(Created 2011-09-01.)

STATIONARY STOCHASTIC PROCESSES | FMS045 |

**Aim**

The student shall aquire a toolbox containing concepts and models for description and handling of stationary stochastic processes within many different areas, such as, signal processing, automatic control, information theory, economics, biology, chemistry, and medicine. The mathematical and statistical elements are therefore illustrated using a wide variety of examples from different areas of application.

The course shall also give the student the ability to identify the presence of stationary processes in other courses in the education, use the knowledge of stationary processes in other courses, and translate the concepts and tools between different courses, building on stationary processes.

*Knowledge and understanding*

For a passing grade the student must

- be able to perform calculations using expectations, variance, covariance, and cross-covariance within and between different stationary processes,
- be able to calculate the relationship between covariance properties in the timedomain and spectral properties in the frequency domain for one and several processes,
- be able to formulate linear filters using covariance and spectral properties,
- be able to estimate covariance function, spectrum, and other parameters in stationary processes using data.

*Skills and abilities*

For a passing grade the student must

- be able to identify natural situations where a stationary process is a suitable mathematical model, e.g., within at least one engineering, science, or economics application,
- be able to formulate a stationary stochastic process model using a concrete problem within the chosen application,
- be able to suggest model parameters, with the help of data,
- be able to interpret the model and translate model concepts to a conclusion regarding the original problem.

*Judgement and approach*

For a passing grade the student must

- be able to read and interpret technical literature with elements of stationary processes within the chosen application,
- be able to describe the model structure and the conclusions,
- be able to describe the possibilities and limitations of stochastic models.

**Contents**

- Models for stochastic dependence.
- Concepts of description of stationary stochastic processes in the time domain: expectation, covariance, and cross-covariance functions.
- Concepts of description of stationary stochastic processes in the frequency domain: effect spectrum, cross spectrum.
- Special processes: Gaussian process, Wiener process, white noise, Gaussian fields in time and space.
- Stochastic processes in linear filters: relationships between in- and out-signals, auto regression and moving average (AR, MA, ARMA), derivation and integration of stochastic processes.
- The basics in statistical signal processing: estimation of expectations, covariance function, and spectrum.
- Application of linear filters: frequency analysis and optimal filters.

**Literature**

Lindgren, G, Rootzén, H, Sandsten, M: Stationary stochastic processes. Lund 2009.

**Code: **0109.
**Name: **Examination.

**Higher education credits: ** 5.
**Grading scale: **TH.
**Assessment:** Written examination.

**Code: **0209.
**Name: **Laboratory Work.

**Higher education credits: ** 1.
**Grading scale: **UG.
**Assessment:** Computer exercises.