6.432 | Spring 2004 | Graduate
Stochastic Processes, Detection, and Estimation


This syllabus is for the Spring 2004 offering of 6.432, taught by Prof. Willsky. Prof. Wornell uses a somewhat different syllabus when he teaches the course.


Prof. Alan S. Willsky

Course Overview

This is a graduate-level introduction to the fundamentals of detection and estimation theory involving signal and system models in which there is some inherent randomness. The concepts that we’ll develop are extraordinarily rich, interesting, and powerful, and form the basis for an enormous range of algorithms used in diverse applications. At the same time, everyone contemplating taking 6.432 should understand that it is an intense, vigorously-paced, and extremely demanding subject - both conceptually and in terms of its workload. However, for those who come prepared with the requisite background, respond to the challenge, and make the serious commitment this subject demands, it is also an extremely rewarding experience!

The material in this course constitutes a common foundation for work in the statistical signal processing, communication, and control areas. Ultimately, the course is about getting you to develop new ways of thinking about signals and systems and solving problems that involve them. The development of the material that forms the basis for 6.432 has historically been very much driven by applications. However, we emphasize that our focus in the course will not be on these applications - which form the basis for entire courses of their own - but on the common problem solving framework that they share. Nevertheless, we will cite various relevant applications as we develop the material and sometimes extract simplified examples from these contexts.


Both 6.011 and 18.06, or their equivalents, are official and important prerequisites for 6.432. In general, this course assumes a fluency in continuous- and discrete-time linear systems, basic probability, and basic linear algebra, as well as an introduction to at least some elementary concepts involving random signals and their manipulation.

Students who attended MIT as undergraduates must have completed 6.011 before enrolling in this class; co-enrollment is not an option. Students from outside MIT are strongly encouraged to consider taking 6.011 before attempting 6.432. While a thorough exposure to all the topics covered in 6.011 is not critical, past experience has shown that students without prior exposure to any of the 6.011 topics had a considerably more difficult time with 6.432 than those who did. It is also worth emphasizing that for those new to MIT there is no shame in taking 6.011 first! In fact, a major function of 6.011 is to ease the transition of non-MIT students into a graduate curriculum which is strongly linked to the MIT undergraduate course sequences.

It is a good idea to have a quick review of these materials at the beginning of the semester. The following text books are very helpful:

6.003: Oppenheim and Willsky. Signals and Systems. Prentice Hall, 1996. ISBN: 0-13-814757-4. 
6.041: Berteskas and Tsitsikilis. Introduction to Probability. Athena Scientific, 2002. ISBN: 1-886529-40-X. 
18.06: Strang. Linear Algebra and its Applications. Harcourt College Publishers, 1998. ISBN: 0-03-010567-6.

In addition, we stress that 6.432 is a graduate subject; undergraduates who have attempted to take 6.432 in the past have generally found the intensity and demands of the subject unmanageable, so undergraduate registration is discouraged. And finally, as important as the specific prerequisites, a high level of maturity, dedication, and commitment to understanding the concepts in depth is expected of all who take the subject.

Course Notes

You’ll need to get a copy of the current version of an evolving set of course notes.

Optional Text

The course has no required or recommended text. However, the optional reference text –

Papoulis, A. Probability, Random Variables, and Stochastic Processes. 3rd ed. McGraw-Hill, 1991. ISBN: 0070484686.

-- will be available for those who want a supplement to the course notes. This book covers many of the same topics we will, but overwhelming student opinion in the past has been that the book is one they tend to refer to much more after the course than during it for a variety of reasons. While you may well find this book a useful addition to your personal library, we will not assume you have access to the book during the term. Several other texts which may be a useful resource to you are listed at the end of this handout.

