LNTwww - User contributions [en]

LNTwww:LNTwww:Imprint for the book "Stochastic Signal Theory"

2025-02-24T13:40:01Z

Höfler: Text replacement - "Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29" to "Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29"

'''Five main chapters with a total of 28 chapters (files);   93 Exercises   ⇒   Two semester hours per week lecture and one semester hour per week exercises      
Development of the German version:   2011–2015.       Development of the English version:   2021.      Last corrections:   November 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Söd88]'''<ref name='Söd88'>Söder, G.:  Aufgabensammlung zu &bdquo;Statistische Methoden der Nachrichtentechnik”. Lehrstuhl für Nachrichtentechnik, TU München, 1988.</ref>;  '''[Söd12]'''<ref name='Söd12'>Söder, G.:  Simulationsmethoden in der Nachrichtentechnik.  Anleitung zum gleichnamigen Praktikum. Lehrstuhl für Nachrichtentechnik, TU München, 2012.</ref>;  –  Textbooks:   '''[Söd93]'''<ref name='Söd93'>Söder, G.:  Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin – Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2.</ref>;    '''[Dav87]'''<ref name='Dav87'>Davenport, W. B.: Probability and random processes. An introduction for applied scientists and engineers. New York NY u.a.: McGraw-Hill, 1987. ISBN 0-07-015440-6</ref>;  '''[PP09]'''<ref name='PP09'>Papoulis, A.; Pillai, S. U.: Probability, random variables, and stochastic processes. 4. Aufl. Boston, Mass.: McGraw-Hill, 2009. ISBN 978-0-07122-661-5</ref>

*Author:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29|'''Günter Söder''']]

*Participating colleagues in alphabetic order:  [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29| '''Benedikt Leible''']],   [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Thomas_Stockhammer_.28at_LNT_from_1995-2004.29|'''Thomas Stockhammer''']],  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|'''Johannes Zangl''']]

*Participating students in chronological order:      Martin Winkler '''(2001)''', Yven Winter, Reinhold Sixt, Jürgen Veitenhansl, Franz Kohl, Bettina Hirner, Ji Li, Markus Elsberger, Thorsten Kalweit, Thomas Großer, David Jobst, Matthias Niller, Veronika Hofmann, Carolin Mirschina, Noah Nagi, Ji Woo Hwang '''(2022)'''

$\text{List of sources:}$

LNTwww:General notes about "Channel Coding"

2025-02-24T13:40:01Z

'''Four main chapters with a total of 22 chapters (files);   98 Exercises   ⇒   Three semester hours per week  lecture and two hours per week  exercises      
Development of the German version:   2011–2017.       Development of the English version:   2022.      Last corrections:   December 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

*Basic materials:   Lecture notes from LNT/LÜT:    '''[Köt08]'''<ref name='Köt08'>Kötter, R.; Mayer, T.; Tüchler, M.; Schreckenbach, F.; Brauchle, J.: Channel Coding. Lecture notes, Lehrstuhl für Nachrichtentechnik, TU München, 2008</ref>;   '''[Liv10]'''<ref name='Liv10'>Liva, G.: Channel Coding. Lecture notes, Lehrstuhl für Nachrichtentechnik, TU München and DLR Oberpfaffenhofen, 2010</ref>;   '''[Liv15]'''<ref name='Liv15'>Liva, G.: Channels Codes for Iterative Decoding. Lecture notes, Lehrstuhl für Nachrichtentechnik, TU München and DLR Oberpfaffenhofen, 2015</ref>   –   Textbooks:   '''[Bos99]'''<ref name='Bos99'>Bossert, M.: Channel Coding for Telecommunications. Chichester: Wiley, 1999. ISBN 978-0-471-98277-7 </ref>;  '''[Cov06]'''<ref name='Cov06'>Cover, T. M.; Thomas, J. A.: Elements of Information Theory. West Sussex: John Wiley & Sons, 2nd Edition, 2006 </ref>;   '''[Hub82]'''<ref name='Hub82'>Huber, J.: Codierung für gedächtnisbehaftete Kanäle. Dissertation – Universität der Bundeswehr München, 1982</ref>

*Authors and involved colleagues in alphabetic order:   [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Ronald_B.C3.B6hnke_.28at_LNT_from_2012-2014.29|'''Ronald Böhnke''']],   [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Joschi_Brauchle_.28at_LNT_from_2007-2015.29|'''Joschi Brauchle''']],   [http://wirelesscoding.org/ '''Gianluigi Liva'''],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

*Participating students in chronologic order:    Martin Winkler '''(2001)''',  Yven Winter,  Ji Li,  Bettina Hirner,  Thomas Großer,  Thorsten Bürgstein,  Martin Völkl,  Dominik Kopp,  Carolin Mirschina,  Jiwoo Hwang,  Noah Nagy '''(2022)'''

$\text{References:}$

Biographies and Bibliographies/LNTwww members from LNT

2025-02-24T13:39:58Z

{{Header|
Untermenü=An LNTwww beteiligte Mitarbeiter und Dozenten
|Nächste Seite= Beteiligte der Professur Leitungsgebundene Übertragungstechnik
|Vorherige Seite=Lehrstuhlinhaber_des_LNT
}}
During the creation of  "LNTwww"  many colleagues at the LNT gave us great support.  In this context, we understand  "LNT"  to mean the  "Lehrstuhl für Nachrichtentechnik"   ⇒   "Chair of Communications Engineering". 

From the  '''scientific staff'''  we would like to especially thank the following (former) colleagues:

==Dr.-Ing. Ronald Böhnke (at LNT from 2012-2014)==
[[File:Ronald.JPG|165px|right|]]
Ronald Böhnke, born in Bremen in 1976, studied Electrical Engineering and Information Technology at the University of Bremen and worked there as a research assistant from September 2002 in the field of Communications Engineering, where he obtained his doctorate on  "Efficient Detection and Adaptive Transmission for MIMO-OFDM Systems".   During his studies, he completed a three-month internship at Intel Corporation's research center in Santa Clara, California.  In addition, he was a visiting scientist at the Fraunhofer Institute for Telecommunications at the Heinrich Hertz Institute in Berlin for two months in 2002.

In September 2010,  Ronald Böhnke became a research associate at the Chair of Communications and Navigation (NAV) at the Technical University of Munich and contributed to the design of a geostationary relay satellite system.  After completing the project, he moved to the LNT in February 2012.  Since June 2014, he has been working in the field of mobile communications at the  "European Research Center"  of Huawei Technologies Düsseldorf GmbH in Munich.

Ronald Böhnke was a member of the German National Academic Foundation and received the Karl Nix Award for the Baccalaureater, the VDE Award for the Diploma and a "Best Paper Award" at the "IEEE International Workshop on Cross-Layer Design 2007".

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*Competent consultant and expert for the books  "Information Theory" and  "Channel Coding". 
*Often he also had to take corrective actions.}}

==Dr.-Ing. Joschi Brauchle (at LNT from 2007-2015)==

[[File:Brauchle.jpg|165px|right|Joschi Brauchle]]

Joschi Brauchle, born 1982 in Bad Reichenhall, studied electrical engineering and information technology at the Technical University of Munich from 2002 to 2007.  After completing his intermediate diploma, he completed a one-year master's degree at the Georgia Institute of Technology in Atlanta, Georgia, USA in 2005/2006.   In his diploma thesis at LNT he worked on  "Soft-input decoding of Reed-Solomon codes in concatenated systems"  and on  "Soft-output list decoding of inner convolutional codes".

After finishing his diploma thesis, Joschi Brauchle was a research assistant of  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|Prof. Kötter]]  and  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28since_2010.29|Prof. Kramer]].   His research was mainly concerned with algebraic coding theory, and in particular with the properties of bivariate interpolation-based decoding schemes for Reed-Solomon codes, as well as with the error correction performance of multidimensional schemes on related code constructions.  In winter 2013, he spent a three-month research stay at the University of Toronto, Canada.  In December 2015, he received his PhD on the topic  "Algebraic Decoding of Reed-Solomon and Related Codes".

In teaching, Joschi Brauchle designed and supervised the exercise for the lecture  "Channel Coding"  in the master's program from 2008 to 2013, for which he was awarded the lecturer prize of the student council for electrical engineering and information technology in 2013.  He supervised various student papers in seminars as well as quite a few bachelor's and master's theses throughout his assistantship. In 2015 he organized the main seminar.

In addition, Joschi Brauchle was jointly responsible for the conception and maintenance of the computer network and all IT systems at the LNT as a system administrator from 2009 to 2015.

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*In his capacity as system administrator, Joschi realized earlier than those responsible for LNTwww that our  "old LNTwww"  was indeed somewhat outdated.
*He put some basic and good thought into porting it to its present wiki form, for which we are all very grateful to him today.
}}

==Dr.-Ing. Klaus Eichin (at LNT from 1972-2011)==

[[File:K_Eichin.gif|165px|right|Klaus Eichin]]

Klaus Eichin was born in Wolfach in 1946.  He studied electrical engineering at the Technical University of Munich from 1965 onwards and obtained the title of Dipl.-Ing. in 1972.  He was awarded a doctorate in engineering in 1984.

*Already during his diploma thesis he was together with his later doctoral supervisor Prof. Karlheinz Tröndle intensively engaged in the topic  "Use of the computer in teaching".
* Afterwards, he worked at the Chair of Communications Engineering at the Technical University of Munich until his retirement in September 2011. In the last years, he focused on the research area of digital mobile radio.
*Furthermore, Dr. Eichin held events for students of the teaching profession at vocational schools (LB) as Academic Director.
 
{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*In 2001 Klaus Eichin was together with his colleague [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29|Günter Söder]] the initiator of this e-learning project and worked intensively on it until 2011.
*Among other things, Klaus was co-author on six of the total nine LNTwww books. }}

==Dr.-Ing. Francisco Javier García Gómez (at LNT from 2016-2021)==
[[File:Javier.jpg|165px|right|]]

Francisco Javier García Gómez received the B.Eng. degree in Telecommunication Technologies and Services in 2014 from the Technical University of Madrid, (Spain), and the M.Sc. degree in Communications Engineering in 2016 from the Technical University of Munich (TUM), Germany.  The topic of his Master’s Thesis was  “Linear and Non-linear Estimation Methods for Single Carrier and Multicarrier Coarsely Quantized MIMO Systems”.

From 2016 to 2021, he was a doctoral researcher under the supervision of Prof. Gerhard Kramer at TUM’s Institute for Communications Engineering (LNT).  His research work was about the information theoretical analysis of the nonlinear optical fiber channel, with the goal of finding bounds on its capacity.

He taught the tutorial of the Bachelor lecture  “Mobile Communications” and the tutorial of the Master lecture   “Advanced Topics in Communications Engineering”. He was also responsible for the organization of Bachelor/Master Thesis Seminars.

 
{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*He has done essential preliminary work to be able to generate the English  "$\rm en.LNTwww.de$"  from  "$\rm www.LNTwww.de$"  with reasonable effort.
*2020/2021, Javier led the student translation team as one of the project managers for the English version. During this time, four books were completed.

}}

==Dr.-Ing. Thomas Hindelang (at LNT from 1994-2000 und 2007-2012)==

[[File:T_Hindelang_Sept15a.jpg|165px|right|]]

Thomas Hindelang received the Dipl.-Ing. and the PhD degree in electrical engineering from Technische Universität München (TUM), Germany, in 1994 and 2001, respectively. He worked as a Research Assistant at the Institute of Communications Engineering from 1994 to 2000, focusing primarily on the field of combined source and channel coding for mobile communications. While on leave from TUM, he served as a consultant at the Communications Research Department of AT&T Labs, Florham Park, NJ, from March to August 1999.
From 2000 to 2008 he has been with Siemens AG and Nokia Siemens Networks GmbH & Co. KG, first in the area of Mobile Devices and later in the area of Mobile Networks. From 2002 to 2006 he actively participated and contributed to the 3GPP standardization body on physical layer issues for the Multimedia Broadcast and Multicast Service (MBMS) and 3G Long Term Evolution (LTE) standards. From 2006 till 2008 he led the group "Baseband Algorithms and Simulations", where he was responsible for the physical layer algorithms and simulations for GSM, UMTS, LTE, and WiMAX hardware development. Since October 2007 he is senior lecturer at TUM, and since November 2008 he works as an examiner in the area "Audio Video Media" at the European Patent Office.
Dr. Hindelang holds approximately 40 patents and is the author of more than 30 published conference and journal papers. He contributed to the book "Advances in Digital Speech Transmission", edited by R. Martin, U. Heute, and C. Antweiler. His research interest includes the entire physical layer from the source until the channel, i.e., speech and video coding, error correction coding, equalization, and channel estimation techniques.

A great hobby of Thomas Hindelang is endurance sports.  As successes are placements in the top 10% at several marathons in Munich and Berlin, the placement in the top third at the mountain runs Jungfrau Marathon (42 km, 1800 vertical meters) and "Swiss Alpine"  (78 km, 2300 vertical meters)  and the time of 10:35 h at the Ironman in Regensburg 2011.

{{BlaueBox|TEXT=
'''His contributions to the LNTwww project''':  
*From 2007 - 2012, Dr. Hindelang held the lecture "Mobilfunk" at the Institute of Communications Engineering as a lecturer.
*His contributions to LNTwww, in particular to the books "Mobile Communications" and "Examples of Communication Systems", also date from this time.}}

==Dr.-Ing. Tobias Lutz (at LNT from 2008-2014)==

[[File:Lutz.jpg|165px|right|Tobias Lutz]]

Tobias Lutz, born in 1980 in Krumbach, studied electrical engineering and information technology at the Technical University of Munich from 2002 to 2008.  After completing his undergraduate studies, he moved to the  "Rensselaer Polytechnic Institute"  (RPI) in Troy, New York, for a one-year guest study in 2005/06.  From 2008 to 2014, Tobias Lutz was a research assistant to  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|Prof. Ralf Kötter]]  and  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29|Prof. Gerhard Kramer]]  at the Institute of Communications Technology (LNT) at the Technical University of Munich.  During this time, he also completed a degree in business mathematics at LMU Munich.

During the first three years of his assistantship, Tobias Lutz worked for the European research project  "N-Crave", where he dealt with theoretical aspects of network coding as well as the implementation of network protocols.

From August 2012 to April 2013, he spent an eight-month research stay at  "Stanford University".  During this time, he attended various courses from the field of mathematics and computer science and engaged in research on memory-based channels.  Regarding teaching, Tobias Lutz organized the basic practical course in communications engineering for four years.  Furthermore, he supervised student work within the framework of various.  In the last third of his assistantship Tobias Lutz was twice supervisor of the lecture  "Advanced Topics in Communications Engineering"  as well as of the lecture  "Information Theory".  In 2012, he received a  "Qualcomm Innovation Fellowship"  in the amount of 10.000 €.

His scientific interest during his LNT time was  "Multi-user Information Theory".  Specifically, he worked on the  "design of timing codes for half-duplex constrained networks"  and their information theoretic analysis.  This was also the topic of his PhD thesis (2014) with Prof. Gerhard Kramer.

{{BlaueBox|TEXT=
'''His contributions to the LNTwww project''':  
* He has been very supportive in the preparation of the book  [[Information theory]], 
*especially in the understandable formulation of not quite simple mathematical relations.}}

==Dr.-Ing. Michael Mecking (at LNT from 1997-2012)==

[[File:Mecking.jpg|165px|right|]]

Michael Mecking, born on 23.2.1972 in Karlsruhe, was a scientific assistant at the LNT from 1997 to 2003.  During this time, he worked on the projects "Receiver techniques for UMTS" and "Access strategies for the uplink of CDMA systems with channel-controlled scheduling" with Siemens.  In 2001 he spent five months at the EPFL in Lausanne with Prof. Bixio Rimoldi.

The focus of his research was multi-user information theory for mobile radio channels.  He completed his PhD thesis on  "Fading Multiple-Access with Channel State Information"  in November 2003.  His thesis advisor was Prof. [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|Joachim Hagenauer]].  The dissertation was awarded the Rohde & Schwarz Prize of the Faculty of Electrical Engineering and Information Technology in 2004.

Since September 2003, Dr. Mecking has been working at BMW AG in various areas of electrics and electronics, first for human-machine interface design and later in electrics/electronics integration.  From 2015 to 2017, he was responsible for change control in the area of electrics/electronics, vehicle dynamics and powertrain at BMW Manufacturing Co in South Carolina, USA.  Since his return to Germany in the summer of 2017, Dr. Mecking has led the department for integration and validation of the electrical/electronic vehicle dynamics and powertrain scopes at BMW's Regensburg and Oxford sites.

{{BlaueBox|TEXT=
'''His contributions to the LNTwww project''':  
*From 2004 to 2012, Dr. Mecking taught the lecture "Information Theory and Source Coding" as a lecturer, for which he was already responsible in 1998-2002.
*His script at that time was a great help for us in writing the book "Information Theory".  We were able to adopt much of it with his approval.}}

==Prof. Dr.-Ing. habil. Günter Söder (at LNT from 1974-2024)==

[[File:G_Soeder.gif|165px|right|Günter Söder]]

Günter Söder was born in Nürnberg on March 21, 1946.  He studied Electrical Engineering and Information Technology from 1964 at the  "Ohm–Polytechnikum Nürnberg"  (today:  Technical University Nuremberg Georg Simon Ohm)  and at the "Technischen Hochschule München"  (today:  Technical University Munich).  He received the academic titles Ing.-grad. (1967),  Dipl.-Ing. (1974),  Dr.-Ing. (1981)  and  Dr.-Ing. habil. (1993).

*Günter Söder worked at the Department of Communications Engineering of the Technical University of Munich from 1974 until his retirement in 2011,  mainly in the fields  "Digital Transmission Systems"  and  "Stochastic Signal Theory".  He held various lectures and practical courses on these topics as Academic Director and Associate Professor (since 2004).  
*He published the textbooks  "Digitale Übertragungssysteme – Theorie, Optimierung und Dimensionierung der Basisbandsysteme"  (1985 by Springer-Verlag Berlin)  and the English version  "Optimization of Digital Transmission Systems"  (1987 by Artech House Inc., Boston),  in each case jointly with his doctoral supervisor,  Prof. Karlheinz Tröndle.  His postdoctoral thesis  "Modellierung, Simulation und Optimierung von Nachrichtensystemen"  was also published by Springer-Verlag in 1993.
*In 1986 Günter Söder was awarded the NTG Prize  (today: Literature Prize of the Information Technology Society)  and in 1992 he and his team of diploma students were awarded the German-Austrian University Software Prize.

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*Günter Söder was together with  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|Klaus Eichin]]  the initiator of our learning tutorial and and is involved in all books as author and editor of the Gerrman version.
*It was of advantage that he has been intensively involved in the creation of educational programs  ("LNTsim"  and  "LNTwin")  since 1984.
*In 2024 he brought this e-learning project, started in 2001, to  "a good end"  (from his subjective point of view)  – thirteen years after his retirement.
*Günter Söder is still responsible for the German version  $($'''www.LNTwww.de'''$)$  and supports his colleagues with the English version  $($'''en.LNTwww.de'''$)$.
}}

==Dr.-Ing. Markus Stinner (at LNT from 2011-2016)==

[[File:Stinner1.jpg|165px|right|Markus Stinner]]

Markus Stinner was born in Ulm in 1986.  He studied electrical engineering at the University of Ulm from 2006 to 2011.  His studies included a semester abroad at the University of Adelaide, Australia in 2008.

After finishing his diploma thesis at the Institute of Telecommunications and Applied Information Technology,  which dealt with  "Partial Unit Memory Codes"  based on Gabidulin codes,  he was a research assistant to  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29|Prof. Kramer]]  at LNT from 2011.  His focus was on the analysis of  "Spatially Coupled Low-Density Parity-Check Codes"  for finite lengths.  He was at ENSEA in Cergy-Pontoise for a research visit in June 2012, at Lund University in Sweden in November 2015, and at EPFL in Lausanne in April 2016.  The topic of his PhD in September 2016 was  "Analysis of Spatially Coupled LDPC Codes on the Binary Erasure Channel".

In teaching, Markus Stinner supervised the  "Basic Laboratory Course on Telecommunications"  for many years and since 2014 the exercises for the lecture  "Channel Codes for Iterative Decoding".  In addition, he was responsible for the lecture  "Communications Technology 1"  at TUM¬Asia in Singapore in 2015.  In 2011/2012, his other tasks at LNT included working on the project  "CONE - Coding for Networks"  in collaboration with Alcatel Lucent Bell Labs, in which efficient channel codes and modulations for wireless links such as backhaul and 5G systems were developed and compared.

More information on Dr. Stinner's career can be found  [https://www.linkedin.com/in/markusstinner '''here'''].

{{BlaueBox|TEXT=
'''His contributions to the LNTwww project''':  
*He was at LNT for several years as a system administrator jointly responsible for design/maintenance of the computer network and all IT systems.
*In 2016, Markus led the porting of the "old LNTwww" (version 2) to the present wiki form (version 3) with a team of students.
*At the end of 2016, Markus left the LNT.  Building on his work, we have been able to release the "new LNTwww" two years after his departure.
*Many thanks for your excellent and forward-looking work.  Without your almost penetrating persistence, this improved version would not have existed.}}

==Dr.-Ing. Thomas Stockhammer (at LNT from 1995-2004)==

[[File:Stocki.jpg|165px|right|Markus Stinner]]

Thomas Stockhammer, born in Traunstein, Germany, in 1971, studied and earned his doctorate at the Institute of Communications Engineering at the Technical University of Munich (TUM). During this time he visited the Rensselear Politechnical Institute (RPI), Troy, NY, USA and the University of California San Diego (UCSD), San Diego, CA, USA as a visiting researcher.

After 10 years as founder and executive director for "Novel Mobile Radio" (NoMoR) research, he joined Qualcomm in 2014 and now acts as Senior Director Technical Standards – working from the scenic Chiemgau in his home office. In his different roles, he coauthored more than 250 research publications, more than 250 patents and thousands of standards contributions. He is the active and has board, leadership and rapporteur positions in 3GPP, DVB, MPEG, IETF, ATSC, CTA, ETSI, Metaverse Standards Forum, SVTA and the DASH-Industry Forum in multimedia communication, TV-distribution, 5G broadcast, content delivery protocols, immersive media representation, adaptive streaming, XR and the Metaverse. Among others, he leads MPEG-I Scene Description efforts, SVTA DASH-IF working group, 3GPP Video and XR activities as well as DVB-5G activities.

He received several awards for work on DASH, media delivery and 5G Broadcast, namely the INCITS Technical Excellence Award 2013, the 3GPP Excellence award 2017, the CTA Technology & Standards Achievement Award 2019 and 2023, ISO/IEC Excellence Award 2024, the DASH-IF Leadership Award 2024, the IEC1906 award as well as an Emmy Inventor Award for in 2022. He is regular speaker and Program Committee Member at events such as IBC, DVB World, Mile High Video, MWS or BroadThinking. In January 2023, he was elevated to IEEE Fellow for his contributions to media delivery and video streaming standards. For more details see: https://www.linkedin.com/in/stockhammer/

{{BlaueBox|TEXT=
'''His contributions to the LNTwww project''':  
*Competent advisor and expert for the book "Theory of Stochastic Signals".}}

==Dr.-Ing. Johannes Zangl (at LNT from 2000-2006)==

[[File:Zangl_2.jpg|165px|right|Johannes Zangl]]

Johannes Zangl, born in Augsburg in 1974, studied electrical engineering and information technology at the TUM and was a research associate at the LNT from 2000.  During this time, he worked on a multi-year DFG project in the priority program "Adaptivity in heterogeneous communication networks with wireless access (AKOM)", among others.  He received his PhD in 2005 on the topic of "Multi-hop networks with channel coding and medium access control (MAC)" under Prof.  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|Joachim Hagenauer]].

In the area of teaching, Johannes Zangl supervised the lectures "Fundamentals of Information Technology" and the "Mobile Communications Lab" for many years, as well as "Channel Coding" in the summer semester 2005.  In addition, he was responsible for the support of the LNT computer network and the mobile communications lab as a system administrator.

From 2006 Dr. Zangl was an employee at Infineon Technologies in Munich and worked as a development engineer in the verification of VDSL2.   In 2008, he moved to Rohde & Schwarz in Munich to the Center of Competence for Digital Signal Processing, where his tasks included the development of FPGA components for radar signal analysis systems, among others.  In June 2014, Dr. Zangl moved in-house to device development for network and spectrum analyzers.

{{BlaueBox|TEXT=
'''His contributions to the LNTwww project''':
*To his great credit, he scanned our crashed hard drive bit by bit for LNTwww parts in 2005, largely salvaging four years of work.
*Since then we know that it is not enough to make a backup, but that it must also be configured correctly.
*He was also a competent contact person and proofreader for the book "Theory of Stochastic Signals".}}

==Dr.-Ing. Georg Zeitler (at LNT from 2007-2012)==

[[File:Zeitler.jpg|165px|right|Johannes Zangl]]

Georg Zeitler, born in Munich in 1982, studied electrical engineering and information technology at the Technical University of Munich from 2002.  After completing his bachelor's thesis, he transferred to the University of Illinois at Urbana-Champaign in 2005 for a two-year master's degree, where he specialized in signal processing and information engineering.  After completing his master's thesis at UIUC on universal prediction of individual sequences, G. Zeitler was a research assistant to Prof. Kötter and Prof. Kramer from 2007 to 2012.  His scientific interests were information-theoretic aspects and the optimization of low-resolution quantizers for intelligence systems.  In spring 2010, he spent a three-month research stay at UIUC.

In teaching, Georg designed and supervised the central exercises and partly the lecture for the course "Communications Engineering 2".  In addition, he supervised student work in the two seminars offered by the LNT and he acted as the students' contact person for industrial contacts (engineering practice in the bachelor's program EI).

From 2007 to 2010, his other tasks at the LNT included working on the research project  "Network Coding for Multihop Relaying", n which quantize-and-forward strategies for relay networks were developed in collaboration with DOCOMO Euro-Labs GmbH.

Since June 2012, Dr. Zeitler has been working at BMW AG in Munich.

{{BlaueBox|TEXT=
'''His contributions to the LNTww project''':
*The main chapter 4 of the LNTwww book [[Digital Signal Transmission]] is largely based on manuscripts by him and Prof. [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|Ralf Kötter]]. }}

{{Display}}

Applets:PDF, CDF and Moments of Special Distributions

2025-02-24T13:39:58Z

{{LntAppletLinkEnDe|wdf-vtf_en|wdf-vtf}}

==Applet Description==
 
The applet presents the description forms of two continuous value random variables  $X$  and  $Y\hspace{-0.1cm}$.  For the red random variable  $X$  and the blue random variable  $Y$,  the following basic forms are available for selection:

* Gaussian distribution, uniform distribution, triangular distribution, exponential distribution, Laplace distribution, Rayleigh distribution, Rice distribution, Weibull distribution, Wigner semicircle distribution, Wigner parabolic distribution, Cauchy distribution.

The following data refer to the random variables  $X$. Graphically represented are
* the probability density function  $f_{X}(x)$  (above) and
* the cumulative distribution function  $F_{X}(x)$  (bottom).

In addition, some integral parameters are output, namely
*the linear mean value  $m_X = {\rm E}\big[X \big]$,
*the second order moment  $P_X ={\rm E}\big[X^2 \big] $,
*the variance  $\sigma_X^2 = P_X - m_X^2$,
*the standard deviation  $\sigma_X$,
*the Charlier skewness  $S_X$,
*the kurtosis  $K_X$.

==Definition and Properties of the Presented Descriptive Variables==
 
In this applet we consider only ''(value–)continuous random variables'', i.e. those whose possible numerical values are not countable.
*The range of values of these random variables is thus in general that of the real numbers  $(-\infty \le X \le +\infty)$.
*However, it is possible that the range of values is limited to an interval:  $x_{\rm min} \le X \le +x_{\rm max}$.
 

===Probability density function (PDF)===
For a continuous random variable  $X$  the probabilities that  $X$  takes on quite specific values  $x$  are zero:  ${\rm Pr}(X= x) \equiv 0$.  Therefore, to describe a continuous random variable, we must always refer to the  ''probability density function''  – in short  $\rm PDF$. 

{{BlaueBox|TEXT=
$\text{Definition:}$  The value of the  »'''probability density function'''«  $f_{X}(x)$  at location  $x$  is equal to the probability that the instantaneous value of the random variable  $x$  lies in an  (infinitesimally small)  interval of width  $Δx$  around  $x_\mu$,  divided by  $Δx$:

:$$f_X(x) = \lim_{ {\rm \Delta} x \hspace{0.05cm}\to \hspace{0.05cm} 0} \frac{ {\rm Pr} \big [x - {\rm \Delta} x/2 \le X \le x +{\rm \Delta} x/2 \big ] }{ {\rm \Delta} x}.$$

}}

This extremely important descriptive variable has the following properties:

*For the probability that the random variable  $X$  lies in the range between  $x_{\rm u}$  and  $x_{\rm o} > x_{\rm u}$: 
:$${\rm Pr}(x_{\rm u} \le X \le x_{\rm o}) = \int_{x_{\rm u}}^{x_{\rm o}} f_{X}(x) \ {\rm d}x.$$
*As an important normalization property,  this yields for the area under the PDF with the boundary transitions  $x_{\rm u} → \hspace{0.1cm} – \hspace{0.05cm} ∞$  and  $x_{\rm o} → +∞$:
:$$\int_{-\infty}^{+\infty} f_{X}(x) \ {\rm d}x = 1.$$
 

===Cumulative distribution function (CDF)===

The  ''cumulative distribution function''  – in short  $\rm CDF$  – provides the same information about the random variable  $X$  as the probability density function.

{{BlaueBox|TEXT=
$\text{Definition:}$  The  »'''cumulative distribution function'''«  $F_{X}(x)$  corresponds to the probability that the random variable  $X$  is less than or equal to a real number  $x$: 
:$$F_{X}(x) = {\rm Pr}( X \le x).$$}}

The CDF has the following characteristics:

*The CDF is computable from the probability density function  $f_{X}(x)$  by integration.  It holds:
:$$F_{X}(x) = \int_{-\infty}^{x}f_X(\xi)\,{\rm d}\xi.$$
*Since the PDF is never negative,  $F_{X}(x)$  increases at least weakly monotonically,  and always lies between the following limits:
:$$F_{X}(x → \hspace{0.1cm} – \hspace{0.05cm} ∞) = 0, \hspace{0.5cm}F_{X}(x → +∞) = 1.$$
*Inversely,  the probability density function can be determined from the CDF by differentiation:
:$$f_{X}(x)=\frac{{\rm d} F_{X}(\xi)}{{\rm d}\xi}\Bigg |_{\hspace{0.1cm}x=\xi}.$$
*For the probability that the random variable  $X$  is in the range between  $x_{\rm u}$  and  $x_{\rm o} > x_{\rm u}$  holds:
:$${\rm Pr}(x_{\rm u} \le X \le x_{\rm o}) = F_{X}(x_{\rm o}) - F_{X}(x_{\rm u}).$$
 

===Expected values and moments===
The probability density function provides very extensive information about the random variable under consideration.  Less,  but more compact information is provided by the so-called  "expected values"  and  "moments".

{{BlaueBox|TEXT=
$\text{Definition:}$  The  »'''expected value'''«  with respect to any weighting function  $g(x)$  can be calculated with the PDF  $f_{\rm X}(x)$  in the following way:
:$${\rm E}\big[g (X ) \big] = \int_{-\infty}^{+\infty} g(x)\cdot f_{X}(x) \,{\rm d}x.$$
Substituting into this equation for  $g(x) = x^k$  we get the  »'''moment of $k$-th order'''«:
:$$m_k = {\rm E}\big[X^k \big] = \int_{-\infty}^{+\infty} x^k\cdot f_{X} (x ) \, {\rm d}x.$$}}

From this equation follows.
*with  $k = 1$  for the  ''first order moment''  or the  ''(linear)  mean'':
:$$m_1 = {\rm E}\big[X \big] = \int_{-\infty}^{ \rm +\infty} x\cdot f_{X} (x ) \,{\rm d}x,$$
*with  $k = 2$  for the  ''second order moment''  or the  ''second moment'':
:$$m_2 = {\rm E}\big[X^{\rm 2} \big] = \int_{-\infty}^{ \rm +\infty} x^{ 2}\cdot f_{ X} (x) \,{\rm d}x.$$

In relation to signals,  the following terms are also common:
* $m_1$  indicates the  ''DC component'';    with respect to the random quantity  $X$  in the following we also write  $m_X$.
* $m_2$  corresponds to the ''signal power''  $P_X$   (referred to the unit resistance  $1 \ Ω$ ) .

For example, if  $X$  denotes a voltage, then according to these equations  $m_X$  has the unit  ${\rm V}$  and the power  $P_X$  has the unit  ${\rm V}^2.$ If the power is to be expressed in "watts"  $\rm (W)$, then  $P_X$  must be divided by the resistance value  $R$. 
 

===Central moments===

Of particular importance in statistics in general are the so-called  ''central moments'' from which many characteristics are derived,

{{BlaueBox|TEXT=
$\text{Definition:}$  The  »'''central moments'''«,  in contrast to the conventional moments, are each related to the mean value  $m_1$  in each case. For these, the following applies with  $k = 1, \ 2,$ ...:

:$$\mu_k = {\rm E}\big[(X-m_{\rm 1})^k\big] = \int_{-\infty}^{+\infty} (x-m_{\rm 1})^k\cdot f_x(x) \,\rm d \it x.$$}}

*For mean-free random variables, the central moments  $\mu_k$  coincide with the noncentral moments  $m_k$. 

*The first order central moment is by definition equal to  $\mu_1 = 0$.

* The noncentral moments  $m_k$  and the central moments  $\mu_k$  can be converted directly into each other.  With  $m_0 = 1$  and  $\mu_0 = 1$  it is valid:
:$$\mu_k = \sum\limits_{\kappa= 0}^{k} \left( \begin{array}{*{2}{c}} k \\ \kappa \\ \end{array} \right)\cdot m_\kappa \cdot (-m_1)^{k-\kappa},$$
:$$m_k = \sum\limits_{\kappa= 0}^{k} \left( \begin{array}{*{2}{c}} k \\ \kappa \\ \end{array} \right)\cdot \mu_\kappa \cdot {m_1}^{k-\kappa}.$$
 
===Some Frequently Used Central Moments===

From the last definition the following additional characteristics can be derived:

{{BlaueBox|TEXT=
$\text{Definition:}$  The  »'''variance'''«  of the considered random variable  $X$  is the second order central moment:
:$$\mu_2 = {\rm E}\big[(X-m_{\rm 1})^2\big] = \sigma_X^2.$$
*The variance  $σ_X^2$  corresponds physically to the  "switching power"  and  »'''standard deviation'''«  $σ_X$  gives the "rms value".
*From the linear and the second moment,  the variance can be calculated according to  ''Steiner's theorem''  in the following way:  $\sigma_X^{2} = {\rm E}\big[X^2 \big] - {\rm E}^2\big[X \big].$}}

{{BlaueBox|TEXT=
$\text{Definition:}$  The  »'''Charlier's skewness'''«  $S_X$  of the considered random variable  $X$  denotes the third central moment related to $σ_X^3$.
*For symmetric probability density function,  this parameter   $S_X$  is always zero.
*The larger  $S_X = \mu_3/σ_X^3$  is,  the more asymmetric is the PDF around the mean  $m_X$.
*For example,  for the exponential distribution the (positive) skewness  $S_X =2$, and this is independent of the distribution parameter  $λ$.}}

{{BlaueBox|TEXT=
$\text{Definition:}$  The  »'''kurtosis'''«  of the considered random variable  $X$  is the quotient  $K_X = \mu_4/σ_X^4$    $(\mu_4:$  fourth-order central moment$)$.
*For a Gaussian distributed random variable this always yields the value  $K_X = 3$.
*This parameter can be used, for example, to check whether a given random variable is actually Gaussian or can at least be approximated by a Gaussian distribution. }}

==Compilation of some Continuous–Value Random Variables==
 
The applet considers the following distributions: 

:Gaussian distribution, uniform distribution, triangular distribution, exponential distribution, Laplace distribution, Rayleigh distribution, Rice distribution, Weibull distribution, Wigner semicircle distribution, Wigner parabolic distribution, Cauchy distribution.

Some of these will be described in detail here.

===Gaussian distributed random variables===

[[File:EN_Sto_T_3_5_S2_v2.png |right|frame|Gaussian random variable:  PDF and CDF]]
'''(1)'''    »'''Probability density function'''«   $($axisymmetric around  $m_X)$
:$$f_X(x) = \frac{1}{\sqrt{2\pi}\cdot\sigma_X}\cdot {\rm e}^{-(X-m_X)^2 /(2\sigma_X^2) }.$$
PDF parameters: 
*$m_X$  (mean or DC component),
*$σ_X$  (standard deviation or rms value).

'''(2)'''    »'''Cumulative distribution function'''«   $($point symmetric around  $m_X)$
:$$F_X(x)= \phi(\frac{\it x-m_X}{\sigma_X})\hspace{0.5cm}\rm with\hspace{0.5cm}\rm \phi (\it x\rm ) = \frac{\rm 1}{\sqrt{\rm 2\it \pi}}\int_{-\rm\infty}^{\it x} \rm e^{\it -u^{\rm 2}/\rm 2}\,\, d \it u.$$

$ϕ(x)$:   Gaussian error integral (cannot be calculated analytically, must be taken from tables).

'''(3)'''    »'''Central moments'''«
:$$\mu_{k}=(k- 1)\cdot (k- 3) \ \cdots \ 3\cdot 1\cdot\sigma_X^k\hspace{0.2cm}\rm (if\hspace{0.2cm}\it k\hspace{0.2cm}\rm even).$$
*Charlier's skewness  $S_X = 0$,  since  $\mu_3 = 0$  $($PDF is symmetric about  $m_X)$.
*Kurtosis  $K_X = 3$,  since  $\mu_4 = 3 \cdot \sigma_X^2$  ⇒   $K_X = 3$  results only for the Gaussian PDF.

'''(4)'''    »'''Further remarks'''«
*The naming is due to the important mathematician, physicist and astronomer Carl Friedrich Gauss.
*If  $m_X = 0$  and  $σ_X = 1$, it is often referred to as the  ''normal distribution''.

*The standard deviation can also be determined graphically from the bell-shaped PDF $f_{X}(x)$   (as the distance between the maximum value and the point of inflection).
*Random quantities with Gaussian WDF are realistic models for many physical physical quantities and also of great importance for communications engineering.
*The sum of many small and independent components always leads to the Gaussian PDF   ⇒   Central Limit Theorem of Statistics   ⇒   Basis for noise processes.
*If one applies a Gaussian distributed signal to a linear filter for spectral shaping, the output signal is also Gaussian distributed.

[[File:Gauss_Signal.png|right|frame| Signal and PDF of a Gaussian noise signal]]
{{GraueBox|TEXT=
$\text{Example 1:}$  The graphic shows a section of a stochastic noise signal  $x(t)$  whose instantaneous value can be taken as a continuous random variable  $X$. From the PDF shown on the right, it can be seen that:
* A Gaussian random variable is present.
*Instantaneous values around the mean  $m_X$  occur most frequently.
*If there are no statistical ties between the samples  $x_ν$  of the sequence, such a signal is also called ''"white noise".''}}

===Uniformly distributed random variables===

[[File:Rechteck_WDF_VTF.png|right|frame|Uniform distribution:  PDF and CDF]]
'''(1)'''    »'''Probability density function'''«

*The probability density function (PDF)  $f_{X}(x)$  is in the range from  $x_{\rm min}$  to  $x_{\rm max}$  constant equal to  $1/(x_{\rm max} - x_{\rm min})$  and outside zero.
*At the range limits for  $f_{X}(x)$  only half the value  (mean value between left and right limit value)  is to be set.

'''(2)'''    »'''Cumulative distribution function'''«

*The cumulative distribution function (CDF) increases in the range from  $x_{\rm min}$  to  $x_{\rm max}$  linearly from zero to  $1$. 

'''(3)'''    »''''''«
*Mean and standard deviation have the following values for the uniform distribution:
:$$m_X = \frac{\it x_ {\rm max} \rm + \it x_{\rm min}}{2},\hspace{0.5cm}
\sigma_X^2 = \frac{(\it x_{\rm max} - \it x_{\rm min}\rm )^2}{12}.$$
*For symmetric PDF   ⇒   $x_{\rm min} = -x_{\rm max}$  the mean value  $m_X = 0$  and the variance  $σ_X^2 = x_{\rm max}^2/3.$
*Because of the symmetry around the mean  $m_X$  the Charlier skewness  $S_X = 0$.
*The kurtosis is with   $K_X = 1.8$  significantly smaller than for the Gaussian distribution because of the absence of PDF outliers.

'''(4)'''    »'''Further remarks'''«

*For modeling transmission systems, uniformly distributed random variables are the exception. An example of an actual (nearly) uniformly distributed random variable is the phase in circularly symmetric interference, such as occurs in  ''quadrature amplitude modulation''  (QAM) schemes.

*The importance of uniformly distributed random variables for information and communication technology lies rather in the fact that, from the point of view of information theory, this PDF form represents an optimum with respect to differential entropy under the constraint of "peak limitation".

*In ''image processing & encoding'', the uniform distribution is often used instead of the actual distribution of the original image, which is usually much more complicated, because the difference in information content between a ''natural image'' and the model based on the uniform distribution is relatively small.

*In the simulation of intelligence systems, one often uses "pseudo-random generators" based on the uniform distribution (which are relatively easy to realize), from which other distributions  (Gaussian distribution, exponential distribution, etc.)  can be easily derived.

===Exponentially distributed random variables===

'''(1)'''    »'''Probability distribution function'''«
[[File:Exponential_WDF_VTF.png|right|frame|Exponential distribution:  PDF and CDF]]

An exponentially distributed random variable  $X$  can only take on non–negative values. For  $x>0$  the PDF has the following shape:
:$$f_X(x)=\it \lambda_X\cdot\rm e^{\it -\lambda_X \hspace{0.05cm}\cdot \hspace{0.03cm} x}.$$
*The larger the distribution parameter  $λ_X$,  the steeper the drop.
*By definition,  $f_{X}(0) = λ_X/2$, which is the average of the left-hand limit  $(0)$  and the right-hand limit  $(\lambda_X)$.

'''(2)'''    »'''Cumulative distribution function'''«

Distribution function PDF, we obtain for  $x > 0$:
:$$F_{X}(x)=1-\rm e^{\it -\lambda_X\hspace{0.05cm}\cdot \hspace{0.03cm} x}.$$

'''(3)'''    »'''Moments and central moments'''«

*The  ''moments''  of the (one-sided) exponential distribution are generally equal to:
:$$m_k = \int_{-\infty}^{+\infty} x^k \cdot f_{X}(x) \,\,{\rm d} x = \frac{k!}{\lambda_X^k}.$$
*From this and from Steiner's theorem we get for mean and standard deviation:
:$$m_X = m_1=\frac{1}{\lambda_X},\hspace{0.6cm}\sigma_X^2={m_2-m_1^2}={\frac{2}{\lambda_X^2}-\frac{1}{\lambda_X^2}}=\frac{1}{\lambda_X^2}.$$
*The PDF is clearly asymmetric here. For the Charlier skewness  $S_X = 2$.
*The kurtosis with   $K_X = 9$  is clearly larger than for the Gaussian distribution, because the PDF foothills extend much further.

