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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 27 - Relating Fourier Transform of Scaling Function to Filter Bank

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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 σ_t², σ_Ω²

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)