ECE502 Mathematics for Communication L T P C 3 1 0 4

Version No. : 1.00

Prerequisite: - Objectives: 1. To make comfortable with high dimensional mathematical analysis in research papers and

advanced signal processing and communication books 2. To develop capacity to understand compact representations and intuitive analysis of lengthy

derivations involving matrices 3. To appreciate the beauty of linear algebra mixed with probability making a simple solution to

complex problems 4. To get geometrical picture of various analysis tools Expected Outcome: After successful completion of the course the students will be 1. Able to develop various compact representations for already existing complex problems 2. Able to develop closed form solutions to various optimization problems in signal, image and

video processing combined with communication issues. 3. Able to quantify performance measures like probability of error, SNR, SINR in advanced

communication scenarios. 4. Able to write a research problem based on their own analysis and notation.

Unit I Basic Matrix Concepts

Linear equations and matrix representations, Determinants .Completing the Square- The scalar case, the matrix case. Vector spaces- Basis and dimension, Norms and inner-products, The Cauchy-Schwarz inequality, The Holder and Murkowski inequalities. Direction of vectors, weighted inner products, Expectation as an inner product, Hilbert and Banach spaces, orthogonal subspaces, null space, column space, row space. Projection matrices. Unit II Some Important Matrix Factorizations The LU factorization, The Cholesky factorization, unitary matrices and the QR factorization Eigen values and Eigenvector- Eigen values and linear systems, The Jordan form, Geometry of quadratic forms and the minimax principle, Extremal quadratic forms subject to linear constraints. Karhunen-Love low-rank approximations. Unit III The Singular Value Decomposition Pseudo inverses and the SVD, Numerically sensitive problems, Rank-reducing approximations, Total least-squares problems. Some Special Matrices and Their Applications –Toeplitz matrix inverses, circulant matrices. Kronecker Products - Some applications of Kronecker products. Unit IV Derivatives and gradients Derivatives of vectors and scalars, products of matrices, powers of a matrix, Modifications for derivatives of complex vectors and matrices Theory of Constrained optimization- Basic definitions, Generalization of chain rule to composite functions, Definitions of constrained optimization, Equality constraints: Legrange multipliers.

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Unit V Probability and random processes Random vectors, transformations, joint moments, joint characteristic function, correlation, covariance matrices- properties. Vector Gaussian, Q-function, Circular complex Gaussian, various transformations Gaussian random vectors, Rayleigh, Rician, Nagakami distributions, probability of error upper bounds for M-ary modulations. Entropy and capacity in the MIMO context with Gaussian noise. Text Books : 1. Mathematical methods and algorithms for signal processing, Todd.K. Moon and Wynne

Stirling, New York, Prentice Hall, 2000. 2. Digital Communications- 5th edition, John G. Proakis, Masoud Salehi, McGraw Hill, 2008. Reference Books:

1. Introduction to Linear Algebra-4th edition, Gilbert Strang, Wellesley-Cambridge press, 2009.

2. Space time block coding for wireless communications, E. Larsson, P. Stoica, Cambridge University press, 2003.

3. Multirate systems and filter banks, P. P. Vaidyanadhan, Pearson Education India, 1993.

4. Statistical and adaptive signal processing: Spectral estimation, signal modeling, adaptive filtering and array processing, Artech House, 2005

Mode of Evaluation: CAT- I & II, Assignments/ Quiz, Term End Examination.

Proceedings of the 29th Academic Council [26.4.2013] 235