'''(4)'''    »'''Further remarks'''«

*The exponential distribution has great importance for reliability studies; in this context, the term "lifetime distribution" is also commonly used.
*In these applications, the random variable is often the time  $t$, that elapses before a component fails.
*Furthermore, it should be noted that the exponential distribution is closely related to the Laplace distribution.

===Laplace distributed random variables===

[[File:Laplace_WDF_VTF.png|right|frame|Laplace distribution:  PDF and CDF]]
'''(1)'''    »'''Probability density function'''«

As can be seen from the graph, the Laplace distribution is a "two-sided exponential distribution":

:$$f_{X}(x)=\frac{\lambda_X} {2}\cdot{\rm e}^ { - \lambda_X \hspace{0.05cm} \cdot \hspace{0.05cm} \vert \hspace{0.05cm} x \hspace{0.05cm} \vert}.$$

* The maximum value here is  $\lambda_X/2$.
*The tangent at  $x=0$  intersects the abscissa at  $1/\lambda_X$, as in the exponential distribution.

'''(2)'''    »'''Cumulative distribution function'''«

:$$F_{X}(x) = {\rm Pr}\big [X \le x \big ] = \int_{-\infty}^{x} f_{X}(\xi) \,\,{\rm d}\xi $$
:$$\Rightarrow \hspace{0.5cm} F_{X}(x) = 0.5 + 0.5 \cdot {\rm sign}(x) \cdot \big [ 1 - {\rm e}^ { - \lambda_X \hspace{0.05cm} \cdot \hspace{0.05cm} \vert \hspace{0.05cm} x \hspace{0.05cm} \vert}\big ] $$
:$$\Rightarrow \hspace{0.5cm} F_{X}(-\infty) = 0, \hspace{0.5cm}F_{X}(0) = 0.5, \hspace{0.5cm} F_{X}(+\infty) = 1.$$

'''(3)'''    »'''Moments and central moments'''«

* For odd  $k$,  the Laplace distribution always gives  $m_k= 0$ due to symmetry. Among others:  Linear mean  $m_X =m_1 = 0$.

* For even  $k$  the moments of Laplace distribution and exponential distribution agree:  $m_k = {k!}/{\lambda^k}$.

* For the variance  $(=$ second order central moment $=$ second order moment$)$  holds:  $\sigma_X^2 = {2}/{\lambda_X^2}$   ⇒   twice as large as for the exponential distribution.

*For the Charlier skewness,  $S_X = 0$ is obtained here due to the symmetric PDF.

*The kurtosis is  $K_X = 6$,  significantly larger than for the Gaussian distribution, but smaller than for the exponential distribution.

'''(4)'''    »'''Further remarks'''«

*The instantaneous values of speech and music signals are Laplace distributed with good approximation. See learning video  [[Wahrscheinlichkeit_und_WDF_(Lernvideo)|"Wahrscheinlichkeit und Wahrscheinlichkeitsdichtefunktion"]],  part 2.
*By adding a Dirac delta function at  $x=0$,  speech pauses can also be modeled.
 
===Brief description of other distributions===
 
$\text{(A) Rayleigh distribution}$     [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading|$\text{More detailed description}$]]

*Probability density function:
:$$f_X(x) =
\left\{ \begin{array}{c} x/\lambda_X^2 \cdot {\rm e}^{- x^2/(2 \hspace{0.05cm}\cdot\hspace{0.05cm} \lambda_X^2)} \\
0 \end{array} \right.\hspace{0.15cm}
\begin{array}{*{1}c} {\rm for}\hspace{0.1cm} x\hspace{-0.05cm} \ge \hspace{-0.05cm}0,
\\ {\rm for}\hspace{0.1cm} x \hspace{-0.05cm}<\hspace{-0.05cm} 0. \\ \end{array}.$$
*Application:     Modeling of the cellular channel (non-frequency selective fading, attenuation, diffraction, and refraction effects only, no line-of-sight).

$\text{(B) Rice distribution}$     [[Mobile_Communications/Non-Frequency-Selective_Fading_With_Direct_Component|$\text{More detailed description}$]]

*Probability density function  $(\rm I_0$  denotes the modified zero-order Bessel function$)$:
:$$f_X(x) = \frac{x}{\lambda_X^2} \cdot {\rm exp} \big [ -\frac{x^2 + C_X^2}{2\cdot \lambda_X^2}\big ] \cdot {\rm I}_0 \left [ \frac{x \cdot C_X}{\lambda_X^2} \right ]\hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k}}{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Application:     Cellular channel modeling (non-frequency selective fading, attenuation, diffraction, and refraction effects only, with line-of-sight).

$\text{(C) Weibull distribution}$     [https://en.wikipedia.org/wiki/Weibull_distribution $\text{More detailed description}$]

*Probability density function:
:$$f_X(x) = \lambda_X \cdot k_X \cdot (\lambda_X \cdot x)^{k_X-1} \cdot {\rm e}^{(\lambda_X \cdot x)^{k_X}}
\hspace{0.05cm}.$$

*Application:     PDF with adjustable skewness $S_X$; exponential distribution  $(k_X = 1)$  and Rayleigh distribution  $(k_X = 2)$  included as special cases.

$\text{(D) Wigner semicircle distribution}$     [https://en.wikipedia.org/wiki/Wigner_semicircle_distribution $\text{More detailed description}$]

*Probability density function:
:$$f_X(x) =
\left\{ \begin{array}{c} 2/(\pi \cdot {R_X}^2) \cdot \sqrt{{R_X}^2 - (x- m_X)^2} \\
0 \end{array} \right.\hspace{0.15cm}
\begin{array}{*{1}c} {\rm for}\hspace{0.1cm} |x- m_X|\hspace{-0.05cm} \le \hspace{-0.05cm}R_X,
\\ {\rm for}\hspace{0.1cm} |x- m_X| \hspace{-0.05cm} > \hspace{-0.05cm} R_X \\ \end{array}.$$
*Application:     PDF of Chebyshev nodes   ⇒   zeros of Chebyshev polynomials from numerics.

$\text{(E) Wigner parabolic distribution}$

*Probability density function:
:$$f_X(x) =
\left\{ \begin{array}{c} 3/(4 \cdot {R_X}^3) \cdot \big ({R_X}^2 - (x- m_X)^2\big ) \\
0 \end{array} \right.\hspace{0.15cm}
\begin{array}{*{1}c} {\rm for}\hspace{0.1cm} |x|\hspace{-0.05cm} \le \hspace{-0.05cm}R_X,
\\ {\rm for}\hspace{0.1cm} |x| \hspace{-0.05cm} > \hspace{-0.05cm} R_X \\ \end{array}.$$
*Application:     PDF of eigenvalues of symmetric random matrices whose dimension approaches infinity.

$\text{(F) Cauchy distribution}$     [[Theory_of_Stochastic_Signals/Further_Distributions#Cauchy_PDF|$\text{More detailed description}$]]

*Probability density function and distribution function:
:$$f_{X}(x)=\frac{1}{\pi}\cdot\frac{\lambda_X}{\lambda_X^2+x^2}, \hspace{2cm} F_{X}(x)={\rm 1}/{2}+{\rm arctan}({x}/{\lambda_X}).$$
*In the Cauchy distribution, all moments  $m_k$  for even  $k$  have an infinitely large value, independent of the parameter  $λ_X$.
*Thus, this distribution also has an infinitely large variance:  $\sigma_X^2 \to \infty$.
*Due to symmetry, for odd  $k$  all moments  $m_k = 0$, if one assumes the "Cauchy Principal Value" as in the program:  $m_X = 0, \ S_X = 0$.
*Example:     The quotient of two Gaussian mean-free random variables is Cauchy distributed. For practical applications the Cauchy distribution has less meaning.

==Exercises==
 
*First, select the number  $(1,\ 2, \text{...} \ )$  of the task to be processed.  The number  "$0$"  corresponds to a  "Reset":  Same setting as at program start.
*A task description is displayed.  The parameter values are adjusted.  Solution after pressing  "Show Solution".
*In the following  $\text{Red}$  stands for the random variable  $X$  and  $\text{Blue}$  for  $Y$.

{{BlueBox|TEXT=
'''(1)'''  Select  $\text{red: Gaussian PDF}\ (m_X = 1, \ \sigma_X = 0.4)$  and  $\text{blue: Rectangular PDF}\ (y_{\rm min} = -2, \ y_{\rm max} = +3)$.  Interpret the  $\rm PDF$  graph.}}

* $\text{Gaussian PDF}$:  The  $\rm PDF$ maximum is equal to  $f_{X}(x = m_X) = \sqrt{1/(2\pi \cdot \sigma_X^2)} = 0.9974 \approx 1$.
* $\text{Rectangular PDF}$:  All  $\rm PDF$ values are equal  $0.2$  in the range  $-2 < y < +3$.  At the edges  $f_Y(-2) = f_Y(+3)= 0.1$  (half value) holds.

{{BlueBox|TEXT=
'''(2)'''  Same setting as for  $(1)$.  What are the probabilities  ${\rm Pr}(X = 0)$,   ${\rm Pr}(0.5 \le X \le 1.5)$,   ${\rm Pr}(Y = 0)$   and  ${\rm Pr}(0.5 \le Y \le 1.5)$ .}}

* ${\rm Pr}(X = 0)={\rm Pr}(Y = 0) \equiv 0$   ⇒   Probability of a discrete random variable to take exactly a certain value.
* The other two probabilities can be obtained by integration over the PDF in the range  $+0.5\ \text{...} \ +\hspace{-0.1cm}1.5$.
* Or:  ${\rm Pr}(0.5 \le X \le 1.5)= F_X(1.5) - F_X(0.5) = 0.8944-0.1056 = 0.7888$. Correspondingly:  ${\rm Pr}(0.5 \le Y \le 1.5)= 0.7-0.5=0.2$.

{{BlueBox|TEXT=
'''(3)'''  Same settings as before.  How must the standard deviation  $\sigma_X$  be changed so that with the same mean  $m_X$  it holds for the second order moment:  $P_X=2$ ?}}

* According to Steiner's theorem:  $P_X=m_X^2 + \sigma_X^2$   ⇒   $\sigma_X^2 = P_X-m_X^2 = 2 - 1^2 = 1 $   ⇒   $\sigma_X = 1$.

{{BlueBox|TEXT=
'''(4)'''  Same settings as before:  How must the parameters  $y_{\rm min}$  and  $y_{\rm max}$  of the rectangular PDF be changed to yield  $m_Y = 0$  and  $\sigma_Y^2 = 0.75$?}}

* Starting from the previous setting  $(y_{\rm min} = -2, \ y_{\rm max} = +3)$  we change  $y_{\rm max}$ until  $\sigma_Y^2 = 0.75$  occurs   ⇒   $y_{\rm max} = 1$.
* The width of the rectangle is now  $3$.  The desired mean   $m_Y = 0$  is obtained by shifting:  $y_{\rm min} = -1.5, \ y_{\rm max} = +1.5$.
* You could also consider that for a mean-free random variable  $(y_{\rm min} = -y_{\rm max})$  the following equation holds:   $\sigma_Y^2 = y_{\rm max}^2/3$.

{{BlueBox|TEXT=
'''(5)'''  For which of the adjustable distributions is the Charlier skewness  $S \ne 0$ ? }}

* The Charlier's skewness denotes the third central moment related to  $σ_X^3$   ⇒  $S_X = \mu_3/σ_X^3$  $($valid for the random variable  $X)$.
* If the PDF  $f_X(x)$  is symmetric around the mean  $m_X$  then the parameter  $S_X$  is always zero.
* Exponential distribution:  $S_X =2$;  Rayleigh distribution:  $S_X =0.631$   $($both independent of  $λ_X)$;   Rice distribution:  $S_X >0$  $($dependent of  $C_X, \ λ_X)$.
* With the Weibull distribution, the Charlier skewness  $S_X$  can be zero, positive or negative,  depending on the PDF parameter  $k_X$.
*  Weibull distribution,  $\lambda_X=0.4$:  With  $k_X = 1.5$  ⇒   PDF is curved to the left  $(S_X > 0)$;   $k_X = 7$  ⇒   PDF is curved to the right  $(S_X < 0)$.

{{BlueBox|TEXT=
'''(6)'''  Select  $\text{Red: Gaussian PDF}\ (m_X = 1, \ \sigma_X = 0.4)$  and  $\text{Blue: Gaussian PDF}\ (m_X = 0, \ \sigma_X = 1)$.  What is the kurtosis in each case?}}

* For each Gaussian distribution the kurtosis has the same value:   $K_X = K_Y =3$.  Therefore,  $K-3$  is called "excess".
*This parameter can be used to check whether a given random variable can be approximated by a Gaussian distribution.

{{BlueBox|TEXT=
'''(7)'''  For which distributions does a significantly smaller kurtosis value result than  $K=3$?  And for which distributions does a significantly larger one?}}

* $K<3$  always results when the PDF values are more concentrated around the mean than in the Gaussian distribution.
* This is true, for example, for the uniform distribution  $(K=1.8)$  and for the triangular distribution  $(K=2.4)$.
* $K>3$,  if the PDF offshoots are more pronounced than for the Gaussian distribution.  Example:  Exponential PDF  $(K=9)$.

{{BlueBox|TEXT=
'''(8)'''  Select  $\text{Red: Exponential PDF}\ (\lambda_X = 1)$  and  $\text{Blue: Laplace PDF}\ (\lambda_Y = 1)$.  Interpret the differences.}}

* The Laplace distribution is symmetric around its mean  $(S_Y=0, \ m_Y=0)$  unlike the exponential distribution  $(S_X=2, \ m_X=1)$.
* The even moments  $m_2, \ m_4, \ \text{...}$  are equal,  for example:  $P_X=P_Y=2$.  But not the variances:  $\sigma_X^2 =1, \ \sigma_Y^2 =2$.
* The probabilities  ${\rm Pr}(|X| < 2) = F_X(2) = 0.864$  and  ${\rm Pr}(|Y| < 2) = F_Y(2) - F_Y(-2)= 0.932 - 0.068 = 0.864$  are equal.
* In the Laplace PDF, the values are more tightly concentrated around the mean than in the exponential PDF:  $K_Y =6 < K_X = 9$.

{{BlueBox|TEXT=
'''(9)'''  Select  $\text{Red: Rice PDF}\ (\lambda_X = 1, \ C_X = 1)$  and  $\text{Blue: Rayleigh PDF}\ (\lambda_Y = 1)$.  Interpret the differences.}}

*  With  $C_X = 0$  the Rice PDF transitions to the Rayleigh PDF.  A larger  $C_X$  improves the performance, e.g., in mobile communications.
*  Both, in  "Rayleigh"  and  "Rice"  the abscissa is the magnitude  $A$  of the received signal.  Favorably, if  ${\rm Pr}(A \le A_0)$  is small  $(A_0$  given$)$.
*  For  $C_X \ne 0$  and equal  $\lambda$  the Rice CDF is below the Rayleigh CDF   ⇒   smaller  ${\rm Pr}(A \le A_0)$  for all  $A_0$.

{{BlueBox|TEXT=
'''(10)'''  Select  $\text{Red: Rice PDF}\ (\lambda_X = 0.6, \ C_X = 2)$.  By which distribution  $F_Y(y)$  can this Rice distribution be well approximated? }}

*  The kurtosis   $K_X = 2.9539 \approx 3$  indicates the Gaussian distribution.   Favorable parameters:  $m_Y = 2.1 > C_X, \ \ \sigma_Y = \lambda_X = 0.6$.
*  The larger tht quotient  $C_X/\lambda_X$  is, the better the Rice PDF is approximated by a Gaussian PDF.
*  For large   $C_X/\lambda_X$  the Rice PDF has no more similarity with the Rayleigh PDF.

{{BlueBox|TEXT=
'''(11)'''  Select  $\text{Red: Weibull PDF}\ (\lambda_X = 1, \ k_X = 1)$  and  $\text{Blue: Weibull PDF}\ (\lambda_Y = 1, \ k_Y = 2)$. Interpret the results. }}

*  The Weibull PDF  $f_X(x)$  is identical to the exponential PDF and  $f_Y(y)$  to the Rayleigh PDF.
*  However, after best fit, the parameters  $\lambda_{\rm Weibull} = 1$  and  $\lambda_{\rm Rayleigh} = 0.7$ differ.
*  Moreover, it holds  $f_X(x = 0) \to \infty$  for  $k_X < 1$.  However, this does not have the affect of infinite moments.

{{BlueBox|TEXT=
'''(12)'''  Select  $\text{Red: Weibull PDF}\ (\lambda_X = 1, \ k_X = 1.6)$  and   $\text{Blue: Weibull PDF}\ (\lambda_Y = 1, \ k_Y = 5.6)$.  Interpret the Charlier skewness. }}

*  One observes:   For the PDF parameter  $k < k_*$  the Charlier skewness is positive and for  $k > k_*$  negative.  It is approximately  $k_* = 3.6$.

{{BlueBox|TEXT=
'''(13)'''  Select  $\text{Red: Semicircle PDF}\ (m_X = 0, \ R_X = 1)$  and  $\text{Blue: Parabolic PDF}\ (m_Y = 0, \ R_Y = 1)$.  Vary the parameter  $R$  in each case. }}

*  The PDF in each case is mean-free and symmetric  $(S_X = S_Y =0)$  with  $\sigma_X^2 = 0.25, \ K_X = 2$  respectively,  $\sigma_Y^2 = 0.2, \ K_Y \approx 2.2$.

==Applet Manual==
 
[[File:Bildschirm_WDF_VTF_neu.png|right|600px|frame|Screenshot of the German version]]
    '''(A)'''     Selection of the distribution  $f_X(x)$  (red curves and output values)

    '''(B)'''     Parameter input for the "red distribution" via slider

    '''(C)'''     Selection of the distribution  $f_Y(y)$  (blue curves and output values)

    '''(D)'''     Parameter input for the "red distribution" via slider

    '''(E)'''     Graphic area for the probability density function (PDF)

    '''(F)'''     Graphic area for the distribution function (CDF)

    '''(G)'''     Numerical output for the "red distribution"

    '''(H)'''     Numerical output for the "blue distribution"

    '''( I )'''     Input of  $x_*$  and  $y_*$  abscissa values for the numerics outputs

    '''(J)'''     Experiment execution area:   task selection

    '''(K)'''     Experiment execution area:   task description

    '''( L)'''     Experiment execution area:   sample solution
 

'''Selection options for''' for  $\rm A$  and  $\rm C$:  

Gaussian distribution,   uniform distribution,   triangular distribution,   exponential distribution,   Laplace distribution,   Rayleigh distribution,  Rice distribution,   Weibull distribution,   Wigner semicircle distribution,   Wigner parabolic distribution,   Cauchy distribution.

The following »'''integral parameters'''« are output  $($with respect to $X)$:  

Linear mean value  $m_X = {\rm E}\big[X \big]$,   second order moment  $P_X ={\rm E}\big[X^2 \big] $,   variance  $\sigma_X^2 = P_X - m_X^2$,   standard deviation  $\sigma_X$,  Charlier's skewness  $S_X$,   kurtosis  $K_X$.

'''In all applets top right''':    Changeable graphical interface design   ⇒   '''Theme''':
* Dark:   black background  (recommended by the authors).
* Bright:   white background  (recommended for beamers and printouts)
* Deuteranopia:   for users with pronounced green–visual impairment
* Protanopia:   for users with pronounced red–visual impairment
 
==About the Authors==
 
This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].
*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29| »Klaus Eichin« ]]).

*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29| »Tasnád Kernetzky« ]] ).

*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|»Carolin Mirschina«]]  in the context of a working student activity. 

*The conversion of this applet was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ $\text{Studienzuschüsse}$]  (TUM Department of Electrical and Computer Engineering).  We thank.

==Once again: Open Applet in new Tab==

{{LntAppletLinkEnDe|wdf-vtf_en|wdf-vtf}}

LNTwww:General notes about "Information Theory"

2025-02-24T13:39:58Z

'''Four main chapters with a total of 13 chapters (files);   71 Exercises   ⇒   Two semester hours per week lecture and one semester hour per week exercises      
Development of the German version:   2011–2015.       Development of the English version:   2021.      Last corrections:   September 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

*Basic materials:   Lecture notes from LNT/LÜT:    '''[Meck09]'''<ref name='Meck09'>Mecking, M.:  Information Theory. Lecture notes, Chair of Communications Engineering, TU München, 2009.</ref>;  '''[Liv15]'''<ref name='Liv15'>Liva, G.:  Channels Codes for Iterative Decoding. Lecture notes, Chair of Communications Engineering, TU München and DLR Oberpfaffenhofen, 2015.</ref>;  '''[Kra16]'''<ref name='Kra16'>Kramer, G.:  Information Theory. Lecture notes, Chair of Communications Engineering, TU München, 2016.</ref>;  '''[Söd14]'''<ref name='Söd14'>Söder, G.:  Wertdiskrete Informationstheorie.  Guidance  (in German)  for the experiment of the same name in the practical course "Simulation of digital transmission systems"; Lecture notes, Chair of Communications Engineering, TU München, 2014. </ref>;  Textbook:  '''[Cov06]'''<ref name='Cov06'>Cover, T. M.; Thomas, J. A.: Elements of Information Theory. West Sussex: John Wiley & Sons, 2nd Edition, 2006. </ref>

*Authors and involved colleagues in alphabetic order:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],   [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|'''Bernhard Göbel''']],   [http://wirelesscoding.org/ '''Gianluigi Liva'''],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Tobias_Lutz_.28at_LNT_from_2008-2014.29|'''Tobias Lutz''']],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Michael_Mecking_.28at_LNT_from_1997-2012.29|'''Michael Mecking''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29|'''Günter Söder''']]

*Participating students in chronologic order:      Martin Winkler '''(2001)''', Yven Winter, Thomas Großer, Stefan Müller, Martin Völkl, Eugen Mehlmann, Alexander Laible, Veronika Hofmann, Noah Nagi, Carolin Mirschina '''(2021)'''

$\text{List of sources:}$

LNTwww:General notes about "Modulation Methods"

2025-02-24T13:39:58Z

'''Four main chapters with a total of 23 chapters (files);   89 Exercises   ⇒   Three semester hours per week lecture and two semester hours per week exercises      
Development of the German version:   2005–2011.       Development of the English version:   2020/21.      Last corrections:   October 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Söd11a]'''<ref name='Söd11a'>Söder, G.:  Analoge Modulationsverfahren. Anleitung zum gleichnamigen V ersuch im Praktikum &bdquo;Simulation digitaler Übertragungssysteme”. Lehrstuhl für Nachrichtentechnik, TU München, 2011.  </ref>;  '''[Söd11b]'''<ref name='Söd11b'>Söder, G.:  Digitale Modulationsverfahren. Anleitung zum gleichnamigen Versuch im Praktikum &bdquo;Simulation digitaler Übertragungssysteme”. Lehrstuhl für Nachrichtentechnik, TU München, 2011. </ref>;  '''[Han16]'''<ref name='Han16'>Hanik, N.:  Nachrichtentechnik 2 (LB): Modulationsverfahren. Vorlesungsmanuskript. Professur Leitungsgebundene Übertragungstechnik, TU München, 2016.</ref>;  '''[Vie16]'''<ref name='Vie16'>Viering, I.: System Aspects in Communications. Vorlesungsmanuskript. Lehrstuhl für Nachrichtentechnik. München: TU München, 2016.</ref>;   '''[Kra17]'''<ref name='Kra17'>Kramer, G.: Nachrichtentechnik 2. Vorlesungsmanuskript, Lehrstuhl für Nachrichtentechnik. München: TU München, 2017.</ref>.   –   Textbooks:   '''[ZP85]'''<ref name='ZP85'>Ziemer, R. E.; Peterson, R. L.: Digital Communications and Spread Spectrum Systems. New York: Macmillan, 1985. ISBN 978-0-02431-670-7</ref>;  '''[Söd93]'''<ref name='Söd93'>Söder, G.: Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin, Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2</ref>;  '''[Kam04]'''<ref name="Kam04">Kammeyer, K.D.:  Nachrichtenübertragung. Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>

*Author:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

* Participating colleagues in alphabetical order:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|'''Klaus Eichin''']],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|'''Johannes Zangl''']],

*Participating students in chronological order:      Martin Winkler '''(2001)''', Yven Winter, Ji Li , Franz Kohl, Bettina Hirner, Thorsten Kalweit, Markus Elsberger, Slim Lamine, Thomas Großer, Johannes Schmidt, David Jobst, Xiaohan Liu, Matthias Riedel, Carolin Mirschina, Sam Reed, JiWoo Hwang '''(2022)'''

$\text{List of sources:}$

LNTwww:Imprint for the book "Channel Coding"

2025-02-24T13:39:57Z

'''Four main chapters with a total of 22 chapters (files) and 175 sections (pages);   98 Exercises   ⇒   Scope:  "$\rm 3L+2E$"      
Development of the German version:   2011–2017.       Development of the English version:   2022.      Last corrections:   October 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:     '''[Köt08]'''<ref name='Köt08'>Kötter, R.; Mayer, T.; Tüchler, M.; Schreckenbach, F.; Brauchle, J.: Channel Coding. Lecture notes, Chair of Communications Engineering, TU München, 2008</ref>;   '''[Liv10]'''<ref name='Liv10'>Liva, G.: Channel Coding. Lecture notes, Chair of Communications Engineering, TU München and DLR Oberpfaffenhofen, 2010</ref>;   '''[Liv15]'''<ref name='Liv15'>Liva, G.: Channels Codes for Iterative Decoding. Lecture notes, Chair of Communications Engineering, TU München and DLR Oberpfaffenhofen, 2015</ref>   –   Textbooks:   '''[Bos99]'''<ref name='Bos99'>Bossert, M.: Channel Coding for Telecommunications. Chichester: Wiley, 1999. ISBN 978-0-471-98277-7 </ref>;  '''[Cov06]'''<ref name='Cov06'>Cover, T. M.; Thomas, J. A.: Elements of Information Theory. West Sussex: John Wiley & Sons, 2nd Edition, 2006 </ref>;   '''[Hub82]'''<ref name='Hub82'>Huber, J.: Codierung für gedächtnisbehaftete Kanäle. Dissertation – Universität der Bundeswehr München, 1982</ref>

*Authors and involved colleagues in alphabetic order:   [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Ronald_B.C3.B6hnke_.28at_LNT_from_2012-2014.29|'''Ronald Böhnke''']],   [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Joschi_Brauchle_.28at_LNT_from_2007-2015.29|'''Joschi Brauchle''']],   [http://wirelesscoding.org/ '''Gianluigi Liva'''],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

*Participating students in chronologic order:    Martin Winkler '''(2001)''',  Yven Winter,  Ji Li,  Bettina Hirner,  Thomas Großer,  Thorsten Bürgstein,  Martin Völkl,  Dominik Kopp,  Carolin Mirschina,  Jiwoo Hwang,  Noah Nagy '''(2022)'''

$\text{References:}$

LNTwww:Imprint for the book "Information Theory"

2025-02-24T13:39:55Z

'''Four main chapters with a total of 13 chapters (files) and 106 sections (pages);   71 Exercises   ⇒   Scope:  "$\rm 2L+1E$"      
Development of the German version:   2011–2015.       Development of the English version:   2021.      Last corrections:   September 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

*Basic materials:   Lecture notes from LNT/LÜT:    '''[Meck09]'''<ref name='Meck09'>Mecking, M.:  Information Theory. Lecture notes, Chair of Communications Engineering, TU München, 2009.</ref>;  '''[Liv15]'''<ref name='Liv15'>Liva, G.:  Channel Codes for Iterative Decoding. Lecture notes, Chair of Communications Engineering, TU München and DLR Oberpfaffenhofen, 2015.</ref>;  '''[Kra16]'''<ref name='Kra16'>Kramer, G.:  Information Theory. Lecture notes, Chair of Communications Engineering, TU München, 2016.</ref>;  '''[Söd14]'''<ref name='Söd14'>Söder, G.:  Wertdiskrete Informationstheorie.  Guidance  (in German)  for the experiment of the same name in the practical course "Simulation of digital transmission systems"; Lecture notes, Chair of Communications Engineering, TU München, 2014. </ref>;  Textbook:  '''[Cov06]'''<ref name='Cov06'>Cover, T. M.; Thomas, J. A.: Elements of Information Theory. West Sussex: John Wiley & Sons, 2nd Edition, 2006. </ref>

*Authors and involved colleagues in alphabetic order:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],   [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|'''Bernhard Göbel''']],   [http://wirelesscoding.org/ '''Gianluigi Liva'''],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Tobias_Lutz_.28at_LNT_from_2008-2014.29|'''Tobias Lutz''']],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Michael_Mecking_.28at_LNT_from_1997-2012.29|'''Michael Mecking''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29|'''Günter Söder''']]

*Participating students in chronological order:      Martin Winkler '''(2001)''', Yven Winter, Reinhold Sixt, Jürgen Veitenhansl, Franz Kohl, Bettina Hirner, Ji Li, Markus Elsberger, Thorsten Kalweit, Thomas Großer, David Jobst, Matthias Niller, Veronika Hofmann, Carolin Mirschina, Noah Nagi, Ji Woo Hwang '''(2022)'''

$\text{References:}$

LNTwww:Imprint for the book "Examples of Communication Systems"

2025-02-24T13:39:55Z

'''Four main chapters with a total of 17 chapters (files) and 164 sections (pages);   38 Exercises   ⇒   Scope:  "$\rm 3L+1E$"      
Development of the German version:   2002–2017.       Development of the English version:   2022/23.      Last corrections:   March 2023

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Hin08]'''<ref name='Hin08'>Hindelang, T.:  Mobile Communications. Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2008.</ref>;  '''[Söd10]'''<ref name='Söd10'>Söder, G.: Mobilfunkkanal.   Instructions for the practical course "Simulation digitaler Übertragungssysteme".   LNT/TUM, 2010. </ref>;   '''[Eich11]'''<ref name='Eich11'>Eichin, K.: Nachrichtensysteme – Kommunikationssysteme (LB).  Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2018.</ref>;  '''[Vie17]'''<ref name='Vie17'>Viering, I.:  System Aspects in Communications. Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2017.</ref>; '''[Kra18]'''<ref name='Kra18'>Kramer, G.:   Mobile Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2018.</ref>  –   Textbook:   '''[EVB01]'''<ref name='EVB01'>Eberspächer, J.; Vögel, H.J.; Bettstetter, C.: Global System for Mobile Communication. 3. Auflage. Stuttgart: Teubner, 2001.</ref>

*Authors in alphabetic order:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29| '''Klaus Eichin''']],  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Thomas_Hindelang_.28at_LNT_from_1994-2000_und_2007-2012.29|'''Thomas Hindelang''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']] 

*Participating students in chronological order:      Martin Winkler  '''(2001)''',  Yven Winter,  Franz Kohl,  Thorsten Kalweit,  Thomas Großer,  Franz-Josef Kaupert,  Hichem Kallel,  Khaled Soussi,  Johannes Schmidt,  Sebastian Seitz,  Alexander Happach,  Stefan Müller,  Noah Nagi,  Carolin Mirschina,  Jiwoo Hwang  '''(2022)'''

$\text{References:}$

LNTwww:Imprint for the book "Linear and Time Invariant Systems"

2025-02-24T13:39:55Z

'''Four main chapters with a total of 12 chapters (files) and 93 sections (pages);   90 Exercises   ⇒   Scope:  "$\rm 2L+1E$"      
Development of the German version:   2002–2017.       Development of the English version:   2020/21.      Last corrections:   December 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| $\text{Gerhard Kramer}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|$\text{Tasnád Kernetzky}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|$\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |$\text{Günter Söder}$]]

*Basic materials for the original German version:     Lecture notes of LNT/LÜT:  '''[Eich03]'''<ref name='Eich03'>Eichin, K.:  Nachrichtentechnik I (LB) – Signaldarstellung.  Lecture notes, Chair of Communications Engineering, TU München, 2003. </ref>;  '''[Han15]'''<ref name='Han15'>Hanik, N.:  Nachrichtentechnik 1 (LB): Signaldarstellung.  Lecture notes, Associate Professorship of Line Transmission Technology, TU München, 2015.</ref>;  '''[Söd12]'''<ref name='Söd12'>Söder, G.:   Simulationsmethoden in der Nachrichtentechnik.  Internship notes, Chair of Communications Engineering, TU München, 2012.</ref>   –  Textbooks:   '''[Söd93]'''<ref name='Söd93'>Söder, G.:  Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin, Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2.</ref>;  '''[Mar94]'''<ref name='Mar94'>Marko, H.:  Methoden der Systemtheorie.  3. Auflage. Berlin – Heidelberg: Springer, 1994.</ref>;  '''[HM09]'''<ref name='HM09'>Haykin, S.; Moher, M.: Communication Systems. 5th edition, Hoboken, N.J.: J. Wiley, 2009. ISBN 978-0-47169-790-9.</ref>

*Authors:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |$\text{Günter Söder}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29| $\text{Klaus Eichin}$]],  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|$\text{Norbert Hanik}$]]

* Participating colleagues in alphabetic order:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]],   [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_Übertragungstechnik#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|$\text{Bernhard Göbel}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|$\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|$\text{Johannes Zangl}$]]

*Participating students in chronological order:      Martin Winkler  '''(2001)''',  Yven Winter,  Franz Kohl,  Thorsten Kalweit,  Markus Elsberger,  Bettina Hirner,  Ji Li,  Thomas Großer,  David Jobst,  Jimmy He,  Xiaohan Liu,  Carolin Mirschina,  Ji Woo Hwang  '''(2021)'''

$\text{References:}$

Biographies and Bibliographies/Chair holders of the LNT since 1962

2025-02-24T13:39:54Z

{{Header
|Untermenü=Lehrstuhlinhaber des LNT|
Vorherige Seite=Buchstaben N - Z|
Nächste Seite=An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten
}}

==Prof. Dr.-Ing. Dr.-Ing. E.h. Hans Marko (1962-1993)==
 
Hans Marko, born on February 24, 1925 in Kronstadt/Siebenbürgen, studied Communications Engineering at the TH Stuttgart and received his doctorate in 1953 under Ernst Feldtkeller.  He then worked at Standard Elektrik Lorenz AG, where he developed one of the first pulse code modulation systems in Germany.  At this time he was already lecturing at the universities of Stuttgart and Karlsruhe.  In 1961, he wrote his post-doctoral thesis on the utilization of telegraph channels for information transmission.

[[File:Marko.png|165px|right|Hans Marko]]

In 1962, at the age of only 37, Hans Marko succeeded Hans Piloty as head of the  "Lehrstuhl für Nachrichtentechnik"  (LNT)  at the  "Technische Hochschule München"  (today:  Technical University of Munich, TUM)  and worked successfully in teaching and research for 31 years until his retirement.  He supervised nine habilitations and 75 doctorates.

The scientific fields he and his institute worked on included

* the application of systems theory in technical, biological and cybernetic systems,

* its multidimensional extension for image processing and pattern recognition,

* the further development of Shannon's information theory to a bidirectional communication theory,

* theoretical investigations and practical realizations of high-rate digital transmission systems over cable and optical fiber.

Hans Marko is the author of several books and more than one hundred publications as well as numerous patents. He has received many high-ranking honors:

# He is a laureate of the  "Nachrichtentechnische Gesellschaft"  (NTG)  and a  "Fellow of the Institute of Electrical and Electronics Engineering"  (IEEE).
#In 1983, he was the first to be awarded the "Karl Küpfmüller Prize" of the Information Technology Society in the VDE.
#In 1985 he received an honorary doctorate from the TH Darmstadt and in 1994 the Cross of Merit of the Federal Republic of Germany.
#He is a founding member of the  "Academia Scientiarium et Artium Europaea"  in Salzburg.

After his retirement in 1993, Hans Marko has always remained connected to his former institute, both to his direct successor  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|Joachim Hagenauer]]  as well as his successors  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|Ralf Kötter]]  and  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29|Gerhard Kramer]].  Of particular note in this context was his participation in a workshop in May 2012, at which, at the age of 87, he discussed his findings on "bidirectional communication" obtained 40 years ago with the pioneer James Massey and other leading researchers in this field.

Professor Hans Marko passed away on Sept. 12, 2017, in Gräfelfing, Germany, at the age of 93.

{{BlaueBox|TEXT=
'''Hans Marko's contribution to the LNTwww'''  results from the fact that our authors  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|Klaus Eichin]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|Norbert Hanik]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29|Günter Söder]]  did their PhD with him. 

Many of the statements in the books 
*"Signal Representation", 
*"Linear Time-Invariant Systems", 
*"Modulation Methods"  and 
*"Digital Signal Transmission" 

are thus indirectly due to Professor Marko.}}

==Prof. Dr.-Ing. Dr.-Ing. E.h. Joachim Hagenauer (1993-2006)==
 
{{BlaueBox|TEXT=
[[File:Hagenauer.jpg|165px|right|Joachim Hagenauer]]
'''Joachim Hagenauer's contribution to the LNTwww'''

*The beginning of the LNTwww falls during Hagenauer's time as chair of the LNT. 

*All the teaching areas considered here were also the content of his courses and those of his PhD students. 

*Many of his PhD students were actively involved as co–authors or experts in the development of LNTwww.
 
'''We thank Professor Joachim Hagenauer for his constant support of our e-learning project'''.}}

[https://www.ce.cit.tum.de/en/lnt/people/professors/hagenauer/ $\text{Hagenauer's Biography from the LNT website}$]

==Prof. Dr. Ralf Kötter (2007-2009)==
 

Ralf Kötter, born on October 10, 1963 in Königstein/Taunus and deceased on February 2, 2009 in Munich, was a German professor in the field of "electrical engineering and information technology" whose numerous works in the field of network coding were of central importance for the further development of mobile communications despite his early death.

[[File:P_ID1790_RalfKoetter1.jpg|165px|right|Ralf Kötter]]

Ralf Kötter studied electrical and communications engineering at Darmstadt Technical University. After graduating in 1990, he subsequently worked at the University of Linköping in the Department of Electrical Engineering until 1996. There he received the degree of Ph.D. (''Teknisk Doktor'') in Electrical Engineering in 1996. In 1996/97, he was a visiting scientist at the IBM Almaden Research Laboratory in San José, California, and subsequently a professor at the Coordinated Science Laboratory and Department of Engineering at the University of Illinois at Urbana-Champaign. In October 2006, he accepted an appointment to the Chair of Communications Engineering in the Department of Electrical Engineering and Information Technology at the Technical University of Munich, succeeding [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|Joachim Hagenauer]].

Ralf Kötter worked in the field of algebraic coding theory and was one of the first scientists to use graph theory to develop error control codes. For his work on decoding Reed-Solomon codes, he was awarded the ''Best Paper Award of the IEEE Information Theory Society'' in 2004. In 2008, he received the ''Best Paper Award of the Signal Processing Society'' for his work on turbo equalization. In addition, he was awarded the 2008 Vodafone Foundation Innovation Award for Research for his "seminal work" on information and coding theory.

Ralf Kötter died at the age of only 45, leaving behind his wife Nuala (who also died of cancer just under 5 years after him) and his then 4-year-old son Finn.

The ''Department for Electrical and Computer Engineering'' at his former university in Illinois established the ''Ralf Koetter Memorial Fund in Electrical and Computer Engineering'' after his death to support students in the department. Ralf's parents Ruth and Hubert Koetter endowed the "Prof. Dr. Ralf Koetter Memorial Award" in 2010, which will be awarded annually until 2023.

'''Important Awards and Honors for Ralf Kötter''':

*IBM Invention Achievement Award (1997).
*NSF CAREER Award (2000)
*IBM Partnership Award (2001)
*Best Paper Award of the IEEE Information Theory Society (2004)
*University of Illinois College of Engineering XEROX Award for Faculty Research (2006)
*Best Paper Award of the Signal Processing Society (2008)
*Innovation Award of the Vodafone Foundation for Research (2008)

[https://www.itsoc.org/news-events/recent-news/koetter-eulogy $\text{IEEE Information Theory Society: In Memoriam Ralf Kötter (2009)}$]

{{BlaueBox|TEXT=
'''Ralf Kötter has been very supportive of the further development of the LNTwww'''  in his unfortunately very short time at LNT. 
*One can clearly recognise his  "scientific handwriting" especially in the books  "Information Theory"  and  "Channel Coding". 

'''We will always keep Ralf in good memory'''.}}

==Prof. Dr. sc. techn. Gerhard Kramer (since 2010)==
 
[[File:Kramer4.jpg|165px|right|Gerhard Kramer]]

Gerhard Kramer, born in 1970 in Winnipeg, Canada, is Alexander von Humboldt Professor and has been full professor of the Department of Communications Engineering (LNT) at the Technical University of Munich (TUM) since 2010 and its Vice President since 2019. He received the B.Sc. in 1991 and the M.Sc. in 1992 in electrical engineering from the University of Manitoba, Winnipeg, Canada. In 1998, he was awarded the Dr. sc. techn. degree (Doctor of Technical Sciences) from the Swiss Federal Institute of Technology (ETH) Zurich.

From 1998 to 2000, Gerhard Kramer worked at Endora Tech AG, Basel, as a communications engineer. From 2000 to 2008 he was a Member of Technical Staff at the Math Center, Bell Laboratories, Alcatel-Lucent in Murray Hill/New Jersey. In 2009, he moved to the University of Southern California (USC) in Los Angeles/California as a professor.

Kramer's research focuses on information theory and communication theory with applications to both wireless and wired networks over copper and fibre respectively.

[https://www.ce.cit.tum.de/en/lnt/people/professors/kramer/ $\text{Kramer's Biography from the LNT website}$]

{{BlaueBox|TEXT=
'''During Gerhard Kramer's tenure (since 2010) the  "German LNTwww"   was completed and he initiated the English version in 2020'''.
*All the teaching areas considered in our learning offer are also covered in Gerhard Kramer's lectures. 
*But not all of his lecture content is presented in the  "LNTwww"   with the same depth and mathematical exactness. 

'''The LNTwww team would like to thank Gerhard Kramer for his great support of our e-learning project'''.}}

{{Display}}

LNTwww:General notes about "Mobile Communications"

2025-02-24T13:39:54Z

'''Four main chapters with a total of 16 chapters (files);   47 Exercises   ⇒   Two semester hours per week lecture and one semester hour per week exercises      
Development of the German version:   2010–2017.       Development of the English version:   2020.      Last corrections:   February 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Hin08]'''<ref name='Hin08'>Hindelang, T.:   Mobile Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2008.</ref>;   '''[Eich11]'''<ref name='Eich11'>Eichin, K.:   Nachrichtensysteme – Kommunikationssysteme (LB).   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2011.</ref>;   '''[Söd10]'''<ref name='Söd10'>Söder, G.: Mobilfunkkanal.   Instructions for the practical course "Simulation digitaler Übertragungssysteme".   LNT/TUM, 2010. </ref>;   '''[Vie17]'''<ref name='Vie17'>Viering, I.:   System Aspects in Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2018</ref>;   '''[Kra18]'''<ref name='Kra18'>Kramer, G.:   Mobile Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2018.</ref>

*Authors:  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]  (responsible for the German version),  [https://www.lntwww.de/Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29 '''Klaus Eichin'''],  and  [https://www.lntwww.de/Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Thomas_Hindelang_.28at_LNT_from_1994-2000_und_2007-2012.29 '''Thomas Hindelang'''].  

