# InfoCoBuild

### 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)

 Introduction

 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)

 References Fundamentals of Wavelets, Filter Banks and Time-Frequency Analysis Instructor: Prof. Vikram M. Gadre, Department of Electrical Engineering, IIT Bombay. This course introduces the idea of wavelets, filter banks and time-frequency analysis.