Date: September 25.2009
Instructor’s office hours: Tuesday 13:00-14:00 pm or by appointment
Instructor’s office no/phone no/e-mail address: 4302/ 3591 /ulugbayazit@itu.edu.tr
Class hours: Thursday 13:30-16:30 pm
Prerequisite:
Course Description :
Random experiments, axioms of probability, techniques of counting, conditional probability, independence, sequential experiments.
Random variables, probability distributions, some important random variables: Bernoulli, binomial, geometric, Poisson, uniform, exponential, Gaussian, gamma. Functions of random variables, expected values, Chebyshev inequality, characteristic functions.
Multiple random variables, joint cdf's and pdf's, independence, conditional probability and conditional expectation, functions of several random variables, expected values of functions of vector random variables, multidimensional Gaussian random variables.Sums of random variables, the law of large numbers, the central limit theorem.
Random processes, distributions, mean, autocorrelation, autocovariance, some important random processes: sum, binomial counting, random walk, Poisson, Wiener, Brownian motion. Stationary random processes, derivatives and integrals of random processes, time averages of random processes and ergodic theorems.
Power spectral density, response of linear systems to random signals.
Course Objectives: Review basic concepts on probability and random variables. Understand generalization of random variable concept to random sequences and random processes. Learn specific types of random sequences and random processes. Analyze random sequences and processes in transform domain. Analyze output of linear time invariant systems to random sequences and inputs in time and transform domains.
Textbook:
Probability and Random Processes
with Applications to
Signal Processing, 3rd Edition
H. Stark, J. W. Woods
Prentice Hall 2002.
References: 1. Probability, Random Variables and Random Signal Principles, 4th Edition
Peyton Z. Peebles, Jr.
Mc Graw Hill
2. Probability, Random Variables and Stochastic Processes, 4th edition
Athanasios Papoulis
Mc Graw Hill
Week Dates Topics Objectives
1 01/10 Basic probability Woods&Stark
concepts, axioms and theorems. Chp. 1.1-1.7
Conditional, total probability.
Bayes theorem.
2 08/10 Statistical Independence, combinatorics Woods&Stark
Bernoulli trials, De-Moivre Laplace Chp. 1.8-1.11
and Poisson approximations to binomial. 2.1-2.4
Random variables, distribution
and density functions.
3 15/10 Random variables, distribution Woods&Stark
and density functions. Some important Chp. 2.1-2.5
random variables: Bernoulli, binomial,
geometric, Poisson, uniform,
exponential, Gaussian.
4 22/10 Functions of random variables, Woods&Stark
expected values, moments, Chebyshev Chp. 3.1-3.2, 4.1,4.3,4.4,4.7
inequality, characteristic functions
5 05/11 Multiple random variables, joint distribution Woods&Stark
and density functions. Conditional density and Chp. 2.6, 4.2, 4.3
distribution functions, conditional expectation,
joint moments.
6 12/11 Discrete random vectors, Woods&Stark
expectation vectors, covariance Chp. 5.1-5.4
matrices and their properties.
Decorrelation of random vectors
7 19/11 Multi-dimensional Gaussian law Woods&Stark
Midterm I (in class) Chp. 5.6
8 03/12 Random sequences: Probability space Woods&Stark
and sequence, examples, Chp. 6.1
statistical specification, distribution
and density functions, discrete valued
random sequence first and second order
statistics and their properties.
Gaussian random sequence.
9 10/12 Random walk, central limit theorem, independent Woods&Stark
increments sequence, stationarity. Chp. 6.1-6.3
Review of LTI systems.
Response of D.T. Linear and LTI systems to
random sequences.
10 17/12 WSS random sequences and power spectral Woods&Stark
density. Markov random sequences and Chp. 6.4-6.5, 7.1
Markov Chains. Random Processes:
Definition, statistical specification, density and
distribution functions,
11 24/12 Midterm II (take home) Woods&Stark
First and second order statistics. Chp. 7.1-7.2
Poisson counting process. Wiener process.
12 31/12 Markov process, Markov chains, Woods&Stark
Markov property, Chapman-Kolmogorov Chp. 7.1-7.2
Equations.
self study C.T. linear systems with random inputs, Woods&Stark
special classes of random processes, Chp. 7.3-7.5
stationarity, WSS processes,
power spectral density.
Stationary processes and LTI systems,
Stationary sequences and LTI systems.
Class Policies :
Homeworks will be assigned for self study but not contribute to final grade. Solutions will be handed out. Problems in the exams may resemble, but will not be identical to those in the homework sets.
Attendance is not required.
Cheating will not be tolerated and will result in the automatic administration of an ‘F’ grade for the course.
Make-up midterm will only be offered to those who miss either midterm exam and who provide a valid excuse.
Grading Policy:
1. Midterm I 30%
2. Midterm II 30%
3. Final 40% (Solutions)