* Participating colleagues in alphabetic order:  [https://www.lntwww.de/Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29 '''Benedikt Leible'''],   [https://www.lntwww.de/Biografien_und_Bibliografien/Externe_Beteiligte_am_LNTwww#Dr.-Ing._Markus_Mummert '''Markus Mummert'''],  [https://www.lntwww.de/Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Markus_Stinner_.28at_LNT_from_2011-2016.29 '''Markus Stinner'''],   [https://www.lntwww.de/Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29 '''Johannes Zangl''']

*Participating students in chronological order:      Martin Winkler '''(2001)''', Yven Winter, Franz Kohl, Bettina Hirner, Ji Li, Thorsten Kalweit, Slim Lamine, Johannes Schmidt, Hedi Abbes, Thomas Großer, Néjib Kchouk, Khaled Soussi, Alexander Happach, Felix Kristl, Martin Völkl, David Ginthör, Hussain Sandhu, Mohamed Ben Ahmed, Mohamed Nabil Babai, Marwen Ben Ammar, Wael Chaouch, Safwen Dridi, Mohamed Mansoor, Ayush Patel, Lukas Wolf, David Jobst, Jimmy He, Xiaohan Liu, Matthias Niller, Veronika Hofmann, André Schulz, Andrés Rosa Aparicio, Fatih Onur Özdemir, Noah Nagi, Carolin Mirschina '''(2021)'''

$\text{List of sources:}$

LNTwww:About LNTwww

2025-02-24T13:39:54Z

==Welcome to the English version of LNTwww==
 
»$\text{https://en.lntwww.de}$«  is an e-learning tutorial for Communications Engineering with nine didactic multimedia textbooks including exercises with solutions,  learning videos,  and interactive applets.  It is offered by the  »[https://www.ce.cit.tum.de/en/lnt/home/ Institute for Communications Engineering]«  of the  »[https://www.tum.de/en/ Technical University of Munich]«. 

:⇒   '''It is freely accessible,  registration is not necessary and no system requirements are needed'''.

The German-language version   »$\text{https://www.lntwww.de}$«   ⇒   »$\rm L$erntutorial für $\rm N$achrichten$\rm T$echnik im $\rm w$orld $\rm w$ide $\rm w$eb«  was created between 2001 – 2021 by members of our Institute.  The toolbar entry  »Deutsch«  takes you to the German original.  In spring 2020 we started the English translation,  and in spring 2023 we finished.

*The current version from 2023 is based on the software  [https://en.wikipedia.org/wiki/MediaWiki »MediaWiki«],  known by the encyclopaedia  »WIKIPEDIA«.   The following is a kind of  »user guide«  to our e–learning project.  Corresponding links to this file  »About LNTwww«  can be found at the bottom of each page between  »Privacy policy« and  »Disclaimer«.

*We consider the present version as final;  an extension is currently not planned.  But of course we will continue to improve detected errors or inaccuracies promptly.  So if you notice any inadequacies regarding content,  presentation or handling,  then please send a detailed message by mail to  »LNTwww@ice.cit.tum.de«.

*On the  [[LNTwww:Information|»Information«]]  page you will find notes about temporary restrictions  $($e.g. in case of unavailability due to service work$)$  and a brief summary of all »LNTwww« features using a few example pages.

We would be pleased if we could arouse your interest in our e-learning project.  We wish you a good learning success.

$\text{Have fun and good luck!}$  

[https://www.ce.cit.tum.de/en/lnt/people/professors/kramer/ $\text{Gerhard Kramer}$'''],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29| $\text{Tasnád Kernetzky}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |$\text{Günter Söder}$]]

Munich,  in spring 2024          

 

==Content==

===(A)   The didactic concept of LNTwww===

At the beginning of the work on  »$\rm LNTwww$«  in 2001,  we gave ourselves the following ten rules.  These still apply today:

'''(1)'''   The teaching area  »Information and Communication Technology«  $\text{(I&K)}$  including associated basic subjects  $($Signal Representation,  Fourier and Laplace Transform,  Stochastic Signal Theory, etc.$)$  is presented in a didactically and multimedia prepared form.

'''(2)'''   Nine subject areas were selected,  each of which is covered by a self-contained book in the scope of a one-semester course with three semester hours per week to five semester hours per week.

'''(3)'''   The target group of our online offer are students of  $\text{I&K}$  technology,  especially of »Communications Engineering«,  as well as practicing engineers  $($Keywords:  »professional training«,  »lifelong learning«$)$.

'''(4)'''   In particular,  the interrelationships between different subfields of our extensive e-leatning offer should also be shown,  which is promoted by a nomenclature that is largely consistent in all books.

'''(5)'''   »$\rm LNTwww$«  offers two modes of learning:   Beginners should proceed sequentially  –   for advanced learners,  use it as a tutorial  $($work through exercises first,  jump to the theory part if deficits are identified$)$.

'''(6)'''   The theory is explained as in a traditional engineering textbook through texts,  graphics,  and mathematical derivations.  In addition,  each chapter includes at least one multimedia module.

'''(7)'''   »$\rm LNTwww$«  shall provide the user with multiple interaction options regarding the selection and presentation of theory chapters,  exercises,  learning videos as well as multimedia and calculation modules.

'''(8)'''   The methodology of hyperlinks typical of the  »world wide web«  is extensively used within
the  »$\rm LNTwww$«  and externally.  This is also intended to show connections between different teaching areas.

'''(9)'''   In order to prevent a user from getting lost in his learning environment and using  »$\rm LNTwww$«  only for  »surfing«,  a purposeful path must be recognizable for him at all times despite certain freedoms.

'''(10)'''  For reasons of sustainability of learning success,  there are possibilities for printing the texts and graphics,  ignoring the fact that today's students generation often devalues this as a  »relapse into the analog age«.
 

===(B)   Content and scope of LNTwww===

»$\rm LNTwww$«  is a virtual course totaling  $\text{36}$  semester hours per week 
*with  $\text{23}$  semester hours per week  (quasi) lectures   ⇒   $\text{23L}$

*and  $\text{13}$  semester hours per week exercises   ⇒   $\text{13E}$. 

It is organized in book form.  Each book contains a one-semester course.  For example,  in the case of the third book,  it is indicated that the book  »Theory of Stochastic Signals«  corresponds to a face-to-face–course with three semester hours per week of  »lecture«  and two semester hours per week of  »exercises«   ⇒   $\text{3L +2E}$.

$\text{Textbooks:}$
# [[Signal_Representation|'''Signal Representation''']]   ⇒   [[LNTwww:General_notes_about_"Signal_Representation"|»Impressum«]],
# [[Linear_and_Time_Invariant_Systems|'''Linear and Time Invariant Systems''']]   ⇒  [[LNTwww:General_notes_about_"Linear_and_Time_Invariant_Systems"|»Impressum«]],
# [[Theory_of_Stochastic_Signals|'''Theory of Stochastic Signals''']]   ⇒  [[LNTwww:General_Notes_about_the_Book_"Stochastic_Signal_Theory"|»Impressum«]],
# [[Information_Theory|'''Information Theory''']]   ⇒  [[LNTwww:General_notes_about_"Information_Theory"|»Impressum«]],
# [[Modulation_Methods|'''Modulation Methods''']]   ⇒  [[LNTwww:General_notes_about_"Modulation_Methods"|»Impressum«]],
# [[Digital_Signal_Transmission|'''Digital Signal Transmission''']]   ⇒  [[LNTwww:General_notes_about_"Digital_Signal_Transmission"|»Impressum«]],
# [[Mobile_Communications|'''Mobile Communications''']]   ⇒  [[LNTwww:General_notes_about_"Mobile_Communications"|»Impressum«]],
# [[Channel_Coding|'''Channel Coding''']]   ⇒  [[LNTwww:General_notes_about_"Channel_Coding"|»Impressum«]],
#[[Examples_of_Communication_Systems|'''Examples of Communication Systems''']]   ⇒  [[LNTwww:General_notes_about_"Examples_of_Communication_Systems"|»Impressum«]].

*The theory pages of all books result in the print version in approx.  $1500$  pages  $($DIN A4$)$  and contain on average one and a half graphics per page. 

*In addition,  LNTwww provides via the link  [[Biographies_and_Bibliographies|»'''Biographies & Bibliography'''«]]  a subject-specific bibliography with approx.  $400$  entries,  plus links to the WIKIPEDIA biographies of important scientists.
 

===(C)   Design and structure of LNTwww===

One can reach the nine reference books and »Biographies & Bibliography«  through the link  [[Book Overview|»'''Book Overview'''«]].  From this interface one can reach the individual books.  
*Each book is divided into several  »'''main chapters'''«, 

*each main chapter into several  »'''chapters'''«,  and

*each chapter includes several  »'''sections'''«.

{{GraueBox|TEXT=
$\text{Example 1:}$ 
We consider the book  [[Signal Representation|»Signal Representation«]].  This contains five  »main chapters«.
*By clicking on the first main chapter  »Basic Terms of Communications Engineering«,  one can get to three  »chapters«.  Each chapter corresponds to a MediaWiki file.

*The exemplary chapter  [[Signal_Representation/Principles_of_Communication|»Principles of Communication«]]  contains ten  »sections»  or  »pages».

*The last two pages are almost the same in all chapters,  namely  »Exercises for the chapter«  and  »References«.}}
 

===(D)   Content overviews for LNTwww===

A brief overview of all books is available on the selection interface  [[Book Overview|»'''Book Overview'''«]].
*More information is provided by the  »first page«  of each book.

*The respective main chapter content can be found in the first subchapter on the first page of each.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The first page  $($title page$)$  of the book  [[Signal_Representation|»Signal Representation«]]  provides the following information:
# A brief summary of the entire book;
# Scope of learning offer:  $2{\rm L} + 1{\rm E}$   ⇒   lecture with two semester hours per week and one additionalhour exercise. 
# Five main chapters,  19 chapters,  127 sections,  58 exercises;
# Links to the five main chapters of the book;
# Links to the associated exercises,  learning videos,  and interactive applets in the book  »Signal Representation«;
# Bibliography for the book;
# The imprint to the book  $($Authors,  other contributors,  materials as a starting point of the book,  referencces$)$.

The content of the first main chapter  »Principles of Communication« can be found on the first page 
[[Signal_Representation/Principles_of_Communication#OVERVIEW_OF_THE_FIRST_MAIN_CHAPTER|»# OVERVIEW OF THE FIRST MAIN CHAPTER #«
]].}}
 
===(E)   LNTwww exercises===

A central role in our didactic concept play »exercises«. We believe that the sensible use of »LNTwww« by a user with previous knowledge should be that he first work on the exercises relating to his actual learning area and only jump to the corresponding theory section when required.

You can find the  »'''exercise overview'''«  for all books  $($approx.  $640$  exercises, approx.  $3100$  subtasks)  on the home page via the link  [[Aufgaben:Aufgabensammlung|»'''Exercises'''«]].  All exercises are structured in the same way:
*Each exercise consists of the »exercise description« and several  »subtasks«.   An exercise is only solved correctly if all subtasks are correct.

* For each exercise there is a detailed  »sample solution«,  sometimes with the indication of several ways to the goal.

* The »exercise types« used are:
# »Single Choice»   ⇒   only one of the  $n$  given answers is correct;      ⇒   Marks of alternative answers:  ${\huge\circ}$
# »Multiple Choice«   ⇒   of the  $n$  given answers, between zero and  $n$  answers can be correct;      ⇒   Marks of alternative answers:  $\square$
# »Arithmetic Task«   ⇒   numerical value query,  possibly with sign;     small deviations  $($usually  $\pm 3\%)$  are allowed when checking real-valued results.

* We distinguish between  »exercises«  $($e.g.  »Exercise 1.1»$)$  and  »additional exercises«  $($e.g.  »Exercise 1.1Z«$)$.
# If you were able to solve all exercises of a chapter without any problems,  we believe that you are familiar with the content of the entire chapter. 
#If you have solved one exercise incorrectly,  you should also work on the following,  usually somewhat easier additional exercise.

{{GraueBox|TEXT=
$\text{Example 3:}$ 
The  $58$  exercises/additional exercises of the first book can be accessed via the link  [https://en.lntwww.de/Category:Signal_Representation:_Exercises »Signal Representation: Exercises«]. 

*From there,  we move on to the individual exercises,  e.g. to  [https://en.lntwww.de/Aufgaben:Exercise_1.1:_Music_Signals »Exercise 1.1: Music Signals«].  This relatively simple exercise consists of
#  one »Single Choice«   ⇒   subtask  '''(1)''',
#  two »Multiple Choice«   ⇒   subtasks  '''(2)''',  '''(3)''',  and
#  one »Arithmetic Task« with two real-valued computational queries   ⇒   subtask  '''(4)'''.

*However,  most of our exercises are not that easy.  Although MediaWiki also calls an arithmetic task  »quiz«,  answering them is usually much more difficult than in the numerous quiz shows on TV.   Because: 
#  There are no predetermined answers in an arithmetic task,  and moreover:
#  Integrals often have to be solved beforehand,  such as in  [[Aufgaben:Exercise_4.4:_Two-dimensional_Gaussian_PDF|»Exercise 4.4: Two-dimensional Gaussian probabilty density function«.]]

*We recommend:  First print the exercise   ⇒   »printable version«  and solve the exercise  offline  before checking  online.
}}

 

 
===(F)   LNTwww learning videos===

You can access approximately  $30$  learning videos via the link  »Videos«  on the start page.  The realization of a learning video required the following individual steps: 
:Writing the script and texts   ⇒   Creating a set of slides with only slight differences between successive slides   ⇒   Voicing texts and audio editing   ⇒   Combining texts and images into a coherent video stream.
#Clicking on this link brings up a; list of all learning videos,  grouped by textbook.  Some videos appear for multiple books.
#After selecting the desired learning video,  a wiki description page appears with a short content description and user interface.
#From here you can start the video in  »mp4«  and  »ogv«  format.  The browser will search for the appropriate format.
#The videos can be played by many browsers  $($Firefox, Chrome, Safari, ...$)$  as well as smartphones and tablets.
#The bottom link provides all available learning videos in alphabetical order.

Note:   All learning videos are with German language.  English translations are not planned.

{{GraueBox|TEXT=
$\text{Example 4:}$ 
We'll take a look at   [[Analoge_und_digitale_Signale_(Lernvideo)|»Analog and digital signals«]]  as an example.  This provides a two-part video in mp4 and ogv format.
*Each video part can be started by single click and paused by another click.

*The playback speed of the videos can be changed:
** Firefox offers a submenu after right-clicking on the video.
** For Google Chrome you can install e.g. the plugin  »Video Speed Controller«.
}}
 

===(G)   LNTwww applets===

Applets have a similar function as laboratories in mathematical-scientific courses:  Supplementing lecture/exercise with independent work by the student on the topic covered. 

You can access the provided interactive applets via the link of the same name on the home page.  It should be noted:
#Clicking on the link  »'''Applets'''«  a list of all applets appears,  grouped by reference books. 
#We distinguish between the newer  $\text{HTML 5/JavaScript}$  applets  $($in the respective lists above$)$  and the older  $\text{SWF}$  applets  $($below$)$. 
#The SWF applets unfortunately do not work on smartphones and tablets.
#After selecting an HTML 5/JS applet  a wiki description page appears with introductory theory section,  exercises to be solved and sample solutions. 
#At the beginning and end of this wiki description page there are links to the actual applet in German resp. English Language.

{{GraueBox|TEXT=
$\text{Example 5:}$ 
The didactic importance of applets shall be proved by  [[Applets:Eye_Pattern_and_Worst-Case_Error_Probability|»Eye Pattern and Worst-Case Error Probability«]]:
*The  »eye diagram«  is a proven transmission engineering tool,  to capture the influence of  »line dispersion«   ⇒   »intersysmbol interference«  on the quality characteristic  »error probability«  of a digital transmission system.  

*Such applets serve the clarification of more difficult facts,  in this example  »the step-by-step construction of the eye diagram from the symbol sequence«.  The program offers a lot of setting possibilities.  However, not every setting brings the user a relevant learning success and even fewer lead to a so-called "Aha! moment". 

*This is why we guide the user specifically through the program on the basis of the experiment.  He has to solve various tasks:  Predict and evaluate results,  Optimize parameters,  etc.

*A top 10% student has of course the possibility,  to set himself tasks going beyond the execution of experiments with the help of the applet and thus to penetrate very deeply into the presented subject matter.

}}

In addition to these  $\approx\hspace{-0.1cm} 30$  HTML 5/JS based applets  we still offer some of our  $\approx\hspace{-0.1cm}50$  older German-language applets,  which are based on  »Shock Wave Flash«  $\rm (SWF)$.  These were programmed for  »Adobe Flash«. 
#Since the Flashplayer browser plugin is no longer supported for security reasons,  these applets must be opened with the  »projector version«.
#You do not need to install the projector version and it will not be integrated into your browser.  So there are no security concerns in this regard.
#On the corresponding wiki pages you can find the projector version of the flash player and of course the applet itself.
 

===(H)   Glossary===

Due to the fact,  that our e–learning project LNTwww was first conceived in German and the wish for an English version came much later,  in the English version the assignment between  »Formula signs«   and  »Designation«  is not quite easy.   What do for example

#  $f_{\rm T}$,
#  $s_{\rm TP}(t)$, 
#  $e$,
#  $E$?

Here the link  [[LNTwww:Glossary|»Glossary«]]  on the home page below can help with the following alphabetically ordered entries:
::  »Formula sign«   ⇒   »German designation«   ⇒   »English designation« 

The file is self-explanatory. A few explanations are given under the last menu item  »Some remarks to the Glossary«.

{{GraueBox|TEXT=
$\text{Example 6:}$  In this file you will find the following entries:
:  $f_{\rm T}$   ⇒   Trägerfrequenz   ⇒   carrier frequency 
:  $s_{\rm TP}(t)$   ⇒   äquivalentes Tiefpass–Sendesignal   ⇒   equivalent low-pass transmitted signal 
:  $e= 2.718281828456$...   ⇒   Eulersche Zahl   ⇒   Eulerian number 
:  $ E$   ⇒   $(1)$ Schwellenwert,  $(2)$ Energie   ⇒   $(1)$ threshold value,  $(2)$ energy 

From the context,  the decision for  $(1)$  or  $(2)$  should be easy. }}

 

===(I)    History of LNTwww===

At the  [https://www.ce.cit.tum.de/en/lnt/home/ »Institute for Communications Engineering«]  $\rm (LNT)$  of the  [https://www.tum.de »Technical University of Munich«]  $\rm (TUM)$  two  teaching software packages  $\text{(LNTsim, LNTwin)}$  were realized by  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29|»Günter Söder«]] from 1984 to 1996, which were used in our practical courses.  Several other universities have also acquired and used these programs.

At the beginning of the first Internet euphoria,  there were inquiries from students whether we could also provide such simulation and demonstration programs online.  After careful consideration  ("Is the expected big effort worth it?")  Günter Söder began 2001 planning the German-language project  »www.LNTwww.de«.  Co-responsible was his colleague  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|»Klaus Eichin«]].  The project was to be completed by 2011 at the latest,  since both would be retiring this year.

The content was derived from their own teaching materials as well as those of his colleague  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|»Norbert Hanik«]]  $($Associate Professor of Line Transmission Technology$)$.  Other lecture material was also taken into account,  which was produced at the Institute of Communications Engineering under the last four chair holders:
::*Professor [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Hans_Marko_.281962-1993.29|»Hans Marko«]]  $($Head of the LNT from 1962 to 1993$)$,
::*Professor [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|»Joachim Hagenauer«]]  $($Head of the LNT from 1993 to 2006$)$,
::*Professor [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|»Ralf Kötter»]]  $($Head of the LNT from 2007 to 2009$)$,  and
::*Professor [https://www.ce.cit.tum.de/en/lnt/people/professors/kramer/ »Gerhard Kramer«]  $($Head of the LNT since 2010$)$.

Just a few dates about progress of the German-language LNTwww project,  eleven years after the planned completion :
* First of all our own platform had to be developed by students  $($Marin Winkler,  Yven Winter$)$.  The authoring system  »LNTwww«  was based on the http server  »Apache«,  the database  »MySQL«,  the script language  »Perl«  and  »Shock Wave Flash«  $\rm (SWF)$  as a basis for multimedia applications   ⇒   version  »LNTwww.v1«  $($2003$)$.

*Work of the following years was online adaptation of the manuscripts,  input into the database with the rather complicated LNTwww syntax,  creation of the graphs as well as conception and realization of multimedia elements.  After completion of all nine textbooks the desired final state was reached   ⇒   version  »LNTwww.v2«  $($2016$)$.

* At the same time,  it became known that  »SWF«  would not longer be supported by relevant manufacturers.  This fact and the criticism heard from some users about the meanwhile too staid design  $($our authoring system was on the level of 2003$)$  were decisive for a new start based on  »MediaWiki«   ⇒   version  »LNTwww.v3«  $($2021$)$.

Finally,  a few sentences about the English LNTwww version. 
*At the beginning of the Corona pandemic and the associated lockdowns,  the call for  »e-learning«  also became louder and louder at the universities,  even from professors who had previously rather rejected this form of teaching. 

* Suddenly,  funds were also made available,  to provide as many e-learning courses as possible in as short a time as possible.  Our chair  »Gerhard Kramer«  therefore already submitted a corresponding application for working student funds in spring 2020,  which was approved within a few weeks.

*In June 2020 we started the English translation with support of the  »DEEPL«  program  $($free version$)$  and finished it in April 2023.  From the LNT staff were involved:  »Javier Garcia Gomez, Tasnád Kernetzky, Benedikt Leible and Günter Söder«.  Of the students involved,  »Noah Nagi«  and  »Jiwoo Hwang«  deserve special mention.
 

===(J)   Acknowledgement===

The Institute for Communications Engineering would like to thank the many people involved in the creation of  $\rm LNTwww$:

*The persons responsible for the German and/or English LNTwww projects:
::[[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |$\text{Günter Söder}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|$\text{Klaus Eichin}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29| $\text{Tasnád Kernetzky}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]].

* The  $($former$)$  LNT/LÜT colleagues,  who contributed as co–authors or experts or supervised student work:
::[[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Ronald_B.C3.B6hnke_.28at_LNT_from_2012-2014.29|$\text{Ronald Böhnke}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Joschi_Brauchle_.28at_LNT_from_2007-2015.29|$\text{Joschi Brauchle}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|$\text{Bernhard Göbel}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|$\text{Norbert Hanik}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Thomas_Hindelang_.28at_LNT_from_1994-2000_und_2007-2012.29|$\text{Thomas Hindelang}$]],   [[Biographies_and_Bibliographies/External_Contributors_to_LNTwww#Dr._Gianluigi_Liva|$\text{Gianluigi Liva}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Tobias_Lutz_.28at_LNT_from_2008-2014.29|$\text{Tobias Lutz}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Michael_Mecking_.28at_LNT_from_1997-2012.29|$\text{Michael Mecking}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Markus_Stinner_.28at_LNT_from_2011-2016.29|$\text{Markus Stinner}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Thomas_Stockhammer_.28at_LNT_from_1995-2004.29|$\text{Thomas Stockhammer}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|$\text{Johannes Zangl}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Georg_Zeitler_.28at_LNT_from_2007-2012.29|$\text{Georg Zeitler}$]].

*The more than  »50 students«,  who have worked on subareas,  designed learning videos and applets or implemented the porting to the MediaWiki version within the framework of Engineering practice,  diploma,  bachelor and master theses or within the framework of a working student activity.

*The  [https://www.https://www.ei.tum.de/en/welcome/ »Department of Electrical and Computer Engineering«]  and the  [https://www.tum.de/en/ »Technical University of Munich«]  for funding working students in the years since 2016 within the framework of the  [https://www.ei.tum.de/studium/studienzuschuesse/ »MoliTUM«]  resp.  [https://www.tum.de/en/studies/teaching/awards-and-competitions/ideas-competition »EXIni«]  programs.

        [https://www.ce.cit.tum.de/en/lnt/people/professors/kramer/ $\text{Gerhard Kramer}$]

LNTwww:Information

2025-02-24T13:39:54Z

{{BlaueBox|TEXT=
$\text{Here some notes on LNTwww}$

#For detailed information about our e–Learning tutorial,  please see the page  »[[LNTwww:About_LNTwww|About LNTwww]]«  –  a kind of &bdquo;user guide”.
#On every LNTwww page,  there is a link to this file at the bottom  $($between »Privacy« and »Disclaimer«$)$.
#If you have difficulties with our terms,  the  »[[LNTwww:Glossary|Glossary]]»  may help. 
#We consider this March 2023 version to be final;  no further revision or expansion is currently planned. 
#Therefore,  there will also be no English translations of the German-language learning videos and SWF applets in the future.
#But,  of course,  we will continue to improve identified errors regarding content,  presentation or handling in a timely manner. 
#Should you notice any such inadequacies,  then please send a detailed message by mail to  »LNTwww@ice.cit.tum.de«.
#Below you will find a compact summary of our e-learning project with useful tips on »how to use LNTwww« sensibly.
}}

==LNTwww: An e-learning project for Communications Engineering==

 
The e-learning project »LNTwww« offered by the  [https://www.ce.cit.tum.de/en/lnt/home/ »TUM Institute for Communications Engineering«]<ref name = 'Link'> Use »CTRL + SHIFT + click« to open the link in new tab. Close the new tab to return to this tab.</ref>  provides nine online courses on the subjects of »Communications Engineering«  $\rm (CE)$  and  »Information and Communication Technology«  $\text{(I&C)}$.  The target group of our e-learning online platform are students of these or similar disciplines as well as practicing engineers and scientists.

The German version  »www.LNTwww.de«  was created from 2001 to 2021.  At the beginning of corona  $($2020$)$ we started the English version, which has been finalized in 2024.  In the following we refer to this version  [https://en.lntwww.de »en.lntwww.de«]<ref name='Link'></ref> ,  whose homepage can be seen in the graphic below.

Here are some features of our e-learning platform:
[[File:LNTwww2023_StartPage_6thJan2024.png|right|frame|Screenshot of the English version  »www.LNTwww.de«. 
Note: (1)  '''LNTwww''' is acronym of the German term »'''L'''erntutorial für '''N'''achrichten'''T'''echnik im '''w'''orld '''w'''ide '''w'''eb«. (2)  '''LNT''' also stands for the German name »'''L'''ehrstuhl für '''N'''achrichten'''T'''echnik« of our chair.]]

# »LNTwww« is freely accessible $($no need for registration$)$. No specific system requirements.
# »LNTwww« uses the free server-based software  [https://en.wikipedia.org/wiki/MediaWiki »MediaWiki«]<ref name='Link'></ref>,  just like »Wikipedia«, the best-known free encyclopedia.
#The  [https://en.lntwww.de/Book_Overview »Book Collection«]<ref name='Link'></ref> link takes you to the nine courses $($which are referred to as »books«$)$ and to the collection »Biographies and Bibliographies«.
#The  [https://en.lntwww.de/Exercises:Exercise_Overview »Exercises«]<ref name='Link'></ref>  link takes you to a list with a total of around  $640$  exercises and  $3100$  subtasks $($each with detailed sample solution$)$.
#About thirty learning videos $($in German language$)$ can get accessed via the  [https://en.lntwww.de/LNTwww:Videos »Videos«]<ref name='Link'></ref>  link. These are grouped according to the individual courses.
#Via the  [https://en.lntwww.de/LNTwww:Applets »Applets«]<ref name='Link'></ref>  link you have access to around thirty applets based on HTML5/JavaScript and some older shockwave flash $($SWF$)$ applets.

Other important project features are summarized in the file »About LNTwww«  $($red marked link$)$,  among others:
* [https://en.lntwww.de/LNTwww:About_LNTwww#.28A.29_The_didactic_concept_of_LNTwww »The didactic concept of LNTwww«]<ref name='Link'></ref>:   These rules from 2001 still apply, although »LNTwww« has had to be continuously adapted to developments on the Internet.

* [https://en.lntwww.de/LNTwww:About_LNTwww#.28B.29_Content_and_scope_of_LNTwww »Content and scope of LNTwww«]<ref name='Link'></ref>:   Our online course corresponds to conventional courses with a total of $36$ semester hours per week of lectures and exercises.

===LNTwww Design and Structure===
 
»LNTwww« has a book structure. Each »course« corresponds to an own »book« that can be selected via »Book Collection«.
*Each book is divided into several  »main chapters«, 
*each main chapter is divided into several  »chapters«,  and
*each chapter comprises several  »sections«.

{{GraueBox|TEXT=
$\text{Example A:}$  To illustrate these statements, here is an example of how to use »LNTwww«:

# After pressing the  [https://en.lntwww.de/Book_Overview »Book Collection«]<ref name='Link'></ref>  button, a selection screen will appear with the nine course-books as well as the book »Biographies and Bibliographies«.
# By selecting the book  [https://en.lntwww.de/Information_Theory »Information Theory«]<ref name='Link'></ref>,  its "start page" will appear with links to the four main chapters and to the corresponding sub-chapters. Furthermore, beside a brief book summary and bibliographical references, links to exercises and to multimedia elements are part of this information page.
# We now select the first main chapter  »Entropy of Discrete Sources«  and of this in turn the first sub-chapter  [https://en.lntwww.de/Information_Theory/Discrete_Memoryless_Sources »Discrete Memoryless Sources«]<ref name='Link'></ref>.  This exemplary sub-chapter explains in eight sections the procedure for calculating the entropy of binary and non-binary sources.
# As in conventional mathematical and technical literature, the facts are illustrated by texts, models, graphs, diagrams, equations and derivations. The last two sections of each sub-chapter contain exercises and references to the topic covered.}}

===LNTwww Exercises===
 
The core elements of our didactic concept are »exercises«. We believe that the sensible use of »LNTwww« by a user with previous knowledge should be to work first on the exercises related to his actual learning area and only jump to the corresponding theory section when required.

All exercises have a similar structure:
*Each exercise consists of the »exercise description« and several  »subtasks«. An exercise will only get valued as solved if all subtasks are completed correctly.

* For each exercise, a detailed »sample solution« exists, sometimes with an indication to different solution paths.

* The »exercise types« used are:
# »Single Choice»   ⇒   only one of the  $n$  given answers is correct;      ⇒   Check Box for alternative answers:  ${\huge\circ}$
# »Multiple Choice«   ⇒   of the  $n$  given answers, between zero and  $n$  answers can be correct;      ⇒   Check Box for alternative answers:  $\square$
# »Arithmetic Task«   ⇒   numerical value query,  possibly with sign;     small deviations  $($usually  $\pm 3\%)$  are allowed when checking real-valued results.

* We distinguish between  »exercises«  $($e.g.  »Exercise 1.1»$)$  and  »additional exercises«  $($e.g.  »Exercise 1.1Z«$)$.
# If you were able to solve all exercises of a chapter without any problems,  we believe that you are familiar with the content of the entire chapter. 
#If you have solved one exercise incorrectly,  you should also work on the following,  usually somewhat easier additional exercise.

{{GraueBox|TEXT=
$\text{Example B:}$ 
The  $58$  exercises/additional exercises of the first book can be accessed via the link  [https://en.lntwww.de/Category:Signal_Representation:_Exercises »Signal Representation: Exercises«]<ref name='Link'></ref>. 

*From there,  we move on to the individual exercises,  e.g. to  [https://en.lntwww.de/Aufgaben:Exercise_1.1:_Music_Signals »Exercise 1.1: Music Signals«]<ref name='Link'></ref>.  This relatively simple exercise consists of
#  one »Single Choice«   ⇒   subtask  '''(1)''',
#  two »Multiple Choice«   ⇒   subtasks  '''(2)''',  '''(3)''',  and
#  one »Arithmetic Task« with two real-valued computational queries   ⇒   subtask  '''(4)'''.

*However,  most of our exercises are not that easy.  Although MediaWiki also calls an arithmetic task  »quiz«,  answering them is usually much more difficult than in the numerous quiz shows on TV.   Because: 
#  There are no predetermined answers in an arithmetic task,  and moreover:
#  Integrals often have to be solved beforehand,  such as in  [https://en.lntwww.de/Aufgaben:Exercise_4.4:_Two-dimensional_Gaussian_PDF »Exercise 4.4: Two-dimensional Gaussian probabilty density function«]<ref name='Link'></ref>.

*We recommend:  First print the exercise   ⇒   »printable version«  and solve the exercise  offline  before checking  online.
}}

===LNTwww Applets===
 
Working with applets in a virtual environment has a similar function to laboratories in mathematical and engineering sciences face-to-face courses:   Supplementing lectures and exercises through independent work by the students on the topic covered.

Starting from the  [https://en.lntwww.de/LNTwww:Applets »Applet«]<ref name='Link'></ref>  button on the homepage, another click takes you to the »Alphabetic list of all HTML5/JS applets $($English language$)$«. All these twenty-four applets have the same structure:
*The »applet description page« on MediaWiki level provides all information about the theoretical background as well as the purpose and handling of the application.

*The HTML5/JavaScript program with graphical user interface takes over the parameter input and the display of the calculated diagrams and numerical results.

*The most important part of an applet is the »questionnaire«, which guides the user through the program. The user has to solve various tasks along the way:     Predict and evaluate results, optimize parameters, etc.

{{GraueBox|TEXT=
$\text{Example C:}$ 
The didactic significance of applets will be demonstrated by applet no. 10:  [https://en.lntwww.de/Applets:Eye_Pattern_and_Worst-Case_Error_Probability »Eye Pattern and Worst-Case Error Probability«]<ref name='Link'></ref>.  We will not go into the detailed explanation of the theoretical background in the »applet description page« here. Just this much:  The »eye diagram« is a proven digital signal transmission tool for quantifying the influence of line dispersion on the quality characteristic »error probability«.

*By pressing »Open applet in new Tab«, the graphical program interface appears, which allows to choose from four coding options and three basic transmission pulse options. Depending on the setting, further parameter values $($as cutoff frequency, rolloff factor, ...$)$ can get determined.   This means that the program offers a large number of setting options. However, not every setting brings the user a relevant learning success and even fewer lead to a so-called "aha effect".

*It was task of the program developers to formulate the »questionnaire» in the lower section $($in this example 14 exercises$)$ and the associated solutions in such a way that the learning success is as great as possible for as many users as possible. A top 10% student naturally has the opportunity to use the applet to set himself tasks that go beyond our questionnaire and thus to delve very deeply into the material presented.

* This applet serves the clarification of difficult facts. Exercise (1), for example, illustrates the step-by-step construction of the eye diagram from the binary symbol sequence for a Gaussian pulse, and Exercise (10) shows the »Overall View« of the eye diagram of a quaternary Nyquist system with rolloff factor $r_f=0.5$.
}}

===LNTwww Learning Videos===
 
The realization of a learning video required a lot of individual steps:  Writing the script and texts   ⇒   Creating a set of slides by gradually adapting the content   ⇒   Voicing texts and audio editing   ⇒   Combining texts and images to a coherent video stream.

All our learning videos $($created between 2003–2015$)$ are only available in German language.  By assuming that the users of the English version also have sufficient knowledge of German language to understand the content of the video sequences, we did not spend any extra effort in time consuming translation activities.

Please note the following points when using the learning videos:

*The link  »Videos«  on the homepage provides a list of all thirty-one learning videos, grouped according to the nine courses. Some videos are part for multiple courses.
*After selecting the desired learning video,  a wiki description page appears with a short content description and the user interface.
*From here you can start the video in  »mp4«  or  »ogv«  format.  The browser will search for the appropriate format.
*Each video part can get started by a single click and paused by another click.
*The videos can can get used in conjunction with many browsers  $($Firefox, Chrome, Safari, ...$)$  as well as with smartphones and tablets.  The playback speed can get changed:  Firefox offers a submenu after right-clicking on the video.  For the use with Google Chrome the plugin  »Video Speed Controller« need to get installed.

{{GraueBox|TEXT=
$\text{Example D:}$ 
We'll take a look at the three-part learning video  [https://en.lntwww.de/Der_AWGN-Kanal_(Lernvideo) »Der AWGN-Kanal«]<ref name='Link'></ref>.  Here you can see the corresponding »video description page« on MediaWiki level. 

At the end of this file you will find the English title »The AWGN Channel « and the English summary:

    $\text{Part 1}$   $($duration 5:59$)$:   $\text{Preliminary remarks}$     $($Common channel models regarding media,  operational equipment,  interference/noise$)$

    $\text{Part 2}$   $($duration 5:14$)$:   $\text{Properties}$  $($additive, white, Gaussian distributed$)$   $\text{and characteristics}$  $($PSD, PDF, variance, standard deviation/rms$)$

    $\text{Part 3}$   $($duration 6:13$)$:   $\text{Calculation/simulation of BER}$  $($bit error rate$)$   $\text{and SNR}$  $($signal-to-noise ratio$)$  $\text{for the optimal binary system}$
}}

===LNTwww Glossary===
 
Due to the fact that our e–learning project was first conceived in German and the wish for an English version came much later on,  the assignment of a  »formula sign«   to the relevant  »designation«  is not quite easy, e.g.   carrier frequency  $f_{\rm T}$,  equivalent low-pass transmitted signal  $s_{\rm TP}(t)$,  threshold value  $E$.

The link  [https://en.lntwww.de/LNTwww:Glossary »Glossary«]<ref name='Link'></ref>  on the homepage can help in this context with the following alphabetically ordered entries:
::  »Formula sign«   ⇒   »German designation«   ⇒   »English designation« 

The file »Glossary« is self-explanatory. A few explanations are available under the last menu item  »Some remarks to the Glossary«.

{{GraueBox|TEXT=
$\text{Example E:}$  In this file you will find the following entries, among others:
:  $f_{\rm T}$   ⇒   Trägerfrequenz   ⇒   carrier frequency 
:  $s_{\rm TP}(t)$   ⇒   äquivalentes Tiefpass–Sendesignal   ⇒   equivalent low-pass transmitted signal 
:  $ E$   ⇒   $(1)$ Schwellenwert,  $(2)$ Energie   ⇒   $(1)$ threshold value,  $(2)$ energy 

From the context,  the decision for  »threshold value«  or  »energy«  should be easy. }}

===Final Remarks===
 
Described is our e-learning project »LNTwww«, which began in 2001 and was finished in 2024 with the completion of the English version. The intention of this article is to make this e-learning course on the subject of »Communications Engineering« a little better known in the university community.

*It is difficult to determine the exact user numbers of »LNTwww«, as there is no fixed entry point. Rather, many paths lead to our platform and you can move around it for different lengths of time.
*Based on a number of indicators $($not explained in detail here$)$, we currently expect at least twenty thousand (longer) hits per year for the German version. Nine months after completion, we are not yet able to provide a reliable number for the English version.
*The comments from our students $($in personal discussions$)$ and other users $($via e-mail$)$ were quite positive, which is why we tackled the English translation in 2020. However, it cannot be denied that we had expected euphoric rather than benevolent criticism.

Let's now take a look at the historical development of »e-learning« worldwide. A number of »virtual universities« have been established in the United States since the 1950s; the communication with users was still by post. In the early 1970s, many European universities were also involved in »computer-based learning«, including a very active research group at »TUM-LNT«, which was led by Professor Karlheinz Tröndle, the doctoral supervisor of the »LNTwww« initiators Klaus Eichin and Günter Söder. As there were no personal computers at the time, a microprocessor system first had to be developed by hardware and software.

The term »e-learning« was first mentioned in 1999 by the educational technology expert [https://en.wikipedia.org/wiki/Elliott_Masie »Elliott Masie«]<ref name='Link'></ref> at the TechLearn conference in Disneyworld. Since 2012, leading US universities have been offering various software platforms for e-learning on the internet under the umbrella term »Massive Open Online Course« $\text{(MOOC)}$.
*The adjective »massive« is intended to indicate that there are a lot of internet-based courses on various STEM subjects $($mathematics, computer science, natural sciences, technology$)$, that these can be used by a large number of people and that they are freely accessible to all interested parties free of charge.
*While an xMOOC is more teacher-centred, the cMOOC is more learner-centred, informal and committed to social media. In contrast to our »LNTwww«, however, registration is required for these MOOC courses.

The following MOOC data was compiled by Jürgen Veitenhansl $($2002 one of our first LNTwww students$)$; they mostly refers to the year 2022:
*The largest user numbers are reported by the Indian platform [https://en.wikipedia.org/wiki/Byju%27s »Byju's«]<ref name='Link'></ref> and the Korean organisation [https://www.khanacademy.org/ »Khana«]<ref name='Link'></ref> with 150 million resp. 137 million. The e-learning provider [https://www.edx.org/courses »edx«]<ref name='Link'></ref>, founded by professors from Harvard University and MIT $($Massachusetts Institute of Technology$)$ had around 100 million users and 2000 courses.
*The US market leader [https://en.wikipedia.org/wiki/Coursera »coursera«]<ref name='Link'></ref> - a foundation of Stanford University - claims even higher figures with more than 100 million users and more than 7000 courses available. Five years earlier, the figures were 40 million resp. 3000 courses.
*The most important MOOCs “coursera” and “edx” have cooperation agreements with over three hundred universities worldwide, including the “Technical University of Munich”. Student groups from these universities can develop their own programs, which can be transferred to the MOOC platform after certification.
*According to the annual »coursera« reports,  this platform alone achieved in 2022 a turnover of more than 500 million US dollars. The more users register for free-of-charge courses, the higher the number of those who opt for a paid course option.
*If some published market data is to be believed, the global e-learning market potential is estimated at more than 100 billion USD, i.e. more and more providers will naturally enter the market in order to obtain the largest possible share of this potential.
*If you compare the courses offered by the most important MOOC platforms »edx« and »edx«, you will see that there are many similarities to our »LNTwww» in terms of 
            $(1)$   basic structure,       $(2)$   didactic concept,       $(3)$  presentation $($theory pages, videos, audios$)$,       $(4)$   exercises for performance assessment.
*We were initially quite dismayed when we recognized these analogies shortly after completing the German version. On the other hand, we felt confirmed that the concept we defined in 2001 could not be entirely wrong.
*But it should also be noted that both »coursera« and »edx« only started in 2012, when 60% of our German version was already finished. We were very surprised that »TUM-LNT« had embraced the topic of »e-learning« a decade earlier than the most important US universities.

{{GraueBox|TEXT=
$\text{Our conclusio:}$ 
#We are well aware that with our total  $($in twenty-four years$)$  sixteen participating professors and doctoral students from  [https://en.lntwww.de/Biographies_and_Bibliographies/LNTwww_members_from_LNT »$\text{LNT}$«]<ref name='Link'></ref> $($Institute for Communications Engineering$)$  and  [https://en.lntwww.de/Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT »$\text{LÜT}$«]<ref name='Link'></ref> $($Professorship Line Transmission Technology$)$  and a total of about fifty students cannot achieve the same number of courses and the same number of participants as the MOOC courses »edx« and »coursera«.
#Nor their professionalism in presentation, especially with the multimedia elements. Many of our applets and videos were created in the years 2001-2010, those of the MOOC courses mostly after 2020. In between, there was a significant further development of the editing tools.
#Even from today's perspective, Gerhard Kramer's decision to tackle the English version after completing the German version was the right one, despite the effort involved, as we did not approach this step as a pure translation, but as a revision of the previously achieved. Both versions benefited from this.
#Of the 7000 MOOC courses mentioned above, some certainly deal with »Communications Engineering«. In the [https://www.tum.de/en/lifelong-learning/innovation-in-teaching-and-continuing-education/moocs-at-tum »TUM MOOC co-operation list«], however, there is no appropriate entry. If you do not find what you are looking for when searching other universities, we recommend »LNTwww«.
#This e-learning project provides nine online courses on the subjects of »Communications Engineering«  $($grouped in another way: twelve courses of three semester hours per week each$)$. The target group of »LNTwww« are students of these or similar disciplines as well as practicing engineers and scientists.
}}

===Acknowledgement===
 
*The authors would like to thank all professors and doctoral students who have invested in addition to their numerous tasks in research and teaching a great deal of time in the development of »LNTwww» . These people are listed by name in the [https://en.lntwww.de/LNTwww:LNTwww_Impressum »Impressum«]<ref name='Link'></ref>, separated by book, as are the basic materials for the respective first versions.

