan, "Multirate Systems and Filter Banks", Pearson Education, Low Price Edition.

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