# EQ1270 Stochastic Signals and Systems 6.0 credits

The course gives a broad overview of modeling using stochastic processes in electrical engineering applications. Formulating problems using mathematical modeling is an important part of the course.

Basics about continuous time an discrete time stochastic processes, especially weakly stationary processes. Definitions of probability distribution and density functions, statistical mean, mean power, variance, autocorrelation function, power spectral density, Gaussian processes and white noise. Linear filtering of stochastic processes.

Ergodicity: Estimation of statistical properties from measurements.

Sampling and reconstruction: Transformations between continuous and discrete time signals. Influence of sampling, sampling theorem. Pulse amplitude modulation. Errrors in the reconstruction of stochastic signals.

Estimation theory: Linear estimation, orthogonality conditions. Prediction and Wiener filtering. Model based signal processing: Linear signal models, AR-models. Spectral estimation.

Application of the above to simpler electrical engineering applications.

### Choose semester and course offering

Choose semester and course offering to see information from the correct course syllabus and course offering.

Headings with content from the Course syllabus EQ1270 (Autumn 2021–) are denoted with an asterisk ( )

## Content and learning outcomes

### Course contents

The course gives a broad introduction to modelling by means of stochastic processes in electrical engineering applications. Problem formulation with mathematical models is an important part of the course.

Basic about time continuous and discrete-time stochastic processes, especially weakly stationary. Definitions such as distribution and density functions, expectation, mean power, variance, autocorrelationfunction, spectral density.Gaussian processes and white noise. Linear filtering of stochastic processes.

The notion of ergodicity: Estimation of the properties of processes from measurements.

Sampling and reconstruction: Transformation between continuous-time and discrete-time signals. Effect of sampling. Sampling theorem. Pulse amplitude modulation. Errors at reconstruction of stochastic signals.

Estimation theory: Linear estimation and the ortogonality principle. Prediction and Wiener filter. Model-based signal processing: Linear signal models, AR models. Spectral estimation.

Application of the above on simple electrotechnical applications.

### Intended learning outcomes

A student who has passed the course should be able to:

• Show basic understanding of properties of stochastic processes.
• Analyse given issues in estimation or optimal filtering.
• Apply mathematical modelling on problems in electrical engineering.
• Demonstrate an understanding of sampling and reconstruction of weakly stationary stochastic processes.
• Develop simple programme code, e.g., by means of the tool Matlab, and use this code to simulate and analyse problems in the area, and report the implementation and the result.
• Use a given or individually formulated mathematical model for solving a given technical problem in the area and analyse the result and its plausibility.

A student who has completed the course with higher grades should in addition to the aims that apply to pass be able to:

• Show good understanding of properties of stochastic processes.
• Analyse given issues in filtering, sampling and reconstruction of weakly stationary stochastic processes.
• Analyse given issues in estimation and optimal filtering.
• Formulate mathematical models that are applicable and relevant at a given problem in the area. When it is missing explicitly given information in the problem, the student should be able to assess and compare different possibilities and make reasonable own assumptions to achieve an adequate modelling.
• Use a given or individually formulated mathematical model for solving a problem in the area e.g. problems that are built-up of several interacting partial problems or such that require more advanced mathematical modelling, and analyse the result and its plausibility.

### Course disposition

No information inserted

## Literature and preparations

### Specific prerequisites

Knowledge in signals and systems in continuous time, 6 higher education credits, equivalent to completed course EQ1110.

Knowledge in signals and systems in discrete time, 6 higher education credits, equivalent to completed course EQ1120.

Active participation in a course offering where the final examination is not yet reported in LADOK is considered equivalent to completion of the course. Registering for a course is counted as active participation. The term 'final examination' encompasses both the regular examination and the first re-examination.

### Recommended prerequisites

Probability theory corresponding to SF1920.

### Equipment

No information inserted

### Literature

No information inserted

## Examination and completion

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

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

### Examination

• PRO1 - Project Work, 1.0 credits, grading scale: P, F
• PRO2 - Project Work, 1.0 credits, grading scale: P, F
• TEN1 - Exam, 4.0 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.

### Opportunity to complete the requirements via supplementary examination

No information inserted

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

No information inserted

### 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 EQ1270

### Offered by

EECS/Intelligent Systems

Technology

### Education cycle

First cycle

EQ2300 Digital Signal Processing

EQ2310 Digital Communications

EQ2321 Speech and audio signal processing

EQ2330 Image and video processing

EQ2341 Pattern recognition and machine learning

### Contact

Magnus Jansson (janssonm@kth.se)

### Supplementary information

In this course, the EECS code of honor applies, see:
http://www.kth.se/en/eecs/utbildning/hederskodex.