*We would also like to thank the more than fifty students who have worked with great dedication at the »LNTwww« project between 2001 and 2023.

*The English translation $($2020–2024$)$ by working students was coordinated by Javier García Gómez, Tasnád Kernetzky and Benedikt Leible. This work was financially supported by the faculty and the university.

          [https://www.ce.cit.tum.de/en/lnt/people/professors/kramer/ $\text{Gerhard Kramer}$''']       [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |$\text{Günter Söder}$]]

===Note===
__NOTOC__

LNTwww:Imprint for the book "Mobile Communications"

2025-02-24T13:39:54Z

'''Four main chapters with a total of 16 chapters (files) and 121 sections (pages);   47 Exercises   ⇒   Scope:  "$\rm 2L+1E$"      
Development of the German version:   2010–2016.       Development of the English version:   2020/21.      Last corrections:   October 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:   '''[Hin08]'''<ref name='Hin08'>Hindelang, T.:   Mobile Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2008.</ref>;   '''[Söd10]'''<ref name='Söd10'>Söder, G.: Mobilfunkkanal.   Instructions for the practical course "Simulation digitaler Übertragungssysteme".   LNT/TUM, 2010. </ref>;   '''[Eich11]'''<ref name='Eich11'>Eichin, K.:   Nachrichtensysteme – Kommunikationssysteme (LB).   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2011.</ref>;  '''[Vie17]'''<ref name='Vie17'>Viering, I.:   System Aspects in Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2018</ref>;   '''[Kra18]'''<ref name='Kra18'>Kramer, G.:   Mobile Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2018.</ref>   –   Textbook:   '''[EVB01]'''<ref name='EVB01'>Eberspächer, J.; Vögel, H.J.; Bettstetter, C.: Global System for Mobile Communication. 3. Auflage. Stuttgart: Teubner, 2001.</ref>

*Authors:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29| '''Klaus Eichin''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Thomas_Hindelang_.28at_LNT_from_1994-2000_und_2007-2012.29| '''Thomas Hindelang''']].  

* Participating colleagues in alphabetic order:  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Markus_Stinner_.28at_LNT_from_2011-2016.29|'''Markus Stinner''']],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29| '''Johannes Zangl''']]

*Participating students in chronological order:  Martin Winkler '''(2001)''', Yven Winter, Thorsten Kalweit, Slim Lamine, Johannes Schmidt, Hedi Abbes, Thomas Großer, Néjib Kchouk, Khaled Soussi, Alexander Happach, Felix Kristl, Martin Völkl, André Schulz, Noah Nagi, Carolin Mirschina, Jiwoo Hwang '''(2023)'''

$\text{References:}$

Applets:Capacity of Memoryless Digital Channels

2025-02-24T13:39:54Z

{{LntAppletLinkEnDe|transinformation_en|transinformation}}

==Applet Description==
 
In this applet,  binary  $(M=2)$  and ternary  $(M=3)$  channel models without memory are considered with  $M$  possible inputs  $(X)$  and  $M$  possible outputs  $(Y)$.  Such a channel is completely determined by the  "probability mass function"  $P_X(X)$  and the matrix  $P_{\hspace{0.01cm}Y\hspace{0.03cm} \vert \hspace{0.01cm}X}(Y\hspace{0.03cm} \vert \hspace{0.03cm} X)$  of the  "transition probabilities".

For these binary and ternary systems,  the following information-theoretic descriptive quantities are derived and clarified:
*the  "source entropy"   $H(X)$  and the  "sink entropy"   $H(Y)$,

*the  "equivocation"   $H(X|Y)$  and the   "irrelevance"   $H(Y|X)$,

*the  "joint entropy"   $H(XY)$  as well as the "mutual information"  $I(X; Y)$,

*the  "channel capacity"   as the decisive parameter of digital channel models without memory:
:$$C = \max_{P_X(X)} \hspace{0.15cm} I(X;Y) \hspace{0.05cm}.$$

These information-theoretical quantities can be calculated both in analytic–closed form or determined simulatively by evaluation of source and sink symbol sequence.

==Theoretical Background==
 
===Underlying model of digital signal transmission ===
 
The set of possible  »'''source symbols'''«  is characterized by the discrete random variable  $X$. 
*In the binary case   ⇒   $M_X= |X| = 2$  holds  $X = \{\hspace{0.05cm}{\rm A}, \hspace{0.15cm} {\rm B} \hspace{0.05cm}\}$  with the probability mass function   $($ $\rm PMF)$   $P_X(X)= \big (p_{\rm A},\hspace{0.15cm}p_{\rm B}\big)$  and the source symbol probabilities  $p_{\rm A}$  and  $p_{\rm B}=1- p_{\rm A}$.

*Accordingly, for a ternary source  ⇒   $M_X= |X| = 3$:     $X = \{\hspace{0.05cm}{\rm A}, \hspace{0.15cm} {\rm B}, \hspace{0.15cm} {\rm C} \hspace{0.05cm}\}$,     $P_X(X)= \big (p_{\rm A},\hspace{0.15cm}p_{\rm B},\hspace{0.15cm}p_{\rm C}\big)$,     $p_{\rm C}=1- p_{\rm A}-p_{\rm B}$.

The set of possible  »'''sink symbols'''«  is characterized by the discrete random variable  $Y$.  These come from the same symbol set as the source symbols   ⇒   $M_Y=M_X = M$.  To simplify the following description, we denote them with lowercase letters, for example, for  $M=3$:    $Y = \{\hspace{0.05cm}{\rm a}, \hspace{0.15cm} {\rm b}, \hspace{0.15cm} {\rm c} \hspace{0.05cm}\}$.

The relationship between the random variables  $X$  and  $Y$  is given by a   »'''discrete memoryless channel model'''«  $($ $\rm DMC)$.  The left graph shows this for  $M=2$  and the right graph for  $M=3$.

[[File:Transinf_1_neu.png|center|frame|  $M=2$  (left) and for  $M=3$  (right).     Please note:  In the right graph not all transitions are labeled]]

The following description applies to the simpler case  $M=2$.  For the calculation of all information theoretic quantities in the next section we need besides  $P_X(X)$  and  $P_Y(Y)$  the two-dimensional probability functions  $($each a  $2\times2$–matrix$)$  of all
#  [[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Conditional_probability|"conditional probabilities"]]   ⇒   $P_{\hspace{0.01cm}Y\hspace{0.03cm} \vert \hspace{0.01cm}X}(Y\hspace{0.03cm} \vert \hspace{0.03cm} X)$   ⇒   given by the DMC model;
#  [[Information_Theory/Some_Preliminary_Remarks_on_Two-Dimensional_Random_Variables#Joint_probability_and_joint_entropy|"joint probabilities"]]  ⇒   $P_{XY}(X,\hspace{0.1cm}Y)$;
#  [[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Inference_probability|"inference probabilities"]]   ⇒   $P_{\hspace{0.01cm}X\hspace{0.03cm} \vert \hspace{0.03cm}Y}(X\hspace{0.03cm} \vert \hspace{0.03cm} Y)$.

{{GraueBox|TEXT=
[[File:Transinf_2.png|right|frame|Considered model of the binary channel]]
$\text{Example 1}$:  We consider the sketched binary channel.
* Let the falsification probabilities be:

:$$\begin{align*}p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}A} & = {\rm Pr}(Y\hspace{-0.1cm} = {\rm a}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm A}) = 0.95\hspace{0.05cm},\hspace{0.8cm}p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}A} = {\rm Pr}(Y\hspace{-0.1cm} = {\rm b}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm A}) = 0.05\hspace{0.05cm},\\
p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}B} & = {\rm Pr}(Y\hspace{-0.1cm} = {\rm a}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm B}) = 0.40\hspace{0.05cm},\hspace{0.8cm}p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}B} = {\rm Pr}(Y\hspace{-0.1cm} = {\rm b}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm B}) = 0.60\end{align*}$$

:$$\Rightarrow \hspace{0.3cm} P_{\hspace{0.01cm}Y\hspace{0.05cm} \vert \hspace{0.05cm}X}(Y\hspace{0.05cm} \vert \hspace{0.05cm} X) =
\begin{pmatrix}
0.95 & 0.05\\
0.4 & 0.6
\end{pmatrix} \hspace{0.05cm}.$$

*Furthermore, we assume source symbols that are not equally probable:

:$$P_X(X) = \big ( p_{\rm A},\ p_{\rm B} \big )=
\big ( 0.1,\ 0.9 \big )
\hspace{0.05cm}.$$

*Thus, for the probability function of the sink we get:

:$$P_Y(Y) = \big [ {\rm Pr}( Y\hspace{-0.1cm} = {\rm a})\hspace{0.05cm}, \ {\rm Pr}( Y \hspace{-0.1cm}= {\rm b}) \big ] = \big ( 0.1\hspace{0.05cm},\ 0.9 \big ) \cdot
\begin{pmatrix}
0.95 & 0.05\\
0.4 & 0.6
\end{pmatrix} $$

:$$\Rightarrow \hspace{0.3cm} {\rm Pr}( Y \hspace{-0.1cm}= {\rm a}) =
0.1 \cdot 0.95 + 0.9 \cdot 0.4 = 0.455\hspace{0.05cm},\hspace{1.0cm}
{\rm Pr}( Y \hspace{-0.1cm}= {\rm b}) = 1 - {\rm Pr}( Y \hspace{-0.1cm}= {\rm a}) = 0.545.$$

*The joint probabilities  $p_{\mu \kappa} = \text{Pr}\big[(X = μ) ∩ (Y = κ)\big]$  between source and sink are:

:$$\begin{align*}p_{\rm Aa} & = p_{\rm a} \cdot p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}A} = 0.095\hspace{0.05cm},\hspace{0.5cm}p_{\rm Ab} = p_{\rm b} \cdot p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}A} = 0.005\hspace{0.05cm},\\
p_{\rm Ba} & = p_{\rm a} \cdot p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}B} = 0.360\hspace{0.05cm},
\hspace{0.5cm}p_{\rm Bb} = p_{\rm b} \cdot p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}B} = 0.540\hspace{0.05cm}.
\end{align*}$$

:$$\Rightarrow \hspace{0.3cm} P_{XY}(X,\hspace{0.1cm}Y) =
\begin{pmatrix}
0.095 & 0.005\\
0.36 & 0.54
\end{pmatrix} \hspace{0.05cm}.$$

* For the inference probabilities one obtains:

:$$\begin{align*}p_{\rm A\hspace{0.03cm}\vert \hspace{0.03cm}a} & = p_{\rm Aa}/p_{\rm a} = 0.095/0.455 = 0.2088\hspace{0.05cm},\hspace{0.5cm}p_{\rm A\hspace{0.03cm}\vert \hspace{0.03cm}b} = p_{\rm Ab}/p_{\rm b} = 0.005/0.545 = 0.0092\hspace{0.05cm},\\
p_{\rm B\hspace{0.03cm}\vert \hspace{0.03cm}a} & = p_{\rm Ba}/p_{\rm a} = 0.36/0.455 = 0.7912\hspace{0.05cm},\hspace{0.5cm}p_{\rm B\hspace{0.03cm}\vert \hspace{0.03cm}b} = p_{\rm Bb}/p_{\rm b} = 0.54/0.545 = 0.9908\hspace{0.05cm}
\end{align*}$$

:$$\Rightarrow \hspace{0.3cm} P_{\hspace{0.01cm}X\hspace{0.05cm} \vert \hspace{0.05cm}Y}(X\hspace{0.05cm} \vert \hspace{0.05cm} Y) =
\begin{pmatrix}
0.2088 & 0.0092\\
0.7912 & 0.9908
\end{pmatrix} \hspace{0.05cm}.$$ }}
 
===Definition and interpretation of various entropy functions ===
 
In the  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgrößen|"theory section"]],  all entropies relevant for two-dimensional random quantities are defined, which also apply to digital signal transmission.  In addition, you will find two diagrams illustrating the relationship between the individual entropies. 
*For digital signal transmission the right representation is appropriate, where the direction from source  $X$  to the sink  $Y$  is recognizable. 
*We now interpret the individual information-theoretical quantities on the basis of this diagram.

[[File:EN_Inf_T_3_3_S2_vers2.png|EN_Inf_T_4_2_S2.png|center|frame|Two information-theoretic models for digital signal transmission.
     Please note:  In the right graph  $H_{XY}$  cannot be represented]]

*The  »'''source entropy'''«  $H(X)$  denotes the average information content of the source symbol sequence.  With the symbol set size  $|X|$  applies:

:$$H(X) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{1}{P_X(X)}\right ] \hspace{0.1cm}
= -{\rm E} \big [ {\rm log}_2 \hspace{0.1cm}{P_X(X)}\big ] \hspace{0.2cm}
=\hspace{0.2cm} \sum_{\mu = 1}^{|X|}
P_X(x_{\mu}) \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{P_X(x_{\mu})} \hspace{0.05cm}.$$

*The  »'''equivocation'''«  $H(X|Y)$  indicates the average information content that an observer who knows exactly about the sink  $Y$  gains by observing the source  $X$ :

:$$H(X|Y) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}X\hspace{-0.01cm}|\hspace{-0.01cm}Y}(X\hspace{-0.01cm} |\hspace{0.03cm} Y)}\right ] \hspace{0.2cm}=\hspace{0.2cm} \sum_{\mu = 1}^{|X|} \sum_{\kappa = 1}^{|Y|}
P_{XY}(x_{\mu},\hspace{0.05cm}y_{\kappa}) \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}X\hspace{-0.01cm}|\hspace{0.03cm}Y}
(\hspace{0.05cm}x_{\mu}\hspace{0.03cm} |\hspace{0.05cm} y_{\kappa})}
\hspace{0.05cm}.$$

*The equivocation is the portion of the source entropy  $H(X)$  that is lost due to channel interference  (for digital channel: transmission errors).  The  »'''mutual information'''«  $I(X; Y)$  remains, which reaches the sink:

:$$I(X;Y) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{P_{XY}(X, Y)}{P_X(X) \cdot P_Y(Y)}\right ] \hspace{0.2cm}=\hspace{0.2cm} \sum_{\mu = 1}^{|X|} \sum_{\kappa = 1}^{|Y|}
P_{XY}(x_{\mu},\hspace{0.05cm}y_{\kappa}) \cdot {\rm log}_2 \hspace{0.1cm} \frac{P_{XY}(x_{\mu},\hspace{0.05cm}y_{\kappa})}{P_{\hspace{0.05cm}X}(\hspace{0.05cm}x_{\mu}) \cdot P_{\hspace{0.05cm}Y}(\hspace{0.05cm}y_{\kappa})}
\hspace{0.05cm} = H(X) - H(X|Y) \hspace{0.05cm}.$$

*The  »'''irrelevance'''«  $H(Y|X)$  indicates the average information content that an observer who knows exactly about the source  $X$  gains by observing the sink  $Y$:

:$$H(Y|X) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}Y\hspace{-0.01cm}|\hspace{-0.01cm}X}(Y\hspace{-0.01cm} |\hspace{0.03cm} X)}\right ] \hspace{0.2cm}=\hspace{0.2cm} \sum_{\mu = 1}^{|X|} \sum_{\kappa = 1}^{|Y|}
P_{XY}(x_{\mu},\hspace{0.05cm}y_{\kappa}) \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}Y\hspace{-0.01cm}|\hspace{0.03cm}X}
(\hspace{0.05cm}y_{\kappa}\hspace{0.03cm} |\hspace{0.05cm} x_{\mu})}
\hspace{0.05cm}.$$

*The  »'''sink entropy'''«  $H(Y)$, the mean information content of the sink.  $H(Y)$  is the sum of the useful mutual information  $I(X; Y)$  and the useless irrelevance  $H(Y|X)$, which comes exclusively from channel errors:

:$$H(Y) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{1}{P_Y(Y)}\right ] \hspace{0.1cm}
= -{\rm E} \big [ {\rm log}_2 \hspace{0.1cm}{P_Y(Y)}\big ] \hspace{0.2cm} =I(X;Y) + H(Y|X)
\hspace{0.05cm}.$$

*The  »'''joint entropy'''«  $H(XY)$  is the average information content of the 2D random quantity  $XY$.  It also describes an upper bound for the sum of source entropy and sink entropy:

:$$H(XY) = {\rm E} \left [ {\rm log} \hspace{0.1cm} \frac{1}{P_{XY}(X, Y)}\right ] = \sum_{\mu = 1}^{|X|} \hspace{0.1cm} \sum_{\kappa = 1}^{|Y|} \hspace{0.1cm}
P_{XY}(x_{\mu}\hspace{0.05cm}, y_{\kappa}) \cdot {\rm log} \hspace{0.1cm} \frac{1}{P_{XY}(x_{\mu}\hspace{0.05cm}, y_{\kappa})}\le H(X) + H(Y) \hspace{0.05cm}.$$

{{GraueBox|TEXT=
[[File:Transinf_2.png|right|frame|Considered model of the binary channel]]
$\text{Example 2}$:  The same requirements as for  [[Applets:Capacity_of_Memoryless_Digital_Channels#Underlying_model_of_digital_signal_transmission|$\text{Example 1}$]] apply: 

'''(1)'''  The source symbols are not equally probable:
:$$P_X(X) = \big ( p_{\rm A},\ p_{\rm B} \big )=
\big ( 0.1,\ 0.9 \big )
\hspace{0.05cm}.$$
'''(2)'''  Let the falsification probabilities be:
:$$\begin{align*}p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}A} & = {\rm Pr}(Y\hspace{-0.1cm} = {\rm a}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm A}) = 0.95\hspace{0.05cm},\hspace{0.8cm}p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}A} = {\rm Pr}(Y\hspace{-0.1cm} = {\rm b}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm A}) = 0.05\hspace{0.05cm},\\
p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}B} & = {\rm Pr}(Y\hspace{-0.1cm} = {\rm a}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm B}) = 0.40\hspace{0.05cm},\hspace{0.8cm}p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}B} = {\rm Pr}(Y\hspace{-0.1cm} = {\rm b}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm B}) = 0.60\end{align*}$$

:$$\Rightarrow \hspace{0.3cm} P_{\hspace{0.01cm}Y\hspace{0.05cm} \vert \hspace{0.05cm}X}(Y\hspace{0.05cm} \vert \hspace{0.05cm} X) =
\begin{pmatrix}
0.95 & 0.05\\
0.4 & 0.6
\end{pmatrix} \hspace{0.05cm}.$$

[[File:Inf_T_1_1_S4_vers2.png|frame|Binary entropy function as a function of  $p$|right]]
*Because of condition  '''(1)'''  we obtain for the source entropy with the  [[Information_Theory/Discrete_Memoryless_Sources#Binary_entropy_function|"binary entropy function"]]  $H_{\rm bin}(p)$: 

:$$H(X) = p_{\rm A} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p_{\rm A}\hspace{0.1cm} } + p_{\rm B} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{p_{\rm B} }= H_{\rm bin} (p_{\rm A}) = H_{\rm bin} (0.1)= 0.469 \ {\rm bit}
\hspace{0.05cm};$$

::$$H_{\rm bin} (p) = p \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p\hspace{0.1cm} } + (1 - p) \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{1 - p} \hspace{0.5cm}{\rm (unit\hspace{-0.15cm}: \hspace{0.15cm}bit\hspace{0.15cm}or\hspace{0.15cm}bit/symbol)}
\hspace{0.05cm}.$$

* Correspondingly, for the sink entropy with PMF  $P_Y(Y) = \big ( p_{\rm a},\ p_{\rm b} \big )=
\big ( 0.455,\ 0.545 \big )$:
:$$H(Y) = H_{\rm bin} (0.455)= 0.994 \ {\rm bit}
\hspace{0.05cm}.$$
*Next, we calculate the joint entropy:
:$$H(XY) = p_{\rm Aa} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p_{\rm Aa}\hspace{0.1cm} }+ p_{\rm Ab} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p_{\rm Ab}\hspace{0.1cm} }+p_{\rm Ba} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p_{\rm Ba}\hspace{0.1cm} }+ p_{\rm Bb} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p_{\rm Bb}\hspace{0.1cm} }$$
:$$\Rightarrow \hspace{0.3cm}H(XY) = 0.095 \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{0.095 } +0.005 \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{0.005 }+0.36 \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{0.36 }+0.54 \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{0.54 }= 1.371 \ {\rm bit}
\hspace{0.05cm}.$$

According to the upper left diagram, the remaining information-theoretic quantities are thus also computable:
[[File:Transinf_4.png|right|frame|Information-theoretic model for  $\text{Example 2}$]]

*the  »'''equivocation'''«:

:$$H(X \vert Y) \hspace{-0.01cm} =\hspace{-0.01cm} H(XY) \hspace{-0.01cm} -\hspace{-0.01cm} H(Y) \hspace{-0.01cm} = \hspace{-0.01cm} 1.371\hspace{-0.01cm} -\hspace{-0.01cm} 0.994\hspace{-0.01cm} =\hspace{-0.01cm} 0.377\ {\rm bit}
\hspace{0.05cm},$$

*the »'''irrelevance'''«:

:$$H(Y \vert X) = H(XY) - H(X) = 1.371 - 0.994 = 0.902\ {\rm bit}
\hspace{0.05cm}.$$

*the  »'''mutual information'''« :

:$$I(X;Y) = H(X) + H(Y) - H(XY) = 0.469 + 0.994 - 1.371 = 0.092\ {\rm bit}
\hspace{0.05cm},$$

The results are summarized in the graph.

Note:  Equivocation and irrelevance could also be computed (but with extra effort) directly from the corresponding probability functions, for example:

:$$H(Y \vert X) = \hspace{-0.2cm} \sum_{(x, y) \hspace{0.05cm}\in \hspace{0.05cm}XY} \hspace{-0.2cm} P_{XY}(x,\hspace{0.05cm}y) \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}Y\hspace{-0.01cm}\vert \hspace{0.03cm}X}
(\hspace{0.05cm}y\hspace{0.03cm} \vert \hspace{0.05cm} x)}= p_{\rm Aa} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}A} } +
p_{\rm Ab} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}A} } +
p_{\rm Ba} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}B} } +
p_{\rm Bb} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}B} } = 0.902 \ {\rm bit} \hspace{0.05cm}.$$}}

{{GraueBox|TEXT=
[[File:Transinf_3.png|right|frame|Considered model of the ternary channel:     Red transitions represent  $p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}A} = p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}B} = p_{\rm c\hspace{0.03cm}\vert \hspace{0.03cm}C} = q$  and     blue ones for  $p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}A} = p_{\rm c\hspace{0.03cm}\vert \hspace{0.03cm}A} =\text{...}= p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}C}= (1-q)/2$]]
$\text{Example 3}$:  Now we consider a transmission system with  $M_X = M_Y = M=3$. 

'''(1)'''  Let the source symbols be equally probable:
:$$P_X(X) = \big ( p_{\rm A},\ p_{\rm B},\ p_{\rm C} \big )=
\big ( 1/3,\ 1/3,\ 1/3 \big )\hspace{0.30cm}\Rightarrow\hspace{0.30cm}H(X)={\rm log_2}\hspace{0.1cm}3 \approx 1.585 \ {\rm bit}
\hspace{0.05cm}.$$
'''(2)'''  The channel model is symmetric   ⇒   the sink symbols are also equally probable:
:$$P_Y(Y) = \big ( p_{\rm a},\ p_{\rm b},\ p_{\rm c} \big )=
\big ( 1/3,\ 1/3,\ 1/3 \big )\hspace{0.30cm}\Rightarrow\hspace{0.30cm}H(Y)={\rm log_2}\hspace{0.1cm}3 \approx 1.585 \ {\rm bit}
\hspace{0.05cm}.$$
'''(3)'''  The joint probabilities are obtained as follows:
:$$p_{\rm Aa}= p_{\rm Bb}= p_{\rm Cc}= q/M,$$
:$$p_{\rm Ab}= p_{\rm Ac}= p_{\rm Ba}= p_{\rm Bc} = p_{\rm Ca}= p_{\rm Cb} = (1-q)/(2M)$$
:$$\Rightarrow\hspace{0.30cm}H(XY) = 3 \cdot p_{\rm Aa} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p_{\rm Aa}\hspace{0.1cm} }+6 \cdot p_{\rm Ab} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p_{\rm Ab}\hspace{0.1cm} }= \
\text{...} \ = q \cdot {\rm log_2}\hspace{0.1cm}\frac{M}{q }+ (1-q) \cdot {\rm log_2}\hspace{0.1cm}\frac{M}{(1-q)/2 }.$$
[[File:Transinf_10.png|right|frame|Some results for  $\text{Example 3}$]]
'''(4)'''  For the mutual information we get after some transformations considering the equation 
:$$I(X;Y) = H(X) + H(Y) - H(XY)\text{:}$$
:$$\Rightarrow \hspace{0.3cm} I(X;Y) = {\rm log_2}\ (M) - (1-q) -H_{\rm bin}(q).$$
* For error-free ternary transfer  $(q=1)$  holds  $I(X;Y) = H(X) = H(Y)={\rm log_2}\hspace{0.1cm}3$.

* With  $q=0.8$  the mutual information already decreases to  $I(X;Y) = 0.663$,  with  $q=0.5$  to  $0.085$  bit.

*The worst case from the point of view of information theory is  $q=1/3$  ⇒   $I(X;Y) = 0$.

*On the other hand, the worst case from the point of view of transmission theory is  $q=0$  ⇒   "not a single transmission symbol arrives correctly"  is not so bad from the point of view of information theory.

* In order to be able to use this good result, however, channel coding is required at the transmitting end. }}
 
===Definition and meaning of channel capacity ===
 
If one calculates the mutual information  $I(X, Y)$  as explained in  $\text{Example 2}$,  then this depends not only on the discrete memoryless channel  $\rm (DMC)$,  but also on the source statistic   ⇒   $P_X(X)$.  Ergo:   '''The mutual information'''  $I(X, Y)$ ''' is not a pure channel characteristic'''.

{{BlaueBox|TEXT=
$\text{Definition:}$  The  »'''channel capacity'''«  introduced by  [https://en.wikipedia.org/wiki/Claude_Shannon $\text{Claude E. Shannon}$]  according to his standard work  [Sha48]<ref name = ''Sha48''>Shannon, C.E.:  A Mathematical Theory of Communication.[[File:Transinf_1_neu.png|center|frame|  $M=2$  (left) and for  $M=3$  (right).     Please note:  In the right graph not all transitions are labeled]] In: Bell Syst. Techn. J. 27 (1948), S. 379-423 und S. 623-656.</ref>:

:$$C = \max_{P_X(X)} \hspace{0.15cm} I(X;Y) \hspace{0.05cm}.$$

*The additional unit  "bit/use"  is often added.  Since according to this definition the best possible source statistics are always the basis:

:⇒   $C$  depends only on the channel properties   ⇒   $P_{Y \vert X}(Y \vert X)$   but not on the source statistics   ⇒   $P_X(X)$.  }}

Shannon needed the quantity  $C$  to formulate the  "Channel Coding Theorem"  –  one of the highlights of the information theory he founded.

{{BlaueBox|TEXT=
$\text{Shannon's Channel Coding Theorem:}$ 
*For every transmission channel with channel capacity  $C > 0$,  there exists  $($at least$)$  one  $(k,\ n)$  block code,  whose  $($block$)$  error probability approaches zero  as long as the code rate  $R = k/n$  is less than or equal to the channel capacity:  
:$$R ≤ C.$$
* The prerequisite for this,  however,  is that the following applies to the block length of this code:  
:$$n → ∞.$$

$\text{Reverse of Shannon's channel coding theorem:}$ 

:If the rate  $R$  of the  $(n$,  $k)$ block code used is larger than the channel capacity  $C$,  then an arbitrarily small block error probability can never be achieved.}}

{{GraueBox|TEXT=
[[File:Transinf_9.png|right|frame|Information-theoretic quantities for different  $p_{\rm A}$  and  $p_{\rm B}= 1- p_{\rm A}$ ]]
$\text{Example 4}$:  We consider the same discrete memoryless channel as in  $\text{Example 2}$. 
*The symbol probabilities  $p_{\rm A} = 0.1$  and  $p_{\rm B}= 1- p_{\rm A}=0.9$  were assumed. 

*The mutual information is  $I(X;Y)= 0.092$  bit/channel use   ⇒   first row,  see fourth column in the table.

The  »'''channel capacity'''«  is the mutual information  $I(X, Y)$  with best possible symbol probabilities     $p_{\rm A} = 0.55$  and  $p_{\rm B}= 1- p_{\rm A}=0.45$:
:$$C = \max_{P_X(X)} \hspace{0.15cm} I(X;Y) = 0.284 \ \rm bit/channel \hspace{0.05cm} access \hspace{0.05cm}.$$

From the table you can see further  $($we do without the additional unit "bit/channel use" in the following$)$:
#The parameter  $p_{\rm A} = 0.1$  was chosen very unfavorably,  because with the present channel the symbol  $\rm A$  is more falsified than  $\rm B$. 
#Already with  $p_{\rm A} = 0.9$  the mutual information results in a somewhat better value:  $I(X; Y)=0.130$.
#For the same reason  $p_{\rm A} = 0.55$,  $p_{\rm B} = 0.45$  gives a slightly better result than equally probable symbols  $(p_{\rm A} = p_{\rm B} =0.5)$.
#The more asymmetric the channel is,  the more the optimal probability function  $P_X(X)$  deviates from the uniform distribution.  Conversely:  If the channel is symmetric,  the uniform distribution is always obtained.}}

The ternary channel of  $\text{Example 3}$  is symmetric.  Therefore here  $P_X(X) = \big ( 1/3,\ 1/3,\ 1/3 \big )$  is optimal for each  $q$–value,  and the mutual information  $I(X;Y)$  given in the result table is at the same time the channel capacity  $C$.

==Exercises==
 
*First, select the number  $(1,\ 2, \text{...} \ )$  of the task to be processed.  The number  "$0$"  corresponds to a "Reset":  Same setting as at program start.
*A task description is displayed.  The parameter values are adjusted.  Solution after pressing "Show Solution".
*Source symbols are denoted by uppercase letters  (binary:  $\rm A$,  $\rm B$),  sink symbols by lowercase letters  ($\rm a$,  $\rm b$).  Error-free transmission:  $\rm A \rightarrow a$.
*For all entropy values, the unit "bit/use" would have to be added.

{{BlueBox|TEXT=
'''(1)'''  Let  $p_{\rm A} = p_{\rm B} = 0.5$  and  $p_{\rm b \vert A} = p_{\rm a \vert B} = 0.1$.  What is the channel model?  What are the entropies  $H(X), \, H(Y)$  and the mutual information  $I(X;\, Y)$?}}

:*  Considered is the BSC model  (Binary Symmetric Channel).  Because of  $p_{\rm A} = p_{\rm B} = 0.5$  it holds for the entropies:  $H(X) = H(Y) = 1$.
:*  Because of  $p_{\rm b \vert A} = p_{\rm a \vert B} = 0.1$  eqivocation and irrelevance are also equal:  $H(X \vert Y) = H(Y \vert X) = H_{\rm bin}(p_{\rm b \vert A}) = H_{\rm bin}(0.1) =0.469$.
:*  The mutual information is  $I(X;\, Y) = H(X) - H(X \vert Y)= 1-H_{\rm bin}(p_{\rm b \vert A}) = 0.531$  and the joint entropy is  $H(XY) =1.469$.

{{BlueBox|TEXT=
'''(2)'''  Let further  $p_{\rm b \vert A} = p_{\rm a \vert B} = 0.1$, but now the symbol probability is  $p_{\rm A} = 0. 9$.  What is the capacity  $C_{\rm BSC}$  of the BSC channel with  $p_{\rm b \vert A} = p_{\rm a \vert B}$?        
Which  $p_{\rm b \vert A} = p_{\rm a \vert B}$  leads to the largest possible channel capacity and which  $p_{\rm b \vert A} = p_{\rm a \vert B}$  leads to the channel capacity  $C_{\rm BSC}=0$?}}

:*  The capacity  $C_{\rm BSC}$  is equal to the maximum mutual information  $I(X;\, Y)$  considering the optimal symbol probabilities.
:*  Due to the symmetry of the BSC model  equally probable symbols  $(p_{\rm A} = p_{\rm B} = 0.5)$  lead to the optimum   ⇒   $C_{\rm BSC}=0.531$.
:*  The best is the  "ideal channel"  $(p_{\rm b \vert A} = p_{\rm a \vert B} = 0)$  ⇒   $C_{\rm BSC}=1$.   The worst BSC channel results with  $p_{\rm b \vert A} = p_{\rm a \vert B} = 0.5$   ⇒   $C_{\rm BSC}=0$.
:*  But also with   $p_{\rm b \vert A} = p_{\rm a \vert B} = 1$  we get  $C_{\rm BSC}=1$.  Here all symbols are inverted, which is information theoretically the same as  $\langle Y_n \rangle \equiv \langle X_n \rangle$.

{{BlueBox|TEXT=
'''(3)'''  Let  $p_{\rm A} = p_{\rm B} = 0.5$,  $p_{\rm b \vert A} = 0.05$  and  $ p_{\rm a \vert B} = 0.4$.   Interpret the results in comparison to the experiment  $(1)$  and to the  $\text{example 2}$  in the theory section.}}

:*  Unlike the experiment  $(1)$  no BSC channel is present here.  Rather, the channel considered here is asymmetric:  $p_{\rm b \vert A} \ne p_{\rm a \vert B}$.
:*  According to  $\text{Example 2}$  it holds for  $p_{\rm A} = 0.1,\ p_{\rm B} = 0.9$:     $H(X)= 0.469$,  $H(Y)= 0.994$,  $H(X \vert Y)=0.377$,  $H(Y \vert X)=0.902$,  $I(X;\vert Y)=0.092$.
:*  Now it holds  $p_{\rm A} = p_{\rm B} = 0.5$  and we get  $H(X)=1,000$,  $H(Y)=0.910$,  $H(X \vert Y)=0.719$,  $H(Y \vert X)=0.629$,  $I(X;\ Y)=0.281$.
:*  All output values depend significantly on  $p_{\rm A}$  and  $p_{\rm B}=1-p_{\rm A}$  except for the conditional probabilities  ${\rm Pr}(Y \vert X)\in \{\hspace{0.05cm}0.95,\ 0.05,\ 0.4,\ 0.6\hspace{0.05cm} \}$.

{{BlueBox|TEXT=
'''(4)'''  Let further  $p_{\rm A} = p_{\rm B}$,  $p_{\rm b \vert A} = 0.05$,  $ p_{\rm a \vert B} = 0.4$.  What differences do you see in terms of analytical calculation and "simulation"  $(N=10000)$.}}

:*  The joint probabilities are  $p_{\rm Aa} =0.475$,  $p_{\rm Ab} =0.025$,  $p_{\rm Ba} =0.200$, $p_{\rm Bb} =0.300$.  Simulation:  Approximation by relative frequencies:
:*  For example, for  $N=10000$:  $h_{\rm Aa} =0.4778$,  $h_{\rm Ab} =0.0264$,  $h_{\rm Ba} =0.2039$, $h_{\rm Bb} =0.2919$.  After pressing  "New sequence"  slightly different values.
:*  For all subsequent calculations, no principal difference between theory and simulation, except  $p \to h$.  Examples: 
:*  $p_{\rm A} = 0.5 \to h_{\rm A}=h_{\rm Aa} + h_{\rm Ab} =0.5042$,  $p_b = 0.325 \to h_{\rm b}=h_{\rm Ab} + h_{\rm Bb} =0. 318$,  $p_{b|A} = 0.05 \to h_{\rm b|A}=h_{\rm Ab}/h_{\rm A} =0.0264/0.5042= 0.0524$, 
:*  $p_{\rm A|b} = 0.0769 \to h_{\rm A|b}=h_{\rm Ab}/h_{\rm b} =0.0264/0.318= 0.0830$.  Thus, this simulation yields  $I_{\rm Sim}(X;\ Y)=0.269$  instead of  $I(X;\ Y)=0.281$.

{{BlueBox|TEXT=
'''(5)'''  Setting according to  $(4)$.  How does  $I_{\rm Sim}(X;\ Y)$  differ from  $I(X;\ Y) = 0.281$  for  $N=10^3$,  $10^4$,  $10^5$ ?  In each case, averaging over ten realizations. }}
:*  $N=10^3$:   $0.232 \le I_{\rm Sim} \le 0.295$,   mean:  $0.263$   #   $N=10^4$:   $0.267 \le I_{\rm Sim} \le 0.293$,   mean:  $0.279$   #   $N=10^5$:   $0.280 \le I_{\rm Sim} \le 0.285$   mean:  $0.282$.
:*  With  $N=10^6$  for this channel, the simulation result differs from the theoretical value by less than  $\pm 0.001$. 

{{BlueBox|TEXT=
'''(6)'''  What is the capacity  $C_6$  of this channel with  $p_{\rm b \vert A} = 0.05$,  $ p_{\rm a \vert B} = 0.4$?  Is the error probability  $0$  possible with the code rate  $R=0.3$? }}

:*  $C_6=0.284$  is the maximum of  $I(X;\ Y)$  for   $p_{\rm A} =0.55$  ⇒  $p_{\rm B} =0. 45$.  Simulation over  ten times  $N=10^5$:  $0.281 \le I_{\rm Sim}(X;\ Y) \le 0.289$.
:*  With the code rate  $R=0.3 > C_6$  an arbitrarily small block error probability is not achievable even with the best possible coding.

{{BlueBox|TEXT=
'''(7)'''  Now let  $p_{\rm A} = p_{\rm B}$,  $p_{\rm b \vert A} = 0$,  $ p_{\rm a \vert B} = 0.5$.   What property does this asymmetric channel exhibit?  What values result for  $H(X)$,  $H(X \vert Y)$,  $I(X;\ Y)$ ? }}

:*  The symbol  $\rm A$  is never falsified, the symbol  $\rm B$  with (information theoretically) maximum falsification probability  $ p_{\rm a \vert B} = 0.5$
:*  The total falsification probability is  $ {\rm Pr} (Y_n \ne X_n)= p_{\rm A} \cdot p_{\rm b \vert A} + p_{\rm B} \cdot p_{\rm a \vert B}= 0.25$  ⇒   about  $25\%$  of the output sink symbols are "purple".
:*  Joint probabilities:  $p_{\rm Aa}= 1/2,\ p_{\rm Ab}= 0,\ p_{\rm Ba}= p_{\rm Bb}= 1/4$,    Inference probabilities:   $p_{\rm A \vert a}= 1,\ p_{\rm B \vert a}= 0,\ p_{\rm A \vert b}= 1/3,\ p_{\rm B \vert b}= 2/3$.
:*  From this we get for equivocation  $H(X \vert Y)=0.689$;   with source entropy  $H(X)= 1$   ⇒   $I(X;\vert Y)=H(X)-H(X \vert Y)=0.311$.

{{BlueBox|TEXT=
'''(8)'''  What is the capacity  $C_8$  of this channel with  $p_{\rm b \vert A} = 0.05$,  $ p_{\rm a \vert B} = 035$?  Is the error probability  $0$  possible with the code rate  $R=0.3$? }}

:*  $C_8=0.326$  is the maximum of  $I(X;\ Y)$  for   $p_{\rm A} =0.55$.  Thus, because of  $C_8 >R=0.3 $  an arbitrarily small block error probability is achievable.
:*  The only difference compared to  $(6)$   ⇒   $C_6=0.284 < 0.3$  is the slightly smaller falsification probability  $ p_{\rm a \vert B} = 0.35$  instead of  $ p_{\rm a \vert B} = 0.4$.

{{BlueBox|TEXT=
'''(9)'''  We consider the ideal ternary channel:  $p_{\rm a \vert A} = p_{\rm b \vert B}=p_{\rm c \vert C}=1$.  What is its capacity  $C_9$?  What is the maximum mutual information displayed by the program? }}

:*  Due to the symmetry of the channel model, equally probable symbols  $(p_{\rm A} = p_{\rm B}=p_{\rm C}=1/3)$  lead to the channel capacity:  $C_9 = \log_2\ (3) = 1.585$.
:*  Since in the program all parameter values can only be entered with a resolution of  $0.05$ , for  $I(X;\ Y)$  this maximum value is not reached.
:*  Possible approximations:  $p_{\rm A} = p_{\rm B}= 0.3, \ p_{\rm C}=0.4$  ⇒   $I(X;\ Y)= 1. 571$     #     $p_{\rm A} = p_{\rm B}= 0.35, \ p_{\rm C}=0.3$  ⇒   $I(X;\ Y)= 1.581$.

{{BlueBox|TEXT=
'''(10)'''  Let the source symbols be (nearly) equally probable.  Interpret the other settings and the results. }}

:*  The falsification probabilities are  $p_{\rm b \vert A} = p_{\rm c \vert B}=p_{\rm a \vert C}=1$   ⇒   no single sink symbol is equal to the source symbol.
:*  This cyclic mapping has no effect on the channel capacity:  $C_{10} = C_9 = 1.585$.  The program returns  ${\rm Max}\big[I(X;\ Y)\big]= 1.581$.

{{BlueBox|TEXT=
'''(11)'''  We consider up to and including  $(13)$  the same ternary source.   What results are obtained for  $p_{\rm b \vert A} = p_{\rm c \vert B}=p_{\rm a \vert C}=0.2$  and  $p_{\rm c \vert A} = p_{\rm a \vert B}=p_{\rm b \vert C}=0$? }}
:*  Each symbol can only be falsified into one of the two possible other symbols.  From  $p_{\rm b \vert A} = p_{\rm c \vert B}=p_{\rm a \vert C}=0.2$  it follows  $p_{\rm a \vert A} = p_{\rm b \vert B}=p_{\rm c \vert C}=0.8$.
:*  This gives us for the maximum mutual information  ${\rm Max}\big[I(X;\ Y)\big]= 0.861$  and for the channel capacity a slightly larger value:  $C_{11} \gnapprox 0.861$.

{{BlueBox|TEXT=
'''(12)'''  How do the results change if each symbol is   $80\%$  transferred correctly and   $10\%$  falsified each in one of the other two symbols? }}
:*  Although the probability of correct transmission is with  $80\%$  as large as in  '''(11)''', here the channel capacity  $C_{12} \gnapprox 0.661$ is smaller.
:*  If one knows for the channel  $(11)$  that  $X = \rm A$  has been falsified, one also knows  $Y = \rm b$.  But not for channel  $(12)$  ⇒   the channel is less favorable.

{{BlueBox|TEXT=
'''(13)'''  Let the falsification probabilities now be  $p_{\rm b \vert A} = p_{\rm c \vert A} = p_{\rm a \vert B} = p_{\rm c \vert B}=p_{\rm a \vert C}=p_{\rm b \vert C}=0.5$.  Interpret this redundancy-free ternary channel. }}
:*  No single sink symbol is equal to its associated source symbol; with respect to the other two symbols, a  $50\hspace{-0.1cm}:\hspace{-0.1cm}50$  decision must be made.
:*  Nevertheless, here the channel capacity is  $C_{13} \gnapprox 0.584$  only slightly smaller than in the previous experiment:  $C_{12} \gnapprox 0.661$.
:*  The channel capacity  $C=0$  results for the redundancy-free ternary channel exactly for the case where all nine falsification probabilities are equal to  $1/3$ .

