# course: System Theory 3: Stochastic Signals

- number:
- 141224
- teaching methods:
- lecture with tutorials
- responsible person:
- Prof. Dr.-Ing. Georg Schmitz
- Lecturers:
- Prof. Dr.-Ing. Georg Schmitz (ETIT), Dr.-Ing. Stefanie Dencks (ETIT)
- language:
- german
- HWS:
- 5
- CP:
- 6
- offered in:
- summer term

## dates in summer term

- start: Tuesday the 21.04.2020
- lecture Tuesdays: from 10:15 to 11.45 o'clock
- tutorial Mondays: from 08:15 to 09.45 o'clock
- lab exercise Fridays: from 10:15 to 11.45 o'clock (every other week)

## Exam

##### All statements pertaining to examination modalities (for the summer/winter term of 2020) are given with reservations. Changes due to new requirements from the university will be announced as soon as possible.

Form of exam: | written |

Registration for exam: | FlexNow |

Date: | 28.07.2020 |

Begin: | 15:30 |

Duration: | 120min |

Rooms : | HZO 20, HZO 50 |

Individual appointments of students to each exam location will be issued by the responsible chair. |

## goals

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.

## content

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

## requirements

keine

## recommended 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

Die Vorlesungs- und Übungsunterlagen werden über Moodle zur Verfügung gestellt. Die notwendigen Informationen erhalten Sie in der ersten Vorlesung.