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Mathematical Methods and Techniques in Signal Processing

Mathematical Methods and Techniques in Signal Processing. Instructor: Prof. Shayan Srinivasa Garani, Department of Electronic Systems Engineering, IISc Bangalore. This course provides an introduction to the foundations of signal processing, focusing on the mathematical aspects for signal processing.

Review of basic signals, systems and signal space: Review of 1-D signals and systems, review of random signals, multidimensional signals, review of vector spaces, inner product spaces, orthogonal projections and related concepts.
Sampling theorems (a peek into Shannon and compressive sampling), Basics of multi-rate signal processing: sampling, decimation and interpolation, sampling rate conversion (integer and rational sampling rates), oversampled processing (A/D and D/A conversion), and introduction to filter banks.
Signal representation: Transform theory and methods (FT and variations, KLT), other transform methods including convergence issues.
Wavelets: Characterization of wavelets, wavelet transform, multi-resolution analysis. (from nptel.ac.in)

Lecture 13 - Inner Products and Induced Norm

Go to the Course Home or watch other lectures:

Lecture 01 - Introduction to Signal Processing

Lecture 02 - Basics of Signals and Systems

Lecture 03 - Linear Time Invariant Systems

Lecture 04 - Modes in a Linear System

Lecture 05 - Introduction to State Space Representation

Lecture 06 - State Space Representation

Lecture 07 - Non-uniqueness of State Space Representation

Lecture 08 - Introduction to Vector Space

Lecture 09 - Linear Independence and Spanning Set

Lecture 10 - Unique Representation Theorem

Lecture 11 - Basis and Cardinality of Basis

Lecture 12 - Norms and Inner Product Spaces

Lecture 13 - Inner Products and Induced Norm

Lecture 14 - Cauchy-Schwarz Inequality

Lecture 15 - Orthonormality

Lecture 16 - Problems on Sum of Subspaces

Lecture 17 - Linear Independence of Orthogonal Vectors

Lecture 18 - Hilbert Space and Linear Transformation

Lecture 19 - Gram-Schmidt Orthonormalization

Lecture 20 - Linear Approximation of Signal Space

Lecture 21 - Gram-Schmidt Orthonormalization of Signals

Lecture 22 - Problems on Orthogonal Complement

Lecture 23 - Problems on Signal Geometry (4-QAM)

Lecture 24 - Basics of Probability and Random Variables

Lecture 25 - Mean and Variance of a Random Variable

Lecture 26 - Introduction to Random Process

Lecture 27 - Statistical Specification of Random Processes

Lecture 28 - Stationarity of Random Processes

Lecture 29 - Problems on Mean and Variance

Lecture 30 - Problems on MAP Detection

Lecture 31 - Fourier Transform of Dirac Comb Sequence

Lecture 32 - Sampling Theorem

Lecture 33 - Basics of Multirate Systems

Lecture 34 - Frequency Representation of Expanders and Decimators

Lecture 35 - Decimation and Interpolation Filters

Lecture 36 - Fractional Sampling Rate Alterations

Lecture 37 - Digital Filter Banks

Lecture 38 - DFT as Filter Bank

Lecture 39 - Noble Identities

Lecture 40 - Polyphase Representation

Lecture 41 - Efficient Architectures for Interpolation and Decimation Filters

Lecture 42 - Problems on Simplifying Multirate Systems using Noble Identities

Lecture 43 - Problems on Designing Synthesis Bank Filters

Lecture 44 - Efficient Architecture for Fractional Decimator

Lecture 45 - Multistage Filter Design

Lecture 46 - Two Channel Filter Banks

Lecture 47 - Amplitude and Phase Distortion in Signals

Lecture 48 - Polyphase Representation of 2-channel Filter Banks, Signal Flow Graphs and Perfect Reconstruction

Lecture 49 - M-channel Filter Banks

Lecture 50 - Polyphase Representation of M-channel Filter Banks

Lecture 51 - Perfect Reconstruction of Signals

Lecture 52 - Nyquist and Half Band Filters

Lecture 53 - Special Filter Banks for Perfect Reconstruction

Lecture 54 - Introduction to Wavelets

Lecture 55 - Multiresolution Analysis and Properties

Lecture 56 - The Haar Wavelet

Lecture 57 - Structure of Subspaces in MRA

Lecture 58 - Haar Decomposition 1

Lecture 59 - Haar Decomposition 2

Lecture 60 - Wavelet Reconstruction

Lecture 61 - Haar Wavelet and Link to Filter Banks

Lecture 62 - Demo on Wavelet Decomposition

Lecture 63 - Problems on Circular Convolution

Lecture 64 - Time Frequency Localization

Lecture 65 - Basic Analysis: Pointwise and Uniform Continuity of Functions

Lecture 66 - Basic Analysis: Convergence of Sequence of Functions

Lecture 67 - Fourier Series and Notions of Convergence

Lecture 68 - Convergence of Fourier Series at a Point of Continuity

Lecture 69 - Convergence of Fourier Series for Piecewise Differentiable Periodic Functions

Lecture 70 - Uniform Convergence of Fourier Series for Piecewise Smooth Periodic Functions

Lecture 71 - Convergence in Norm of Fourier Series

Lecture 72 - Convergence of Fourier Series for All Square Integrable Periodic Functions

Lecture 73 - Problems on Limits of Integration of Periodic Functions

Lecture 74 - Matrix Calculus

Lecture 75 - Karhunen-Loeve (KL) Transform

Lecture 76 - Applications of KL Transform

Lecture 77 - Demo on KL Transform