Courses » Foundations of Wavelets and Multirate Digital Signal Processing

# Foundations of Wavelets and Multirate Digital Signal Processing

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

PRE-REQUISITES
Exposure to Signals and systems, Some basic Engineering Mathematics

COURSE INSTRUCTOR

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 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 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-domain
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
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.

Certification exam:

• The exam is optional for a fee. Exams will be on 26 March, 2017.
• Time: Shift 1: 9 AM-12 Noon; Shift 2: 2 PM-5PM
• Any one shift can be chosen to write the exam for a course.
• 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.

Certificate:

• Final score will be calculated as : 25% assignment score + 75% final exam score
• 25% assignment score is calculated as 25% of average of 4 weeks course: Best 3 out of 4 assignments.
• E-Certificate will be given to those who register and write the exam and score greater than or equal to 40% final score. 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.