System Theory 3 – Stochastic Signals
Course given by
Prof. Dr.-Ing. Georg Schmitz
Course number
141224
Language
German
Credit Points
6
Hours per week
5
Click here to go to
Moodle
Contents
In signal processing dealing with noise and formulating models for signals with random fluctuations as speech or images is often a central task. The mathematical model for such signals are random processes. To treat such signals, profound knowledge of probability theory and random variables is a prerequisite. This course teaches the mathematical methods that are needed and based on that treats estimation theory and detection theory as the two main topics.
Introduction
Definition of stochastic processes
Probability distributions and densities for stochastic processes
Moment functions, autocovariance, crosscovariance, autocorrelation, crosscorrelation
Properties of covarianve and correlation functions, stationarity and ergodicity, power spectral density, white noise processes
Detection theory
Binary decisions, Bayes-test, Maximum-a-posteriori (MAP) test, Maximum-Likelihood-test, MiniMax-test
Receiver-Operating-Characteristics (ROC)
Parameter Estimation
Estimates and estimators
Bias, consistency, Cramér-Rao Lower Bound, efficiency
Least squares estimators and Maximum Likelihood estimators
Random signals and systems
Transfer by LTI systems
Linear processes (AR, MA, ARMA)
Yule-Walker-equations
Wiener-filter
Statistics for random processes
Estimation of the covariance function
Spectral estimation with the periodogram
Parameter estimation for linear processes
Lectures
Room
HID
Lesson begins
10:15
Lesson ends
11:45
First lesson is on
Tuesday, 14.04.2026
Excercises
Room
ID 04/459 & 04/471
Excercise begins
08:15
Excercise ends
09:45
First excercise is on
Monday, 20.04.2026
Exam
Type of exam
Written exam
Exam date
to be defined
Duration of exam
120 minutes
Registration for the exam
FlexNow
Practical Excercises
Room
CIP-Pool 2
Excercise begins
10:15
Excercise ends
11:45
First excercise is on
Friday, 24.04.2026
Objectives
The students have subject-specific knowledge of the mathematical treatment of stochastic models for discrete and continuous measured signals. The students understand the necessity of stochastic signal models and their relation to practical problems (measurement accuracy, reliability). They have the qualification to analyze signal transmission and processing problems for random signals, to propose suitable solution methods, to explain them and to implement them in practice. The students know and understand in particular relevant methods for parameter estimation in signal processing and are able to transfer and apply them to new problems. Through the exercises and computer exercises the students are enabled to apply the acquired knowledge in a small team in practice, to explain and evaluate solution approaches and to represent them with supporting arguments. The important basic nomenclature of stochastic signals is also translated to English, so that students are able to understand the international literature in the field of statistical signal processing.
Requirements
none
Prior knowledge
Contents of the courses in System Theory 1 and 2
Literature
- Kay, Steven M. “Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory”, Prentice Hall, 1993
- Kay, Steven M. “Fundamentals of Statistical Signal Processing, Volume II: Detection Theory “, Prentice Hall, 1998
- Kay, Steven M. “Fundamentals of Statistical Signal Processing, Volume III: Practical Algorithm Development “, Prentice Hall, 2013
- Kay, Steven M. “Intuitive Probability and Random Processes using MATLAB”, Prentice Hall, 2005
- Mertins, Alfred “Signaltheorie”, Springer, 2013 http://www.springer.com/de/book/9783834813947
- Kroschel, Kristian, Rigoll, Grhard, Schuller, Björn W. “Statistische Informationstechnik”, Springer Verlag, 2011 http://www.springer.com/de/book/9783642159534
- Hänsler, Eberhard “Statistische Signale. Grundlagen und Anwendungen”, Springer, 2001
- Böhme, Johann F. “Stochastische Signale”, Teubner Verlag, 1998
Miscellaneous
Registration is carried out via the E-Learning Portal Moodle of the Ruhr-Universität Bochum. The required information is provided in the first lecture.