### Mathematical Methods and Techniques in Signal Processing. Instructor: Prof. Shayan Srinivasa Garani, Department of Electronic Systems Engineering, IISc Bangalore.

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### Description

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**)

### Course Curriculum

- Lecture 01 – Introduction to Signal Processing Unlimited
- Lecture 02 – Basics of Signals and Systems Unlimited
- Lecture 03 – Linear Time Invariant Systems Unlimited
- Lecture 04 – Modes in a Linear System Unlimited
- Lecture 05 – Introduction to State Space Representation Unlimited
- Lecture 06 – State Space Representation Unlimited
- Lecture 07 – Non-uniqueness of State Space Representation Unlimited
- Lecture 08 – Introduction to Vector Space Unlimited
- Lecture 09 – Linear Independence and Spanning Set Unlimited
- Lecture 10 – Unique Representation Theorem Unlimited
- Lecture 11 – Basis and Cardinality of Basis Unlimited
- Lecture 12 – Norms and Inner Product Spaces Unlimited
- Lecture 13 – Inner Products and Induced Norm Unlimited
- Lecture 14 – Cauchy-Schwarz Inequality Unlimited
- Lecture 15 – Orthonormality Unlimited
- Lecture 16 – Problems on Sum of Subspaces Unlimited
- Lecture 17 – Linear Independence of Orthogonal Vectors Unlimited
- Lecture 18 – Hilbert Space and Linear Transformation Unlimited
- Lecture 19 – Gram-Schmidt Orthonormalization Unlimited
- Lecture 20 – Linear Approximation of Signal Space Unlimited
- Lecture 21 – Gram-Schmidt Orthonormalization of Signals Unlimited
- Lecture 22 – Problems on Orthogonal Complement Unlimited
- Lecture 23 – Problems on Signal Geometry (4-QAM) Unlimited
- Lecture 24 – Basics of Probability and Random Variables Unlimited
- Lecture 25 – Mean and Variance of a Random Variable Unlimited
- Lecture 26 – Introduction to Random Process Unlimited
- Lecture 27 – Statistical Specification of Random Processes Unlimited
- Lecture 28 – Stationarity of Random Processes Unlimited
- Lecture 29 – Problems on Mean and Variance Unlimited
- Lecture 30 – Problems on MAP Detection Unlimited
- Lecture 31 – Fourier Transform of Dirac Comb Sequence Unlimited
- Lecture 32 – Sampling Theorem Unlimited
- Lecture 33 – Basics of Multirate Systems Unlimited
- Lecture 34 – Frequency Representation of Expanders and Decimators Unlimited
- Lecture 35 – Decimation and Interpolation Filters Unlimited
- Lecture 36 – Fractional Sampling Rate Alterations Unlimited
- Lecture 37 – Digital Filter Banks Unlimited
- Lecture 38 – DFT as Filter Bank Unlimited
- Lecture 39 – Noble Identities Unlimited
- Lecture 40 – Polyphase Representation Unlimited
- Lecture 41 – Efficient Architectures for Interpolation and Decimation Filters Unlimited
- Lecture 42 – Problems on Simplifying Multirate Systems using Noble Identities Unlimited
- Lecture 43 – Problems on Designing Synthesis Bank Filters Unlimited
- Lecture 44 – Efficient Architecture for Fractional Decimator Unlimited
- Lecture 45 – Multistage Filter Design Unlimited
- Lecture 46 – Two Channel Filter Banks Unlimited
- Lecture 47 – Amplitude and Phase Distortion in Signals Unlimited
- Lecture 48 – Polyphase Representation of 2-channel Filter Banks, Signal Flow Graphs and … Unlimited
- Lecture 49 – M-channel Filter Banks Unlimited
- Lecture 50 – Polyphase Representation of M-channel Filter Banks Unlimited
- Lecture 51 – Perfect Reconstruction of Signals Unlimited
- Lecture 52 – Nyquist and Half Band Filters Unlimited
- Lecture 53 – Special Filter Banks for Perfect Reconstruction Unlimited
- Lecture 54 – Introduction to Wavelets Unlimited
- Lecture 55 – Multiresolution Analysis and Properties Unlimited
- Lecture 56 – The Haar Wavelet Unlimited
- Lecture 57 – Structure of Subspaces in MRA Unlimited
- Lecture 58 – Haar Decomposition 1 Unlimited
- Lecture 59 – Haar Decomposition 2 Unlimited
- Lecture 60 – Wavelet Reconstruction Unlimited
- Lecture 61 – Haar Wavelet and Link to Filter Banks Unlimited
- Lecture 62 – Demo on Wavelet Decomposition Unlimited
- Lecture 63 – Problems on Circular Convolution Unlimited
- Lecture 64 – Time Frequency Localization Unlimited
- Lecture 65 – Basic Analysis: Pointwise and Uniform Continuity of Functions Unlimited
- Lecture 66 – Basic Analysis: Convergence of Sequence of Functions Unlimited
- Lecture 67 – Fourier Series and Notions of Convergence Unlimited
- Lecture 68 – Convergence of Fourier Series at a Point of Continuity Unlimited
- Lecture 69 – Convergence of Fourier Series for Piecewise Differentiable Periodic Functions Unlimited
- Lecture 70 – Uniform Convergence of Fourier Series for Piecewise Smooth Periodic Functions Unlimited
- Lecture 71 – Convergence in Norm of Fourier Series Unlimited
- Lecture 72 – Convergence of Fourier Series for All Square Integrable Periodic Functions Unlimited
- Lecture 73 – Problems on Limits of Integration of Periodic Functions Unlimited
- Lecture 74 – Matrix Calculus Unlimited
- Lecture 75 – Karhunen-Loeve (KL) Transform Unlimited
- Lecture 76 – Applications of KL Transform Unlimited
- Lecture 77 – Demo on KL Transform Unlimited

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