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Chair for Medical Engineering

System Theory 3 – Stochastic Signals

This course is 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

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Inhalt

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

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Lesson begins

10:15

Lesson ends

11:45

First lesson is on

Tuesday, 14.04.2026

Excercises

Room

ID 04/459 & 04/471

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Excercise begins

08:15

Excercise ends

09:45

First excercise is on

Monday, 20.04.2026

Exam

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Type of exam

Written exam

Exam date

to be defined

Duration of exam

120 minutes

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Registration for the exam

FlexNow

Practical Excercises

Room

CIP-Pool 2

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Excercise begins

10:15

Excercise ends

11:45

First excercise is on

Friday, 24.04.2026

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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.
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Requirements

none

Prior knowledge

Con­tents of the cour­ses in Sys­tem Theo­ry 1 and 2

Literature

  1. Kay, Ste­ven M. “Fun­da­men­tals of Sta­tis­ti­cal Si­gnal Pro­ces­sing, Vo­lu­me I: Esti­ma­ti­on Theo­ry”, Pren­ti­ce Hall, 1993
  2. Kay, Ste­ven M. “Fun­da­men­tals of Sta­tis­ti­cal Si­gnal Pro­ces­sing, Vo­lu­me II: De­tec­tion Theo­ry “, Pren­ti­ce Hall, 1998
  3. Kay, Ste­ven M. “Fun­da­men­tals of Sta­tis­ti­cal Si­gnal Pro­ces­sing, Vo­lu­me III: Prac­tical Al­go­rithm De­ve­lop­ment “, Pren­ti­ce Hall, 2013
  4. Kay, Ste­ven M. “In­tui­ti­ve Pro­ba­bi­li­ty and Ran­dom Pro­ces­ses using MAT­LAB”, Pren­ti­ce Hall, 2005
  5. Mertins, Al­fred “Si­gnal­theo­rie”, Sprin­ger, 2013 http://​www.​springer.​com/​de/​book/​9783834813947
  6. Kro­schel, Kris­ti­an, Ri­goll, Grhard, Schul­ler, Björn W. “Sta­tis­ti­sche In­for­ma­ti­ons­tech­nik”, Sprin­ger Ver­lag, 2011 http://​www.​springer.​com/​de/​book/​9783642159534
  7. Häns­ler, Eber­hard “Sta­tis­ti­sche Si­gna­le. Grund­la­gen und An­wen­dun­gen”, Sprin­ger, 2001
  8. Böhme, Jo­hann F. “Sto­chas­ti­sche Si­gna­le”, Teub­ner Ver­lag, 1998
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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.