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September 25, 2023

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Description

Foundations of Wavelets and Multirate Digital Signal Processing. Instructor: Prof. Vikram M. Gadre, Department of Electrical Engineering, IIT Bombay.

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. (from nptel.ac.in)

Course Curriculum

  • Lecture 01 – Introduction Unlimited
  • Lecture 02 – Origin of Wavelets Unlimited
  • Lecture 03 – Haar Wavelet Unlimited
  • Lecture 04 – Dyadic Wavelet Unlimited
  • Lecture 05 – Dilates and Translates of Haar Wavelet Unlimited
  • Lecture 06 – L2 Norm of a Function Unlimited
  • Lecture 07 – Piecewise Constant Representation of a Function Unlimited
  • Lecture 08 – Ladder of Subspaces Unlimited
  • Lecture 09 – Scaling Function of Haar Wavelet Unlimited
  • Lecture 10 – Demonstration: Piecewise Constant Approximation of Functions Unlimited
  • Lecture 11 – Vector Representation of Sequences Unlimited
  • Lecture 12 – Properties of Norm Unlimited
  • Lecture 13 – Parseval’s Theorem Unlimited
  • Lecture 14 – Equivalence of Functions and Sequences Unlimited
  • Lecture 15 – Angle between Functions and their Decomposition Unlimited
  • Lecture 16 – Additional Information on Direct Sum Unlimited
  • Lecture 17 – Introduction to Filter Banks Unlimited
  • Lecture 18 – Haar Analysis Filter Bank in Z-Domain Unlimited
  • Lecture 19 – Haar Synthesis Filter Bank in Z-Domain Unlimited
  • Lecture 20 – Moving from Z-Domain to Frequency Domain Unlimited
  • Lecture 21 – Frequency Response of Haar Analysis Lowpass Filter Bank Unlimited
  • Lecture 22 – Frequency Response of Haar Analysis Highpass Filter Bank Unlimited
  • Lecture 23 – Ideal Two-band Filter Bank Unlimited
  • Lecture 24 – Disqualification of Ideal Filter Bank Unlimited
  • Lecture 25 – Realizable Two-band Filter Bank Unlimited
  • Lecture 26 – Demonstration: Discrete Wavelet Transform (DWT) of Images Unlimited
  • Lecture 27 – Relating Fourier Transform of Scaling Function to Filter Bank Unlimited
  • Lecture 28 – Fourier Transform of Scaling Function Unlimited
  • Lecture 29 – Construction of Scaling and Wavelet Functions from Filter Bank Unlimited
  • Lecture 30 – Demonstration: Constructing Scaling and Wavelet Functions Unlimited
  • Lecture 31 – Conclusive Remarks and Future Prospects Unlimited

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