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.

• 8 Lectures
• 6 sets of weekly homeworks. Individual solutions.
• Two project assignments; one with written report, one with oral presentation+submitted software implementation. Groups of 2 students.

See the modules in Canvas for detailed dates.

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

EQ2300 Digital Signal Processing, EQ2401 Adaptive Signal Processing

Material from several sources:

• 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, 2022.
• Selected journal articles

All available electronically through KTH library!

Matlab and/or Python, for the homework and projects.

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

Funka - compensatory support for students with disabilities

Please inform the course coordinator if you have special needs, and show your certificate from Funka.

• Support measures under code R (i.e. adjustments related to space, time, and physical circumstances, e.g. longer writing time) are always granted.
• Support measures under code P (pedagogical measures) may be granted or rejected by the examiner after you have applied for this in accordance with KTH rules. Support measures under code P are usually always granted for courses given at EECS.

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.

•                     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.

Individual solutions are required for all the weekly homeworks.

Each submission (full homework set/project), is graded as a whole according to the following two criteria.

Technical/mathematical content, 1-3 points:

1. point: More than 50% of the difficulties have been solved, at least by providing a valid approach and working out some details.
2. points: More than 75% of the difficulties have been solved, at least by providing a valid approach and working out some details. In addition, at least two of the problems have been solved fully.
3. points: Full solutions are provided for all problems and only minor details are flawed/missing

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):

1. point: The presentation manages to convey the main ideas of the solved problems, but not fully convincing in all details.
2. points: Clear and convincing presentation of the solved problems.

To summarize, each approved submission gets between 1+1=2 points and 3+2=5 points.

The final grade is based on 6 homeworks + 2 projects, where project points are weighted by 2 ⇒ maximum (6 + 2 · 2) · 5 = 50 points.

• 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.

Code of Honour

In this course, the EECS code of honor applies.