{{BlueBox|TEXT=
'''(14)'''  What is the capacity $C_{14}$ of the ternary channel with  $p_{\rm b \vert A} = p_{\rm a \vert B}= 0$  and  $p_{\rm c \vert A} = p_{\rm c \vert B} = p_{\rm a \vert C}=p_{\rm b \vert C}=0. 1$  ⇒   $p_{\rm a \vert A} = p_{\rm b \vert B}=0.9$,   $p_{\rm c \vert C} =0.8$? }}
:*  With the default  $p_{\rm A}=p_{\rm B}=0.2$   ⇒   $p_{\rm C}=0.6$  we get  $I(X;\ Y)= 0.738$.  Now we are looking for "better" symbol probabilities.
:*  From the symmetry of the channel, it is obvious that  $p_{\rm A}=p_{\rm B}$  is optimal.  The channel capacity  $C_{14}=0.995$  is obtained for  $p_{\rm A}=p_{\rm B}=0.4$   ⇒   $p_{\rm C}=0.2$.
:*  Example:  Ternary transfer if the middle symbol  $C$  can be distorted in two directions, but the outer symbols can only be distorted in one direction at a time.
 

==Applet Manual==
 
[[File:Anleitung_transinformation.png|left|600px|frame|Screenshot of the German version]]
 
    '''(A)'''     Select whether  "analytically"  or  "by simulation"

    '''(B)'''     Setting of the parameter  $N$  for the simulation

    '''(C)'''     Option to select "binary source"  or  "ternary source"

    '''(D)'''     Setting of the symbol probabilities

    '''(E)'''     Setting of the transition probabilities

    '''(F)'''     Numerical output of different probabilities

    '''(G)'''     Two diagrams with the information theoretic quantities

    '''(H)'''     Output of an exemplary source symbol sequence

    '''(I)'''     Associated simulated sink symbol sequence

    '''(J)'''     Exercise area: Selection, questions, sample solutions
 

==About the Authors==
 
This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].
*The first version was created in 2010 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Martin_V.C3.B6lkl_.28Diplomarbeit_LB_2010.29|Martin Völkl]]  as part of his diploma thesis with “FlashMX – Actionscript”.  Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29|Günter Söder]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|Klaus Eichin]].

*In 2020 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Veronika_Hofmann_.28Ingenieurspraxis_Math_2020.29|Veronika Hofmann]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|Benedikt Leible]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|Tasnád Kernetzky]].

*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  in the context of a working student activity. 

*The conversion of this applet was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ $\text{Studienzuschüsse}$]  $($TUM Department of Electrical and Computer Engineering$)$.  $\text{Many thanks}$.

==Once again: Open Applet in new Tab==

{{LntAppletLinkEnDe|transinformation_en|transinformation}}

==References==

LNTwww:Imprint for the book "Modulation Methods"

2025-02-24T13:39:53Z

'''Five main chapters with a total of 23 chapters (files) and 192 sections (pages);   89 Exercises   ⇒     ⇒   Scope:  "$\rm 3L+2E$"      
Development of the German version:   2005–2011.       Development of the English version:   2020/21.      Last corrections:   October 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Söd14]'''<ref name='Söd14'>Söder, G.:  Analoge & Digitale Modulationsverfahren. Praktikum &bdquo;Simulation digitaler Übertragungssysteme”. Lehrstuhl für Nachrichtentechnik, TU München, 2014. </ref>;  '''[Han16]'''<ref name='Han16'>Hanik, N.:  Nachrichtentechnik 2 (LB): Modulationsverfahren. Vorlesungsmanuskript. Professur Leitungsgebundene Übertragungstechnik, TU München, 2016.</ref>;  '''[Vie16]'''<ref name='Vie16'>Viering, I.: System Aspects in Communications. Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2016.</ref>;   '''[Kra17]'''<ref name='Kra17'>Kramer, G.: Nachrichtentechnik 2. Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2017.</ref>.   –   Textbooks:   '''[ZP85]'''<ref name='ZP85'>Ziemer, R. E.; Peterson, R. L.: Digital Communications and Spread Spectrum Systems. New York: Macmillan, 1985. ISBN 978-0-02431-670-7</ref>;  '''[Kam04]'''<ref name="Kam04">Kammeyer, K.D.:  Nachrichtenübertragung. Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>

*Authors:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|'''Klaus Eichin''']],

* Participating colleagues in alphabetical order:   [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|'''Johannes Zangl''']],

*Participating students in chronological order:      Martin Winkler '''(2001)''',  Yven Winter,  Ji Li ,  Franz Kohl,  Bettina Hirner,  Thorsten Kalweit,  Markus Elsberger,  Slim Lamine,  Thomas Großer,  Johannes Schmidt,  David Jobst,  Xiaohan Liu,  Matthias Riedel,  Carolin Mirschina,  Sam Reed,  JiWoo Hwang '''(2022)'''

$\text{References:}$

Applets:Frequency & Impulse Responses

2025-02-24T13:39:51Z

{{LntAppletLinkEnDe|frequImpResp_en|frequImpResp}}

==Applet Description==
 

Real and symmetric low-pass frquency responses  $H(f)$  and the corresponding impulse responses  $h(t)$  are shown,  namely
#Gaussian low-pass,
#rectangular low-pass,
#triangular low-pass,
#trapezoidal low-pass,
#cosine rolloff low-pass   ⇒   »raised-cosine low-pass«,
#cosine rolloff squared low-pass   ⇒   »cosine-square low-pass«.

It should be noted:
* The functions  $H(f)$  resp.  $h(t)$  are shown for up to two parameter sets in one diagram each.

* The red curves and numbers apply to the left parameter set,  the blue ones to the right parameter set.

* The abscissas  $t$  $($time$)$  and  $f$  $($frequency$)$  as well as the ordinates  $H(f)$  and  $h(t)$  are normalized in each case.

*We use here the function  ${\rm si}(x)=\sin(x)/x)$.  The relation with the function  ${\rm sinc}(x)=\sin(\pi x)/(\pi x)$  is:  ${\rm sinc}(x)={\rm si}(x/\pi).$

*For the last two filters,  we use in the applet the former labels  "cosine rolloff low-pass"  and  "cosine rolloff squared low-pass".

==Theoretical background==
 
===Frequency response  $H(f)$  and impulse response  $h(t)$===

*The  [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function|$\text{frequency response}$]]  $($or the  "transfer function"$)$  $H(f)$  of a linear time-invariant transmission system gives the ratio between the output spectrum  $Y(f)$  and that of the input spectrum  $X(f)$:
:$$H(f) = \frac{Y(f)}{X(f)}.$$
*If the transmission behavior at low frequencies is better than at higher frequencies,  it is called a  »'''low-pass'''«.

*The properties of  $H(f)$  are expressed in the time domain by the  [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Impulse_response|$\text{impulse response}$]]  $h(t)$.   According to the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral|$\text{second Fourier integral}$]]  holds:
:$$h(t)={\rm IFT} [H(f)] = \int_{-\infty}^{+\infty}H(f)\cdot {\rm e}^{+{\rm j}2\pi f t}\hspace{0.15cm} {\rm d}f\hspace{1cm}
{\rm IFT}\hspace{-0.1cm}: \rm Inverse \ Fourier \ transform.$$
*The inverse direction is described by the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|$\text{first Fourier integral}$]]:
:$$H(f)={\rm FT} [h(t)] = \int_{-\infty}^{+\infty}h(t)\cdot {\rm e}^{-{\rm j}2\pi f t}\hspace{0.15cm} {\rm d}t\hspace{1cm}
\rm FT\hspace{-0.1cm}: \ Fourier\ transform.$$
*In all examples we use real and even functions.  Thus:
:$$h(t)=\int_{-\infty}^{+\infty}H(f)\cdot \cos(2\pi ft) \hspace{0.15cm} {\rm d}f \ \circ\!\!-\!\!-\! \!\!-\!\!\bullet\ \ H(f)=\int_{-\infty}^{+\infty}h(t)\cdot \cos(2\pi ft) \hspace{0.15cm} {\rm d}t .$$
*For a quadripole  $[$meaning:  $X(f)$  and  $Y(f)$  have equal units$]$:   $Y(f)$  is dimensionless. 

*The unit of impulse response is  $\rm 1/s$.  It is true that  $\rm 1/s = 1 \ Hz$,  but the unit  "Hertz"  is unusual in this context.

*The relationship between this applet and the similarly constructed applet  [[Applets:Pulses_and_Spectra|"Pulses and Spectra"]]  is based on the  [[Signal_Representation/Fourier_Transform_Theorems#Duality_Theorem|$\text{Duality Theorem}$]].

*All times are normalized to a normalization time  $T$  and all frequencies are normalized to  $1/T  \ \Rightarrow$  the numerical values of   $h(t)$  still have to be divided by  $T$.

{{GraueBox|TEXT=
$\text{Example:}$  If one sets a rectangular low-pass with height  $K_1 = 1$  and equivalent bandwidth  $\Delta f_1 = 1$, 
*so the frequency response  $H_1(f)=1$  in the range  $-1 < f < 1$  and zero outside this range. 

*The impulse response  $h_1(t)$  is  $\rm si$–shaped with  $h_1(t= 0) = 1$  and the first zero at  $t=1$.

If a rectangular low-pass with  $K = 1.5$  and  $\Delta f = 2 \ \rm kHz$  should to be simulated,  where the normalization time is  $T= 1 \ \rm ms$,  Then:
*The first zero is at  $t=0.5\ \rm ms$  and the impulse response maximum is  $h(t= 0) = 3 \cdot 10^3 \ \rm 1/s$.}}

===Gaussian low-pass ===

*The Gaussian low-pass with height  $K$  and  $($equivalent$)$  bandwidth  $\Delta f$  reads:
:$$H(f)=K\cdot {\rm e}^{-\pi\hspace{0.05cm}\cdot\hspace{0.05cm}(f/\Delta f)^2}.$$
*The equivalent bandwidth  $\Delta f$  is obtained from the equal-area rectangle.

*The value at  $f = \Delta f/2$  is smaller by a factor  $\approx 0.456$  than the value at  $f=0$.

*For the impulse response one obtains according to the inverse Fourier transform:
:$$h(t)=K\cdot \Delta f \cdot {\rm e}^{-\pi(t\hspace{0.05cm}\cdot\hspace{0.05cm} \Delta f)^2} .$$

*The smaller  $\Delta f$,  the wider and lower is the impulse response   ⇒   [[Signal_Representation/Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth|$\text{Reciprocity theorem of bandwidth and impulse duration}$]].

*Both  $H(f)$  and  $h(t)$  are not exactly equal to zero at any value of  $f$  resp.  $t$.

*However,  for practical applications,  the Gaussian pulse can be assumed to be limited in time and frequency. 

*For example,  $h(t)$  has already dropped to less than  $0.1\% $  of its maximum at  $t=1.5 \cdot \Delta t$.
 
===Rectangular low-pass ===
*The rectangular low-pass with height  $K$  and  $($equivalent$)$  bandwidth  $\Delta f$  reads:
:$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}K \\ K /2 \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{20}c} {\rm{for}} \\ {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.05cm} f\hspace{0.05cm} \right| < \Delta f/2,} \\ {\left| \hspace{0.05cm}f\hspace{0.05cm} \right| = \Delta f/2,} \\ {\left|\hspace{0.05cm} f \hspace{0.05cm} \right| > \Delta f/2.} \\ \end{array}$$
*The  $\pm \Delta f/2$  value lies midway between the left-hand and right-hand limits.

*For the impulse response  $h(t)$  one obtains according to the laws of the inverse Fourier transform  $($"2nd Fourier integral"$)$:
:$$h(t)=K\cdot \Delta f \cdot {\rm si}(\pi\cdot \Delta f \cdot t) \quad \text{with} \quad {\rm si}(x)={\sin(x)}/{x}.$$
*The  $h(t)$  value at  $t=0$  is equal to the square area of the frequency response.

*The impulse response has zeros at equidistant intervals  $1/\Delta f$.

*The integral over the impulse response  $h(t)$  is equal to the frequency response  $H(f)$  at frequency  $f=0$,  thus is equal to  $K$.
 

===Triangular low-pass===

*The triangular low-pass with height  $K$  and  $($equivalent$)$  bandwidth  $\Delta f$  reads:
:$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}K\cdot \Big(1-\frac{|f|}{\Delta f}\Big) \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{20}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.05cm} f\hspace{0.05cm} \right| < \Delta f,} \\ {\left| \hspace{0.05cm}f\hspace{0.05cm} \right| \ge \Delta f.} \\ \end{array}$$
*The absolute physical bandwidth  $B$   ⇒  [positive frequencies only]   is also equal  $\Delta f$,  thus is as large as for the rectangular low-pass.

*For the impulse response  $h(t)$  one obtains according to the second Fourier transform:
:$$h(t)=K\cdot \Delta f \cdot {\rm si}^2(\pi\cdot \Delta f \cdot t) \quad \text{with} \quad {\rm si}(x)={\sin(x)}/{x}={\rm sinc}(x/\pi).$$
*$H(f)$  can be represented as a convolution of two rectangular functions  $($each with width  $\Delta f)$.

*It follows:  $h(t)$  contains instead of the  ${\rm si}$–function the  ${\rm si}^2$–function.

*$h(t)$  thus also exhibits zeros at equidistant intervals  $1/\Delta f$.

*The asymptotic decay of  $h(t)$  occurs here with  $1/t^2$,  while for comparison in the case of the rectangular low-pass  $h(t)$  decays with  $1/t$.

 

===Trapezoidal low-pass===

*The trapezoidal low-pass with height  $K$  and the two corner frequencies  $f_1$  and  $f_2$  reads:
:$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}K \\ K\cdot \frac{f_2-|f|}{f_2-f_1} \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{20}c} {\rm{for}} \\ {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.05cm} f\hspace{0.05cm} \right| \le f_1,} \\ {f_1\le \left| \hspace{0.05cm}f\hspace{0.05cm} \right| \le f_2,} \\ {\left|\hspace{0.05cm} f \hspace{0.05cm} \right| \ge f_2.} \\ \end{array}$$

*For the equivalent bandwidth  $($equal-area rectangle$)$  the following applies:  $\Delta f = f_1+f_2$.

*The rolloff factor  $($in the frequency domain$)$  characterizes the slope:
:$$r=\frac{f_2-f_1}{f_2+f_1}.$$
*The special case  "$r=0$"  corresponds to the rectangular low-pass and the special case  "$r=1$"  to the triangular low-pass.

*For the impulse response,  according to the inverse Fourier back transform,  we obtain:
:$$h(t)=K\cdot \delta f \cdot {\rm si}(\pi\cdot \delta f \cdot t)\cdot {\rm si}(\pi \cdot r \cdot \delta f \cdot t) \quad \text{with} \quad {\rm si}(x)={\sin(x)}/{x}.$$
*The asymptotic decay of  $h(t)$  lies between  $1/t$  $($for rectangular low-pass or  $r=0)$  and  $1/t^2$  $($for triangular low-pass or  $r=1)$.
 

===Cosine rolloff low-pass ===

Note:   In English-language literature,  this filter is often also referred to as  »'''raised-cosine low-pass'''«.

*The cosine-rolloff low-pass with height  $K$  and the two corner frequencies  $f_1$  and  $f_2$  reads:

:$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}K \\ K\cdot \cos^2\Big(\frac{|f|-f_1}{f_2-f_1}\cdot {\pi}/{2}\Big) \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{20}c} {\rm{for}} \\ {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.05cm} f\hspace{0.05cm} \right| \le f_1,} \\ {f_1\le \left| \hspace{0.05cm}f\hspace{0.05cm} \right| \le f_2,} \\ {\left|\hspace{0.05cm} f \hspace{0.05cm} \right| \ge f_2.} \\ \end{array}$$

*For the equivalent bandwidth  $($equal-area rectangle$)$  the following applies:  $\Delta f = f_1+f_2$.

*The rolloff factor  $($in the frequency domain$)$  characterizes the slope:
:$$r=\frac{f_2-f_1}{f_2+f_1}.$$
*The special case  "$r=0$"  corresponds to the rectangular low-pass and the special case  "$r=1$"  to the cosine-square low-pass.

*For the impulse response,  according to the inverse Fourier transform,  we obtain:
:$$h(t)=K\cdot \Delta f \cdot \frac{\cos(\pi \cdot r\cdot \Delta f \cdot t)}{1-(2\cdot r\cdot \Delta f \cdot t)^2} \cdot {\rm si}(\pi \cdot \Delta f \cdot t).$$
*The larger the rolloff factor  $r$,  the faster decreases  $h(t)$  asymptotically with  $t$.
 

===Cosine rolloff squared low-pass ===

*This is a special case of the raised-cosine low-pass and results from it for  $r=1 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}f_1=0,\ f_2= \Delta f$:

:$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}K\cdot \cos^2\Big(\frac{|f|\hspace{0.05cm}\cdot\hspace{0.05cm} \pi}{2\hspace{0.05cm}\cdot\hspace{0.05cm} \Delta f}\Big) \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{20}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.05cm} f\hspace{0.05cm} \right| < \Delta f,} \\ {\left| \hspace{0.05cm}f\hspace{0.05cm} \right| \ge \Delta f.} \\ \end{array}$$

*For the impulse response one obtains according to the Fourier inverse transform:
:$$h(t)=K\cdot \Delta f \cdot {\pi}/{4}\cdot \big [{\rm sinc}(\Delta f\cdot t +0.5)+{\rm sinc}(\Delta f\cdot t -0.5)\big ]\cdot {\rm sinc}(\Delta f \cdot t)\quad \text{with} \quad {\rm sinc}(x)={\sin(x)}/{(\pi x})={\rm si}(\pi x).$$

*Because of the last  ${\rm sinc}$–function,  $h(t)=0$  for all multiples of  $T=1/\Delta f$   ⇒   The equidistant zero crossings of the cosine rolloff low-pass are preserved.
*Due to the expression in parentheses,  $h(t)$  now shows further zero crossings at  $t=\pm1.5 T$,  $\pm2.5 T$,  $\pm3.5 T$, ... 
*For  $t=\pm T/2$  the impulse response has the value  $K\cdot \Delta f/2$.
*The asymptotic decay of  $h(t)$  runs in this special case with  $1/t^3$.

==Exercises==
 

* First select the number (1, ... , 6) of the exercise.  The number 0 corresponds to a "Reset":  Same setting as at the program start. 
* A description of the exercise will be displayed.  The parameter values are adjusted.  Solution after pressing "Show solution". 
* "Red" corresponds to the first parameter set   ⇒   $H_1(f)   \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\   h_1(t)$,  and "Blue" corresponds to the second parameter set   ⇒   $H_2(f)   \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\   h_2(t)$. 
* Values smaller than  $0.0005$  are set to zero in the program.

 

{{BlaueBox|TEXT=
'''(1)'''   Compare the  »red Gaussian low-pass«  $(K_1 = 1, \Delta f_1 = 1)$  to the  »blue rectangular low-pass«  $(K_2 = 1, \Delta f_2 = 1)$.  Questions: 
        '''(a)'''  Which output signals  $y(t)$  result from the signal  $x(t) = 2 \cdot \cos (2\pi f_0 t -\varphi_0)$  with  $f_0 = 0.5$? 
        '''(b)'''  What are the differences between the two low-pass filters with  $f_0 = 0.5 \pm f_\varepsilon$  and  $f_\varepsilon \ne 0, \ f_\varepsilon \to 0$?}}

:'''(a)'''  It holds  $y(t) = A \cdot \cos (2\pi f_0 t -\varphi_0)$  with  $A = 2 \cdot H(f = f_0) \ \Rightarrow \ A_1 = 0.912, \ A_2 = 1,000$.  The phase  $\varphi_0$  remains unchanged. 
:'''(b)'''  For  red  $ A_1 = 0.912$  is still valid.  For  blue  it holds  $A_2 = 0$  for  $f_0 = 0.5000\text{...}001$  and  $A_2 = 2$  for  $f_0 = 0.4999\text{...}999$.

{{BlaueBox|TEXT=
'''(2)'''   Leave the settings unchanged.  Which low-pass  $H(f)$  fulfills the first or the second Nyquist criterion?         Here  $H(f)$  denotes the total frequency response of transmitter,
channel and recettion filter.}}

* First Nyquist criterion:  The impulse response  $h(t)$  must have equidistant zero crossings at the (normalized) times  $t = 1,\ 2$, ... 
* The impulse response  $h(t) = {\rm sinc}(\delta f \cdot t)$  of the rectangular low-pass filter fulfils this criterion with  $\Delta f = 1$. 
* In contrast, the first Nyquist criterion is never fulfilled for the Gaussian low-pass and there is always impulse interference. 
* The second Nyquist criterion is met by neither the rectangular low-pass nor the Gaussian low-pass.

{{BlaueBox|TEXT=
'''(3)'''   Compare the  »red rectangular low-pass«  $(K_1 = 0.5, \Delta f_1 = 2)$  to the  »blue rectangular low-pass«  $(K_2 = 1, \Delta f_2 = 1)$.  Then vary  $\Delta f_1$  between  $2$  ...  $0.5$.}}

* With  $\Delta f_1 = 2$  the zeros of  $h_1(t)$  are multiples  of  $0.5$   ⇒   $h_1(t)$   will decay twice as fast as  $h_2(t)$. 
* With the present setting,  $h_1(t = 0) = h_2(t = 0)$ holds,  since the rectangular areas of  $H_1(f)$  and  $H_2(f)$  are equal. 
* By decreasing  $\Delta f_1$,  the impulse response  $h_1(t)$  becomes wider and lower.  With  $\Delta f_1 = 0.5$,  $h_1(t)$ is twice as wide as  $h_2(t)$,  but simultaneously by a factor  $4$  lower.

{{BlaueBox|TEXT=
'''(4)'''   Compare the  »red trapezoidal low-pass«  $(K_1 = 1, \ \Delta f_1 = 1, \ r_1 = 0.5)$  with  »blue rectangular low-pass«   $(K_2 = 1, \ \Delta f_2 = 1)$.  $r_1$  varies between  $0$  ...  $1$. }}

* With  $r_1 = 0.5$  the  followers/precursors of  $h_1(t)$  for the "trapezoid" are less than for the "rectangle" due to the flatter edge drop . 
* With smaller  $r_1$  followers & precursors increase.  With  $r_1= 0$  the trapezoidal is equal to the rectangular low-pass   ⇒   $h(t)= {\rm si}(\pi \cdot t/T)$. 
* With larger  $r_1$  followers & precursors become smaller.  With  $r_1= 1$  the trapezoidal is equal to the triangular low-pass   ⇒   $h(t)= {\rm si}^2(\pi \cdot t/T)$.

{{BlaueBox|TEXT=
'''(5)'''   Compare the  »trapezoidal low-pass«  $(K_1 = 1, \ \Delta f_1 = 1, \ r_1 = 0.5)$  to the 
»cosine-rolloff low-pass«  $(K_2 = 1, \ \Delta f_2 = 1, \ r_2 = 0.5)$.         Vary  $r_2$  between  $0$  and  $1$.  Interpret the impulse response for  $r_2 = 0.75$.  Which low-pass satisfies the first Nyquist criterion?}}

* With  $r_1 = r_2= 0.5$  the edge drop of  $H_2(f)$  is steeper by the frequency  $f = 0.5$  than the edge drop of  $H_1(f)$. 
* With the same rolloff  $r= 0.5$  the impulse response  $h_2(t)$  for  $t > 1$  has larger magnitudes than  $h_1(t)$. 
* With  $r_1 = 0.5$  and  $r_2 = 0.75$  $H_1(f) \approx H_2(f)$  holds and therefore also  $h_1(t)
\approx h_2(t)$.  
* $H_1(f)$  and  $H_2(f)$  both fulfill the first Nyquist criterion:  Both functions are point-symmetrical around the "Nyquist point". 
* Because of  $\Delta f = 1$  both  $h_1(t)$  and  $h_2(t)$  have zero crossings at  $\pm 1$,  $\pm 2$   ⇒   in each case maximum vertical eye opening.

{{BlaueBox|TEXT=
'''(6)'''   Compare the  »cosine-square low-pass«  $(K_1 = 1, \ \ \Delta f_1 = 1)$  with the  »cosine-rolloff low-pass«  $(K_2 = 1, \ \ \Delta f_2 = 1,\ r_2 = 0.5)$.        
Vary  $r_2$  between  $0$  and  $1$.  Interpret the results.  Which low-pass satisfies the second Nyquist criterion?
}}

* $H_1(f)$  is a special case of the cosine-rolloff low-pass with rolloff  $r_2 =1$.  The first Nyquist criterion is also fulfilled with  $r_2 \ne 1$. 
* According to the second Nyquist criterion  $h(t)$  must also have zeros at  $t=\pm 1.5$,  $\pm 2.5$,  $\pm 3.5$, ...  $($ but not, however, at  $t = \pm 0.5)$. 
* For the cosine-square low-pass,  $h_1(t=\pm 0.5) = 0.5$  and it therefore holds  $h_1(t=\pm 1) = h_1(t=\pm 1.5) = h_1(t=\pm 2)= h_1(t=\pm 2.5) = \text{...} =0$. 
* Only the cosine-square low-pass fulfils the first and second Nyquist criteria simultaneously:  Maximum vertical and horizontal eye opening.

==Applet Manual==
[[File:Frequenz.png|right]]
    '''(A)'''     Theme (changeable graphical user interface design)
:* Dark:   dark background  (recommended by the authors)
:* Bright:   white background  (recommended for beamers and printouts)
:* Deuteranopia:   for users with pronounced green visual impairment
:* Protanopia:   for users with pronounced red visual impairment

    '''(B)'''     Preselection for frequency response  $H_1(f)$  (red curve)

    '''(C)'''     Parameter definition for  $H_1(f)$ 

    '''(D)'''     Numeric output for  $H_1(f_*)$  and  $h_1(t_*)$

    '''(E)'''     Preselection for frequency response  $H_2(f)$   (blue curve)

    '''(F)'''     Parameter definition for  $H_2(f)$ 

    '''(G)'''     Numeric output for  $H_2(f_*)$  and  $h_2(t_*)$

    '''(H)'''     Setting the frequency  $f_*$  for the numeric output

    '''(I)'''      Setting the time  $t_*$  for the numeric output

    '''(J)'''     Graphic field for the frequency domain

    '''(K)'''     Graphic field for the time domain

    '''(L)'''     Selection of the exercise according to the numbers

    '''(M)'''     Task description and questions

    '''(N)'''     Show and hide sample solution

==About the authors==

This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].
*The first version was created in 2005 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|»Ji Li«]]  as part of her diploma thesis with  “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29| »Günter Söder«]]).

*In 2017 the program was redesigned by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#David_Jobst_.28Ingenieurspraxis_Math_2017.29|»David Jobst«]]  (Ingenieurspraxis_Math,  Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|»Tasnád Kernetzky«]] ) via "HTML5".

*Last revision and English version 2020 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|»Carolin Mirschina«]]  in the context of a working student activity. 

==Once again: Open Applet in new Tab==
{{LntAppletLinkEnDe|frequImpResp_en|frequImpResp}}

LNTwww:Imprint for the book "Theory of Stochastic Signals"

2025-02-24T13:39:47Z

'''Five main chapters with a total of 28 chapters (files) and 166 sections (pages);   93 Exercises  ⇒   Scope:  "$\rm 3L+2E$"      
Development of the German version:   2011–2015.       Development of the English version:   2021.      Last corrections:   November 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Söd88]'''<ref name='Söd88'>Söder, G.:  Aufgabensammlung zu &bdquo;Statistische Methoden der Nachrichtentechnik”. Lecture notes, Chair of Communications Engineering, TU München, 1988.</ref>;  '''[Söd12]'''<ref name='Söd12'>Söder, G.:  Simulationsmethoden in der Nachrichtentechnik.  Internship notes, Chair of Communications Engineering, TU München, 2012</ref>;  –  Textbooks:   '''[Söd93]'''<ref name='Söd93'>Söder, G.:  Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin – Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2.</ref>;    '''[Dav87]'''<ref name='Dav87'>Davenport, W. B.: Probability and random processes. An introduction for applied scientists and engineers. New York NY u.a.: McGraw-Hill, 1987. ISBN 0-07-015440-6</ref>;  '''[PP09]'''<ref name='PP09'>Papoulis, A.; Pillai, S. U.: Probability, random variables, and stochastic processes. 4. Aufl. Boston, Mass.: McGraw-Hill, 2009. ISBN 978-0-07122-661-5</ref>

*Author:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29|'''Günter Söder''']]

*Participating colleagues in alphabetic order:  [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29| '''Benedikt Leible''']],   [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Thomas_Stockhammer_.28at_LNT_from_1995-2004.29|'''Thomas Stockhammer''']],  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|'''Johannes Zangl''']]

*Participating students in chronological order:      Martin Winkler '''(2001)''', Yven Winter, Reinhold Sixt, Jürgen Veitenhansl, Franz Kohl, Bettina Hirner, Ji Li, Markus Elsberger, Thorsten Kalweit, Thomas Großer, David Jobst, Matthias Niller, Veronika Hofmann, Carolin Mirschina, Noah Nagi, Ji Woo Hwang '''(2022)'''

$\text{References:}$

LNTwww:Imprint for the book "Signal Representation"

2025-02-24T13:39:46Z

'''Five main chapters with a total of 19 chapters (files) and 127 sections (pages);   58 Exercises   ⇒   Scope:  "$\rm 2L+1E$"      
Development of the German version:   2002–2017.       Development of the English version:   2020/21.      Last corrections:   December 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| $\text{Gerhard Kramer}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|$\text{Tasnád Kernetzky}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|$\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |$\text{Günter Söder}$]]

*Basic materials for the original German version:     Lecture notes of LNT/LÜT:  '''[Eich03]'''<ref name='Eich03'>Eichin, K.:  Nachrichtentechnik I (LB) – Signaldarstellung.  Lecture notes, Chair of Communications Engineering, TU München, 2003. </ref>;  '''[Han15]'''<ref name='Han15'>Hanik, N.:  Nachrichtentechnik 1 (LB): Signaldarstellung.  Lecture notes, Associate Professorship of Line Transmission Technology, TU München, 2015.</ref>;  '''[Söd12]'''<ref name='Söd12'>Söder, G.:   Simulationsmethoden in der Nachrichtentechnik.  Internship notes, Chair of Communications Engineering, TU München, 2012.</ref>   –  Textbooks:   '''[Söd93]'''<ref name='Söd93'>Söder, G.:  Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin, Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2.</ref>;  '''[Mar94]'''<ref name='Mar94'>Marko, H.:  Methoden der Systemtheorie.  3. Auflage. Berlin – Heidelberg: Springer, 1994.</ref>;  '''[HM09]'''<ref name='HM09'>Haykin, S.; Moher, M.: Communication Systems. 5th edition, Hoboken, N.J.: J. Wiley, 2009. ISBN 978-0-47169-790-9.</ref>

*Authors:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |$\text{Günter Söder}$]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29| $\text{Klaus Eichin}$]]

* Participating colleagues in alphabetic order:   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]]  and  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|$\text{Norbert Hanik}$]]

*Participating students in chronological order:      Martin Winkler  '''(2001)''',  Yven Winter,  Franz Kohl,  Roland Kiefl,  Thorsten Kalweit,  Markus Elsberger,  Bettina Hirner,  Ji Li,  Slim Lamine,  Thomas Großer,  Jimmy He,  Xiaohan Liu,  Noah Nagi,  Carolin Mirschina,  Jiwoo Hwang  '''(2022)'''

$\text{References:}$

Applets:Discrete Fouriertransform and Inverse

2025-02-24T13:39:44Z

{{LntAppletLinkEnDe|dft_en|dft}}

==Applet Description==
 
The conventional  »Fourier Transform«  $\rm (FT)$  allows the calculation of the spectral function  $X(f)$  of a time-continuous signal  $x(t)$. 

In contrast,  the  »Discrete Fourier Transform«  $\rm (DFT)$  is limited to a time-discrete signal,  represented by  $N$  time domain coefficients   $d(\nu)$  with indices  $\nu = 0, \text{...} , N\hspace{-0.1cm}-\hspace{-0.1cm}1$,  which can be interpreted as equidistant samples of the time-continuous signal  $x(t)$.

If the  [[Signal_Representation/Discrete-Time_Signal_Representation#Sampling_theorem|»sampling theorem«]]  is fulfilled, the DFT algorithm likewise allows the calculation of  $N$  frequency domain coefficients  $D(\mu)$  with indices  $\mu = 0, \text{...} , N\hspace{-0.1cm}-\hspace{-0.1cm}1$.  These are equidistant samples of the frequency-continuous spectrum  $X(f)$.

*The applet illustrates the properties of the  $\text{DFT:}\hspace{0.3cm}d(\nu)\hspace{0.1cm} \Rightarrow \hspace{0.1cm} D(\mu)$  by using the example  $N=16$.  The default   $d(\nu)$ assignments for the DFT are:

: $\rm (a)$  According to the input field,  $\rm (b)$  Constant signal,  $\rm (c)$  Complex exponential time function ,  $\rm (d)$  Harmonic oscillation  $($with  phase  $\varphi = 45^\circ)$,
: $\rm (e)$  Cosine signal  $($one period$)$,  $\rm (f)$  Sinusoidal signal  $($one period$)$,  $\rm (g)$  Cosine signal  $($two periods$)$,  $\rm (h)$  Alternating time coefficients,
: $\rm (i)$  Dirac delta impulse,  $\rm (j)$  Rectangular pulse ,  $\rm (k)$  Triangular pulse,  $\rm (l)$  Gaussian pulse.

*Possible  $D(\mu)$ assignments for the Inverse Discrete Fourier Transform   ⇒   $\text{IDFT:}\hspace{0.3cm}D(\mu)\hspace{0.1cm} \Rightarrow \hspace{0.1cm} d(\nu)$  are:

: $\rm (A)$  According to the input field,  $\rm (B)$  Constant spectrum,  $\rm (C)$  Complex exponential function (of frequency),  $\rm (D)$  Equivalent to setting  $\rm (d)$  in the time domain,
: $\rm (E)$  Cosine spectrum  $($one frequency period$)$,  $\rm (F)$  Sinusoidal spectrum  $($one frequency period$)$,  $\rm (G)$  Cosine spectrum  $($two frequency periods$)$, 
: $\rm (H)$  Alternating spectral coefficients,  $\rm (I)$  Dirac delta spectrum,  $\rm (J)$  Rectangular spectrum,  $\rm (K)$  Triangular spectrum,  $\rm (L)$  Gaussian spectrum.

The applet uses the framework  [https://en.wikipedia.org/wiki/Plotly »Plot.ly«].

==Theoretical Background==
 
===Arguments for the discrete realization of the Fourier transform===

The  »'''Fourier transform'''«  according to the conventional description for continuous-time signals has an infinitely high selectivity due to the unlimited extension of the integration interval and is therefore an ideal theoretical tool for spectral analysis.

If the spectral components  $X(f)$  of a time function  $x(t)$  are to be determined numerically,  the general transformation equations

:$$\begin{align*}X(f) & = \int_{-\infty
}^{+\infty}x(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi f t}\hspace{0.1cm} {\rm d}t\hspace{0.5cm} \Rightarrow\hspace{0.5cm} \text{Forward transformation}\hspace{0.7cm} \Rightarrow\hspace{0.5cm} \text{First Fourier integral}
\hspace{0.05cm},\\
x(t) & = \int_{-\infty
}^{+\infty}\hspace{-0.15cm}X(f) \cdot {\rm e}^{\hspace{0.05cm}+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi f t}\hspace{0.1cm} {\rm d}f\hspace{0.35cm} \Rightarrow\hspace{0.5cm}
\text{Backward transformation}\hspace{0.4cm} \Rightarrow\hspace{0.5cm} \text{Second Fourier integral}
\hspace{0.05cm}\end{align*}$$

are unsuitable for two reasons:
*The equations apply exclusively to continuous-time signals.  With digital computers or signal processors,  however,  only discrete-time signals can be processed.

*For a numerical evaluation of the two Fourier integrals it is necessary to limit the respective integration interval to a finite value.

{{BlaueBox|TEXT=
$\text{This results in the following consequence:}$ 

A  »'''continuous-time signal'''«  must undergo two processes before the numerical determination of its spectral properties, viz.

*  »'''sampling'''«  for discretization,  and

*  »'''windowing'''«  to limit the integration interval.}}

In the following,  starting from an aperiodic time function  $x(t)$  and the corresponding Fourier spectrum  $X(f)$,  a discrete-time and discrete-frequency description suitable for computer processing is presented.

===Time discretization – periodization in the frequency domain===

The following graphs uniformly show the time domain on the left and the frequency domain on the right.  Without restriction of generality,  $x(t)$  and  $X(f)$  are real and Gaussian,  respectively.

[[File:P_ID1132__Sig_T_5_1_S2_neu.png|center|frame|Discretization in the time domain – periodization in the frequency domain]]

One can describe the sampling of the time signal  $x(t)$  by multiplication with a Dirac delta pulse  $p_{\delta}(t)$.  This results in the time signal sampled at distance  $T_{\rm A}$: 

:$${\rm A}\{x(t)\} = \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot x(\nu \cdot T_{\rm A})\cdot
\delta (t- \nu \cdot T_{\rm A}
)\hspace{0.05cm}.$$

We now transform this sampled signal  $\text{A}\{ x(t)\}$  into the frequency domain. The multiplication of the Dirac delta pulse  $p_{\delta}(t)$  with  $x(t)$  corresponds in the frequency domain to the convolution of  $P_{\delta}(f)$  with  $X(f)$.  The periodized spectrum  $\text{P}\{ X(f)\}$ is obtained,  where  $f_{\rm P}$  is the frequency period of the function  $\text{P}\{ X(f)\}$: 

:$${\rm A}\{x(t)\} \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} {\rm P}\{X(f)\} = \sum_{\mu = - \infty }^{+\infty}
X (f- \mu \cdot f_{\rm P} )\hspace{0.5cm} {\rm with }\hspace{0.5cm}f_{\rm
P}= {1}/{T_{\rm A}}\hspace{0.05cm}.$$

*We call the sampled signal  $\text{A}\{ x(t)\}$.

*The  »'''frequency period'''«  is denoted by  $f_{\rm P}$ = $1/T_{\rm A}$. 

The graph above shows the functional relationship described here.  It should be noted:
#The frequency period  $f_{\rm P}$  was deliberately chosen small here so that the overlap of the spectra to be summed can be clearly seen.
#In practice,  due to the sampling theorem,  $f_{\rm P}$  should be at least twice as large as the largest frequency contained in the signal  $x(t)$. 
#If this is not fulfilled,  »'''aliasing'''«  must be expected.

===Frequency discretization – periodization in the time domain===

The discretization of  $X(f)$  can also be described by a multiplication with a Dirac delta pulse. The result is the spectrum sampled at distance  $f_{\rm A}$: 

:$${\rm A}\{X(f)\} = X(f) \cdot \sum_{\mu = - \infty }^{+\infty}
f_{\rm A} \cdot \delta (f- \mu \cdot f_{\rm A } ) = \sum_{\mu = - \infty }^{+\infty}
f_{\rm A} \cdot X(\mu \cdot f_{\rm A } ) \cdot\delta (f- \mu \cdot f_{\rm A } )\hspace{0.05cm}.$$

*Transforming the frequency Dirac delta pulse used here  $($with pulse weights  $f_{\rm A})$  into the time domain,  we obtain with  $T_{\rm P} = 1/f_{\rm A}$:

:$$\sum_{\mu = - \infty }^{+\infty}
f_{\rm A} \cdot \delta (f- \mu \cdot f_{\rm A } ) \hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}
\sum_{\nu = - \infty }^{+\infty}
\delta (t- \nu \cdot T_{\rm P } ) \hspace{0.05cm}.$$

*The multiplication with  $X(f)$  corresponds in the time domain to the convolution with  $x(t)$. The signal  $\text{P}\{ x(t)\}$ periodized at distance  $T_{\rm P}$  is obtained:

:$${\rm A}\{X(f)\} \hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}
{\rm P}\{x(t)\} = x(t) \star \sum_{\nu = - \infty }^{+\infty}
\delta (t- \nu \cdot T_{\rm P } )= \sum_{\nu = - \infty }^{+\infty}
x (t- \nu \cdot T_{\rm P } ) \hspace{0.05cm}.$$

{{GraueBox|TEXT=
[[File:P_ID1134__Sig_T_5_1_S3_neu.png|right|frame|Discretization in the frequency domain – periodization in the time domain]]
$\text{Example 1:}$ 
This relationship is illustrated in the graph:

#Due to the coarse frequency rastering,  this example results in a relatively small value for the time period  $T_{\rm P}$.  
#Therefore,  the  $($blue$)$  periodized time signal  $\text{P}\{ x(t)\}$  differs significantly from  $x(t)$ due to overlaps.}}

===Finite signal representation===

One arrives at the so-called  »'''finite signal representation'''« 
*when both the time function  $x(t)$  and

*the spectral function  $X(f)$

[[File:P_ID1135__Sig_T_5_1_S4_neu.png|right|frame|Finite signals of the Discrete Fourier Transform]]

are specified exclusively by their sample values:

The graph is to be interpreted as follows:
*In the left graph the function  $\text{A}\{ \text{P}\{ x(t)\}\}$  is drawn in blue.  It is obtained by sampling the periodized time function  $\text{P}\{ x(t)\}$  with equidistant Dirac delta pulses in the distance  $T_{\rm A} = 1/f_{\rm P}$.

*In the right graph the function  $\text{P}\{ \text{A}\{ X(f)\}\}$  is drawn in green.  This results from periodization  $($with  $f_{\rm P})$  of the sampled spectral function  $\{ \text{A}\{ X(f)\}\}$.

*There is also a Fourier correspondence between the blue finite signal and the green finite signal,  as follows:

:$${\rm A}\{{\rm P}\{x(t)\}\} \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} {\rm P}\{{\rm A}\{X(f)\}\} \hspace{0.05cm}.$$

{{BlaueBox|TEXT=
However,  the Dirac delta lines of the periodic continuation  $\text{P}\{ \text{A}\{ X(f)\}\}$  of the sampled spectral function fall into the same frequency grid as those of  $\text{A}\{ X(f)\}$  only if the frequency period  $f_{\rm P}$  is an integer multiple  $(N)$  of the frequency sampling interval  $f_{\rm A}$. 

When using the finite signal representation,  the following condition must always be fulfilled,  where in practice a power of two is usually used for the natural number  $N$  $($the graph above is based on the value  $N = 8)$:

:$$f_{\rm P} = N \cdot f_{\rm A} \hspace{0.5cm} \Rightarrow\hspace{0.5cm} {1}/{T_{\rm A} }= N \cdot f_{\rm A} \hspace{0.5cm} \Rightarrow\hspace{0.5cm}
N \cdot f_{\rm A}\cdot T_{\rm A} = 1\hspace{0.05cm}.$$}}

If the condition  $N \cdot f_{\rm A} \cdot T_{\rm A} = 1$  is satisfied, the order of periodization and sampling can be interchanged. Thus:

:$${\rm A}\{{\rm P}\{x(t)\}\} = {\rm P}\{{\rm A}\{x(t)\}\}\hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm}
{\rm P}\{{\rm A}\{X(f)\}\} = {\rm A}\{{\rm P}\{X(f)\}\}\hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Conclusions:}$ 
#The time function  $\text{P}\{ \text{A}\{ x(t)\}\}$  has the period  $T_{\rm P} = N \cdot T_{\rm A}$.
#The period in the frequency domain is  $f_{\rm P} = N \cdot f_{\rm A}$.
#For the description of the discretized time and frequency course  $N$  '''»complex numerical values«''' in the form of pulse weights are sufficient in each case.}}

{{GraueBox|TEXT=
$\text{Example 2:}$ 
A time-limited  $($pulse-like$)$  signal  $x(t)$  is present in sampled form,  where the distance between two samples is  $T_{\rm A} = 1\, {\rm µ s}$: 

*After a discrete Fourier transform with  $N = 512$  the spectrum  $X(f)$  is available as samples with the distance  $f_{\rm A} = (N \cdot T_{\rm A})^{–1} \approx 1.953\,\text{kHz} $. 

