Single Random Variables:
Definition of random variables, cumulative distribution function continuous and discrete random variables; probability mass function, probability density functions and properties; Expectations, Characteristic functions, Functions of single Random Variables, Conditioned Random variables. Application exercises to Some special distributions: Uniform, Exponential, Laplace, Gaussian; Binomial, and Poisson distribution.
(Chapter 4 Text 1)
Multiple Random variables:
Concept, Two variable CDF and PDF, Two Variable expectations (Correlation, orthogonality, Independent), Two variable transformation, Two Gaussian Random variables, Sum of two independent Random Variables, Sum of IID Random Variables – Central limit Theorem and law of large numbers, Conditional joint Probabilities, Application exercises to Chi-square RV, Student-T RV, Cauchy and Rayleigh RVs. (Chapter 5 Text 1)
Random Processes:
Ensemble, PDF, Independence, Expectations, Stationarity, Correlation Functions (ACF, CCF, Addition, and Multiplication), Ergodic Random Processes, Power Spectral Densities (Wiener Khinchin, Addition and Multiplication of RPs, Cross spectral densities), Linear Systems (output Mean, Cross correlation and Auto correlation of Input and output), Exercises with Noise. (Chapter 6 Text 1)
Vector Spaces:
Vector spaces and Null subspaces, Rank and Row reduced form, Independence, Basis and dimension, Dimensions of the four subspaces, Rank-Nullity Theorem, Linear Transformations
Orthogonality:
Orthogonal Vectors and Subspaces, Projections and Least squares, Orthogonal Bases and Gram- Schmidt Orthogonalization procedure. (Refer Chapters 2 and 3 Text 2)
Determinants:
Properties of Determinants, Permutations and Cofactors. (Refer Chapter 4, Text 2)
Eigenvalues and Eigen vectors:
Review of Eigenvalues and Diagonalization of a Matrix, Special Matrices (Positive Definite, Symmetric) and their properties, Singular Value Decomposition. (Refer Chapter 5, Text 2)
Course Outcomes:
After studying this course, students will be able to:
• Identify and associate Random Variables and Random Processes in Communication events.
• Analyze and model the Random events in typical communication events to extract quantitative statistical parameters.
• Analyze and model typical signal sets in terms of a basis function set of Amplitude, phase and frequency.
• Demonstrate by way of simulation or emulation the ease of analysis employing basis functions, statistical representation and Eigen values.
Question paper pattern:
• Examination will be conducted for 100 marks with question paper containing 10 full questions, each of 20 marks.
• Each full question can have a maximum of 4 sub questions.
• There will be 2 full questions from each module covering all the topics of the module.
• Students will have to answer 5 full questions, selecting one full question from each module.
• The total marks will be proportionally reduced to 60 marks as SEE marks is 60.
Text Books:
1. Richard H Williams, “Probability, Statistics and Random Processes for Engineers” Cengage Learning, 1st Edition, 2003, ISBN 13: 978-0-534- 36888-3, ISBN 10: 0-534-36888-3.
2. Gilbert Strang, “Linear Algebra and its Applications”, Cengage Learning, 4th Edition, 2006, ISBN 97809802327
Reference Books:
1. Hwei P. Hsu, “Theory and Problems of Probability, Random Variables, and Random Processes” Schaums Outline Series, McGraw Hill. ISBN 10: 0-07- 030644-3.
2. K. N. HariBhat, K Anitha Sheela, Jayant Ganguly, “Probability Theory and Stochastic Processes for Engineers”, Cengage Learning India, 2019, ISBN: Not in book