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# Foundations of Wavelets and Multirate Digital Signal Processing

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### Overview

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.

### Syllabus

Week 1
Introduction
Origin of Wavelets
Haar Wavelet
Dilates and Translates of Haar Wavelets
L2 Norm of a Function

Week 2
Piecewise Constant Representation of a Function
Scaling Function for Haar Wavelet
Vector Representation of Sequences
Properties of Norm
Parseval's Theorem

Week 3
Equivalence of sequences and functions
Angle between functions and decomposition of functions in terms of decomposition of sequences
Introduction to Filter Bank
Haar Analysis Filter Bank in Z-domain
Haar Synthesis Filter Bank in Z-domain

Week 4
Moving from Z-domain to frequency domain
Frequency response of Haar Analysis Filter bank
Power complementary and Magnitude Complementary of Haar Filter Bank
Ideal two-band filter bank
Disqualification of Ideal filter bank
Realizable two-band filter bank

Week 5
Relating Fourier transform of φ (t) and filter bank
Fourier transform of scaling function φ (t)
Construction of φ (t) and ψ (t) from filter bank.

### Taught by 