Random variables & stochastic processes

Course overview

The primary goal of this course is to introduce the principles of random signals and to provide tools where by one can deal with systems involving such signals. In real physical system, there are many undesirable signals which are random in nature. For example: In a radio astronomer’s receiver, noise interferes with the desired signal from outer space. In a sonar system, randomly generated sea sounds give rise to a noise that interferes with the desired echoes. There is also number of desirable random signals for example: the bits in a computer bit stream appear to fluctuate randomly with time between 0 and 1 state, thereby creating a random signal. From above examples it is clear that random signals represent the behavior of more fundamental underlying random phenomena. All these phenomena must be described in some probabilistic way. Thus the entire course deals with this concept.

There are actually two things to be considered in characterizing random signals. One is how to describe any one of a variety of random phenomena; another is how to bring time into the problem so as to create the random signals of interest. To accomplish the first item, the mathematical concepts are introduced in UNIT-I and UNIT-II that are sufficiently general they can apply to any suitably defined random phenomena. To accomplish the second item, another mathematical concept called random process is introduced in UNIT-III. Spectral characteristics of random processes are introduced in UNIT-IV. Finally, the UNIT-V deals with passing of random signals through linear time-invariant (LTI) systems, and classification of random processes based on spectral characteristics.

prerequisites

  • Basic knowledge of linear algebra.

  • Fundamentals of Signals & Systems

this course will be useful in study of

  • Analog and Digital Communications

  • Wireless Communications

  • Radar Signal Processing

  • Advanced Digital Signal Processing

  • Statistical Signal Processing

  • Detection and Estimation Theory

  • Adaptive Signal Processing

course contents

UNIT I

THE RANDOM VARIABLE: Introduction, Review of Probability Theory, Definition of a Random Variable, Conditions for a Function to be a Random Variable, Discrete, Continuous and Mixed Random Variables, Distribution and Density functions, Properties, Binomial, Poisson, Uniform, Gaussian, Exponential, Rayleigh, Conditional Distribution, Conditional Density, Properties.

OPERATION ON ONE RANDOM VARIABLE - EXPECTATIONS: Introduction, Expected Value of a Random Variable, Function of a Random Variable, Moments about the Origin, Central Moments, Variance and Skew, Chebychev’s Inequality, Characteristic Function, Moment Generating Function, Transformations of a Random Variable: Monotonic Transformations for a Continuous Random Variable, Non-monotonic Transformations of Continuous Random Variable.

UNIT II

MULTIPLE RANDOM VARIABLES: Vector Random Variables, Joint Distribution Function, Properties of Joint Distribution, Marginal Distribution Functions, Conditional Distribution and Density, Statistical Independence, Sum of Two Random Variables, Sum of Several Random Variables, Central Limit Theorem: Unequal Distribution, Equal Distributions. OPERATIONS ON MULTIPLE RANDOM VARIABLES: Joint Moments about the Origin, Joint Central Moments, Joint Characteristic Functions, Jointly Gaussian Random Variables: Two Random Variables case, N Random Variables case, Properties, Transformations of Multiple Random Variables, Linear Transformations of Gaussian Random Variables.

UNIT III

RANDOM PROCESSES–TEMPORAL CHARACTERISTICS: The Random Process Concept, Classification of Processes, Deterministic and Nondeterministic Processes, Distribution and Density Functions, Concept of Stationarity and Statistical Independence. First-Order Stationary Processes, Second-order and Wide-Sense Stationarity, Nth-order and Strict -Sense Stationarity, Time Averages and Ergodicity, Autocorrelation Function and its Properties, Cross-Correlation Function and its Properties, Covariance Functions, Gaussian Random Processes, Poisson Random Process.

UNIT IV

RANDOM PROCESSES-SPECTRAL CHARACTERISTICS: The Power Density Spectrum: Properties, Relationship between Power Density Spectrum and Autocorrelation Function, The Cross-Power Density Spectrum, Properties, Relationship between Cross-Power Density Spectrum and Cross-Correlation Function.

UNIT V

LINEAR SYSTEMS WITH RANDOM INPUTS: Random Signal Response of Linear Systems: System Response–Convolution, Mean and Mean-squared Value of System Response, Autocorrelation Function of Response, Cross-Correlation Functions of Input and Output, Spectral Characteristics of System Response: Power Density Spectrum of Response, Cross-Power Density Spectra of Input and Output, Band pass, Band-Limited and Narrowband Processes, Properties.

Course outcomes

After completion of the course, the student will be able to

  • mathematically model the random phenomena and solve simple probabilistic problems.

  • identify different types of random variables and compute statistical averages of these random variables.

  • characterize the random processes in the time and frequency domains.

  • analyze the LTI systems with random inputs.

Weekly Schedule

Mon 11:00 AM to 12 Noon

Tue 09:00 AM to 10:00 AM

Wed ---

Thu 10:00 AM to 11:00 AM

Fri 03:00 PM to 04:00 PM

Sat ---