# InfoCoBuild

## 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)

 Introduction to Signal Processing

 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

 References 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.