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• 7:53 PM, Saturday, 16 Oct 2021

 Course Postgraduate Semester Sem. I Subject Code AVD612 Subject Title Mathematical Methods for Signal Processing

##### Syllabus

Linear Algebra: Basic analysis and topology. Vector spaces, linear operators and matrices. Solution to system of linear equations ‐ linear, independence basis dimension ‐ orthogonal basis (QR/LU/SVD/ED/GS) diagonalizability, isomorphic vector spaces, Matrix transformation and decomposition, least square solution: under determined/over determined.

Bayesian : MMSE/MAP/IMMSE

Optimization : Langrauge multiplier, Matched filter, Gradient descent ‐ derivative, convex/nonconvex sets and optimization, Kehn ‐ Tucker method, linear programming/dynamic programming.

Vectors: Representation and Dot products, Matrices: Matrix Multiplication, Transposes, Inverses, Gaussian Elimination, factorization, rank of a matrix, Vector spaces: Column and row spaces, Solving Ax=0 and Ax=b, Independence, basis, dimension, linear transformations, Orthogonality: Orthogonal vectors and subspaces, projection and least squares, Gram-Schmidt orthogonalization, Determinants: Determinant formula, cofactors, inverses and volume, Eigenvalues and Eigenvectors: characteristic polynomial, Diagonalization, Hermitian and Unitary matrices, Spectral theorem, Change of basis, Positive definite matrices and singular value decomposition, Linear transformations

Review of Probability: Basic set theory and set algebra, basic axioms of probability, Conditional Probability, Random variables ‐ PDF/PMF/CDF ‐ Properties, Bayes theorem/Law of total probability, random vectors ‐ marginal/joint/conditional density functions, transformation of Random Variables, characteristic/moment generating functions, Random sums of Random variables, Law of Large numbers (strong and Weak), Limit theorems ‐ convergence types, Inequalities ‐ Chebyshev/Markov/Chernoff bounds.

Random processes: classification of random processes, wide sense stationary processes, autocorrelation function and power spectral density and their properties. Examples of random process models - Gaussian/Markov Random process, Random processes through LTI systems.

##### Text Books

Same as Reference

##### References

1.Introduction to linear algebra - Gilbert Strang, SIAM, 2016.

2.Introduction to probability - Bertsekas and Tsitsiklis, Athena, 2008

3.Probability and Random processes for Electrical Engineers, Leon Garcia Addison Wesley, 2nd edition, 1994

4.Probability and Random Processes, Geoffrey Grimmett, David Stirzaker, 3rd Edition, Oxford University Press,2001.

5.Probability and Stochastic Process, Roy D Yates, David J Goodman, 2nd edition Wiley, 2010