SRI VIVEKA INSTITUTE OF TECHNOLOGYMADALAVARIGUDAM, KRISHNA Dt. Department of Electronics and communication Engineering FIRST SEMESTER 2010-11 Course Handout Date: 14.06.2010 Course Title Probability Theory and Stochastic Processes 1. Course description: In this course, the basic aim is to provide a strong background in probability theory, complete knowledge in statistical methods and random processes, with adequate number of solved problems. The concepts offer clear and concise coverage of the theories of probability, random variables, and random signals, including the response of linear networks to random waveforms and noise, random signals. 2. Scope and Objective of the course: The need of knowledge of random process is encountered in every practical communication system. The main objective of this subject is the tool for analyzing the information bearing signal which contains a random interference component, noise. The mathematical discipline that deals with the statistical characterization of random signals is Probability theory. 3.Textbooks: T1. 1. Probability, Random Variables & Random Signal Principles - Peyton Z. Peebles, TMH, 4th Edition, T2. 2. Probability, Random Variables and Stochastic Processes – Athanasios Papoulis and S. Unnikrishna Pillai, PHI, 4th Edition, 2002. T3. 3Probability,Stastistics and Random Processes – K.Murugesan and P.Gurusamy 4. Reference books: R1. 1. Communication Systems Analog & Digital – R.P. Singh and S.D. Sapre, TMH, 1995. R2. 2. Probability and Random Processes with Application to Signal Processing – Henry Stark and John W. Woods, Pearson Education, 3rd Edition. R3. 3. Probability Methods of Signal and System Analysis. George R. Cooper, Clave D. MC Gillem, Oxford, 3rd Edition, 1999. R4. 4. Statistical Theory of Communication - S.P. Eugene Xavier, New Age Publications, 2003. R5. 5. Signals, Systems & Communications - B.P. Lathi, B.S. Publications, 2003. 5.Course Plan: Lec. No. 1-10 Learning Objectives PROBABILITY Topics to be covered Probability introduced through Sets and Relative Frequency: Experiments and Sample Spaces, Discrete and Continuous Sample Spaces, Events, probability Definitions and Axioms, Mathematical Model of Experiments, Probability as a Relative Frequency, Joint Probability, Conditional Probability, Total Probability, Bayes’ Theorem, Independent Events: Definition of a Random Variable, Conditions for a Function to be a Random Variable, Discrete and Continuous, Mixed Random Variable, Distribution and Density functions, Properties, Binomial, Poisson, Uniform, Gaussian, Exponential, Rayleigh, Conditional Distribution, Methods of defining Conditioning Event, Conditional Density, Properties. References Ch 1 of T1, Ch 1 & 2 of T2, Ch 1 of T3 11-20 THE RANDOM VARIABLE Ch 2 of T1, Ch2 & 4 of T3 Sum of Two Random Variables. Vector Random Variables. Relationship between Cross-Power Spectrum and Cross-Correlation Function Ch 3 of T1. Equal Distributions. Variance and Skew. (N-Order) and Strict-Sense Stationarity. Classification of Processes. Function of a Random Variable. Ch 5 of T1 Ch 6 of T1. Central Limit Theorem. The Power Spectrum: Properties. Transformations of a Random Variable: Monotonic Transformations for a Continuous Random Variable. Transformation of a Discrete Random Variable. Gaussian Random Processes. Expected Value of a Function of Random Variables: Joint Moments about the Origin. Time Averages and Ergodicity. Properties. Nonmonotonic Transformations of Continuous Random Variable. N Random Variable case.Ch 11 of T3 Ch 7 of T1. Distribution and Density Functions. Unequal Distribution. Second. Correlation-Ergodic Processes. Properties. Joint Central Moments. Moment Generating Function. Joint Characteristic Functions. Sum of Several Random Variables. Joint Distribution Function. Autocorrelation Function and Its Properties. Deterministic and Nondeterministic Processes.Ch 3 of T3 Ch 4 of T1. Relationship between Power Spectrum and Autocorrelation Function. Marginal Distribution Functions. Covariance Functions. Central Moments.. Properties of Joint Distribution. Characteristic Function. Chebychev’s Inequality. Jointly Gaussian Random Variables: Two Random Variables case. Conditional Distribution and Density – Point Conditioning. Transformations of Multiple Random Variables.21-31 OPERATION ON ONE RANDOM VARIABLE – EXPECTATIONS 32-40 MULTIPLE RANDOM VARIABLES 41-50 OPERATIONS ON MULTIPLE RANDOM VARIABLES 51-60 RANDOM PROCESSES – TEMPORAL CHARACTERISTICS 61-69 RANDOM PROCESSES – SPECTRAL CHARACTERISTICS Introduction.. Linear Transformations of Gaussian Random Variables. The Random Process Concept. concept of Stationarity and Statistical Independence. Conditional Distribution and Density – Interval conditioning. Poisson Random Process. CrossCorrelation Function and Its Properties.Order and Wide-Sense Stationarity.Ch 11 of T3 . (Proof not expected). Statistical Independence. The Cross-Power Density Spectrum. First-Order Stationary Processes. Moments about the Origin. Expected Value of a Random Variable. MeanErgodic Processes. Band-Limited and Narrowband Processes. Average Noise Figure of cascaded networks. Band pass. Notices: Concerning the course will be displayed on Department Notice Board.Evaluation scheme EC No MID-I Evaluation Component Descriptive-1 Internal Quiz-1 Open Book Tests (at the end of each unit) Online Quiz-1 Descriptive-2 Internal Quiz-2 Open Book Tests (at the end of each unit) Online Quiz -2 Duration minutes 90 30 30 20 90 30 30 20 Marks 10 2 2 (each) = 4x2=8M 20 10 2 2 (each) = 4x2=8M 20 Date Time Venue Will be Announced later Continuous --- Will be Announced later Will be Announced later Continuous Will be Announced later MID-II 08. Cross-Correlation Functions of Input and Output. 7. Average Noise Figures. Effective Noise Temperature. Spectral Characteristics of System Response: Power Density Spectrum of Response. Cross-Power Density Spectrums of Input and Output. Modeling of Noise Sources: Resistive (Thermal) Noise Source. System Response – Convolution. Assignments: Comprises of Reading and/or Home assignments. INSTRUCTOR-IN-CHARGE . Arbitrary Noise Sources. Details will be announced in the class from time to time and also will be uploaded in college website.70-78 LINEAR SYSTEMS WITH RANDOM INPUTS Random Signal Response of Linear Systems: Ch 8 of T1. Mean and Meansquared Value of System Response. autocorrelation Function of Response. Properties. 6.