EE 6349: Stochastic Processes

Summer 2004

Instructor	Aria Nosratinia, ECSS 4.208 Tel: 972-883-2894
Time	Mon-Wed 5:00-6:50pm
Place	Room EC 2.110
Textbook	Stark and Woods: Probability and Random Processes with applications to signal processing, Prentice-Hall
Grading	Midterm-1 (15%), Midterm-2 (30%), Final Exam (40%), Homeworks (10%), class participation (5%)
Prerequisite	An undergraduate level course in probability theory is required, as well as knowledge of undergraduate-level calculus and linear algebra.
Office Hours	Mon-Wed 4:00-4:50pm
TA	Kripasagar Venkat, Office hours: Tue-Thu 5:00-6:00pm, room ECSN 4.528

Many processes are either random, or our limited knowledge of the parameters of the system and/or the signal makes a random formulation the best engineering choice. Stochastic modeling is one of the most powerful tools in engineering, and is used in many different areas, from signal and image processing to control theory, economic models, and design of computer networks. It may interest you to know that many engineering feats have been made possible largely on the basis of our knowledge of random processes, including radio and television reception, the hard disk on the computer and the Audio Compact Disc and DVD, as well as the guidance system of intercontinental ballistic missiles, the weather radar you see in the evening news, and the cell phones and pagers that many of you may use daily. This course explores the fundamentals of stochastic processes from an engineering point of view.

Probability Review
- Basic axioms, Borel fields, probability measures, probability spaces
- Conditional probability and independence
- Probability distribution and density
- Random variables
- Expectation
- Conditional expectation and its properties
Sequences of Random Variables
- Inequalities: Cauchy-Schwartz, Jensen, Markov, Chebychev
- Different notions of convergence
- Laws of large numbers
- Dominated convergence theorem
- Central Limit theorem, Liapunov theorem
Random (Stochastic) Processes
- Sample path, finite dimensional distributions
- Second order r.v.'s, moments, mean, autocovariance
- Stationary random processes
- Power spectral density
- Gaussian processes
- Multivariate random processes
- Separability
- Ergodicity
- Orthogonal representations, the Karhunen-Loeve expansion
Continuity of Random Processes
- In probability, in the mean, a.s., surely
- Sample path continuity (Kolmogorov Condition)
Special Types of Random Processes
- Markov (Chapman-Kolmogorov Equation)
- Martingales
- Independent increment
- Wiener (Brownian motion)
- Poisson
- White noise and spectral representation of WSS processes
Random Signals in Linear Systems
- Causal and non-causal systems
- Spectral Factorization, whitening filters
- Time domain realizations
Filtering, Prediction and smoothing
- The role of orthogonality principle
- Noncausal and causal Wiener filtering
- The discrete-time Kalman filter
- Relationship between Wiener and Kalman filters

Aria Nosratinia

Last modified: August 2002

EE 6349: Stochastic Processes

Summer 2004

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