Foundations of Wavelets and Multirate Digital Signal Processing

By Prof. V. M. Gadre   |   IIT Bombay
Learners enrolled: 75
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

INTENDED AUDIENCE: Students of BE/ME/MSc/PhD, Both UG/PG can take this course

PREREQUISITES: Exposure to Signals and systems, Some basic Engineering Mathematics

Summary
 Course Status : Upcoming Course Type : Elective Duration : 4 weeks Start Date : 23 Jan 2023 End Date : 17 Feb 2023 Exam Date : 26 Mar 2023 IST Enrollment Ends : 30 Jan 2023 Category : Electrical, Electronics and Communications Engineering Credit Points : 1 Level : Undergraduate/Postgraduate

Course layout

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 of Haar Wavelet
Demonstration: Piecewise constant approximation of functions
Vector Representation of Sequences
Properties of Norm
Parseval's Theorem
Week 3: Equivalence of functions & sequences
Angle between Functions & their Decomposition
Introduction to Filter Bank
Haar Analysis Filter Bank in Z-domain
Haar Synthesis Filter Bank in Z-omain
Week 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
Week 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

Books and references

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.

Instructor bio

IIT Bombay
Prof. Vikram M. Gadre is currently a Professor at Department of Electrical Engineering, IIT Bombay. He received his Undergraduate degree, along with President’s Gold Medal for cumulative performance during his B.Tech, from IIT Delhi in 1989. He received his PhD degree in Electrical Engineering from Indian Institute of Technology, Delhi in 1994.
His research interests are Communication and signal processing, with emphasis on multiresolution and multi-rate signal processing, especially wavelets and filter banks: theory and applications. He is known for his unique way of teaching for which he received Award for Excellence in Teaching four times from IIT Bombay.
His other recognitions and awards include: S.S.I. Varshney Award from the Systems Society of India (S.S.I) (2011), IIT Bombay Research Paper Award (2008), Felicitation from Society for Cancer Research and Communication (SCRAC), India (2006), Sixth SVC Aiya Memorial Award for Telecom Education from IETE Pune Centre (2005), 11th IETE Prof K Sreenivasan Memorial Award(2004), INAE Young Engineer Award from the Indian National Academy of Engineers (2001), Student Journal Award of the IETE(1994), Adarsh Ratna Bhagat Award from National Service Scheme,IIT Delhi(1992)

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 26 March 2023 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 3 assignments out of the total 4 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Bombay .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team