*Increasing the DFT parameter to  $N= 2048$ results  in a finer frequency grid with  $f_{\rm A} \approx 488\,\text{Hz}$.}}

===Discrete Fourier Transform===

From the conventional  »first Fourier integral«

:$$X(f) =\int_{-\infty
}^{+\infty}x(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm} f \hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}t$$

»discretization«  $(\text{d}t \to T_{\rm A}$,  $t \to \nu \cdot T_{\rm A}$,  $f \to \mu \cdot f_{\rm A}$,  $T_{\rm A} \cdot f_{\rm A} = 1/N)$  yields the sampled and periodized spectral function

:$${\rm P}\{X(\mu \cdot f_{\rm A})\} = T_{\rm A} \cdot \sum_{\nu = 0 }^{N-1}
{\rm P}\{x(\nu \cdot T_{\rm A})\}\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm} \cdot \hspace{0.05cm}\nu \hspace{0.05cm}
\cdot \hspace{0.05cm}\mu /N} \hspace{0.05cm}.$$

It is taken into account that due to the discretization the periodized functions have to be used in each case.

For reasons of a simplified notation we now make the following substitutions:
*Let the  $N$  »'''time-domain coefficients'''«  be associated with the indexing variable  $\nu = 0$, ... , $N - 1$:
:$$d(\nu) =
{\rm P}\left\{x(t)\right\}{\big|}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm A}}\hspace{0.05cm}.$$
*Let the  $N$  »'''frequency-domain coefficients'''«  be associated with the indexing variable  $\mu = 0,$ ... , $N$ – 1:
:$$D(\mu) = f_{\rm A} \cdot
{\rm P}\left\{X(f)\right\}{\big|}_{f \hspace{0.05cm}= \hspace{0.05cm}\mu \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm A}}\hspace{0.05cm}.$$
*Abbreviation for the  »'''complex rotation factor'''«  depending on  $N$  is written:
:$$w = {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N}
= \cos \left( {2 \pi}/{N}\right)-{\rm j} \cdot \sin \left( {2 \pi}/{N}\right)
\hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Definition:}$  The term  »'''Discrete Fourier Transform'''«  $\rm (DFT)$  means the calculation of the  $N$  spectral coefficients  $D(\mu)$  from the  $N$  signal coefficients  $d(\nu)$:
[[File:P_ID2730__Sig_T_5_1_S5_neu.png|right|frame|On the definition of the Discrete Fourier Transform  $\rm (DFT)$  with  $N=8$]]

:$$D(\mu) = \frac{1}{N} \cdot \sum_{\nu = 0 }^{N-1}
d(\nu)\cdot {w}^{\hspace{0.05cm}\nu \hspace{0.03cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}. $$

In the graph you can see in an example
#the  $N = 8$  signal coefficients  $d(\nu)$  at the blue filling,
#the  $N = 8$  spectral coefficients  $D(\mu)$  at the green filling.}}

===Inverse Discrete Fourier Transform===

The  »Inverse Discrete Fourier Transform«  describes the  »second Fourier integral«:

:$$\begin{align*}x(t) & = \int_{-\infty
}^{+\infty}X(f) \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm} f \hspace{0.05cm}\cdot \hspace{0.05cm}
t}\hspace{0.1cm} {\rm d}f\end{align*}$$

in discretized form:  
:$$d(\nu) =
{\rm P}\left\{x(t)\right\}{\big|}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm
A}}\hspace{0.01cm}.$$

{{BlaueBox|TEXT=
$\text{Definition:}$  The term  »'''Inverse Discrete Fourier Transform«'''  $\rm (IDFT)$  refers to the calculation of the signal coefficients  $d(\nu)$  from the spectral coefficients  $D(\mu)$:
[[File:P_ID2731__Sig_T_5_1_S6_neu.png|right|frame|For the definition of the IDFT with  $N=8$]]
:$$d(\nu) = \sum_{\mu = 0 }^{N-1}
D(\mu) \cdot {w}^{-\nu \hspace{0.03cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}.$$

With the indexing variables 
*$\nu = 0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1$,
*$\mu = 0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1$,:

then holds:
#$d(\nu) =
{\rm P}\left\{x(t)\right\}{\big \vert}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm
A} }\hspace{0.01cm},$
#$D(\mu) = f_{\rm A} \cdot
{\rm P}\left\{X(f)\right\}{\big \vert}_{f \hspace{0.05cm}= \hspace{0.05cm}\mu \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm A} }
\hspace{0.01cm},$
#$w = {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N}
\hspace{0.01cm}.$}}

A comparison between DFT and IDFT shows that exactly the same algorithm can be used. The only differences of the IDFT compared to the DFT are:
*The exponent of the rotation factor must be applied with different sign.

*With the IDFT the division by  $N$  is omitted.

===Interpretation of DFT and IDFT===

The graph shows the discrete coefficients in the time and frequency domain together with the periodized continuous-time functions.

[[File:P_ID1136__Sig_T_5_1_S7_neu.png|right|frame|Time and frequency domain coefficients of the DFT]]

When using DFT or IDFT,  it should be noted:
*According to the above definitions,  the DFT coefficients  $d(ν)$  and  $D(\mu)$  always have the unit of the time function.

*Dividing  $D(\mu)$  by  $f_{\rm A}$  gives the spectral value  $X(\mu \cdot f_{\rm A})$.

*The spectral coefficients  $D(\mu)$  must always be set complex to be able to consider also odd time functions.

*In order to be able to transform band–pass signals in the equivalent low–pass range,  complex time coefficients  $d(\nu)$ are usually used.

*The basic interval for  $\nu$  and  $\mu$  is usually defined as the range from  $0$  to  $N - 1$,  as in the above diagram.

*With the complex-valued number sequences 
**$\langle \hspace{0.03cm}d(\nu)\hspace{0.03cm}\rangle = \langle \hspace{0.03cm}d(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , d(N-1) \hspace{0.03cm}\rangle$,
**$\langle \hspace{0.03cm}D(\mu)\hspace{0.03cm}\rangle = \langle \hspace{0.03cm}D(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , D(N-1) \hspace{0.03cm}\rangle$: 

:⇒   DFT and IDFT are symbolized similar to the conventional Fourier transform:
::$$\langle \hspace{0.03cm} D(\mu)\hspace{0.03cm}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.03cm} d(\nu) \hspace{0.03cm}\rangle \hspace{0.05cm}.$$
*If the function  $x(t)$  is already limited to the range  $0 \le t \lt N \cdot T_{\rm A}$,  then the time coefficients output by the IDFT directly indicate the samples of the time function:  
:$$d(\nu) = x(\nu \cdot T_{\rm A}).$$
*If  $x(t)$  is shifted with respect to the basic interval,  one has to choose the assignment between  $x(t)$  and the coefficients  $d(\nu)$  as shown in  $\text{Example 3}$. 

{{GraueBox|TEXT=
$\text{Example 3:}$ 
The upper graph shows the asymmetric triangular pulse  $x(t)$  whose absolute width is smaller than  $T_{\rm P} = N \cdot T_{\rm A}$.

[[File:EN_Sig_T_5_1_S7b_neu.png|right|frame|On assigning of the DFT coefficients with  $N=8$]]

The sketch below shows the assigned DFT coefficients valid for  $N = 8$:

*For  $\nu = 0,\hspace{0.05cm}\text{...} \hspace{0.05cm} , N/2 = 4$,    $d(\nu) = x(\nu \cdot T_{\rm A})$   is valid:

:$$d(0) = x (0)\hspace{0.05cm}, \hspace{0.15cm}
d(1) = x (T_{\rm A})\hspace{0.05cm}, \hspace{0.15cm}
d(2) = x (2T_{\rm A})\hspace{0.05cm}, $$
:$$d(3) = x (3T_{\rm A})\hspace{0.05cm}, \hspace{0.15cm}
d(4) = x (4T_{\rm A})\hspace{0.05cm}.$$
*The coefficients  $d(5)$,  $d(6)$  and  d$(7)$  are to be set as follows:

:$$d(\nu) = x \big ((\nu\hspace{-0.05cm} - \hspace{-0.05cm} N ) \cdot T_{\rm A}\big ) $$

:$$ \Rightarrow \hspace{0.2cm}d(5) = x (-3T_{\rm A})\hspace{0.05cm}, \hspace{0.35cm}
d(6) = x (-2T_{\rm A})\hspace{0.05cm}, \hspace{0.35cm}
d(7) = x (-T_{\rm A})\hspace{0.05cm}.$$ }}

 

==Exercises==
 
[[File:Aufgaben_2D-Gauss.png|right]]
* First select the number  $($1, ...$)$  of the exercise. 
* A description of the exercise will be displayed.
*The parameter values are adjusted. 
*Solution after pressing  »Show solution«. 
*The number  "$0$"  corresponds to a  »Reset«:
#Same setting as at program start.
#Output of a  »reset text«  with further explanations about the applet.

{{BlaueBox|TEXT=
'''(1)'''  New setting:  DFT of signal  $\rm (b)$:  Constant signal.  Interpret the result in the frequency domain.  What is the analogon of the conventional Fourier transform?}}

:* All coefficients in the time domain are  $d(\nu)=1$.  Thus all  $D(\mu)=0$  with the exception of  $\textrm{Re}[D(0)]=1$. 
:* This corresponds to the conventional  $($time-continuous$)$  Fourier Transform:  $x(t)=A\hspace{0.15cm} \circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm} X(f)=A\cdot \delta (f=0)$  with  $A=1$.

{{BlaueBox|TEXT=
'''(2)'''  Assume the obtained  $D(\mu)$  field and shift all coefficients one entry down.  Which time function does the IDFT provide?
}}

:* Now all  $D(\mu)=0$,  except for  $\textrm{Re}[D(1)]=1$.  The result in the time domain is a complex exponential function.
:* The real part of the  $d(\nu)$  field shows a cosine and the imaginary part a sine function.  For each function one can see one period respectively.

{{BlaueBox|TEXT=
'''(3)'''  Add the following coefficient to the current  $D(\mu)$ field:  $\textrm{Im}[D(1)]=1$.  What are the differences compared to  '''(2)'''  in the time domain?
 }}
:* On the one hand,  a phase shift of two support values can now be detected for the real and the imaginary parts.  This corresponds to the phase  $\varphi = 45^\circ$.
:* On the other hand,  the amplitudes of the real and the imaginary part were each increased by the factor  $\sqrt{2}$.

{{BlaueBox|TEXT=
'''(4)'''  Set the  $D(\mu)$ field  to zero except for  $\textrm{Re}[D(1)]=1$.  Which additional  $D(\mu)$  coefficient yields a real  $d(\nu)$  field?
}}

:* By trial and error,  one can see that  $\textrm{Re}[D(15)]=1$  must apply additionally.  Then the  $d(\nu)$  field describes a cosine.
:* The following applies to the conventional  $($time continuous$)$  Fourier transform:  $x(t)=2\cdot \cos(2\pi \cdot f_0 \cdot t)\hspace{0.15cm}\circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}
X(f)=\delta (f-f_0)+\delta (f+f_0)$.
:* The entry  $D(1)$  is representative of the frequency  $f_0$  and due to the periodicity with  $N=16$  the frequency  $-f_0$  is expressed by  $D(15)=D(-1)$.

{{BlaueBox|TEXT=
'''(5)'''  According to the IDFT in the  $d(\nu)$  field, by which  $D(\mu)$  field does one obtain a real cosine function with the amplitude  $A=1$?}}

:* Like the conventional Fourier transform the discrete Fourier Transform is linear  ⇒   $D(1)=D(15)=0.5$.

{{BlaueBox|TEXT=
'''(6)'''  New setting:  DFT of signal  $\rm (e)$:  Cosine signal and subsequent signal shifts.  What are the effects of these shifts in the frequency domain?
}}

:* A shift in the time domain changes the cosine signal to a  "harmonic oscillation"  with arbitrary phase.
:* The  $D(\mu)$  field is still zero except for  $D(1)$  and  $D(15)$.  The absolute values  $|D(1)|$  and  $|D(15)|$  also remain the same.
:* The only change concerns the phase,  i.e. the different distribution of the absolute values between the real and imaginary part.

{{BlaueBox|TEXT=
'''(7)'''  New setting:  DFT of signal  $\rm (f)$:  Sinusoidal signal.  Interpret the result in the frequency domain.  What is the analogon of the conventional Fourier Transform?
}}

:* The sine signal results from the cosine signal by applying four time shifts.  Therefore all statements of  '''(6)'''  are still valid.
:* For the conventional (time continuous) Fourier transform it holds that  $x(t)= \sin(2\pi \cdot f_0 \cdot t)\hspace{0.15cm}\circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}
X(f)=j/2 \cdot [\delta (f+f_0)-\delta (f-f_0)]$.
:* The coefficient  $D(1)$   $\Rightarrow$  $($frequency:  $+f_0)$  is imaginary and has the imaginary part  $-0.5$.  Accordingly,  $\textrm{Im}[D(15)]=+0.5$   ⇒   $($frequency:  $-f_0)$  applies.

{{BlaueBox|TEXT=
'''(8)'''  New setting:  DFT of signal  $\rm (g)$:  Cosine signal (two periods).  Interpret the result in comparison to exercise  '''(5)'''.
}}

:* Here the time continuous Fourier transform reads  $x(t)=\cos(2\pi \cdot (2 f_0) \cdot t)\hspace{0.15cm}\circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}X(f)=0.5 \cdot \delta (f- 2 f_0)+0.5 \cdot \delta (f+ 2 f_0)$.
:* $D(2)$  is representative of the frequency  $2 f_0$.  Due to the periodicity,  $D(14)=D(-2)$:   $D(2)=D(14)=0.5$ is representative of the frequency  $-2 f_0$.

{{BlaueBox|TEXT=
'''(9)'''  Now examine the case DFT of a sinodial signal (two periods).  Which modifications do you need to make in the time domain?  Interpret the result.
}}

:* The desired signal can be obtained from the DFT of signal  $\rm (g)$:  Cosine signal (two periods) with two shifts.  With the result of  '''(7)''':  Four shifts.
:* The DFT result is accordingly  $\textrm{Im}[D(2)]=-0.5$  and  $\textrm{Im}[D(14)]=+0.5$.

{{BlaueBox|TEXT=
'''(10)'''  New setting: DFT of signal  $\rm (h)$:  Alternating time coefficients.  Interpret the DFT result.}}

:* Here, the time continuous Fourier transform is given by:  $x(t)=\cos(2\pi \cdot (8 f_0) \cdot t)\hspace{0.15cm}\circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}
X(f)=0.5 \cdot \delta (f- 8 f_0)+0.5 \cdot \delta (f+ 8 f_0)$.
:* $8 f_0$ is the highest frequency that can be displayed with  $N=16$  in the DFT.  There are only two sampled values per period, namely $+1$ and $-1$.
:* Difference to exercise  '''(5)''':  $D(1)=0.5$  now becomes  $D(8)=0.5$.  Likewise,  $D(15)=0.5$  is shifted to  $D(8)=0.5$.  Final result:  $D(8)=1$.

{{BlaueBox|TEXT=
'''(11)'''  What are the differences between the two settings DFT from signal  $\rm (i)$:  Dirac delta impulse and   IDFT from spectrum  $\rm (I)$:  Dirac delta spectrum?
}}

:* None! In the first case, all coefficients are  $D(\mu)=1$ (real);  in the second case, however, equivalently  $d(\nu)=1$ (real).

{{BlaueBox|TEXT=
'''(12)'''  Are there differences in shifting the real  "$1$"  in the according input fields by one place at a time, that is for  $d(\nu = 1)=1$  and  $D(\mu = 1)=1$?
}}

:* The first case  $\Rightarrow$  $\textrm{Re}[d(\nu = 1)]=1$  results in the complex exponential function in the frequency domain given by  $X(f)= \textrm{e}^{-{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2 \pi\hspace{0.05cm}\cdot\hspace{0.05cm} f/f_0}$  with negative sign.
:* The second case  $\Rightarrow$  $\textrm{Re}[D(\mu = 1)]=1$ results in the complex exponential function in the time domain given by  $x(t)= \textrm{e}^{+{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2 \pi\hspace{0.05cm}\cdot\hspace{0.05cm} f_0\cdot t}$  with positive sign.
:* Note:  With  $\textrm{Re}[D(\mu=15)]=1$  the result in the time domain would also be a complex exponential function  $x(t)= \textrm{e}^{-{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2 \pi\hspace{0.05cm}\cdot\hspace{0.05cm} f_0\hspace{0.05cm}\cdot\hspace{0.05cm} t}$ with negative sign.

{{BlaueBox|TEXT=
'''(13)'''  New setting: DFT of signal  $\rm (k)$:  Triangle pulse.  Interpret the  $d(\nu)$  assignment under the assumption  $T_\textrm{A} = 1$ ms.}}

:* Change the display to "absolute value".  $x(t)$ is symmetrical around  $t=0$  and extends from  $-8 \cdot T_\textrm{A} = -8$  ms to  $+8 \cdot T_\textrm{A}= +8$  ms.
:* $d(\nu)$  assignment:  $d(0)=x(0)=1$,  $d(1)=x(T_\textrm{A})=0.875$, ... ,  $d(8)=x(8 T_\textrm{A})=0$,  $d(9)=x(-7 T_\textrm{A})=0.125$, ... ,  $d(15)=x(-T_\textrm{A})=0.875$.

{{BlaueBox|TEXT=
'''(14)'''  Same setting as  '''(13)'''.  Interpret the DFT result, especially the coefficients  $D(0)$,  $D(1)$,  $D(2)$  and  $D(15)$.}}

:* In the frequency range  $D(0)$  stands for the frequency  $f=0$  and  $D(1)$  and  $D(15)$  for the frequencies  $\pm f_\textrm{A}$.  It holds that  $f_\textrm{A}= 1/ (N\cdot T_\textrm{A})=62.5$  Hz.
:* For the value of the continuous spectrum at $f=0$ the following applies:   $X(f=0)=D(0)/f_\textrm{A} = 0.5/ (0.0625$ kHz$)=8\cdot \textrm{kHz}^{-1}$.
:* The first zero of the  $\textrm{si}^2$–shaped spectrum  $X(f)$  occurs at  $2\cdot f_\textrm{A} = 125$ Hz.  The other zeros are equidistant.

{{BlaueBox|TEXT=
'''(15)'''  New setting: DFT of signal  $\rm (i)$:  Rectangular pulse.  Interpret the displayed results.}}
:* The set (symmetrical) rectangle extends over  $\pm 4 \cdot T_\textrm{A}$.  At the edges, the time coefficients are only half as large:  $d(4)=d(12)=0.5$.
:* The further statements of  '''(14)'''  also apply to this  $\textrm{si}$–shaped spectrum  $X(f)$.

{{BlaueBox|TEXT=
'''(16)'''  Same setting as for  '''(15)'''.  Which modifications need to be made in the  $d(\nu)$  field,
to have the duration of the rectangle   $\Rightarrow$   $\pm 2 \cdot T_\textrm{A}$.
}}
:* $d(0) = d(1) = d(15) =1, \ d(2) = d(14) = 0.5$. All other time coefficients zero  ⇒   first zero of the  ${\rm si}$ spectrum at  $4 \cdot f_{\rm A}= 250\text{ Hz}$.

{{BlaueBox|TEXT=
'''(17)'''  New setting:  IDFT of spectrum  $\rm (L)$:  Gaussian spectrum.  Interpret the result in the time domain.}}
:* Here, the time function  $x(t)$  is Gaussian with the maximum  $x(t=0)=4$.  For the spectrum the following applies:  $X(f=0)=D(0)/f_\textrm{A} = 16 \cdot \textrm{kHz}^{-1}$.
:* The equivalent duration of the pulse is  $\Delta t = X(f=0)/x(t=0)=4\text{ ms}$.  The inverse value gives the equivalent bandwidth  $\Delta f = 1/\Delta t = 250\text{ Hz}$.

==Applet Manual==
 
[[File:Anleitung_DFT_endgültig.png|left|600px|frame|Screenshot of the German version]]
    '''(A)'''     Time domain (input and result field)

    '''(B)'''     '''(A)''' representation numerical, graphical, magnitude

    '''(C)'''     frequency domain (input and result field)

    '''(D)'''     '''(C)''' representation numerical, graphical, magnitude

    '''(E)'''     Selection: DFT  $(t \to f)$  or IDFT  $(f \to t)$

    '''(F)'''     Given  $d(\nu)$ assignments (if DFT), or

                    Given  $D(\mu)$ assignments (if IDFT)

    '''(G)'''     Set input field to zero

    '''(H)'''     Move input field cyclically down (or up)

    '''( I )'''     Range for experiment execution:   exercise selection

    '''(J)'''     Range for experiment execution:   exercise definition

    '''(K)'''     Range for experiment execution:   show sample solution
 
*Given  $d(\nu)$ assignments (for DFT):

:(a)  corresponding number field,  (b)  DC signal,  (c)  Complex exponential function of time,  (d)  Harmonic oscillation  $($phase  $\varphi = 45^\circ)$,
:(e)  Cosine signal (one period),  (f)  Sine signal (one period),  (g)  Cosine signal (two periods), (h)  Alternating time coefficients,
:  (i)  Dirac delta pulse,  (j)  Rectangular pulse,  (k)  Triangular pulse,  (l)  Gaussian pulse.

*Given  $D(\mu)$ assignments (for IDFT):

:(A)  corresponding number field,  (B)  Constant spectrum,  (C)  Complex exponential function of frequency,  (D)  equivalent to setting (d) in time domain ,
:(E)  Cosine signal (one frequency period),  (F)  Sine signal (one frequency period),  (G)  Cosine signal (two frequency periods),  (H)  Alternating spectral coefficients,
:(I)  Dirac delta spectrum,  (J)  Rectangular spectrum,  (K)  Triangular spectrum,  (L)  Gaussian spectrum.

==About the Authors==

This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
*The first version was created in 2008 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|»Thomas Großer«]]  as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29|Günter Söder]]).

*Last revision and English version 2020/2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|»Carolin Mirschina«]]  in the context of a working student activity. 

The conversion of this applet to HTML 5 was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ Studienzuschüsse]  ("study grants") of the TUM Faculty EI. We thank them.

==Once again: Open Applet in new Tab==

{{LntAppletLinkEnDe|dft_en|dft}}

LNTwww:Imprint for the book "Digital Signal Transmission"

2025-02-24T13:39:41Z

'''Five main chapters with a total of 26 chapters (files) and 201 sections (pages);   90 Exercises   ⇒   Scope:  "$\rm 3L+2E$"      
Development of the German version:   2007–2016.       Development of the English version:   2020/21.      Last corrections:   October 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:   '''[Söd11]'''<ref name='Söd11'>Söder, G.:  Simulation digitaler Übertragungssysteme. Internship notes, Chair of Communications Engineering, TU München, 2011. </ref>;  '''[Söd12]'''<ref name='Söd12'>Söder, G.:  Simulationsmethoden in der Nachrichtentechnik. Internship notes, Chair of Communications Engineering, TU München, 2012.</ref>;   '''[Han17]'''<ref name='Han17'>Hanik, N.:  Leitungsgebundene Übertragungstechnik. Lecture notes, Associate Professorship of Line Transmission Technology, TU München, 2017.</ref>;  '''[Kra17]'''<ref name='Kra17'>Kramer, G.: Nachrichtentechnik 2. Lecture notes, Chair of Communications Engineering, TU München, 2017.</ref>.   –   Textbooks:   '''[ST85]'''<ref name='ST85'> Söder, G.; Tröndle, K.: Digitale Übertragungssysteme - Theorie, Optimierung & Dimensionierung der Basisbandsysteme. Berlin – Heidelberg: Springer, 1985. </ref>;  '''[TS87]'''<ref name='TS87'> Tröndle, K.; Söder, G.: Optimization of Digital Transmission Systems. Boston – London: Artech House, 1987, ISBN: 0-89006-225-0. </ref>;  '''[Söd93]'''<ref name='Söd93'>Söder, G.: Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin, Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2</ref>;  '''[Kam04]'''<ref name="Kam04">Kammeyer, K.D.:  Nachrichtenübertragung. Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>.

*Author:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29 |'''Günter Söder''']]

* Participating colleagues in alphabetical order:    [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|'''Klaus Eichin''']],   [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|'''Bernhard Göbel''']],  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']], [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|'''Johannes Zangl''']],

*Participating students in chronological order:      Martin Winkler '''(2001)''',  Yven Winter,  Ji Li ,  Franz Kohl,  Bettina Hirner,  Thorsten Kalweit,  Markus Elsberger,  Thomas Großer,  Johannes Schmidt,  Martin Völkl,  Xiaohan Liu,  Carolin Mirschina,  JiWoo Hwang '''(2022)'''

$\text{References:}$

Biographies and Bibliographies/LNTwww members from LNT

2025-02-24T13:37:49Z

Höfler:

{{Header|
Untermenü=An LNTwww beteiligte Mitarbeiter und Dozenten
|Nächste Seite= Beteiligte der Professur Leitungsgebundene Übertragungstechnik
|Vorherige Seite=Lehrstuhlinhaber_des_LNT
}}
During the creation of  "LNTwww"  many colleagues at the LNT gave us great support.  In this context, we understand  "LNT"  to mean the  "Lehrstuhl für Nachrichtentechnik"   ⇒   "Chair of Communications Engineering". 

From the  '''scientific staff'''  we would like to especially thank the following (former) colleagues:

==Dr.-Ing. Ronald Böhnke (at LNT from 2012-2014)==
[[File:Ronald.JPG|165px|right|]]
Ronald Böhnke, born in Bremen in 1976, studied Electrical Engineering and Information Technology at the University of Bremen and worked there as a research assistant from September 2002 in the field of Communications Engineering, where he obtained his doctorate on  "Efficient Detection and Adaptive Transmission for MIMO-OFDM Systems".   During his studies, he completed a three-month internship at Intel Corporation's research center in Santa Clara, California.  In addition, he was a visiting scientist at the Fraunhofer Institute for Telecommunications at the Heinrich Hertz Institute in Berlin for two months in 2002.

In September 2010,  Ronald Böhnke became a research associate at the Chair of Communications and Navigation (NAV) at the Technical University of Munich and contributed to the design of a geostationary relay satellite system.  After completing the project, he moved to the LNT in February 2012.  Since June 2014, he has been working in the field of mobile communications at the  "European Research Center"  of Huawei Technologies Düsseldorf GmbH in Munich.

Ronald Böhnke was a member of the German National Academic Foundation and received the Karl Nix Award for the Baccalaureater, the VDE Award for the Diploma and a "Best Paper Award" at the "IEEE International Workshop on Cross-Layer Design 2007".

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*Competent consultant and expert for the books  "Information Theory" and  "Channel Coding". 
*Often he also had to take corrective actions.}}

==Dr.-Ing. Joschi Brauchle (at LNT from 2007-2015)==

[[File:Brauchle.jpg|165px|right|Joschi Brauchle]]

Joschi Brauchle, born 1982 in Bad Reichenhall, studied electrical engineering and information technology at the Technical University of Munich from 2002 to 2007.  After completing his intermediate diploma, he completed a one-year master's degree at the Georgia Institute of Technology in Atlanta, Georgia, USA in 2005/2006.   In his diploma thesis at LNT he worked on  "Soft-input decoding of Reed-Solomon codes in concatenated systems"  and on  "Soft-output list decoding of inner convolutional codes".

After finishing his diploma thesis, Joschi Brauchle was a research assistant of  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|Prof. Kötter]]  and  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28since_2010.29|Prof. Kramer]].   His research was mainly concerned with algebraic coding theory, and in particular with the properties of bivariate interpolation-based decoding schemes for Reed-Solomon codes, as well as with the error correction performance of multidimensional schemes on related code constructions.  In winter 2013, he spent a three-month research stay at the University of Toronto, Canada.  In December 2015, he received his PhD on the topic  "Algebraic Decoding of Reed-Solomon and Related Codes".

In teaching, Joschi Brauchle designed and supervised the exercise for the lecture  "Channel Coding"  in the master's program from 2008 to 2013, for which he was awarded the lecturer prize of the student council for electrical engineering and information technology in 2013.  He supervised various student papers in seminars as well as quite a few bachelor's and master's theses throughout his assistantship. In 2015 he organized the main seminar.

In addition, Joschi Brauchle was jointly responsible for the conception and maintenance of the computer network and all IT systems at the LNT as a system administrator from 2009 to 2015.

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*In his capacity as system administrator, Joschi realized earlier than those responsible for LNTwww that our  "old LNTwww"  was indeed somewhat outdated.
*He put some basic and good thought into porting it to its present wiki form, for which we are all very grateful to him today.
}}

==Dr.-Ing. Klaus Eichin (at LNT from 1972-2011)==

[[File:K_Eichin.gif|165px|right|Klaus Eichin]]

Klaus Eichin was born in Wolfach in 1946.  He studied electrical engineering at the Technical University of Munich from 1965 onwards and obtained the title of Dipl.-Ing. in 1972.  He was awarded a doctorate in engineering in 1984.

*Already during his diploma thesis he was together with his later doctoral supervisor Prof. Karlheinz Tröndle intensively engaged in the topic  "Use of the computer in teaching".
* Afterwards, he worked at the Chair of Communications Engineering at the Technical University of Munich until his retirement in September 2011. In the last years, he focused on the research area of digital mobile radio.
*Furthermore, Dr. Eichin held events for students of the teaching profession at vocational schools (LB) as Academic Director.
 
{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*In 2001 Klaus Eichin was together with his colleague [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|Günter Söder]] the initiator of this e-learning project and worked intensively on it until 2011.
*Among other things, Klaus was co-author on six of the total nine LNTwww books. }}

==Dr.-Ing. Francisco Javier García Gómez (at LNT from 2016-2021)==
[[File:Javier.jpg|165px|right|]]

Francisco Javier García Gómez received the B.Eng. degree in Telecommunication Technologies and Services in 2014 from the Technical University of Madrid, (Spain), and the M.Sc. degree in Communications Engineering in 2016 from the Technical University of Munich (TUM), Germany.  The topic of his Master’s Thesis was  “Linear and Non-linear Estimation Methods for Single Carrier and Multicarrier Coarsely Quantized MIMO Systems”.

From 2016 to 2021, he was a doctoral researcher under the supervision of Prof. Gerhard Kramer at TUM’s Institute for Communications Engineering (LNT).  His research work was about the information theoretical analysis of the nonlinear optical fiber channel, with the goal of finding bounds on its capacity.

He taught the tutorial of the Bachelor lecture  “Mobile Communications” and the tutorial of the Master lecture   “Advanced Topics in Communications Engineering”. He was also responsible for the organization of Bachelor/Master Thesis Seminars.

 
{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*He has done essential preliminary work to be able to generate the English  "$\rm en.LNTwww.de$"  from  "$\rm www.LNTwww.de$"  with reasonable effort.
*2020/2021, Javier led the student translation team as one of the project managers for the English version. During this time, four books were completed.

}}

==Dr.-Ing. Thomas Hindelang (at LNT from 1994-2000 und 2007-2012)==

[[File:T_Hindelang_Sept15a.jpg|165px|right|]]

Thomas Hindelang received the Dipl.-Ing. and the PhD degree in electrical engineering from Technische Universität München (TUM), Germany, in 1994 and 2001, respectively. He worked as a Research Assistant at the Institute of Communications Engineering from 1994 to 2000, focusing primarily on the field of combined source and channel coding for mobile communications. While on leave from TUM, he served as a consultant at the Communications Research Department of AT&T Labs, Florham Park, NJ, from March to August 1999.
From 2000 to 2008 he has been with Siemens AG and Nokia Siemens Networks GmbH & Co. KG, first in the area of Mobile Devices and later in the area of Mobile Networks. From 2002 to 2006 he actively participated and contributed to the 3GPP standardization body on physical layer issues for the Multimedia Broadcast and Multicast Service (MBMS) and 3G Long Term Evolution (LTE) standards. From 2006 till 2008 he led the group "Baseband Algorithms and Simulations", where he was responsible for the physical layer algorithms and simulations for GSM, UMTS, LTE, and WiMAX hardware development. Since October 2007 he is senior lecturer at TUM, and since November 2008 he works as an examiner in the area "Audio Video Media" at the European Patent Office.
Dr. Hindelang holds approximately 40 patents and is the author of more than 30 published conference and journal papers. He contributed to the book "Advances in Digital Speech Transmission", edited by R. Martin, U. Heute, and C. Antweiler. His research interest includes the entire physical layer from the source until the channel, i.e., speech and video coding, error correction coding, equalization, and channel estimation techniques.

A great hobby of Thomas Hindelang is endurance sports.  As successes are placements in the top 10% at several marathons in Munich and Berlin, the placement in the top third at the mountain runs Jungfrau Marathon (42 km, 1800 vertical meters) and "Swiss Alpine"  (78 km, 2300 vertical meters)  and the time of 10:35 h at the Ironman in Regensburg 2011.

{{BlaueBox|TEXT=
'''His contributions to the LNTwww project''':  
*From 2007 - 2012, Dr. Hindelang held the lecture "Mobilfunk" at the Institute of Communications Engineering as a lecturer.
*His contributions to LNTwww, in particular to the books "Mobile Communications" and "Examples of Communication Systems", also date from this time.}}

==Dr.-Ing. Tobias Lutz (at LNT from 2008-2014)==

[[File:Lutz.jpg|165px|right|Tobias Lutz]]

Tobias Lutz, born in 1980 in Krumbach, studied electrical engineering and information technology at the Technical University of Munich from 2002 to 2008.  After completing his undergraduate studies, he moved to the  "Rensselaer Polytechnic Institute"  (RPI) in Troy, New York, for a one-year guest study in 2005/06.  From 2008 to 2014, Tobias Lutz was a research assistant to  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|Prof. Ralf Kötter]]  and  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29|Prof. Gerhard Kramer]]  at the Institute of Communications Technology (LNT) at the Technical University of Munich.  During this time, he also completed a degree in business mathematics at LMU Munich.

During the first three years of his assistantship, Tobias Lutz worked for the European research project  "N-Crave", where he dealt with theoretical aspects of network coding as well as the implementation of network protocols.

From August 2012 to April 2013, he spent an eight-month research stay at  "Stanford University".  During this time, he attended various courses from the field of mathematics and computer science and engaged in research on memory-based channels.  Regarding teaching, Tobias Lutz organized the basic practical course in communications engineering for four years.  Furthermore, he supervised student work within the framework of various.  In the last third of his assistantship Tobias Lutz was twice supervisor of the lecture  "Advanced Topics in Communications Engineering"  as well as of the lecture  "Information Theory".  In 2012, he received a  "Qualcomm Innovation Fellowship"  in the amount of 10.000 €.

His scientific interest during his LNT time was  "Multi-user Information Theory".  Specifically, he worked on the  "design of timing codes for half-duplex constrained networks"  and their information theoretic analysis.  This was also the topic of his PhD thesis (2014) with Prof. Gerhard Kramer.

{{BlaueBox|TEXT=
'''His contributions to the LNTwww project''':  
* He has been very supportive in the preparation of the book  [[Information theory]], 
*especially in the understandable formulation of not quite simple mathematical relations.}}

==Dr.-Ing. Michael Mecking (at LNT from 1997-2012)==

[[File:Mecking.jpg|165px|right|]]

Michael Mecking, born on 23.2.1972 in Karlsruhe, was a scientific assistant at the LNT from 1997 to 2003.  During this time, he worked on the projects "Receiver techniques for UMTS" and "Access strategies for the uplink of CDMA systems with channel-controlled scheduling" with Siemens.  In 2001 he spent five months at the EPFL in Lausanne with Prof. Bixio Rimoldi.

The focus of his research was multi-user information theory for mobile radio channels.  He completed his PhD thesis on  "Fading Multiple-Access with Channel State Information"  in November 2003.  His thesis advisor was Prof. [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|Joachim Hagenauer]].  The dissertation was awarded the Rohde & Schwarz Prize of the Faculty of Electrical Engineering and Information Technology in 2004.

Since September 2003, Dr. Mecking has been working at BMW AG in various areas of electrics and electronics, first for human-machine interface design and later in electrics/electronics integration.  From 2015 to 2017, he was responsible for change control in the area of electrics/electronics, vehicle dynamics and powertrain at BMW Manufacturing Co in South Carolina, USA.  Since his return to Germany in the summer of 2017, Dr. Mecking has led the department for integration and validation of the electrical/electronic vehicle dynamics and powertrain scopes at BMW's Regensburg and Oxford sites.

{{BlaueBox|TEXT=
'''His contributions to the LNTwww project''':  
*From 2004 to 2012, Dr. Mecking taught the lecture "Information Theory and Source Coding" as a lecturer, for which he was already responsible in 1998-2002.
*His script at that time was a great help for us in writing the book "Information Theory".  We were able to adopt much of it with his approval.}}

==Prof. Dr.-Ing. habil. Günter Söder (at LNT from 1974-2024)==

[[File:G_Soeder.gif|165px|right|Günter Söder]]

Günter Söder was born in Nürnberg on March 21, 1946.  He studied Electrical Engineering and Information Technology from 1964 at the  "Ohm–Polytechnikum Nürnberg"  (today:  Technical University Nuremberg Georg Simon Ohm)  and at the "Technischen Hochschule München"  (today:  Technical University Munich).  He received the academic titles Ing.-grad. (1967),  Dipl.-Ing. (1974),  Dr.-Ing. (1981)  and  Dr.-Ing. habil. (1993).

*Günter Söder worked at the Department of Communications Engineering of the Technical University of Munich from 1974 until his retirement in 2011,  mainly in the fields  "Digital Transmission Systems"  and  "Stochastic Signal Theory".  He held various lectures and practical courses on these topics as Academic Director and Associate Professor (since 2004).  
*He published the textbooks  "Digitale Übertragungssysteme – Theorie, Optimierung und Dimensionierung der Basisbandsysteme"  (1985 by Springer-Verlag Berlin)  and the English version  "Optimization of Digital Transmission Systems"  (1987 by Artech House Inc., Boston),  in each case jointly with his doctoral supervisor,  Prof. Karlheinz Tröndle.  His postdoctoral thesis  "Modellierung, Simulation und Optimierung von Nachrichtensystemen"  was also published by Springer-Verlag in 1993.
*In 1986 Günter Söder was awarded the NTG Prize  (today: Literature Prize of the Information Technology Society)  and in 1992 he and his team of diploma students were awarded the German-Austrian University Software Prize.

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*Günter Söder was together with  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|Klaus Eichin]]  the initiator of our learning tutorial and and is involved in all books as author and editor of the Gerrman version.
*It was of advantage that he has been intensively involved in the creation of educational programs  ("LNTsim"  and  "LNTwin")  since 1984.
*In 2024 he brought this e-learning project, started in 2001, to  "a good end"  (from his subjective point of view)  – thirteen years after his retirement.
*Günter Söder is still responsible for the German version  $($'''www.LNTwww.de'''$)$  and supports his colleagues with the English version  $($'''en.LNTwww.de'''$)$.
}}

==Dr.-Ing. Markus Stinner (at LNT from 2011-2016)==

[[File:Stinner1.jpg|165px|right|Markus Stinner]]

Markus Stinner was born in Ulm in 1986.  He studied electrical engineering at the University of Ulm from 2006 to 2011.  His studies included a semester abroad at the University of Adelaide, Australia in 2008.

After finishing his diploma thesis at the Institute of Telecommunications and Applied Information Technology,  which dealt with  "Partial Unit Memory Codes"  based on Gabidulin codes,  he was a research assistant to  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29|Prof. Kramer]]  at LNT from 2011.  His focus was on the analysis of  "Spatially Coupled Low-Density Parity-Check Codes"  for finite lengths.  He was at ENSEA in Cergy-Pontoise for a research visit in June 2012, at Lund University in Sweden in November 2015, and at EPFL in Lausanne in April 2016.  The topic of his PhD in September 2016 was  "Analysis of Spatially Coupled LDPC Codes on the Binary Erasure Channel".

In teaching, Markus Stinner supervised the  "Basic Laboratory Course on Telecommunications"  for many years and since 2014 the exercises for the lecture  "Channel Codes for Iterative Decoding".  In addition, he was responsible for the lecture  "Communications Technology 1"  at TUM¬Asia in Singapore in 2015.  In 2011/2012, his other tasks at LNT included working on the project  "CONE - Coding for Networks"  in collaboration with Alcatel Lucent Bell Labs, in which efficient channel codes and modulations for wireless links such as backhaul and 5G systems were developed and compared.

More information on Dr. Stinner's career can be found  [https://www.linkedin.com/in/markusstinner '''here'''].

{{BlaueBox|TEXT=
'''His contributions to the LNTwww project''':  
*He was at LNT for several years as a system administrator jointly responsible for design/maintenance of the computer network and all IT systems.
*In 2016, Markus led the porting of the "old LNTwww" (version 2) to the present wiki form (version 3) with a team of students.
*At the end of 2016, Markus left the LNT.  Building on his work, we have been able to release the "new LNTwww" two years after his departure.
*Many thanks for your excellent and forward-looking work.  Without your almost penetrating persistence, this improved version would not have existed.}}

==Dr.-Ing. Thomas Stockhammer (at LNT from 1995-2004)==

[[File:Stocki.jpg|165px|right|Markus Stinner]]

Thomas Stockhammer, born in Traunstein, Germany, in 1971, studied and earned his doctorate at the Institute of Communications Engineering at the Technical University of Munich (TUM). During this time he visited the Rensselear Politechnical Institute (RPI), Troy, NY, USA and the University of California San Diego (UCSD), San Diego, CA, USA as a visiting researcher.

After 10 years as founder and executive director for "Novel Mobile Radio" (NoMoR) research, he joined Qualcomm in 2014 and now acts as Senior Director Technical Standards – working from the scenic Chiemgau in his home office. In his different roles, he coauthored more than 250 research publications, more than 250 patents and thousands of standards contributions. He is the active and has board, leadership and rapporteur positions in 3GPP, DVB, MPEG, IETF, ATSC, CTA, ETSI, Metaverse Standards Forum, SVTA and the DASH-Industry Forum in multimedia communication, TV-distribution, 5G broadcast, content delivery protocols, immersive media representation, adaptive streaming, XR and the Metaverse. Among others, he leads MPEG-I Scene Description efforts, SVTA DASH-IF working group, 3GPP Video and XR activities as well as DVB-5G activities.

