# EQ2801 Optimal Filtering 7.5 credits

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.

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## Content and learning outcomes

### Course contents^{}

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.

### Intended learning outcomes^{}

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.

### Course disposition

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## Literature and preparations

### Specific prerequisites^{}

For single course students: 180hp and English B or equivalent

### Recommended prerequisites

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

EQ2300 Digital Signal Processing, EQ2401 Adaptive Signal Processing

### Equipment

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### Literature

D. Simon "Optimal State Estimation", Wiley, 2006, or similarly (to be announced before course start).

## Examination and completion

If the course is discontinued, students may request to be examined during the following two academic years.

### Grading scale^{}

### Examination^{}

- 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.

### Other requirements for final grade^{}

• 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.

### Opportunity to complete the requirements via supplementary examination

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### Opportunity to raise an approved grade via renewed examination

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### Examiner

### Ethical approach^{}

- 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.

## Further information

### Course web

Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.

Course web EQ2801### Offered by

### Main field of study^{}

### Education cycle^{}

### Add-on studies

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