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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