Lectures and Recitations

Lectures: 2 sessions / week, 1.5 hours / session

Recitations: 1 session / week, 1 hour / session

Recitations begin this week, two days after Lecture 1. Recitation provides an opportunity to develop many topics in the subject more fully than lecture allows and to develop a number of important additional insights into the material. Like the lectures, recitations are an integral part of the subject and attendance, while not formally recorded, is assumed. Both lectures and recitations work best (and are most fun) when they’re highly interactive, so your participation is both strongly encouraged and important. Remember that asking questions both in class and in office hours is a sign of engagement in the material, not an admission of weakness!

Problem Sets

There will be 11 problem sets (1-11), corresponding to a quasi-weekly schedule. Due to holidays and quizzes, some problem sets will be due in lecture on Tuesdays and some in lecture on Thursdays. Please consult the attached schedule and plan your time accordingly. You are expected to do all the assigned problems, though a randomly chosen subset will be actually graded. Don’t be misled by the relatively few points assigned to homework grades in the final grade calculation! While the grade you get on your homework is only a minor component of your final grade, working through (and, yes, often struggling at length with!) the homework is a crucial part of the learning process and will invariably have a major impact on your understanding of the material (and, in turn, your exam performances and final grade!). Also, sincere effort on problem sets provides us with a positive, qualitative sense of your involvement with the course material that may be of value when we assign final grades.

From this perspective, course “bibles” from previous terms are self-defeating and are not to be used. However, moderate collaboration in the form of joint problem solving with one or two classmates is permitted provided your writeup is your own. In making up the exams, we will assume that you have worked all the assigned problems.

Occasionally, some problems on the homework may be labeled “(practice).” You can make use of these optional problems when you think you might benefit from (and have time for!) some extra practice with the material. These problems are not to be turned in (i.e., we don’t grade them), although solutions will be provided along with those for the regular problems.

Problem sets must be handed in by the end of the class in which they are due. Problem set solutions will be available at the end of the due date’s lecture. Consequently, late problem sets cannot be seriously evaluated.


There will be two exams during the semester and a final exam during the scheduled final exam period. Both exams will be two hours long. The scheduled times for these exams are:

Exam #1 due on Lecture 11 
Exam #2 due on Lecture 20

Note that each exam includes the lecture time for that day plus an additional half hour prior to the regular lecture time. The additional time in both exams has been added to minimize the effects of time pressure in the exam. Make sure to reserve these exam times in your schedule. You will have three hours for the final exam. The exams will all be closed book. However, you will be allowed to bring one 8.5 x 11-inch sheet of notes (both sides) to Exam #1, two 8.5 x 11-inch sheets of notes (both sides) to Exam #2, and three 8.5 x 11-inch sheets of notes (both sides) to the Final Exam.

Course Grade

The final grade in the course is based upon our best assessment of your understanding of the material during the semester. Roughly, the weights used in grade assignment will be:

Exam #1 25%
Exam #2 25%
Final Exam 40%
Homework 10%

However, other factors such as interaction with the staff and participation in lecture and recitation can make a significant difference in the final grade. In general, the process of assigning a final grade involves considerable discussion among the staff; very often a careful review of the final exam is involved to examine the kinds of mistakes that were made. Although the focus of the course is obviously learning, not grades, we know the final grade is important to you, and we want you to know that we take the process seriously.

Topics Covered

The core of this subject is comprised of the following topics, each of which corresponds to one chapter of the notes.

I. Probability, Random Vectors, Vector Spaces 
Reading: Chapter 1 of Course Notes

II. Detection Theory, Decision Theory, and Hypothesis Testing 
Reading: Chapter 2 of Course Notes

III. Parameter Estimation 
Reading: Chapter 3 of Course Notes

IV. Stochastic Processes and Systems 
Reading: Chapter 4 of Course Notes

V. Karhunen-Loève and Sampled Signal Expansions
Reading: Topic 5 of Course Notes

VI. Detection and Estimation from Waveform Observations 
Reading: Topic 6 of Course Notes

VII. Wiener and Kalman Filtering, Advanced Topics
Reading: Topic 7 of Course Notes