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Fundamentals of Wavelets, Filter Banks and Time-Frequency Analysis

Fundamentals of Wavelets, Filter Banks and Time-Frequency Analysis. Instructor: Prof. Vikram M. Gadre, Department of Electrical Engineering, IIT Bombay. The 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. (from nptel.ac.in)

Lecture 61 - Discontent with the Optimal Function


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Lecture 01 - Introduction
Lecture 02 - Origin of Wavelets
Lecture 03 - Haar Wavelet
Lecture 04 - Dyadic Wavelet
Lecture 05 - Dilates and Translates of Haar Wavelets
Lecture 06 - L2 Norm of a Function
Lecture 07 - Piecewise Constant Representation of a Function
Lecture 08 - Ladder of Subspaces
Lecture 09 - Scaling Function for Haar Wavelet
Lecture 10 - Demonstration: Piecewise Constant Approximation of Functions
Lecture 11 - Vector Representation of Sequences
Lecture 12 - Properties of Norm
Lecture 13 - Parseval's Theorem
Lecture 14 - Equivalence of Sequences and Functions
Lecture 15 - Angle between Functions and their Decomposition
Lecture 16 - Demonstration: Additional Information on Direct Sum
Lecture 17 - Introduction to Filter Banks
Lecture 18 - Haar Analysis Filter Bank in Z-Domain
Lecture 19 - Haar Synthesis Filter Bank in Z-Domain
Lecture 20 - Moving from Z-Domain to Frequency Domain
Lecture 21 - Frequency Response of Haar Analysis Lowpass Filter Bank
Lecture 22 - Frequency Response of Haar Analysis Highpass Filter Bank
Lecture 23 - Ideal Two-band Filter Bank
Lecture 24 - Disqualification of Ideal Filter Bank
Lecture 25 - Realizable Two-band Filter Bank
Lecture 26 - Demonstration: Discrete Wavelet Transform (DWT) of Images
Lecture 27 - Relating Fourier Transform of Scaling Function to Filter Bank
Lecture 28 - Fourier Transform of Scaling Function
Lecture 29 - Construction of Scaling and Wavelet Functions from Filter Bank
Lecture 30 - Demonstration: Constructing Scaling and Wavelet Functions
Lecture 31 - Introduction to Upsampling and Downsampling as Multirate Operations
Lecture 32 - Upsampling by a General Factor M-a Z-Domain Analysis
Lecture 33 - Downsampling by a General Factor M-a Z-Domain Analysis
Lecture 34 - Z-Domain Analysis of 2 Channel Filter Bank
Lecture 35 - Effect of X (-Z) in Time Domain and Aliasing
Lecture 36 - Consequences of Aliasing and Simple Approach to Avoid It
Lecture 37 - Revisiting Aliasing and the Idea of Perfect Reconstruction
Lecture 38 - Applying Perfect Reconstruction and Alias Cancellation on Haar MRA
Lecture 39 - Instruction to Daubechies Family of MRA
Lecture 40 - Power Complementarity of Low-pass Filter
Lecture 41 - Applying Perfect Reconstruction Condition to Obtain Filter Coefficient
Lecture 42 - Effect of Minimum Phase Requirement on Filter Coefficients
Lecture 43 - Building Compactly Supported Scaling Functions
Lecture 44 - Second Member of Daubechies Family
Lecture 45 - Fourier Transform Analysis of Haar Scaling and Wavelet Functions
Lecture 46 - Revisiting Fourier Transform and Parseval's Theorem
Lecture 47 - Transform Analysis of Haar Wavelet Function
Lecture 48 - Nature of Haar Scaling and Wavelet Functions in Frequency Domain
Lecture 49 - The Idea of Time-Frequency Resolution
Lecture 50 - Some Thoughts on Ideal Time-Frequency Domain Behavior
Lecture 51 - Defining Probability Density Function
Lecture 52 - Defining Mean, Variance and Containment in a Given Domain
Lecture 53 - Example: Haar Scaling Function
Lecture 54 - Variance from a Slightly Different Perspective
Lecture 55 - Signal Transformations: Effect on Mean and Variance
Lecture 56 - Time-Bandwidth Product and its Properties
Lecture 57 - Simplification of Time-Bandwidth Formulae
Lecture 58 - Evaluating and Bounding σt2, σΩ2
Lecture 59 - Evaluation of Time-Bandwidth Product
Lecture 60 - Optimal Function in the Sense of Time-Bandwidth Product
Lecture 61 - Discontent with the Optimal Function
Lecture 62 - Journey from Infinite to Finite Time-Bandwidth Product of Haar Scaling Function
Lecture 63 - More Insights about Time-Bandwidth Product
Lecture 64 - Time-Frequency Plane
Lecture 65 - Tilling the Time-Frequency Plane
Lecture 66 - STFT: Conditions for Valid Windows
Lecture 67 - STFT: Time Domain and Frequency Domain Formulations
Lecture 68 - STFT: Duality in the Interpretations
Lecture 69 - Continuous Wavelet Transform (CWT)
Lecture 70 - Tools in 1-D Data Analysis (STFT and CWT)