Dr.Pushkar Patwardhan Senior Design Engineer Tensilica, Pune

Dr. Aditya Abhyankar Professor Vishwakarma Institute of Information Technology

Acknowledgement

ACKNOWLEDGEMENTS
The development of the course material for this course has been and continues to be, a joint effort by my students, our reviewers, the officers and staff at NPTEL/C-DEEP and me. It is my privilege to acknowledge the efforts and significant contributions by many people, who have helped complete this educational endeavour.
The officers and staff in the NPTEL Project and in C-DEEP, have, together, provided unflinching support to the creation of this course. I am especially grateful to Shri Arun Kalwankar, Ms. Sangeeta Shrivastava, Shri Tarun Negi and his Team of Staff at NPTEL/ C-DEEP for providing all the logistic and administrative support required to complete this endeavour.
I am grateful to the two reviewers, Dr. Pushkar Patwardhan and Dr. Aditya Abhyankar, who have, very generously, given their time and put in a great deal of effort to go through the lectures carefully and provided constructive inputs in improving the contents and supporting material.
Many students, who are working for their Masters' Dissertation or Ph.D. Thesis with my supervision, have contributed significantly, to creating the three quadrants other than the video lecture itself. These include the following people and I would like to thank each of them personally for their work: Nawaj Shikalgar (who also led and co-ordinated this effort for over one year), Prasad Chaphekar, Parmeshwar Birajadar, Omkar Gune, Abhishek Pandey, Venkateswararao Cherukuri, Shivraj Patil, Sanket Deshmukh, Arpit Agrawal, Satyaprakash Pareek, Abhishek Men, Kamlendra Chandra, Shireesh Agrawal and Shubham Bansal.
I would also like to thank the following students who reviewed and embellished the lectures, listening to them shortly after the video lectures were recorded in the early part of the year 2010: Ritesh Kolte, Ujjwal Kumar, Dinesh Bhati, Kunal Shah, Rishabh Iyer, Ronak Shah, Ruttika Maheshwari, M. Ratnababu, Siddharth M., Mohit Dandekar, Soham Basu, Hrishikesh M. Sulay, P.R.R.K. Tiruamalesu Manda, Rohan Shah, K. L. Srinivas, Vinod Kumar Lohar, Arka Chowdhury, Debashish Bhattacharya, Toney Sebastian, Mehul Shah, Rithesh Kini, Anwesha Roy.
Students, who registered for the course on Wavelets in the Spring Semester of Academic Year 2011-2012, have also had a major role in providing further embellishments and inputs. I would like to place, on record, my gratitude for this entire class for doing so.
Without meaning to, if I have inadvertently omitted the name of someone, to whom I owe gratitude in this endeavour, my heartfelt apologies. I take this opportunity to rectify that faux pas and complete this note, by expressing my gratitude to many other people for their direct and indirect help in completing this task.
Prof. Vikram M. Gadre, Department of Electrical Engineering, IIT Bombay.

The subject of wavelets has received considerable attention over the last twenty years, with contributions coming from researchers in electrical engineering, mathematics and physics.

The word "wavelet" refers to a little wave, and implies functions that are reasonably localized both in the Time Domain and in the Fourier Domain.

The idea stems from the limitation posed by the Uncertainty Principle, which puts a limit on simultaneous localization in the time and frequency domains. As in the case of the Uncertainty Principle of Physics, the implications are seen more when one would like to make a microscopic analysis of signals.

In a number of signal processing situations, one does indeed need to look at local features: in fact, the requirement of simultaneous localization is far more widespread than often perceived. For example, there are many situations in audio, image and video where, for the purpose of analysis, one very often wishes to focus one's attention on a specific time/ space range and frequency range simultaneously.

A number of problems in digital communication also point to the implications of this uncertainty, and the need to address it suitably. The origin of the wavelet transform is in trying to achieve this to the best extent possible while working within the limits posed by the uncertainty principle. In fact, one may relate the idea of the wavelet transform to the use of a positional notation in the context of real numbers.

The wavelet transform allows a generalization of the positional notation for the context of functions. In fact, another aspect of the whole subject is multiresolution analysis - the process of analyzing phenomena and information with a scale/
fineness, matched to the content being analyzed.

This issue has important implications in waveform and signal synthesis and design, in data compression, in the analysis of signals coming from geophysical sources and biomedical sources, in locating and analyzing singularities in signals and functions, in interpolation and in many other areas.

The whole idea of wavelets manifests itself differently in many different disciplines although the basic principles remain the same. The aim of this course is to introduce the idea of wavelets, and the related notions of time-frequency analysis, of time-scale analysis, and to describe the manner in which technical developments related to wavelets have led to numerous applications.