He received several awards for work on DASH, media delivery and 5G Broadcast, namely the INCITS Technical Excellence Award 2013, the 3GPP Excellence award 2017, the CTA Technology & Standards Achievement Award 2019 and 2023, ISO/IEC Excellence Award 2024, the DASH-IF Leadership Award 2024, the IEC1906 award as well as an Emmy Inventor Award for in 2022. He is regular speaker and Program Committee Member at events such as IBC, DVB World, Mile High Video, MWS or BroadThinking. In January 2023, he was elevated to IEEE Fellow for his contributions to media delivery and video streaming standards. For more details see: https://www.linkedin.com/in/stockhammer/

{{BlaueBox|TEXT=
'''His contributions to the LNTwww project''':  
*Competent advisor and expert for the book "Theory of Stochastic Signals".}}

==Dr.-Ing. Johannes Zangl (at LNT from 2000-2006)==

[[File:Zangl_2.jpg|165px|right|Johannes Zangl]]

Johannes Zangl, born in Augsburg in 1974, studied electrical engineering and information technology at the TUM and was a research associate at the LNT from 2000.  During this time, he worked on a multi-year DFG project in the priority program "Adaptivity in heterogeneous communication networks with wireless access (AKOM)", among others.  He received his PhD in 2005 on the topic of "Multi-hop networks with channel coding and medium access control (MAC)" under Prof.  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|Joachim Hagenauer]].

In the area of teaching, Johannes Zangl supervised the lectures "Fundamentals of Information Technology" and the "Mobile Communications Lab" for many years, as well as "Channel Coding" in the summer semester 2005.  In addition, he was responsible for the support of the LNT computer network and the mobile communications lab as a system administrator.

From 2006 Dr. Zangl was an employee at Infineon Technologies in Munich and worked as a development engineer in the verification of VDSL2.   In 2008, he moved to Rohde & Schwarz in Munich to the Center of Competence for Digital Signal Processing, where his tasks included the development of FPGA components for radar signal analysis systems, among others.  In June 2014, Dr. Zangl moved in-house to device development for network and spectrum analyzers.

{{BlaueBox|TEXT=
'''His contributions to the LNTwww project''':
*To his great credit, he scanned our crashed hard drive bit by bit for LNTwww parts in 2005, largely salvaging four years of work.
*Since then we know that it is not enough to make a backup, but that it must also be configured correctly.
*He was also a competent contact person and proofreader for the book "Theory of Stochastic Signals".}}

==Dr.-Ing. Georg Zeitler (at LNT from 2007-2012)==

[[File:Zeitler.jpg|165px|right|Johannes Zangl]]

Georg Zeitler, born in Munich in 1982, studied electrical engineering and information technology at the Technical University of Munich from 2002.  After completing his bachelor's thesis, he transferred to the University of Illinois at Urbana-Champaign in 2005 for a two-year master's degree, where he specialized in signal processing and information engineering.  After completing his master's thesis at UIUC on universal prediction of individual sequences, G. Zeitler was a research assistant to Prof. Kötter and Prof. Kramer from 2007 to 2012.  His scientific interests were information-theoretic aspects and the optimization of low-resolution quantizers for intelligence systems.  In spring 2010, he spent a three-month research stay at UIUC.

In teaching, Georg designed and supervised the central exercises and partly the lecture for the course "Communications Engineering 2".  In addition, he supervised student work in the two seminars offered by the LNT and he acted as the students' contact person for industrial contacts (engineering practice in the bachelor's program EI).

From 2007 to 2010, his other tasks at the LNT included working on the research project  "Network Coding for Multihop Relaying", n which quantize-and-forward strategies for relay networks were developed in collaboration with DOCOMO Euro-Labs GmbH.

Since June 2012, Dr. Zeitler has been working at BMW AG in Munich.

{{BlaueBox|TEXT=
'''His contributions to the LNTww project''':
*The main chapter 4 of the LNTwww book [[Digital Signal Transmission]] is largely based on manuscripts by him and Prof. [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|Ralf Kötter]]. }}

{{Display}}

LNTwww:Imprint for the book "Modulation Methods"

2025-02-03T14:06:41Z

Höfler: Text replacement - "'''Benedikt Leible'''" to " $\text{Benedikt Leible}$"

'''Five main chapters with a total of 23 chapters (files) and 192 sections (pages);   89 Exercises   ⇒     ⇒   Scope:  "$\rm 3L+2E$"      
Development of the German version:   2005–2011.       Development of the English version:   2020/21.      Last corrections:   October 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Söd14]'''<ref name='Söd14'>Söder, G.:  Analoge & Digitale Modulationsverfahren. Praktikum &bdquo;Simulation digitaler Übertragungssysteme”. Lehrstuhl für Nachrichtentechnik, TU München, 2014. </ref>;  '''[Han16]'''<ref name='Han16'>Hanik, N.:  Nachrichtentechnik 2 (LB): Modulationsverfahren. Vorlesungsmanuskript. Professur Leitungsgebundene Übertragungstechnik, TU München, 2016.</ref>;  '''[Vie16]'''<ref name='Vie16'>Viering, I.: System Aspects in Communications. Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2016.</ref>;   '''[Kra17]'''<ref name='Kra17'>Kramer, G.: Nachrichtentechnik 2. Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2017.</ref>.   –   Textbooks:   '''[ZP85]'''<ref name='ZP85'>Ziemer, R. E.; Peterson, R. L.: Digital Communications and Spread Spectrum Systems. New York: Macmillan, 1985. ISBN 978-0-02431-670-7</ref>;  '''[Kam04]'''<ref name="Kam04">Kammeyer, K.D.:  Nachrichtenübertragung. Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>

*Authors:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|'''Klaus Eichin''']],

* Participating colleagues in alphabetical order:   [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|'''Johannes Zangl''']],

*Participating students in chronological order:      Martin Winkler '''(2001)''',  Yven Winter,  Ji Li ,  Franz Kohl,  Bettina Hirner,  Thorsten Kalweit,  Markus Elsberger,  Slim Lamine,  Thomas Großer,  Johannes Schmidt,  David Jobst,  Xiaohan Liu,  Matthias Riedel,  Carolin Mirschina,  Sam Reed,  JiWoo Hwang '''(2022)'''

$\text{References:}$

LNTwww:General notes about "Modulation Methods"

2025-02-03T14:06:41Z

Höfler: Text replacement - "'''Benedikt Leible'''" to " $\text{Benedikt Leible}$"

'''Four main chapters with a total of 23 chapters (files);   89 Exercises   ⇒   Three semester hours per week lecture and two semester hours per week exercises      
Development of the German version:   2005–2011.       Development of the English version:   2020/21.      Last corrections:   October 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Söd11a]'''<ref name='Söd11a'>Söder, G.:  Analoge Modulationsverfahren. Anleitung zum gleichnamigen V ersuch im Praktikum &bdquo;Simulation digitaler Übertragungssysteme”. Lehrstuhl für Nachrichtentechnik, TU München, 2011.  </ref>;  '''[Söd11b]'''<ref name='Söd11b'>Söder, G.:  Digitale Modulationsverfahren. Anleitung zum gleichnamigen Versuch im Praktikum &bdquo;Simulation digitaler Übertragungssysteme”. Lehrstuhl für Nachrichtentechnik, TU München, 2011. </ref>;  '''[Han16]'''<ref name='Han16'>Hanik, N.:  Nachrichtentechnik 2 (LB): Modulationsverfahren. Vorlesungsmanuskript. Professur Leitungsgebundene Übertragungstechnik, TU München, 2016.</ref>;  '''[Vie16]'''<ref name='Vie16'>Viering, I.: System Aspects in Communications. Vorlesungsmanuskript. Lehrstuhl für Nachrichtentechnik. München: TU München, 2016.</ref>;   '''[Kra17]'''<ref name='Kra17'>Kramer, G.: Nachrichtentechnik 2. Vorlesungsmanuskript, Lehrstuhl für Nachrichtentechnik. München: TU München, 2017.</ref>.   –   Textbooks:   '''[ZP85]'''<ref name='ZP85'>Ziemer, R. E.; Peterson, R. L.: Digital Communications and Spread Spectrum Systems. New York: Macmillan, 1985. ISBN 978-0-02431-670-7</ref>;  '''[Söd93]'''<ref name='Söd93'>Söder, G.: Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin, Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2</ref>;  '''[Kam04]'''<ref name="Kam04">Kammeyer, K.D.:  Nachrichtenübertragung. Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>

*Author:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

* Participating colleagues in alphabetical order:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|'''Klaus Eichin''']],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|'''Johannes Zangl''']],

*Participating students in chronological order:      Martin Winkler '''(2001)''', Yven Winter, Ji Li , Franz Kohl, Bettina Hirner, Thorsten Kalweit, Markus Elsberger, Slim Lamine, Thomas Großer, Johannes Schmidt, David Jobst, Xiaohan Liu, Matthias Riedel, Carolin Mirschina, Sam Reed, JiWoo Hwang '''(2022)'''

$\text{List of sources:}$

LNTwww:LNTwww:Imprint for the book "Stochastic Signal Theory"

2025-02-03T14:06:36Z

Höfler: Text replacement - "'''Benedikt Leible'''" to " $\text{Benedikt Leible}$"

'''Five main chapters with a total of 28 chapters (files);   93 Exercises   ⇒   Two semester hours per week lecture and one semester hour per week exercises      
Development of the German version:   2011–2015.       Development of the English version:   2021.      Last corrections:   November 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Söd88]'''<ref name='Söd88'>Söder, G.:  Aufgabensammlung zu &bdquo;Statistische Methoden der Nachrichtentechnik”. Lehrstuhl für Nachrichtentechnik, TU München, 1988.</ref>;  '''[Söd12]'''<ref name='Söd12'>Söder, G.:  Simulationsmethoden in der Nachrichtentechnik.  Anleitung zum gleichnamigen Praktikum. Lehrstuhl für Nachrichtentechnik, TU München, 2012.</ref>;  –  Textbooks:   '''[Söd93]'''<ref name='Söd93'>Söder, G.:  Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin – Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2.</ref>;    '''[Dav87]'''<ref name='Dav87'>Davenport, W. B.: Probability and random processes. An introduction for applied scientists and engineers. New York NY u.a.: McGraw-Hill, 1987. ISBN 0-07-015440-6</ref>;  '''[PP09]'''<ref name='PP09'>Papoulis, A.; Pillai, S. U.: Probability, random variables, and stochastic processes. 4. Aufl. Boston, Mass.: McGraw-Hill, 2009. ISBN 978-0-07122-661-5</ref>

*Author:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|'''Günter Söder''']]

*Participating colleagues in alphabetic order:  [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29| '''Benedikt Leible''']],   [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Thomas_Stockhammer_.28at_LNT_from_1995-2004.29|'''Thomas Stockhammer''']],  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|'''Johannes Zangl''']]

*Participating students in chronological order:      Martin Winkler '''(2001)''', Yven Winter, Reinhold Sixt, Jürgen Veitenhansl, Franz Kohl, Bettina Hirner, Ji Li, Markus Elsberger, Thorsten Kalweit, Thomas Großer, David Jobst, Matthias Niller, Veronika Hofmann, Carolin Mirschina, Noah Nagi, Ji Woo Hwang '''(2022)'''

$\text{List of sources:}$

LNTwww:Imprint for the book "Channel Coding"

2025-02-03T14:06:31Z

Höfler: Text replacement - "'''Benedikt Leible'''" to " $\text{Benedikt Leible}$"

'''Four main chapters with a total of 22 chapters (files) and 175 sections (pages);   98 Exercises   ⇒   Scope:  "$\rm 3L+2E$"      
Development of the German version:   2011–2017.       Development of the English version:   2022.      Last corrections:   October 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:     '''[Köt08]'''<ref name='Köt08'>Kötter, R.; Mayer, T.; Tüchler, M.; Schreckenbach, F.; Brauchle, J.: Channel Coding. Lecture notes, Chair of Communications Engineering, TU München, 2008</ref>;   '''[Liv10]'''<ref name='Liv10'>Liva, G.: Channel Coding. Lecture notes, Chair of Communications Engineering, TU München and DLR Oberpfaffenhofen, 2010</ref>;   '''[Liv15]'''<ref name='Liv15'>Liva, G.: Channels Codes for Iterative Decoding. Lecture notes, Chair of Communications Engineering, TU München and DLR Oberpfaffenhofen, 2015</ref>   –   Textbooks:   '''[Bos99]'''<ref name='Bos99'>Bossert, M.: Channel Coding for Telecommunications. Chichester: Wiley, 1999. ISBN 978-0-471-98277-7 </ref>;  '''[Cov06]'''<ref name='Cov06'>Cover, T. M.; Thomas, J. A.: Elements of Information Theory. West Sussex: John Wiley & Sons, 2nd Edition, 2006 </ref>;   '''[Hub82]'''<ref name='Hub82'>Huber, J.: Codierung für gedächtnisbehaftete Kanäle. Dissertation – Universität der Bundeswehr München, 1982</ref>

*Authors and involved colleagues in alphabetic order:   [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Ronald_B.C3.B6hnke_.28at_LNT_from_2012-2014.29|'''Ronald Böhnke''']],   [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Joschi_Brauchle_.28at_LNT_from_2007-2015.29|'''Joschi Brauchle''']],   [http://wirelesscoding.org/ '''Gianluigi Liva'''],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Participating students in chronologic order:    Martin Winkler '''(2001)''',  Yven Winter,  Ji Li,  Bettina Hirner,  Thomas Großer,  Thorsten Bürgstein,  Martin Völkl,  Dominik Kopp,  Carolin Mirschina,  Jiwoo Hwang,  Noah Nagy '''(2022)'''

$\text{References:}$

LNTwww:Imprint for the book "Theory of Stochastic Signals"

2025-02-03T14:06:31Z

Höfler: Text replacement - "'''Benedikt Leible'''" to " $\text{Benedikt Leible}$"

'''Five main chapters with a total of 28 chapters (files) and 166 sections (pages);   93 Exercises  ⇒   Scope:  "$\rm 3L+2E$"      
Development of the German version:   2011–2015.       Development of the English version:   2021.      Last corrections:   November 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Söd88]'''<ref name='Söd88'>Söder, G.:  Aufgabensammlung zu &bdquo;Statistische Methoden der Nachrichtentechnik”. Lecture notes, Chair of Communications Engineering, TU München, 1988.</ref>;  '''[Söd12]'''<ref name='Söd12'>Söder, G.:  Simulationsmethoden in der Nachrichtentechnik.  Internship notes, Chair of Communications Engineering, TU München, 2012</ref>;  –  Textbooks:   '''[Söd93]'''<ref name='Söd93'>Söder, G.:  Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin – Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2.</ref>;    '''[Dav87]'''<ref name='Dav87'>Davenport, W. B.: Probability and random processes. An introduction for applied scientists and engineers. New York NY u.a.: McGraw-Hill, 1987. ISBN 0-07-015440-6</ref>;  '''[PP09]'''<ref name='PP09'>Papoulis, A.; Pillai, S. U.: Probability, random variables, and stochastic processes. 4. Aufl. Boston, Mass.: McGraw-Hill, 2009. ISBN 978-0-07122-661-5</ref>

*Author:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|'''Günter Söder''']]

*Participating colleagues in alphabetic order:  [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29| '''Benedikt Leible''']],   [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Thomas_Stockhammer_.28at_LNT_from_1995-2004.29|'''Thomas Stockhammer''']],  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|'''Johannes Zangl''']]

*Participating students in chronological order:      Martin Winkler '''(2001)''', Yven Winter, Reinhold Sixt, Jürgen Veitenhansl, Franz Kohl, Bettina Hirner, Ji Li, Markus Elsberger, Thorsten Kalweit, Thomas Großer, David Jobst, Matthias Niller, Veronika Hofmann, Carolin Mirschina, Noah Nagi, Ji Woo Hwang '''(2022)'''

$\text{References:}$

LNTwww:General notes about "Information Theory"

2025-02-03T14:06:31Z

Höfler: Text replacement - "'''Benedikt Leible'''" to " $\text{Benedikt Leible}$"

'''Four main chapters with a total of 13 chapters (files);   71 Exercises   ⇒   Two semester hours per week lecture and one semester hour per week exercises      
Development of the German version:   2011–2015.       Development of the English version:   2021.      Last corrections:   September 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials:   Lecture notes from LNT/LÜT:    '''[Meck09]'''<ref name='Meck09'>Mecking, M.:  Information Theory. Lecture notes, Chair of Communications Engineering, TU München, 2009.</ref>;  '''[Liv15]'''<ref name='Liv15'>Liva, G.:  Channels Codes for Iterative Decoding. Lecture notes, Chair of Communications Engineering, TU München and DLR Oberpfaffenhofen, 2015.</ref>;  '''[Kra16]'''<ref name='Kra16'>Kramer, G.:  Information Theory. Lecture notes, Chair of Communications Engineering, TU München, 2016.</ref>;  '''[Söd14]'''<ref name='Söd14'>Söder, G.:  Wertdiskrete Informationstheorie.  Guidance  (in German)  for the experiment of the same name in the practical course "Simulation of digital transmission systems"; Lecture notes, Chair of Communications Engineering, TU München, 2014. </ref>;  Textbook:  '''[Cov06]'''<ref name='Cov06'>Cover, T. M.; Thomas, J. A.: Elements of Information Theory. West Sussex: John Wiley & Sons, 2nd Edition, 2006. </ref>

*Authors and involved colleagues in alphabetic order:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],   [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|'''Bernhard Göbel''']],   [http://wirelesscoding.org/ '''Gianluigi Liva'''],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Tobias_Lutz_.28at_LNT_from_2008-2014.29|'''Tobias Lutz''']],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Michael_Mecking_.28at_LNT_from_1997-2012.29|'''Michael Mecking''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|'''Günter Söder''']]

*Participating students in chronologic order:      Martin Winkler '''(2001)''', Yven Winter, Thomas Großer, Stefan Müller, Martin Völkl, Eugen Mehlmann, Alexander Laible, Veronika Hofmann, Noah Nagi, Carolin Mirschina '''(2021)'''

$\text{List of sources:}$

LNTwww:Imprint for the book "Information Theory"

2025-02-03T14:06:30Z

Höfler: Text replacement - "'''Benedikt Leible'''" to " $\text{Benedikt Leible}$"

'''Four main chapters with a total of 13 chapters (files) and 106 sections (pages);   71 Exercises   ⇒   Scope:  "$\rm 2L+1E$"      
Development of the German version:   2011–2015.       Development of the English version:   2021.      Last corrections:   September 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials:   Lecture notes from LNT/LÜT:    '''[Meck09]'''<ref name='Meck09'>Mecking, M.:  Information Theory. Lecture notes, Chair of Communications Engineering, TU München, 2009.</ref>;  '''[Liv15]'''<ref name='Liv15'>Liva, G.:  Channel Codes for Iterative Decoding. Lecture notes, Chair of Communications Engineering, TU München and DLR Oberpfaffenhofen, 2015.</ref>;  '''[Kra16]'''<ref name='Kra16'>Kramer, G.:  Information Theory. Lecture notes, Chair of Communications Engineering, TU München, 2016.</ref>;  '''[Söd14]'''<ref name='Söd14'>Söder, G.:  Wertdiskrete Informationstheorie.  Guidance  (in German)  for the experiment of the same name in the practical course "Simulation of digital transmission systems"; Lecture notes, Chair of Communications Engineering, TU München, 2014. </ref>;  Textbook:  '''[Cov06]'''<ref name='Cov06'>Cover, T. M.; Thomas, J. A.: Elements of Information Theory. West Sussex: John Wiley & Sons, 2nd Edition, 2006. </ref>

*Authors and involved colleagues in alphabetic order:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],   [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|'''Bernhard Göbel''']],   [http://wirelesscoding.org/ '''Gianluigi Liva'''],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Tobias_Lutz_.28at_LNT_from_2008-2014.29|'''Tobias Lutz''']],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Michael_Mecking_.28at_LNT_from_1997-2012.29|'''Michael Mecking''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|'''Günter Söder''']]

*Participating students in chronological order:      Martin Winkler '''(2001)''', Yven Winter, Reinhold Sixt, Jürgen Veitenhansl, Franz Kohl, Bettina Hirner, Ji Li, Markus Elsberger, Thorsten Kalweit, Thomas Großer, David Jobst, Matthias Niller, Veronika Hofmann, Carolin Mirschina, Noah Nagi, Ji Woo Hwang '''(2022)'''

$\text{References:}$

LNTwww:Imprint for the book "Mobile Communications"

2025-02-03T14:06:30Z

Höfler: Text replacement - "'''Benedikt Leible'''" to " $\text{Benedikt Leible}$"

'''Four main chapters with a total of 16 chapters (files) and 121 sections (pages);   47 Exercises   ⇒   Scope:  "$\rm 2L+1E$"      
Development of the German version:   2010–2016.       Development of the English version:   2020/21.      Last corrections:   October 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:   '''[Hin08]'''<ref name='Hin08'>Hindelang, T.:   Mobile Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2008.</ref>;   '''[Söd10]'''<ref name='Söd10'>Söder, G.: Mobilfunkkanal.   Instructions for the practical course "Simulation digitaler Übertragungssysteme".   LNT/TUM, 2010. </ref>;   '''[Eich11]'''<ref name='Eich11'>Eichin, K.:   Nachrichtensysteme – Kommunikationssysteme (LB).   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2011.</ref>;  '''[Vie17]'''<ref name='Vie17'>Viering, I.:   System Aspects in Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2018</ref>;   '''[Kra18]'''<ref name='Kra18'>Kramer, G.:   Mobile Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2018.</ref>   –   Textbook:   '''[EVB01]'''<ref name='EVB01'>Eberspächer, J.; Vögel, H.J.; Bettstetter, C.: Global System for Mobile Communication. 3. Auflage. Stuttgart: Teubner, 2001.</ref>

*Authors:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29| '''Klaus Eichin''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Thomas_Hindelang_.28at_LNT_from_1994-2000_und_2007-2012.29| '''Thomas Hindelang''']].  

* Participating colleagues in alphabetic order:  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Markus_Stinner_.28at_LNT_from_2011-2016.29|'''Markus Stinner''']],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29| '''Johannes Zangl''']]

*Participating students in chronological order:  Martin Winkler '''(2001)''', Yven Winter, Thorsten Kalweit, Slim Lamine, Johannes Schmidt, Hedi Abbes, Thomas Großer, Néjib Kchouk, Khaled Soussi, Alexander Happach, Felix Kristl, Martin Völkl, André Schulz, Noah Nagi, Carolin Mirschina, Jiwoo Hwang '''(2023)'''

$\text{References:}$

LNTwww:Imprint for the book "Digital Signal Transmission"

2025-02-03T14:06:30Z

Höfler: Text replacement - "'''Benedikt Leible'''" to " $\text{Benedikt Leible}$"

'''Five main chapters with a total of 26 chapters (files) and 201 sections (pages);   90 Exercises   ⇒   Scope:  "$\rm 3L+2E$"      
Development of the German version:   2007–2016.       Development of the English version:   2020/21.      Last corrections:   October 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:   '''[Söd11]'''<ref name='Söd11'>Söder, G.:  Simulation digitaler Übertragungssysteme. Internship notes, Chair of Communications Engineering, TU München, 2011. </ref>;  '''[Söd12]'''<ref name='Söd12'>Söder, G.:  Simulationsmethoden in der Nachrichtentechnik. Internship notes, Chair of Communications Engineering, TU München, 2012.</ref>;   '''[Han17]'''<ref name='Han17'>Hanik, N.:  Leitungsgebundene Übertragungstechnik. Lecture notes, Associate Professorship of Line Transmission Technology, TU München, 2017.</ref>;  '''[Kra17]'''<ref name='Kra17'>Kramer, G.: Nachrichtentechnik 2. Lecture notes, Chair of Communications Engineering, TU München, 2017.</ref>.   –   Textbooks:   '''[ST85]'''<ref name='ST85'> Söder, G.; Tröndle, K.: Digitale Übertragungssysteme - Theorie, Optimierung & Dimensionierung der Basisbandsysteme. Berlin – Heidelberg: Springer, 1985. </ref>;  '''[TS87]'''<ref name='TS87'> Tröndle, K.; Söder, G.: Optimization of Digital Transmission Systems. Boston – London: Artech House, 1987, ISBN: 0-89006-225-0. </ref>;  '''[Söd93]'''<ref name='Söd93'>Söder, G.: Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin, Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2</ref>;  '''[Kam04]'''<ref name="Kam04">Kammeyer, K.D.:  Nachrichtenübertragung. Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>.

*Author:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

* Participating colleagues in alphabetical order:    [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|'''Klaus Eichin''']],   [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|'''Bernhard Göbel''']],  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']], [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|'''Johannes Zangl''']],

*Participating students in chronological order:      Martin Winkler '''(2001)''',  Yven Winter,  Ji Li ,  Franz Kohl,  Bettina Hirner,  Thorsten Kalweit,  Markus Elsberger,  Thomas Großer,  Johannes Schmidt,  Martin Völkl,  Xiaohan Liu,  Carolin Mirschina,  JiWoo Hwang '''(2022)'''

$\text{References:}$

LNTwww:General notes about "Mobile Communications"

2025-02-03T14:06:30Z

Höfler: Text replacement - "'''Benedikt Leible'''" to " $\text{Benedikt Leible}$"

'''Four main chapters with a total of 16 chapters (files);   47 Exercises   ⇒   Two semester hours per week lecture and one semester hour per week exercises      
Development of the German version:   2010–2017.       Development of the English version:   2020.      Last corrections:   February 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Hin08]'''<ref name='Hin08'>Hindelang, T.:   Mobile Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2008.</ref>;   '''[Eich11]'''<ref name='Eich11'>Eichin, K.:   Nachrichtensysteme – Kommunikationssysteme (LB).   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2011.</ref>;   '''[Söd10]'''<ref name='Söd10'>Söder, G.: Mobilfunkkanal.   Instructions for the practical course "Simulation digitaler Übertragungssysteme".   LNT/TUM, 2010. </ref>;   '''[Vie17]'''<ref name='Vie17'>Viering, I.:   System Aspects in Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2018</ref>;   '''[Kra18]'''<ref name='Kra18'>Kramer, G.:   Mobile Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2018.</ref>

*Authors:  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]  (responsible for the German version),  [https://www.lntwww.de/Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29 '''Klaus Eichin'''],  and  [https://www.lntwww.de/Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Thomas_Hindelang_.28at_LNT_from_1994-2000_und_2007-2012.29 '''Thomas Hindelang'''].  

* Participating colleagues in alphabetic order:  [https://www.lntwww.de/Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29 '''Benedikt Leible'''],   [https://www.lntwww.de/Biografien_und_Bibliografien/Externe_Beteiligte_am_LNTwww#Dr.-Ing._Markus_Mummert '''Markus Mummert'''],  [https://www.lntwww.de/Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Markus_Stinner_.28at_LNT_from_2011-2016.29 '''Markus Stinner'''],   [https://www.lntwww.de/Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29 '''Johannes Zangl''']

*Participating students in chronological order:      Martin Winkler '''(2001)''', Yven Winter, Franz Kohl, Bettina Hirner, Ji Li, Thorsten Kalweit, Slim Lamine, Johannes Schmidt, Hedi Abbes, Thomas Großer, Néjib Kchouk, Khaled Soussi, Alexander Happach, Felix Kristl, Martin Völkl, David Ginthör, Hussain Sandhu, Mohamed Ben Ahmed, Mohamed Nabil Babai, Marwen Ben Ammar, Wael Chaouch, Safwen Dridi, Mohamed Mansoor, Ayush Patel, Lukas Wolf, David Jobst, Jimmy He, Xiaohan Liu, Matthias Niller, Veronika Hofmann, André Schulz, Andrés Rosa Aparicio, Fatih Onur Özdemir, Noah Nagi, Carolin Mirschina '''(2021)'''

$\text{List of sources:}$

LNTwww:General notes about "Channel Coding"

2025-02-03T14:06:30Z

Höfler: Text replacement - "'''Benedikt Leible'''" to " $\text{Benedikt Leible}$"

'''Four main chapters with a total of 22 chapters (files);   98 Exercises   ⇒   Three semester hours per week  lecture and two hours per week  exercises      
Development of the German version:   2011–2017.       Development of the English version:   2022.      Last corrections:   December 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials:   Lecture notes from LNT/LÜT:    '''[Köt08]'''<ref name='Köt08'>Kötter, R.; Mayer, T.; Tüchler, M.; Schreckenbach, F.; Brauchle, J.: Channel Coding. Lecture notes, Lehrstuhl für Nachrichtentechnik, TU München, 2008</ref>;   '''[Liv10]'''<ref name='Liv10'>Liva, G.: Channel Coding. Lecture notes, Lehrstuhl für Nachrichtentechnik, TU München and DLR Oberpfaffenhofen, 2010</ref>;   '''[Liv15]'''<ref name='Liv15'>Liva, G.: Channels Codes for Iterative Decoding. Lecture notes, Lehrstuhl für Nachrichtentechnik, TU München and DLR Oberpfaffenhofen, 2015</ref>   –   Textbooks:   '''[Bos99]'''<ref name='Bos99'>Bossert, M.: Channel Coding for Telecommunications. Chichester: Wiley, 1999. ISBN 978-0-471-98277-7 </ref>;  '''[Cov06]'''<ref name='Cov06'>Cover, T. M.; Thomas, J. A.: Elements of Information Theory. West Sussex: John Wiley & Sons, 2nd Edition, 2006 </ref>;   '''[Hub82]'''<ref name='Hub82'>Huber, J.: Codierung für gedächtnisbehaftete Kanäle. Dissertation – Universität der Bundeswehr München, 1982</ref>

*Authors and involved colleagues in alphabetic order:   [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Ronald_B.C3.B6hnke_.28at_LNT_from_2012-2014.29|'''Ronald Böhnke''']],   [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Joschi_Brauchle_.28at_LNT_from_2007-2015.29|'''Joschi Brauchle''']],   [http://wirelesscoding.org/ '''Gianluigi Liva'''],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Participating students in chronologic order:    Martin Winkler '''(2001)''',  Yven Winter,  Ji Li,  Bettina Hirner,  Thomas Großer,  Thorsten Bürgstein,  Martin Völkl,  Dominik Kopp,  Carolin Mirschina,  Jiwoo Hwang,  Noah Nagy '''(2022)'''

$\text{References:}$

LNTwww:Imprint for the book "Examples of Communication Systems"

2025-02-03T14:06:21Z

Höfler: Text replacement - "'''Benedikt Leible'''" to " $\text{Benedikt Leible}$"

'''Four main chapters with a total of 17 chapters (files) and 164 sections (pages);   38 Exercises   ⇒   Scope:  "$\rm 3L+1E$"      
Development of the German version:   2002–2017.       Development of the English version:   2022/23.      Last corrections:   March 2023

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Hin08]'''<ref name='Hin08'>Hindelang, T.:  Mobile Communications. Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2008.</ref>;  '''[Söd10]'''<ref name='Söd10'>Söder, G.: Mobilfunkkanal.   Instructions for the practical course "Simulation digitaler Übertragungssysteme".   LNT/TUM, 2010. </ref>;   '''[Eich11]'''<ref name='Eich11'>Eichin, K.: Nachrichtensysteme – Kommunikationssysteme (LB).  Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2018.</ref>;  '''[Vie17]'''<ref name='Vie17'>Viering, I.:  System Aspects in Communications. Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2017.</ref>; '''[Kra18]'''<ref name='Kra18'>Kramer, G.:   Mobile Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2018.</ref>  –   Textbook:   '''[EVB01]'''<ref name='EVB01'>Eberspächer, J.; Vögel, H.J.; Bettstetter, C.: Global System for Mobile Communication. 3. Auflage. Stuttgart: Teubner, 2001.</ref>

*Authors in alphabetic order:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29| '''Klaus Eichin''']],  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Thomas_Hindelang_.28at_LNT_from_1994-2000_und_2007-2012.29|'''Thomas Hindelang''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']] 

*Participating students in chronological order:      Martin Winkler  '''(2001)''',  Yven Winter,  Franz Kohl,  Thorsten Kalweit,  Thomas Großer,  Franz-Josef Kaupert,  Hichem Kallel,  Khaled Soussi,  Johannes Schmidt,  Sebastian Seitz,  Alexander Happach,  Stefan Müller,  Noah Nagi,  Carolin Mirschina,  Jiwoo Hwang  '''(2022)'''

$\text{References:}$

LNTwww:About LNTwww

2025-02-03T13:58:31Z

Höfler: Text replacement - " $\text{Benedikt Leible}$" to " $\text{Benedikt Leible}$"

==Welcome to the English version of LNTwww==
 
»$\text{https://en.lntwww.de}$«  is an e-learning tutorial for Communications Engineering with nine didactic multimedia textbooks including exercises with solutions,  learning videos,  and interactive applets.  It is offered by the  »[https://www.ce.cit.tum.de/en/lnt/home/ Institute for Communications Engineering]«  of the  »[https://www.tum.de/en/ Technical University of Munich]«. 

:⇒   '''It is freely accessible,  registration is not necessary and no system requirements are needed'''.

The German-language version   »$\text{https://www.lntwww.de}$«   ⇒   »$\rm L$erntutorial für $\rm N$achrichten$\rm T$echnik im $\rm w$orld $\rm w$ide $\rm w$eb«  was created between 2001 – 2021 by members of our Institute.  The toolbar entry  »Deutsch«  takes you to the German original.  In spring 2020 we started the English translation,  and in spring 2023 we finished.

*The current version from 2023 is based on the software  [https://en.wikipedia.org/wiki/MediaWiki »MediaWiki«],  known by the encyclopaedia  »WIKIPEDIA«.   The following is a kind of  »user guide«  to our e–learning project.  Corresponding links to this file  »About LNTwww«  can be found at the bottom of each page between  »Privacy policy« and  »Disclaimer«.

*We consider the present version as final;  an extension is currently not planned.  But of course we will continue to improve detected errors or inaccuracies promptly.  So if you notice any inadequacies regarding content,  presentation or handling,  then please send a detailed message by mail to  »LNTwww@ice.cit.tum.de«.

*On the  [[LNTwww:Information|»Information«]]  page you will find notes about temporary restrictions  $($e.g. in case of unavailability due to service work$)$  and a brief summary of all »LNTwww« features using a few example pages.

We would be pleased if we could arouse your interest in our e-learning project.  We wish you a good learning success.

$\text{Have fun and good luck!}$  

[https://www.ce.cit.tum.de/en/lnt/people/professors/kramer/ $\text{Gerhard Kramer}$'''],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29| $\text{Tasnád Kernetzky}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |$\text{Günter Söder}$]]

Munich,  in spring 2024          

 

==Content==

===(A)   The didactic concept of LNTwww===

At the beginning of the work on  »$\rm LNTwww$«  in 2001,  we gave ourselves the following ten rules.  These still apply today:

'''(1)'''   The teaching area  »Information and Communication Technology«  $\text{(I&K)}$  including associated basic subjects  $($Signal Representation,  Fourier and Laplace Transform,  Stochastic Signal Theory, etc.$)$  is presented in a didactically and multimedia prepared form.

'''(2)'''   Nine subject areas were selected,  each of which is covered by a self-contained book in the scope of a one-semester course with three semester hours per week to five semester hours per week.

'''(3)'''   The target group of our online offer are students of  $\text{I&K}$  technology,  especially of »Communications Engineering«,  as well as practicing engineers  $($Keywords:  »professional training«,  »lifelong learning«$)$.

'''(4)'''   In particular,  the interrelationships between different subfields of our extensive e-leatning offer should also be shown,  which is promoted by a nomenclature that is largely consistent in all books.

'''(5)'''   »$\rm LNTwww$«  offers two modes of learning:   Beginners should proceed sequentially  –   for advanced learners,  use it as a tutorial  $($work through exercises first,  jump to the theory part if deficits are identified$)$.

'''(6)'''   The theory is explained as in a traditional engineering textbook through texts,  graphics,  and mathematical derivations.  In addition,  each chapter includes at least one multimedia module.

'''(7)'''   »$\rm LNTwww$«  shall provide the user with multiple interaction options regarding the selection and presentation of theory chapters,  exercises,  learning videos as well as multimedia and calculation modules.

'''(8)'''   The methodology of hyperlinks typical of the  »world wide web«  is extensively used within
the  »$\rm LNTwww$«  and externally.  This is also intended to show connections between different teaching areas.

'''(9)'''   In order to prevent a user from getting lost in his learning environment and using  »$\rm LNTwww$«  only for  »surfing«,  a purposeful path must be recognizable for him at all times despite certain freedoms.

'''(10)'''  For reasons of sustainability of learning success,  there are possibilities for printing the texts and graphics,  ignoring the fact that today's students generation often devalues this as a  »relapse into the analog age«.
 

===(B)   Content and scope of LNTwww===

»$\rm LNTwww$«  is a virtual course totaling  $\text{36}$  semester hours per week 
*with  $\text{23}$  semester hours per week  (quasi) lectures   ⇒   $\text{23L}$

*and  $\text{13}$  semester hours per week exercises   ⇒   $\text{13E}$. 

It is organized in book form.  Each book contains a one-semester course.  For example,  in the case of the third book,  it is indicated that the book  »Theory of Stochastic Signals«  corresponds to a face-to-face–course with three semester hours per week of  »lecture«  and two semester hours per week of  »exercises«   ⇒   $\text{3L +2E}$.

$\text{Textbooks:}$
# [[Signal_Representation|'''Signal Representation''']]   ⇒   [[LNTwww:General_notes_about_"Signal_Representation"|»Impressum«]],
# [[Linear_and_Time_Invariant_Systems|'''Linear and Time Invariant Systems''']]   ⇒  [[LNTwww:General_notes_about_"Linear_and_Time_Invariant_Systems"|»Impressum«]],
# [[Theory_of_Stochastic_Signals|'''Theory of Stochastic Signals''']]   ⇒  [[LNTwww:General_Notes_about_the_Book_"Stochastic_Signal_Theory"|»Impressum«]],
# [[Information_Theory|'''Information Theory''']]   ⇒  [[LNTwww:General_notes_about_"Information_Theory"|»Impressum«]],
# [[Modulation_Methods|'''Modulation Methods''']]   ⇒  [[LNTwww:General_notes_about_"Modulation_Methods"|»Impressum«]],
# [[Digital_Signal_Transmission|'''Digital Signal Transmission''']]   ⇒  [[LNTwww:General_notes_about_"Digital_Signal_Transmission"|»Impressum«]],
# [[Mobile_Communications|'''Mobile Communications''']]   ⇒  [[LNTwww:General_notes_about_"Mobile_Communications"|»Impressum«]],
# [[Channel_Coding|'''Channel Coding''']]   ⇒  [[LNTwww:General_notes_about_"Channel_Coding"|»Impressum«]],
#[[Examples_of_Communication_Systems|'''Examples of Communication Systems''']]   ⇒  [[LNTwww:General_notes_about_"Examples_of_Communication_Systems"|»Impressum«]].

*The theory pages of all books result in the print version in approx.  $1500$  pages  $($DIN A4$)$  and contain on average one and a half graphics per page. 

*In addition,  LNTwww provides via the link  [[Biographies_and_Bibliographies|»'''Biographies & Bibliography'''«]]  a subject-specific bibliography with approx.  $400$  entries,  plus links to the WIKIPEDIA biographies of important scientists.
 

===(C)   Design and structure of LNTwww===

One can reach the nine reference books and »Biographies & Bibliography«  through the link  [[Book Overview|»'''Book Overview'''«]].  From this interface one can reach the individual books.  
*Each book is divided into several  »'''main chapters'''«, 

*each main chapter into several  »'''chapters'''«,  and

*each chapter includes several  »'''sections'''«.

{{GraueBox|TEXT=
$\text{Example 1:}$ 
We consider the book  [[Signal Representation|»Signal Representation«]].  This contains five  »main chapters«.
*By clicking on the first main chapter  »Basic Terms of Communications Engineering«,  one can get to three  »chapters«.  Each chapter corresponds to a MediaWiki file.

*The exemplary chapter  [[Signal_Representation/Principles_of_Communication|»Principles of Communication«]]  contains ten  »sections»  or  »pages».

*The last two pages are almost the same in all chapters,  namely  »Exercises for the chapter«  and  »References«.}}
 

===(D)   Content overviews for LNTwww===

A brief overview of all books is available on the selection interface  [[Book Overview|»'''Book Overview'''«]].
*More information is provided by the  »first page«  of each book.

*The respective main chapter content can be found in the first subchapter on the first page of each.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The first page  $($title page$)$  of the book  [[Signal_Representation|»Signal Representation«]]  provides the following information:
# A brief summary of the entire book;
# Scope of learning offer:  $2{\rm L} + 1{\rm E}$   ⇒   lecture with two semester hours per week and one additionalhour exercise. 
# Five main chapters,  19 chapters,  127 sections,  58 exercises;
# Links to the five main chapters of the book;
# Links to the associated exercises,  learning videos,  and interactive applets in the book  »Signal Representation«;
# Bibliography for the book;
# The imprint to the book  $($Authors,  other contributors,  materials as a starting point of the book,  referencces$)$.

The content of the first main chapter  »Principles of Communication« can be found on the first page 
[[Signal_Representation/Principles_of_Communication#OVERVIEW_OF_THE_FIRST_MAIN_CHAPTER|»# OVERVIEW OF THE FIRST MAIN CHAPTER #«
]].}}
 
===(E)   LNTwww exercises===

A central role in our didactic concept play »exercises«. We believe that the sensible use of »LNTwww« by a user with previous knowledge should be that he first work on the exercises relating to his actual learning area and only jump to the corresponding theory section when required.

You can find the  »'''exercise overview'''«  for all books  $($approx.  $640$  exercises, approx.  $3100$  subtasks)  on the home page via the link  [[Aufgaben:Aufgabensammlung|»'''Exercises'''«]].  All exercises are structured in the same way:
*Each exercise consists of the »exercise description« and several  »subtasks«.   An exercise is only solved correctly if all subtasks are correct.

* For each exercise there is a detailed  »sample solution«,  sometimes with the indication of several ways to the goal.

* The »exercise types« used are:
# »Single Choice»   ⇒   only one of the  $n$  given answers is correct;      ⇒   Marks of alternative answers:  ${\huge\circ}$
# »Multiple Choice«   ⇒   of the  $n$  given answers, between zero and  $n$  answers can be correct;      ⇒   Marks of alternative answers:  $\square$
# »Arithmetic Task«   ⇒   numerical value query,  possibly with sign;     small deviations  $($usually  $\pm 3\%)$  are allowed when checking real-valued results.

* We distinguish between  »exercises«  $($e.g.  »Exercise 1.1»$)$  and  »additional exercises«  $($e.g.  »Exercise 1.1Z«$)$.
# If you were able to solve all exercises of a chapter without any problems,  we believe that you are familiar with the content of the entire chapter. 
#If you have solved one exercise incorrectly,  you should also work on the following,  usually somewhat easier additional exercise.

{{GraueBox|TEXT=
$\text{Example 3:}$ 
The  $58$  exercises/additional exercises of the first book can be accessed via the link  [https://en.lntwww.de/Category:Signal_Representation:_Exercises »Signal Representation: Exercises«]. 

*From there,  we move on to the individual exercises,  e.g. to  [https://en.lntwww.de/Aufgaben:Exercise_1.1:_Music_Signals »Exercise 1.1: Music Signals«].  This relatively simple exercise consists of
#  one »Single Choice«   ⇒   subtask  '''(1)''',
#  two »Multiple Choice«   ⇒   subtasks  '''(2)''',  '''(3)''',  and
#  one »Arithmetic Task« with two real-valued computational queries   ⇒   subtask  '''(4)'''.

