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# Fundamentals of Wavelets, Filter Banks and Time Frequency Analysis

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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, filter banks and time-frequency analysis. Haar wavelets have 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. Then the relation between wavelets and Multirate filter banks, from the point of view of implementation is explained.

### Syllabus

Week 1

Module 1
Lecture 1. Introduction
Lecture 2. Origin of Wavelets
Lecture 3. Haar Wavelet

Module 2
Lecture 2. Dilates and Translates of Haar Wavelets
Lecture 3.L2 Norm of a Function

Module 3
Lecture 1.Piecewise Constant Representation of a Function
Lecture 3.Scaling Function for Haar Wavelet Demo:

Demonstration: Piecewise constant approximation of functions

Week 2
Module 4
Lecture 1. Vector Representation of Sequences
Lecture 2. Properties of Norm
Lecture 3. Parseval's Theorem

Module 5
Lecture 1. Equivalence of sequences and functions
Lecture 2. Angle between Functions & their Decomposition

Module 6
Lecture 1.Introduction to filter banks
Lecture 2.Haar Analysis Filter Bank in Z-domain
Lecture 3.Haar Synthesis Filter Bank in Z-domain.

Module 7
Lecture 1.Moving from Z-domain to frequency domain
Lecture 2.Frequency Response of Haar Analysis Low pass Filter bank
Lecture 3.Frequency Response of Haar Analysis High pass Filter bank

Week 3
Module 8
Lecture 1.Ideal two-band filter bank
Lecture 2.Disqualification of Ideal filter bank
Lecture 3.Realizable two-band filter bank
Demo: Demonstration: DWT of images

Module 9
Lecture 1.Relating Fourier transform of scaling function to filter bank
Lecture 2.Fourier transform of scaling function
Lecture 3.Construction of scaling and wavelet functions from filter bank

Demo: Demonstration: Constructing scaling and wavelet functions.

Module 10
Lecture 1.Introduction to upsampling and down sampling as Multirate operations
Lecture 2.Up sampling by a general factor M- a Z-domain analysis.
Lecture 3.Down sampling by a general factor M- a Z-domain analysis.

Week 4
Module 11
Lecture 1.Z domain analysis of 2 channel filter bank.
Lecture 2.Effect of X (-Z) in time domain and aliasing.
Lecture 3.Consequences of aliasing and simple approach to avoid it

Module 12
Lecture 1.Revisiting aliasing and the Idea of perfect reconstruction
Lecture 2.Applying perfect reconstruction and alias cancellation on Haar MRA
Lecture 3.Introduction to Daubechies family of MRA.

Week 5
Module 13
Lecture 1.Power Complementarity of low pass filter
Lecture 2.Applying perfect reconstruction condition to obtain filter coefficient

Module 14
Lecture 1.Effect of minimum phase requirement on filter coefficients
Lecture 2.Building compactly supported scaling functions
Lecture 3.Second member of Daubechies family.

Week 6
Module 15
Lecture 1.Fourier transform analysis of Haar scaling and Wavelet functions
Lecture 2.Revisiting Fourier Transform and Parseval's theorem
Lecture 3.Transform Analysis of Haar Wavelet function

Module 16
Lecture 1.Nature of Haar scaling and Wavelet functions in frequency domain
Lecture 2.The Idea of Time-Frequency Resolution.
Lecture 3.Some thoughts on Ideal time- frequency domain behavior

Week 7

Module 17
Lecture 1.Defining Probability Density function
Lecture 2.Defining Mean, Variance and “containment in a given domain”
Lecture 3.Example: Haar Scaling function
Lecture 4.Variance from a slightly different perspective

Module 18
Lecture 1.Signal transformations: effect on mean and variance
Lecture 2.Time-Bandwidth product and its properties.
Lecture 3.Simplification of Time-Bandwidth formulae

Module 19
Lecture 1. Introduction
Lecture 2.Evaluation of Time-Bandwidth product
Lecture 3.Optimal function in the sense of Time-Bandwidth product

Week 8
Module 20
Lecture 1.Discontent with the “Optimal function”.
Lecture 2.Journey from infinite to finite Time-Bandwidth product of Haar scaling function
Lecture 3.More insights about Time-Bandwidth product
Lecture 4.Time-frequency plane
Lecture 5.Tiling the Time-frequency plane

Module 21
Lecture 1.STFT: Conditions for valid windows
Lecture 2.STFT: Time domain and frequency domain formulations.
Lecture 3.STFT: Duality in the interpretations
Lecture 4.Continuous Wavelet Transform (CWT)

Conclusive Remarks and Future Prospects

1. Michael W. Frazier, "An Introduction to Wavelets through Linear Algebra”, Springer, 1999.
2. Stephane Mallat, "A Wavelet Tour of Signal Processing", Academic Press, Elsevier, 1998, 1999, Second Edition.
3. http://nptel.ac.in/courses/117101001/: The lecture series on Wavelets and Multirate Digital Signal Processing created by Prof. Vikram M. Gadre in NPTEL.
4. 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.
5. P.P. Vaidyanathan, "Multirate Systems and Filter Banks", Pearson Education, Low Price Edition.