A discussion on multirate filter banks will also form an important part of the course. The relation between wavelets and multirate systems will be brought out; to illustrate
how wavelets may actually be realized in practice.

Module No.

Topics

1

A Beginning with some practical situations, which call for multiresolution/ multiscale analysis - and how time-frequency analysis and wavelets arise from them. Examples: Image Compression, Wideband Correlation Processing, Magnetic Resonance Imaging, Digital Communication.

2

Piecewise constant approximation - the Haar wavelet.

Building up the concept of dyadic Multiresolution Analysis (MRA).

3

Relating dyadic MRA to filter banks.

A review of discrete signal processing.

Elements of multirate systems and two-band filter bank design for dyadic wavelets.

4

Families of wavelets: Orthogonal and biorthogonal wavelets.

Daubechies' family of wavelets in detail.

Vanishing moments and regularity.

Conjugate Quadrature Filter Banks (CQF) and their design.

Dyadic MRA more formally .

Data compression - fingerprint compression standards, JPEG-2000 standards.

5

The Uncertainty Principle: and its implications: the fundamental issue in this subject - the problem and the challenge that Nature imposes.

The importance of the Gaussian function: the Gabor Transform and its generalization; time, frequency and scale - their interplay.

The Continuous Wavelet Transform (CWT).

Condition of admissibility and its implications.

Application of the CWT in wideband correlation processing.

6

Journey from the CWT to the DWT: Discretization in steps.

Discretization of scale - generalized filter bank.

Discretization of translation - generalized output sampling.

Discretization of time/ space (independent variable) - sampled inputs.

7

Going from piecewise linear to piecewise polynomial.

The class of spline wavelets - a case for infinite impulse response (IIR) filter banks.

8

Variants of the wavelet transform and its implementational structures.

The wavepacket transform.

Computational efficiency in realizing filter banks - Polyphase components.

The lattice structure.

The lifting scheme.

9

An exploration of applications (this will be a joint effort between the instructor and the class).

Examples: Transient analysis; singularity detection; Biomedical signal processing applications; Geophysical signal analysis applications; Efficient signal design and realization: wavelet based modulation and demodulation; Applications in mathematical approximation; Applications to the solution of some differential equations; Applications in computer graphics and computer vision; Relation to the ideas of fractals and fractal phenomena.

Basic engineering mathematics courses should have been done, typically in the first and second year of a typical four-year engineering programme.

A course on Basic Signal and Systems Theory/Applications and a course on first principles of Digital Signal Processing should have been completed.

Howard L. Resnikoff, Raymond O. Wells, Ă˘€śWavelet Analysis: The Scalable Structure of InformationĂ˘€ť, Springer, 1998: available in Indian Edition.

Raghuveer M. Rao, Ajit S. Bopardikar, Ă˘€śWavelet Transforms: Introduction to

K. P. Soman, K. I. Ramachandran, "Insight Into Wavelets - From Theory to Practice", Prentice Hall of India, Eastern Economy Edition, Prentice Hall of India Private Limited, M-97, Connaught Circus, New Delhi - 110 001, Copyright 2004, ISBN Number 81-203-2650-4.

Michael W. Frazier, Ă˘€śAn Introduction to Wavelets Through Linear AlgebraĂ˘€ť, Springer, ISBN 3-540-780-75-0, c 1999, Reprint for sale in India, Pakistan, Bangladesh, Nepal, Sri Lanka only, at a reasonable price.

P. P. Vaidyanathan, Ă˘€śMultirate Systems and Filter BanksĂ˘€ť, Pearson Education, Low Price Edition, ISBN 81 Ă˘€“ 7758 Ă˘€“ 942 Ă˘€“ 3. This book introduces multirate systems and filter banks, making the connection to wavelets, very elegantly and systematically. It is written in a very lucid and student- friendly manner. The authorĂ˘€™s own contributions to the field of multirate systems and filter banks are phenomenal, and well respected.

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, Mathematics, Copyright 1998, ISBN 81-7371-450-9, Published by Universities Press (India) Private Limited, 3-5-819, Hyderguda, Hyderabad 500 029 (AP), India.

Stephen Welstead, Fractal and Wavelet Image Compression Techniques, Prentice Hall of India, New Delhi Ă˘€“ Eastern Economy Edition, ISBN 81-203-2827-2, c 1999 by Society of Photo-Optical Instrumentation Engineers (SPIE).

George Bachman, Lawrence Narici, Edward Beckenstein, Ă˘€śFourier and Wavelet Analysis, Springer International Edition (SIE), c 2000, Indian Edition, ISBN 81-8128-276-0. This book introduces Fourier and
Wavelet analysis in a mathematically rigorous manner Ă˘€“ and could be

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