This course gives thorough knowledge of linear estimation theory. The main theme of the course is optimal linear estimation, Kalman and Weiner filtering, which are systematic methods to solve estimation problems with applications in several technical disciplines, for example in telecommunications, automatic control and signal processing but also in other disciplines, such as econometrics and statistics.The course also provides an introduction to optimal filtering for non-linear systems. The course assumes familiarity with basic concepts from matrix theory, stochastic processes, and linear systems theory. The course is directed towards the students who intend to work with development and research within these fields.

The following topics are covered; Basic estimation theory, time discrete and time continuous Wiener filters, time discrete Kalman filters, properties of Wiener and Kalman filters, smoothing, Extended Kalman filters, sigma-point filters and particle filters.

After successfully completing the course, the student should be able to

•                     Understand to which type of estimation problems linear estimation can be applied.

•                     Understand the relationship between computational complexity, filter structure, and performance.

•                     Understand the relationship between optimal filtering, linear estimation, and Wiener/Kalman filtering.

•                     Approach estimation problems in a systematic way.

•                     Compute, analyze, and modify state space models.

•                     Derive and manipulate the time discrete and time continuous Wiener filter equations and compute the Wiener filter for a given estimation problem.

•                     Derive and manipulate the time discrete Kalman filter equations and compute the Kalman filter for a given estimation problem.

•                     Analyze properties of optimal filters.

•                     Implement Wiener and Kalman filters (time discrete) and state space models using Matlab.

•                     Simulate state space models and optimal filters, analyze the results, optimize the filter performance, and provide a written report on the findings.

•                     Know about common methods for optimal filtering in the case of non-Gaussian noise or non-linear models, such as Extended Kalman filter, sigma point filtering and particle filtering.

•                    Formulate logical arguments, orally and in writing, in a way that is considered valid in scientific publications and presentations within the topic area.

The course is organized jointly with the PhD level course FEM3200.

Lectures

During the 8 weekly lectures, we go through the theory and together discuss important concepts, using clicker questions and group discussions. Therefore, the lectures are primarily offered in real life at campus, but (somewhat primitively) recorded lectures will also be available as a complement.

Weekly homework

As a student, you are expected to spend a large part of the effort in the course, on solving weekly homework problems (in total 6 sets of homework problems). You may discuss the problems with other students in the course, but the handed in solutions should be formulated individually.

Projects

Two home projects should be solved during the course. The tasks involve a combination of pen/paper derivations and Matlab implementations. The project assignments may be solved and reported by groups of no more than 2 students. For Project 1, a short written report should be handed in, whereas Project 2 is reported orally.

Time and location, see Schema HT-2022-276

EQ1220/EQ1270 Signal Theory/Stochastic Signals and Systems, or equivalent

EQ2300 Digital Signal Processing, EQ2401 Adaptive Signal Processing

The material in the course will be taken from several text books, all of them
being available as E-books through KTH Library:

• D. Simon, Optimal State Estimation, John Wiley & Sons, 2006
• B. D. O. Anderson, J. B. Moore, Optimal Filtering, Dover Publications, 2012.
• S. Theodoridis, Machine Learning; A Bayesian and Optimization Perspective, 2nd ed., Academic press, 2020.

In addition, a few research articles will be used (also available through KTH Library).

The projects and homework require some implementation work, preferably in Matlab.

Students at KTH with a permanent disability can get support during studies from Funka:

Funka - compensatory support for students with disabilities

A, B, C, D, E, FX, F

• INL1 - Homework assignments, 4.5 credits, Grading scale: A, B, C, D, E, FX, F
• PRO1 - Project assignment, 1.5 credits, Grading scale: A, B, C, D, E, FX, F
• PRO2 - Project assignment, 1.5 credits, Grading scale: A, B, C, D, E, FX, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

Homework assignments ( INL1 )

There are 6 sets of weekly homework. 

You may discuss the problems with other students in the course, but the handed in homework solutions should be formulated individually.

Project assignments 1 and 2 (PRO1 and PRO2)

The task of Project 1, will be based on the material from Lectures 1-4 and will therefore be made available at Lecture 4. The task of Project 2 will be based on material from Lecture 7 (and 8) and will be made available at Lecture 7. 

The project assignments may be solved and reported by groups of 2 students (or individually, though it's recommended to work two and two). The projects involves a combination and paper/pen calculations and numerical implementation and evaluations.

For Project 1, a short written report documenting the solution and results should be handed in, before Lecture 7. For Project 2, the implementation should be handed in and the results are reported orally, at latest during the first week of study period 2.

•                     PRO1 – Project assignment, 1.5, grade scale: A, B, C, D, E, FX, F

•                     PRO2 – Project assignment, 1.5, grade scale: A, B, C, D, E, FX, F

•                     INL1 – Homework assignments, 4.5, grade scale: A, B, C, D, E, FX, F

Final grade based on 70% from INL1 and 15% each from PRO1 and PRO2, respectively.

The course requires significant individual effort. Solving the homework problems requires good familiarity with the theory but also an ability to formulate a practical problem using suitable mathematical models and applying the theory to these. The written presentation of solutions and project also provide training in the ability to formulate logical arguments in a way that is considered valid in scientific publications. One of the project assignments is presented in a technical report, the other one in an oral presentation.

The different homework sets and projects cover different parts of the intended learning outcomes. The basic idea of the grading is therefore that an acceptable level has to be reached for all the homework sets and both projects, in order to obtain a passing grade. Also, since the ability to formulate valid mathematical arguments (orally and written) is an important part of the intended learning outcomes, this is judged separately and contributes to the final grade. The details are as follows.

For each homework set and for each of the projects, you obtain points according to the following criteria.

Mathematical/technical content, 1-3 points:

Presentation, 1-2 points:

(only judged if at least 1 point was obtained for the technical content, and only judged for the problems or parts of the project that have been ``solved'' according to above).

These points are summed up and result in a grade according to the table below. When calculating the number of points for the final grade, each of the project points are weighted by a factor two, so that the maximum is 6·5+2(5+5)=50 points. Please note that at least 1+1 points is needed for each of the 6 homework sets and for each of the projects, to obtain a passing grade. Note also that only the grading scale A-D is used for the projects, so that a passing grade for the projects give a D (or higher). 

Students who have passed all but one of the homeworks+projects get the chance to do an extra assignment to pass the course. The corresponding homework/project will then get the minimum passing number of points, whereafter the final grade is calculated as described above.

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.