Graduate Course in Random Processes, Fall Semester 2004,
(School of Electrical and Computer Engineering)
The course will be held in the format of a study group, where most of the
time will be spent on discussions and clarifications. There are also
problem sessions every week with active participations of the group
members. Furthermore four computer projects will be demonstrated in class
and thoroughly discussed. For active participants there will be an oral
examination at the end of the course and for those who have not taken
active part in the class discussions there will be a written exam, as
well as an oral one.
Teachers:
Jacques de Maré (JdM),
demare@math.chalmers.se,
Jenny Andersson (JA),
jennya@math.chalmers.se,
Mats Viberg (MV),
viberg@s2.chalmers.se
Registration: Agneta Kinnander, agnetak@s2.chalmers.se at Signal Processing is in charge
of the administration of the course.
Literature: GS: G. R. Grimmett and D. R. Stirzaker (2001), "Probability and
random processes", Oxford University Press, 3rd edition, ISBN 0198572220
(paperback) approximately SEK 450.
The book can be bougth from some internet store, for example Bokus.
Vitor H. Nascimento and Ali H. Sayed, "On the learning mechanism of adaptive filters", IEEe Transactions on Signal Processing, vol 48, no 6, june 2000.
Xiaodong Wang, Rong Chen, Jun S. Liu, "Monte Carlo bayesian signal processing for wireless communications", Journal of VLSI Signal Processing 30, 89-105, 2002.
Tentative program
The class will be held on Wednesdays
Room: S1 on the second floor at the
School of Mathematical Sciences, Eklandagatan 86.
1000-1145 Wednesday, September 1, S1 : Introduction and course overview (JdM,
JA, MV)
0800-1145 Wednesday, September 8, S1: Events, probabilities and random
variables (GS: 1-2); Discrete random variables (GS: 3.1-3.6) (JdM, JA)
0800-1145 Wednesday, September 15, S1: Discrete random variables
(GS: 3.7-3.10); Continuous random variables (GS: 4) (JdM, JA)
0800-1145 Wednesday, September 22, S1: Fourier transforms (GS: 5.6-5.10);
Markov chains (GS: 6.1-6.6) (JdM, JA)
0800-1145 Wednesday, September 29, S1: Continuous time Markov chains (GS: 6.8-6.9);
Convergence of random variables (GS: 7.1-7.3)
(JdM, JA, MV)
0800-1145 Wednesday, October 6, S1: Law of large numbers, prediction,
martingales and conditional expectation (GS: 7.4-7.10) (JdM, JA)
0800-1145 Wednesday October 13, S1: Random processes (GS: 8) (JdM, JA)
0800-1145 Wednesday October 20: No class today!
0800-1145 Wednesday October 27, S1: Stationary processes (GS:
9) (JdM, JA, MV)
Exercises and problems
Here follows a list of the problems which will be discussed during the
meetings. (Look here for the right
pagenumbers and problemnumbers if you have the second edition of the book).
- September 8
Chapter 1, GS
Exercises: p.4 (1), p.8 (2), p.12 (3), p.14 (2)
Problems p.21: 3, 4, 10, 11
Chapter 2, GS
Exercises: p.32 (1), p.35 (2), p.41 (2) (4)
Problems p.43: 7, 9, 14, 16
Chapter 3, GS
Exercises: p.55 (1) (6), p.59 (6), p.62 (2)
- September 15
Chapter 3, GS
Exercises:, p.69 (1) (6), p.75 (5), p.83 (2)
Problems p.83: 4, 6, 13a, 16
Chapter 4, GS
Exercises: p.103 (4), p.118 (4)
Problems p.140: 3, 7, 17
- September 22
Chapter 5.6-5.10, GS
Exercises: p. 181 (1), p. 188 (1),
Chapter 6.1-6.5, GS
Exercises: p. 223 (2), p. 225 (2), p. 236 (4)
- September 29
Chapter 6.8-6.9, GS
Exercises:
p. 255 (1),
p. 264 (1), (2), (3),
Problems p.299: 22, p. 301: 31
Chapter 7.1-7.3, GS
Exercises:
p. 307 (2), p. 317 (1), (2), p.323 (2),
Problems p.355: 9
- October 6
Chapter 7.4-7.10, GS
Exercises:
p. 328 (1), p. 331 (1), (2), (3), p. 349 (1), (2), p 338 (1)
Problems p. 354: (13)
- October 13
Chapter 8, GS
Problems p. 373: 1, 3, 4, 5
- October 27
Chapter 9, GS
Exercises: p379 (1),
p386 (2), p393 (2)
Computer projects
- Computer project 1
Generate random numbers from (a) the Rayleigh, and (b) the Gaussian
distribution. Plot histograms and do quantile plots.
Generate also points in the plane from the bivariate Gaussian
distribution for different correlations. How big must the
correlation be in order to make the dependence clearly visible in
the plot?
Computer project 2
The Monte-Carlo method for solving difficult integrals
was developed more than 50 years ago. With modern computer
power it is increasingly being used in practice. An
exciting application is to implement optimal Bayesian
estimators requiring integration over complicated
distributions. A key component for doing this is to
be able to generate samples from such distributions.
A useful method is the "Rejection method", described in
Example 5, G&S page 123-124. Your task is to implement the
method in Matlab, and verify that it works. The desired
distribution is f=N(0,sigma), with sigma<1. To generate
test samples we use the standard N(0,1) distribution.
Try sigma=1/2 and sigma=1/10. Compare to a direct
generation using scaled N(0,1) variables. What determines
the number of test variables one has to generate, i.e. when
can we expect the method to "work well"? (This is of course
a toy example, since N(0,sigma) is not a difficult distribution.
Feel free to try out more interesting cases!)
- Computer project 3
Choose a non-negative continuous distribution of a random variable
X. Generate n independent copies X1, X2,...,Xn of X and plot the n
points X1, X1+X2,...,Sn=X1+X2+...+Xn on the real line. Choose a random
number Y uniformly in the interval [0;Sn]. Simulate the length of the
random subinterval which happens to contain Y and calculate thereafter
its theoretical distribution.
- Computer project 4
Let {X(t): t>0} be a differentiable, stationary Gaussian process with
unit variance and mean zero. The process crosses the zero level at the
random time points t1, t2,... . Estimate, by means of simulation, the
distribution of the random variable X'(tk), where X' denotes the
derivative of X. Give an analytical derivation of the distribution of
X'(tk).
Last modified: Tue Oct 1 07:52:20 MET DST 2002