*However,  most of our exercises are not that easy.  Although MediaWiki also calls an arithmetic task  »quiz«,  answering them is usually much more difficult than in the numerous quiz shows on TV.   Because: 
#  There are no predetermined answers in an arithmetic task,  and moreover:
#  Integrals often have to be solved beforehand,  such as in  [[Aufgaben:Exercise_4.4:_Two-dimensional_Gaussian_PDF|»Exercise 4.4: Two-dimensional Gaussian probabilty density function«.]]

*We recommend:  First print the exercise   ⇒   »printable version«  and solve the exercise  offline  before checking  online.
}}

 

 
===(F)   LNTwww learning videos===

You can access approximately  $30$  learning videos via the link  »Videos«  on the start page.  The realization of a learning video required the following individual steps: 
:Writing the script and texts   ⇒   Creating a set of slides with only slight differences between successive slides   ⇒   Voicing texts and audio editing   ⇒   Combining texts and images into a coherent video stream.
#Clicking on this link brings up a; list of all learning videos,  grouped by textbook.  Some videos appear for multiple books.
#After selecting the desired learning video,  a wiki description page appears with a short content description and user interface.
#From here you can start the video in  »mp4«  and  »ogv«  format.  The browser will search for the appropriate format.
#The videos can be played by many browsers  $($Firefox, Chrome, Safari, ...$)$  as well as smartphones and tablets.
#The bottom link provides all available learning videos in alphabetical order.

Note:   All learning videos are with German language.  English translations are not planned.

{{GraueBox|TEXT=
$\text{Example 4:}$ 
We'll take a look at   [[Analoge_und_digitale_Signale_(Lernvideo)|»Analog and digital signals«]]  as an example.  This provides a two-part video in mp4 and ogv format.
*Each video part can be started by single click and paused by another click.

*The playback speed of the videos can be changed:
** Firefox offers a submenu after right-clicking on the video.
** For Google Chrome you can install e.g. the plugin  »Video Speed Controller«.
}}
 

===(G)   LNTwww applets===

Applets have a similar function as laboratories in mathematical-scientific courses:  Supplementing lecture/exercise with independent work by the student on the topic covered. 

You can access the provided interactive applets via the link of the same name on the home page.  It should be noted:
#Clicking on the link  »'''Applets'''«  a list of all applets appears,  grouped by reference books. 
#We distinguish between the newer  $\text{HTML 5/JavaScript}$  applets  $($in the respective lists above$)$  and the older  $\text{SWF}$  applets  $($below$)$. 
#The SWF applets unfortunately do not work on smartphones and tablets.
#After selecting an HTML 5/JS applet  a wiki description page appears with introductory theory section,  exercises to be solved and sample solutions. 
#At the beginning and end of this wiki description page there are links to the actual applet in German resp. English Language.

{{GraueBox|TEXT=
$\text{Example 5:}$ 
The didactic importance of applets shall be proved by  [[Applets:Eye_Pattern_and_Worst-Case_Error_Probability|»Eye Pattern and Worst-Case Error Probability«]]:
*The  »eye diagram«  is a proven transmission engineering tool,  to capture the influence of  »line dispersion«   ⇒   »intersysmbol interference«  on the quality characteristic  »error probability«  of a digital transmission system.  

*Such applets serve the clarification of more difficult facts,  in this example  »the step-by-step construction of the eye diagram from the symbol sequence«.  The program offers a lot of setting possibilities.  However, not every setting brings the user a relevant learning success and even fewer lead to a so-called "Aha! moment". 

*This is why we guide the user specifically through the program on the basis of the experiment.  He has to solve various tasks:  Predict and evaluate results,  Optimize parameters,  etc.

*A top 10% student has of course the possibility,  to set himself tasks going beyond the execution of experiments with the help of the applet and thus to penetrate very deeply into the presented subject matter.

}}

In addition to these  $\approx\hspace{-0.1cm} 30$  HTML 5/JS based applets  we still offer some of our  $\approx\hspace{-0.1cm}50$  older German-language applets,  which are based on  »Shock Wave Flash«  $\rm (SWF)$.  These were programmed for  »Adobe Flash«. 
#Since the Flashplayer browser plugin is no longer supported for security reasons,  these applets must be opened with the  »projector version«.
#You do not need to install the projector version and it will not be integrated into your browser.  So there are no security concerns in this regard.
#On the corresponding wiki pages you can find the projector version of the flash player and of course the applet itself.
 

===(H)   Glossary===

Due to the fact,  that our e–learning project LNTwww was first conceived in German and the wish for an English version came much later,  in the English version the assignment between  »Formula signs«   and  »Designation«  is not quite easy.   What do for example

#  $f_{\rm T}$,
#  $s_{\rm TP}(t)$, 
#  $e$,
#  $E$?

Here the link  [[LNTwww:Glossary|»Glossary«]]  on the home page below can help with the following alphabetically ordered entries:
::  »Formula sign«   ⇒   »German name«   ⇒   »English name« 

The file is self-explanatory. A few explanations are given under the last menu item  »Some remarks to the Glossary«.

{{GraueBox|TEXT=
$\text{Example 6:}$  In this file you will find the following entries:
:  $f_{\rm T}$   ⇒   Trägerfrequenz   ⇒   carrier frequency 
:  $s_{\rm TP}(t)$   ⇒   äquivalentes Tiefpass–Sendesignal   ⇒   equivalent low-pass transmitted signal 
:  $e= 2.718281828456$...   ⇒   Eulersche Zahl   ⇒   Eulerian number 
:  $ E$   ⇒   $(1)$ Schwellenwert,  $(2)$ Energie   ⇒   $(1)$ threshold value,  $(2)$ energy 

From the context,  the decision for  $(1)$  or  $(2)$  should be easy. }}

 

===(I)    History of LNTwww===

At the  [https://www.ce.cit.tum.de/en/lnt/home/ »Institute for Communications Engineering«]  $\rm (LNT)$  of the  [https://www.tum.de »Technical University of Munich«]  $\rm (TUM)$  two  teaching software packages  $\text{(LNTsim, LNTwin)}$  were realized by  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|»Günter Söder«]] from 1984 to 1996, which were used in our practical courses.  Several other universities have also acquired and used these programs.

At the beginning of the first Internet euphoria,  there were inquiries from students whether we could also provide such simulation and demonstration programs online.  After careful consideration  ("Is the expected big effort worth it?")  Günter Söder began 2001 planning the German-language project  »www.LNTwww.de«.  Co-responsible was his colleague  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|»Klaus Eichin«]].  The project was to be completed by 2011 at the latest,  since both would be retiring this year.

The content was derived from his own teaching materials as well as those of his colleague  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|»Norbert Hanik«]]  $($Associate Professor of Line Transmission Technology$)$.  Other lecture material was also taken into account,  which was produced at the Institute of Communications Engineering under the last four chair holders:
::*Professor [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Hans_Marko_.281962-1993.29|»Hans Marko«]]  $($Head of the LNT from 1962 to 1993$)$,
::*Professor [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|»Joachim Hagenauer«]]  $($Head of the LNT from 1993 to 2006$)$,
::*Professor [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|»Ralf Kötter»]]  $($Head of the LNT from 2007 to 2009$)$,  and
::*Professor [https://www.ce.cit.tum.de/en/lnt/people/professors/kramer/ »Gerhard Kramer«]  $($Head of the LNT since 2010$)$.

Just a few dates about progress of the German-language LNTwww project,  eleven years after the planned completion :
* First of all our own platform had to be developed by students  $($Marin Winkler,  Yven Winter$)$.  The authoring system  »LNTwww«  was based on the http server  »Apache«,  the database  »MySQL«,  the script language  »Perl«  and  »Shock Wave Flash«  $\rm (SWF)$  as a basis for multimedia applications   ⇒   version  »LNTwww.v1«  $($2003$)$.

*Work of the following years was online adaptation of the manuscripts,  input into the database with the rather complicated LNTwww syntax,  creation of the graphs as well as conception and realization of multimedia elements.  After completion of all nine textbooks the desired final state was reached   ⇒   version  »LNTwww.v2«  $($2016$)$.

* At the same time,  it became known that  »SWF«  would not longer be supported by relevant manufacturers.  This fact and the criticism heard from some users about the meanwhile too staid design  $($our authoring system was on the level of 2003$)$  were decisive for a new start based on  »MediaWiki«   ⇒   version  »LNTwww.v3«  $($2021$)$.

Finally,  a few sentences about the English LNTwww version. 
*At the beginning of the Corona pandemic and the associated lockdowns,  the call for  »e-learning«  also became louder and louder at the universities,  even from professors who had previously rather rejected this form of teaching. 

* Suddenly,  funds were also made available,  to provide as many e-learning courses as possible in as short a time as possible.  Our chair  »Gerhard Kramer«  therefore already submitted a corresponding application for working student funds in spring 2020,  which was approved within a few weeks.

*In June 2020 we started the English translation with support of the  »DEEPL«  program  $($free version$)$  and finished it in April 2023.  From the LNT staff were involved:  »Javier Garcia Gomez, Tasnád Kernetzky, Benedikt Leible and Günter Söder«.  Of the students involved,  »Noah Nagi«  and  »Jiwoo Hwang«  deserve special mention.
 

===(J)   Acknowledgement===

The Institute for Communications Engineering would like to thank the many people involved in the creation of  $\rm LNTwww$:

*The persons responsible for the German and/or English LNTwww projects:
::[[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |$\text{Günter Söder}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|$\text{Klaus Eichin}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29| $\text{Tasnád Kernetzky}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]].

* The  $($former$)$  LNT/LÜT colleagues,  who contributed as co–authors or experts or supervised student work:
::[[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Ronald_B.C3.B6hnke_.28at_LNT_from_2012-2014.29|$\text{Ronald Böhnke}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Joschi_Brauchle_.28at_LNT_from_2007-2015.29|$\text{Joschi Brauchle}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|$\text{Bernhard Göbel}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|$\text{Norbert Hanik}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Thomas_Hindelang_.28at_LNT_from_1994-2000_und_2007-2012.29|$\text{Thomas Hindelang}$]],   [[Biographies_and_Bibliographies/External_Contributors_to_LNTwww#Dr._Gianluigi_Liva|$\text{Gianluigi Liva}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Tobias_Lutz_.28at_LNT_from_2008-2014.29|$\text{Tobias Lutz}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Michael_Mecking_.28at_LNT_from_1997-2012.29|$\text{Michael Mecking}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Markus_Stinner_.28at_LNT_from_2011-2016.29|$\text{Markus Stinner}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Thomas_Stockhammer_.28at_LNT_from_1995-2004.29|$\text{Thomas Stockhammer}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|$\text{Johannes Zangl}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Georg_Zeitler_.28at_LNT_from_2007-2012.29|$\text{Georg Zeitler}$]].

*The more than  »50 students«,  who have worked on subareas,  designed learning videos and applets or implemented the porting to the MediaWiki version within the framework of Engineering practice,  diploma,  bachelor and master theses or within the framework of a working student activity.

*The  [https://www.https://www.ei.tum.de/en/welcome/ »Department of Electrical and Computer Engineering«]  and the  [https://www.tum.de/en/ »Technical University of Munich«]  for funding working students in the years since 2016 within the framework of the  [https://www.ei.tum.de/studium/studienzuschuesse/ »MoliTUM«]  resp.  [https://www.tum.de/en/studies/teaching/awards-and-competitions/ideas-competition »EXIni«]  programs.

        [https://www.ce.cit.tum.de/en/lnt/people/professors/kramer/ $\text{Gerhard Kramer}$]

Biographies and Bibliographies/LNTwww members from LÜT

2025-02-03T13:53:01Z

Höfler:

{{Header|
Untermenü=Beteiligte der Professur Leitungsgebundene Übertragungstechnik
|Vorherige Seite= An LNTwww beteiligte Mitarbeiter und Dozenten|
Nächste Seite= An LNTwww beteiligte Studierende
}}

== The Professorship »Line Transmission Technology«==

The subject area  »'''Line Transmission Technology'''«  $\rm (LÜT)$  was established in 2004,  when the former LNT doctoral student  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|»Norbert Hanik«]]  returned to TU Munich and was appointed as its head.  In 2014,  this field became the  »Associate Professorship of Line Transmission Technology«.  More information on the  [https://www.ce.cit.tum.de/en/lnt/research/associate-professorship-of-line-transmission-technology/ »'''LÜT homepage'''«].

==Prof. Dr.-Ing. Norbert Hanik (at LNT from 1989-1995, at LÜT since 2004)==

[[File:n_hanik.jpg|165px|right|Norbert Hanik]]

Norbert Hanik was born in 1962 in the Bavarian town of Wemding in the Donau-Ries region and studied at the Faculty of Electrical Engineering and Information Technology at the Technical University of Munich from 1983 onwards, specializing in communications engineering.  In 1995, he received his doctorate from Prof. [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Hans_Marko_.281962-1993.29|Hans Marko]] at the LNT on »Nonlinear effects in optical signal transmission«.  He then worked at the Technology Center of Deutsche Telekom AG in the field of optical transmission technology,  since 1999 as head of the research group »System Concepts of Photonic Networks«.  In 2002, he was a visiting professor at the COM Research Center of the Technical University of Denmark (TUD) in Copenhagen.

With effect from April 1, 2004, Norbert Hanik was appointed to the (current) professorship for »Line Transmission Technology« at the Faculty of Electrical Engineering and Information Technology at TUM.  He thus returned to his home chair after nine years in Berlin.  After the death of our chair holder Prof. Ralf Kötter,  Norbert Hanik was appointed acting head of the LNT in Spring 2009.

His research focuses on modeling, simulation and optimization of components, subsystems and transmission links of optical transmission systems and optical networks.

[https://www.ce.cit.tum.de/en/lnt/people/professors/hanik/ $\text{Biography of Norbert Hanik on the LÜT homepage}$]

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*Professor Hanik has been very supportive of the development of our learning tutorial and he has always been an extremely competent technical advisor.
*He was co-author on »Linear and Time-Invariant Systems« and on single chapters of »Digital Signal Transmission» and »Examples of Communication Systems«.
*In particular, the initiators of »$\rm LNTwww$« would like to thank Norbert  for his early and versatile use of our learning tutorial in his lectures.
}}

==Dr.-Ing. Bernhard Göbel (at LÜT from 2004-2010)==

[[File:bernhard.jpg|165px|right|Bernhard Göbel]]

Bernhard Göbel, born in Munich in 1978, finished his studies of electrical engineering and information technology at the Technical University of Munich in 2004 after semesters abroad in Southampton and Princeton with a diploma thesis on the investigation of genetic diseases using information theory.

From autumn 2004 until the end of 2010, Bernhard Göbel was an assistant to Prof.  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|Norbert Hanik]]  in the department of "Line Transmission Technology".  After a research stay at Bell Labs in New Jersey, he received his PhD in 2010 on the topic of "Information-theoretical properties of fiber-optic communication channels".  In addition to supervising courses, his other responsibilities included managing the CITPER project,  which was initiated by the European Union.

After completing his doctorate, Dr. Göbel moved to Volkswagen AG in Wolfsburg,  where he began training as a patent attorney.  In 2014, he returned to Munich and is now working for BMW AG.

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*Bernhard was always consulted by us when the authors realized that some things could be done better with "MATLAB" than without.
*Furthermore, he was an expert advisor for several tutorial videos and interaction modules, for example "Attenuation of Copper Cables", "Time Response of Copper Cables" and "Viterbi Receivers".
*We would also like to thank Bernhard for making our learning tutorial known to many students of the TU Munich as an exercise assistant for "Line Transmission Technology".}}

==Dr.-Ing. Tasnád Kernetzky (at LÜT from 2014-2022)==

[[File:Tasnad.png|165px|right|Tasnád Kernetzky]]

Tasnád Kernetzky was born in 1987 in Marosvásárhely  (today: Târgu Mureș, Romania).  He studied Electrical Engineering and Information Technology at the Technical University of Munich from 2009 and graduated in 2014 with a master thesis on the transmission characteristics of  "Powerline Communication" (PLC) systems.

Since December 2014, he has been working as a Ph.D. student with Prof. [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|Norbert Hanik]] in the professorship "Line Transmission Technology systems" – initially in cooperation with SIEMENS AG continued on the topic  "PLC".  The focus of his later work was on the simulation and optimization of the nonlinear optical process "four wave mixing" in multi-mode optical waveguides.  He completed his Ph.D. thesis with the topic  »Numerical Optimization of Ultra-Broadband Wavelength Conversion in Nonlinear Optical Waveguides«  in October 2023.

In teaching, Tasnád was responsible for the exercises for the lecture  "Fundamentals of Information Technology (LB)"  by Prof. Hanik.  Besides, he organized the  "Advanced Seminar Digital Communication Systems".

From 2016-2022, Tasnád had been involved in the system administrator of the chair computers.

{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  

For many years Tasnád was intensively involved in the LNTwww team as a system/web administrator,  and is still one of the project leaders without whom nothing works:

*In 2016, he took over as successor to  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Markus_Stinner_.28at_LNT_from_2011-2016.29|Markus Stinner]]  assisted the student team in porting the  "old LNTwww"  to the present wiki form (version 3).
*He completed the pending move of the wiki to a new server in 2018, and also the associated update–work on the wiki.
*He converted the learning videos to modern formats (mp4, ogv).  These can now be played by many browsers, but also by smartphones.
*He was supervisor and contact person for all student work on porting the interactive applets to HTML5.
*He has done essential preliminary work to be able to generate the English  »$\rm en.LNTwww.de$«  version from  »$\rm www.LNTwww.de$«  with reasonable effort.}}

==Dr.-Ing. Benedikt Leible (at LÜT from 2017-2024)==

[[File:Leible.png|165px|right|Tasnád Kernetzky]]

Benedikt Leible, born in Kempten in 1988, studied electrical engineering and information technology at the Technical University of Ulm (Bachelor) and at the Technical University of Stuttgart (Master) from 2010. He graduated in 2016 with a master's thesis on »Parallelization of Channel Decoders for 5G Communication Systems«. 

Since February 2017, he was working as a PhD student with Prof. Norbert Hanik in the professorship »Line Transmission Technology».  Furthermore, he wass responsible for the supervision of the lecture »Physical Layer Methods« and also conducts the corresponding tutorial.
He completed his Ph.D. thesis with the topic  »Fiber optic communication using nonlinear Fourier transform«  in December 2024.
 
{{BlueBox|TEXT=
'''His contributions to the LNTwww project''':  
*He was supervisor of students who programmed interactive HTML5/JS applets for the LNTwww in their Bachelor Thesis/Engineering Practice.
*From 2021, Benedikt led the conversion to the English version  »[https://en.lntwww.de/Home $\text{https://en.lntwww.de}$]«  by the student translation team. }}

{{Display}}

LNTwww:Imprint for the book "Information Theory"

2025-02-03T13:49:31Z

Höfler: Text replacement - "[[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29|" to "[[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|"

'''Four main chapters with a total of 13 chapters (files) and 106 sections (pages);   71 Exercises   ⇒   Scope:  "$\rm 2L+1E$"      
Development of the German version:   2011–2015.       Development of the English version:   2021.      Last corrections:   September 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|'''Benedikt Leible''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials:   Lecture notes from LNT/LÜT:    '''[Meck09]'''<ref name='Meck09'>Mecking, M.:  Information Theory. Lecture notes, Chair of Communications Engineering, TU München, 2009.</ref>;  '''[Liv15]'''<ref name='Liv15'>Liva, G.:  Channel Codes for Iterative Decoding. Lecture notes, Chair of Communications Engineering, TU München and DLR Oberpfaffenhofen, 2015.</ref>;  '''[Kra16]'''<ref name='Kra16'>Kramer, G.:  Information Theory. Lecture notes, Chair of Communications Engineering, TU München, 2016.</ref>;  '''[Söd14]'''<ref name='Söd14'>Söder, G.:  Wertdiskrete Informationstheorie.  Guidance  (in German)  for the experiment of the same name in the practical course "Simulation of digital transmission systems"; Lecture notes, Chair of Communications Engineering, TU München, 2014. </ref>;  Textbook:  '''[Cov06]'''<ref name='Cov06'>Cover, T. M.; Thomas, J. A.: Elements of Information Theory. West Sussex: John Wiley & Sons, 2nd Edition, 2006. </ref>

*Authors and involved colleagues in alphabetic order:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],   [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|'''Bernhard Göbel''']],   [http://wirelesscoding.org/ '''Gianluigi Liva'''],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Tobias_Lutz_.28at_LNT_from_2008-2014.29|'''Tobias Lutz''']],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Michael_Mecking_.28at_LNT_from_1997-2012.29|'''Michael Mecking''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|'''Günter Söder''']]

*Participating students in chronological order:      Martin Winkler '''(2001)''', Yven Winter, Reinhold Sixt, Jürgen Veitenhansl, Franz Kohl, Bettina Hirner, Ji Li, Markus Elsberger, Thorsten Kalweit, Thomas Großer, David Jobst, Matthias Niller, Veronika Hofmann, Carolin Mirschina, Noah Nagi, Ji Woo Hwang '''(2022)'''

$\text{References:}$

LNTwww:Imprint for the book "Channel Coding"

2025-02-03T13:49:21Z

'''Four main chapters with a total of 22 chapters (files) and 175 sections (pages);   98 Exercises   ⇒   Scope:  "$\rm 3L+2E$"      
Development of the German version:   2011–2017.       Development of the English version:   2022.      Last corrections:   October 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|'''Benedikt Leible''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:     '''[Köt08]'''<ref name='Köt08'>Kötter, R.; Mayer, T.; Tüchler, M.; Schreckenbach, F.; Brauchle, J.: Channel Coding. Lecture notes, Chair of Communications Engineering, TU München, 2008</ref>;   '''[Liv10]'''<ref name='Liv10'>Liva, G.: Channel Coding. Lecture notes, Chair of Communications Engineering, TU München and DLR Oberpfaffenhofen, 2010</ref>;   '''[Liv15]'''<ref name='Liv15'>Liva, G.: Channels Codes for Iterative Decoding. Lecture notes, Chair of Communications Engineering, TU München and DLR Oberpfaffenhofen, 2015</ref>   –   Textbooks:   '''[Bos99]'''<ref name='Bos99'>Bossert, M.: Channel Coding for Telecommunications. Chichester: Wiley, 1999. ISBN 978-0-471-98277-7 </ref>;  '''[Cov06]'''<ref name='Cov06'>Cover, T. M.; Thomas, J. A.: Elements of Information Theory. West Sussex: John Wiley & Sons, 2nd Edition, 2006 </ref>;   '''[Hub82]'''<ref name='Hub82'>Huber, J.: Codierung für gedächtnisbehaftete Kanäle. Dissertation – Universität der Bundeswehr München, 1982</ref>

*Authors and involved colleagues in alphabetic order:   [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Ronald_B.C3.B6hnke_.28at_LNT_from_2012-2014.29|'''Ronald Böhnke''']],   [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Joschi_Brauchle_.28at_LNT_from_2007-2015.29|'''Joschi Brauchle''']],   [http://wirelesscoding.org/ '''Gianluigi Liva'''],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Participating students in chronologic order:    Martin Winkler '''(2001)''',  Yven Winter,  Ji Li,  Bettina Hirner,  Thomas Großer,  Thorsten Bürgstein,  Martin Völkl,  Dominik Kopp,  Carolin Mirschina,  Jiwoo Hwang,  Noah Nagy '''(2022)'''

$\text{References:}$

LNTwww:General notes about "Information Theory"

2025-02-03T13:49:21Z

'''Four main chapters with a total of 13 chapters (files);   71 Exercises   ⇒   Two semester hours per week lecture and one semester hour per week exercises      
Development of the German version:   2011–2015.       Development of the English version:   2021.      Last corrections:   September 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|'''Benedikt Leible''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials:   Lecture notes from LNT/LÜT:    '''[Meck09]'''<ref name='Meck09'>Mecking, M.:  Information Theory. Lecture notes, Chair of Communications Engineering, TU München, 2009.</ref>;  '''[Liv15]'''<ref name='Liv15'>Liva, G.:  Channels Codes for Iterative Decoding. Lecture notes, Chair of Communications Engineering, TU München and DLR Oberpfaffenhofen, 2015.</ref>;  '''[Kra16]'''<ref name='Kra16'>Kramer, G.:  Information Theory. Lecture notes, Chair of Communications Engineering, TU München, 2016.</ref>;  '''[Söd14]'''<ref name='Söd14'>Söder, G.:  Wertdiskrete Informationstheorie.  Guidance  (in German)  for the experiment of the same name in the practical course "Simulation of digital transmission systems"; Lecture notes, Chair of Communications Engineering, TU München, 2014. </ref>;  Textbook:  '''[Cov06]'''<ref name='Cov06'>Cover, T. M.; Thomas, J. A.: Elements of Information Theory. West Sussex: John Wiley & Sons, 2nd Edition, 2006. </ref>

*Authors and involved colleagues in alphabetic order:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],   [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|'''Bernhard Göbel''']],   [http://wirelesscoding.org/ '''Gianluigi Liva'''],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Tobias_Lutz_.28at_LNT_from_2008-2014.29|'''Tobias Lutz''']],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Michael_Mecking_.28at_LNT_from_1997-2012.29|'''Michael Mecking''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|'''Günter Söder''']]

*Participating students in chronologic order:      Martin Winkler '''(2001)''', Yven Winter, Thomas Großer, Stefan Müller, Martin Völkl, Eugen Mehlmann, Alexander Laible, Veronika Hofmann, Noah Nagi, Carolin Mirschina '''(2021)'''

$\text{List of sources:}$

LNTwww:Imprint for the book "Signal Representation"

2025-02-03T13:49:21Z

'''Five main chapters with a total of 19 chapters (files) and 127 sections (pages);   58 Exercises   ⇒   Scope:  "$\rm 2L+1E$"      
Development of the German version:   2002–2017.       Development of the English version:   2020/21.      Last corrections:   December 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| $\text{Gerhard Kramer}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|$\text{Tasnád Kernetzky}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|$\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |$\text{Günter Söder}$]]

*Basic materials for the original German version:     Lecture notes of LNT/LÜT:  '''[Eich03]'''<ref name='Eich03'>Eichin, K.:  Nachrichtentechnik I (LB) – Signaldarstellung.  Lecture notes, Chair of Communications Engineering, TU München, 2003. </ref>;  '''[Han15]'''<ref name='Han15'>Hanik, N.:  Nachrichtentechnik 1 (LB): Signaldarstellung.  Lecture notes, Associate Professorship of Line Transmission Technology, TU München, 2015.</ref>;  '''[Söd12]'''<ref name='Söd12'>Söder, G.:   Simulationsmethoden in der Nachrichtentechnik.  Internship notes, Chair of Communications Engineering, TU München, 2012.</ref>   –  Textbooks:   '''[Söd93]'''<ref name='Söd93'>Söder, G.:  Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin, Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2.</ref>;  '''[Mar94]'''<ref name='Mar94'>Marko, H.:  Methoden der Systemtheorie.  3. Auflage. Berlin – Heidelberg: Springer, 1994.</ref>;  '''[HM09]'''<ref name='HM09'>Haykin, S.; Moher, M.: Communication Systems. 5th edition, Hoboken, N.J.: J. Wiley, 2009. ISBN 978-0-47169-790-9.</ref>

*Authors:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |$\text{Günter Söder}$]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29| $\text{Klaus Eichin}$]]

* Participating colleagues in alphabetic order:   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]]  and  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|$\text{Norbert Hanik}$]]

*Participating students in chronological order:      Martin Winkler  '''(2001)''',  Yven Winter,  Franz Kohl,  Roland Kiefl,  Thorsten Kalweit,  Markus Elsberger,  Bettina Hirner,  Ji Li,  Slim Lamine,  Thomas Großer,  Jimmy He,  Xiaohan Liu,  Noah Nagi,  Carolin Mirschina,  Jiwoo Hwang  '''(2022)'''

$\text{References:}$

LNTwww:Imprint for the book "Modulation Methods"

2025-02-03T13:49:21Z

'''Five main chapters with a total of 23 chapters (files) and 192 sections (pages);   89 Exercises   ⇒     ⇒   Scope:  "$\rm 3L+2E$"      
Development of the German version:   2005–2011.       Development of the English version:   2020/21.      Last corrections:   October 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|'''Benedikt Leible''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Söd14]'''<ref name='Söd14'>Söder, G.:  Analoge & Digitale Modulationsverfahren. Praktikum &bdquo;Simulation digitaler Übertragungssysteme”. Lehrstuhl für Nachrichtentechnik, TU München, 2014. </ref>;  '''[Han16]'''<ref name='Han16'>Hanik, N.:  Nachrichtentechnik 2 (LB): Modulationsverfahren. Vorlesungsmanuskript. Professur Leitungsgebundene Übertragungstechnik, TU München, 2016.</ref>;  '''[Vie16]'''<ref name='Vie16'>Viering, I.: System Aspects in Communications. Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2016.</ref>;   '''[Kra17]'''<ref name='Kra17'>Kramer, G.: Nachrichtentechnik 2. Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2017.</ref>.   –   Textbooks:   '''[ZP85]'''<ref name='ZP85'>Ziemer, R. E.; Peterson, R. L.: Digital Communications and Spread Spectrum Systems. New York: Macmillan, 1985. ISBN 978-0-02431-670-7</ref>;  '''[Kam04]'''<ref name="Kam04">Kammeyer, K.D.:  Nachrichtenübertragung. Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>

*Authors:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|'''Klaus Eichin''']],

* Participating colleagues in alphabetical order:   [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|'''Benedikt Leible''']],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|'''Johannes Zangl''']],

*Participating students in chronological order:      Martin Winkler '''(2001)''',  Yven Winter,  Ji Li ,  Franz Kohl,  Bettina Hirner,  Thorsten Kalweit,  Markus Elsberger,  Slim Lamine,  Thomas Großer,  Johannes Schmidt,  David Jobst,  Xiaohan Liu,  Matthias Riedel,  Carolin Mirschina,  Sam Reed,  JiWoo Hwang '''(2022)'''

$\text{References:}$

LNTwww:Imprint for the book "Digital Signal Transmission"

2025-02-03T13:49:21Z

'''Five main chapters with a total of 26 chapters (files) and 201 sections (pages);   90 Exercises   ⇒   Scope:  "$\rm 3L+2E$"      
Development of the German version:   2007–2016.       Development of the English version:   2020/21.      Last corrections:   October 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|'''Benedikt Leible''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:   '''[Söd11]'''<ref name='Söd11'>Söder, G.:  Simulation digitaler Übertragungssysteme. Internship notes, Chair of Communications Engineering, TU München, 2011. </ref>;  '''[Söd12]'''<ref name='Söd12'>Söder, G.:  Simulationsmethoden in der Nachrichtentechnik. Internship notes, Chair of Communications Engineering, TU München, 2012.</ref>;   '''[Han17]'''<ref name='Han17'>Hanik, N.:  Leitungsgebundene Übertragungstechnik. Lecture notes, Associate Professorship of Line Transmission Technology, TU München, 2017.</ref>;  '''[Kra17]'''<ref name='Kra17'>Kramer, G.: Nachrichtentechnik 2. Lecture notes, Chair of Communications Engineering, TU München, 2017.</ref>.   –   Textbooks:   '''[ST85]'''<ref name='ST85'> Söder, G.; Tröndle, K.: Digitale Übertragungssysteme - Theorie, Optimierung & Dimensionierung der Basisbandsysteme. Berlin – Heidelberg: Springer, 1985. </ref>;  '''[TS87]'''<ref name='TS87'> Tröndle, K.; Söder, G.: Optimization of Digital Transmission Systems. Boston – London: Artech House, 1987, ISBN: 0-89006-225-0. </ref>;  '''[Söd93]'''<ref name='Söd93'>Söder, G.: Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin, Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2</ref>;  '''[Kam04]'''<ref name="Kam04">Kammeyer, K.D.:  Nachrichtenübertragung. Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>.

*Author:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

* Participating colleagues in alphabetical order:    [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|'''Klaus Eichin''']],   [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|'''Bernhard Göbel''']],  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']], [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|'''Benedikt Leible''']],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|'''Johannes Zangl''']],

*Participating students in chronological order:      Martin Winkler '''(2001)''',  Yven Winter,  Ji Li ,  Franz Kohl,  Bettina Hirner,  Thorsten Kalweit,  Markus Elsberger,  Thomas Großer,  Johannes Schmidt,  Martin Völkl,  Xiaohan Liu,  Carolin Mirschina,  JiWoo Hwang '''(2022)'''

$\text{References:}$

LNTwww:About LNTwww

2025-02-03T13:49:11Z

==Welcome to the English version of LNTwww==
 
»$\text{https://en.lntwww.de}$«  is an e-learning tutorial for Communications Engineering with nine didactic multimedia textbooks including exercises with solutions,  learning videos,  and interactive applets.  It is offered by the  »[https://www.ce.cit.tum.de/en/lnt/home/ Institute for Communications Engineering]«  of the  »[https://www.tum.de/en/ Technical University of Munich]«. 

:⇒   '''It is freely accessible,  registration is not necessary and no system requirements are needed'''.

The German-language version   »$\text{https://www.lntwww.de}$«   ⇒   »$\rm L$erntutorial für $\rm N$achrichten$\rm T$echnik im $\rm w$orld $\rm w$ide $\rm w$eb«  was created between 2001 – 2021 by members of our Institute.  The toolbar entry  »Deutsch«  takes you to the German original.  In spring 2020 we started the English translation,  and in spring 2023 we finished.

*The current version from 2023 is based on the software  [https://en.wikipedia.org/wiki/MediaWiki »MediaWiki«],  known by the encyclopaedia  »WIKIPEDIA«.   The following is a kind of  »user guide«  to our e–learning project.  Corresponding links to this file  »About LNTwww«  can be found at the bottom of each page between  »Privacy policy« and  »Disclaimer«.

*We consider the present version as final;  an extension is currently not planned.  But of course we will continue to improve detected errors or inaccuracies promptly.  So if you notice any inadequacies regarding content,  presentation or handling,  then please send a detailed message by mail to  »LNTwww@ice.cit.tum.de«.

*On the  [[LNTwww:Information|»Information«]]  page you will find notes about temporary restrictions  $($e.g. in case of unavailability due to service work$)$  and a brief summary of all »LNTwww« features using a few example pages.

We would be pleased if we could arouse your interest in our e-learning project.  We wish you a good learning success.

$\text{Have fun and good luck!}$  

[https://www.ce.cit.tum.de/en/lnt/people/professors/kramer/ $\text{Gerhard Kramer}$'''],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29| $\text{Tasnád Kernetzky}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_L%C3%9CT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_from_2017-2024.29| $\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |$\text{Günter Söder}$]]

Munich,  in spring 2024          

 

==Content==

===(A)   The didactic concept of LNTwww===

At the beginning of the work on  »$\rm LNTwww$«  in 2001,  we gave ourselves the following ten rules.  These still apply today:

'''(1)'''   The teaching area  »Information and Communication Technology«  $\text{(I&K)}$  including associated basic subjects  $($Signal Representation,  Fourier and Laplace Transform,  Stochastic Signal Theory, etc.$)$  is presented in a didactically and multimedia prepared form.

'''(2)'''   Nine subject areas were selected,  each of which is covered by a self-contained book in the scope of a one-semester course with three semester hours per week to five semester hours per week.

'''(3)'''   The target group of our online offer are students of  $\text{I&K}$  technology,  especially of »Communications Engineering«,  as well as practicing engineers  $($Keywords:  »professional training«,  »lifelong learning«$)$.

'''(4)'''   In particular,  the interrelationships between different subfields of our extensive e-leatning offer should also be shown,  which is promoted by a nomenclature that is largely consistent in all books.

'''(5)'''   »$\rm LNTwww$«  offers two modes of learning:   Beginners should proceed sequentially  –   for advanced learners,  use it as a tutorial  $($work through exercises first,  jump to the theory part if deficits are identified$)$.

'''(6)'''   The theory is explained as in a traditional engineering textbook through texts,  graphics,  and mathematical derivations.  In addition,  each chapter includes at least one multimedia module.

'''(7)'''   »$\rm LNTwww$«  shall provide the user with multiple interaction options regarding the selection and presentation of theory chapters,  exercises,  learning videos as well as multimedia and calculation modules.

'''(8)'''   The methodology of hyperlinks typical of the  »world wide web«  is extensively used within
the  »$\rm LNTwww$«  and externally.  This is also intended to show connections between different teaching areas.

'''(9)'''   In order to prevent a user from getting lost in his learning environment and using  »$\rm LNTwww$«  only for  »surfing«,  a purposeful path must be recognizable for him at all times despite certain freedoms.

'''(10)'''  For reasons of sustainability of learning success,  there are possibilities for printing the texts and graphics,  ignoring the fact that today's students generation often devalues this as a  »relapse into the analog age«.
 

===(B)   Content and scope of LNTwww===

»$\rm LNTwww$«  is a virtual course totaling  $\text{36}$  semester hours per week 
*with  $\text{23}$  semester hours per week  (quasi) lectures   ⇒   $\text{23L}$

*and  $\text{13}$  semester hours per week exercises   ⇒   $\text{13E}$. 

It is organized in book form.  Each book contains a one-semester course.  For example,  in the case of the third book,  it is indicated that the book  »Theory of Stochastic Signals«  corresponds to a face-to-face–course with three semester hours per week of  »lecture«  and two semester hours per week of  »exercises«   ⇒   $\text{3L +2E}$.

$\text{Textbooks:}$
# [[Signal_Representation|'''Signal Representation''']]   ⇒   [[LNTwww:General_notes_about_"Signal_Representation"|»Impressum«]],
# [[Linear_and_Time_Invariant_Systems|'''Linear and Time Invariant Systems''']]   ⇒  [[LNTwww:General_notes_about_"Linear_and_Time_Invariant_Systems"|»Impressum«]],
# [[Theory_of_Stochastic_Signals|'''Theory of Stochastic Signals''']]   ⇒  [[LNTwww:General_Notes_about_the_Book_"Stochastic_Signal_Theory"|»Impressum«]],
# [[Information_Theory|'''Information Theory''']]   ⇒  [[LNTwww:General_notes_about_"Information_Theory"|»Impressum«]],
# [[Modulation_Methods|'''Modulation Methods''']]   ⇒  [[LNTwww:General_notes_about_"Modulation_Methods"|»Impressum«]],
# [[Digital_Signal_Transmission|'''Digital Signal Transmission''']]   ⇒  [[LNTwww:General_notes_about_"Digital_Signal_Transmission"|»Impressum«]],
# [[Mobile_Communications|'''Mobile Communications''']]   ⇒  [[LNTwww:General_notes_about_"Mobile_Communications"|»Impressum«]],
# [[Channel_Coding|'''Channel Coding''']]   ⇒  [[LNTwww:General_notes_about_"Channel_Coding"|»Impressum«]],
#[[Examples_of_Communication_Systems|'''Examples of Communication Systems''']]   ⇒  [[LNTwww:General_notes_about_"Examples_of_Communication_Systems"|»Impressum«]].

*The theory pages of all books result in the print version in approx.  $1500$  pages  $($DIN A4$)$  and contain on average one and a half graphics per page. 

*In addition,  LNTwww provides via the link  [[Biographies_and_Bibliographies|»'''Biographies & Bibliography'''«]]  a subject-specific bibliography with approx.  $400$  entries,  plus links to the WIKIPEDIA biographies of important scientists.
 

===(C)   Design and structure of LNTwww===

One can reach the nine reference books and »Biographies & Bibliography«  through the link  [[Book Overview|»'''Book Overview'''«]].  From this interface one can reach the individual books.  
*Each book is divided into several  »'''main chapters'''«, 

*each main chapter into several  »'''chapters'''«,  and

*each chapter includes several  »'''sections'''«.

{{GraueBox|TEXT=
$\text{Example 1:}$ 
We consider the book  [[Signal Representation|»Signal Representation«]].  This contains five  »main chapters«.
*By clicking on the first main chapter  »Basic Terms of Communications Engineering«,  one can get to three  »chapters«.  Each chapter corresponds to a MediaWiki file.

*The exemplary chapter  [[Signal_Representation/Principles_of_Communication|»Principles of Communication«]]  contains ten  »sections»  or  »pages».

*The last two pages are almost the same in all chapters,  namely  »Exercises for the chapter«  and  »References«.}}
 

===(D)   Content overviews for LNTwww===

A brief overview of all books is available on the selection interface  [[Book Overview|»'''Book Overview'''«]].
*More information is provided by the  »first page«  of each book.

*The respective main chapter content can be found in the first subchapter on the first page of each.

{{GraueBox|TEXT=
$\text{Example 2:}$ 
The first page  $($title page$)$  of the book  [[Signal_Representation|»Signal Representation«]]  provides the following information:
# A brief summary of the entire book;
# Scope of learning offer:  $2{\rm L} + 1{\rm E}$   ⇒   lecture with two semester hours per week and one additionalhour exercise. 
# Five main chapters,  19 chapters,  127 sections,  58 exercises;
# Links to the five main chapters of the book;
# Links to the associated exercises,  learning videos,  and interactive applets in the book  »Signal Representation«;
# Bibliography for the book;
# The imprint to the book  $($Authors,  other contributors,  materials as a starting point of the book,  referencces$)$.

The content of the first main chapter  »Principles of Communication« can be found on the first page 
[[Signal_Representation/Principles_of_Communication#OVERVIEW_OF_THE_FIRST_MAIN_CHAPTER|»# OVERVIEW OF THE FIRST MAIN CHAPTER #«
]].}}
 
===(E)   LNTwww exercises===

A central role in our didactic concept play »exercises«. We believe that the sensible use of »LNTwww« by a user with previous knowledge should be that he first work on the exercises relating to his actual learning area and only jump to the corresponding theory section when required.

You can find the  »'''exercise overview'''«  for all books  $($approx.  $640$  exercises, approx.  $3100$  subtasks)  on the home page via the link  [[Aufgaben:Aufgabensammlung|»'''Exercises'''«]].  All exercises are structured in the same way:
*Each exercise consists of the »exercise description« and several  »subtasks«.   An exercise is only solved correctly if all subtasks are correct.

* For each exercise there is a detailed  »sample solution«,  sometimes with the indication of several ways to the goal.

* The »exercise types« used are:
# »Single Choice»   ⇒   only one of the  $n$  given answers is correct;      ⇒   Marks of alternative answers:  ${\huge\circ}$
# »Multiple Choice«   ⇒   of the  $n$  given answers, between zero and  $n$  answers can be correct;      ⇒   Marks of alternative answers:  $\square$
# »Arithmetic Task«   ⇒   numerical value query,  possibly with sign;     small deviations  $($usually  $\pm 3\%)$  are allowed when checking real-valued results.

* We distinguish between  »exercises«  $($e.g.  »Exercise 1.1»$)$  and  »additional exercises«  $($e.g.  »Exercise 1.1Z«$)$.
# If you were able to solve all exercises of a chapter without any problems,  we believe that you are familiar with the content of the entire chapter. 
#If you have solved one exercise incorrectly,  you should also work on the following,  usually somewhat easier additional exercise.

{{GraueBox|TEXT=
$\text{Example 3:}$ 
The  $58$  exercises/additional exercises of the first book can be accessed via the link  [https://en.lntwww.de/Category:Signal_Representation:_Exercises »Signal Representation: Exercises«]. 

*From there,  we move on to the individual exercises,  e.g. to  [https://en.lntwww.de/Aufgaben:Exercise_1.1:_Music_Signals »Exercise 1.1: Music Signals«].  This relatively simple exercise consists of
#  one »Single Choice«   ⇒   subtask  '''(1)''',
#  two »Multiple Choice«   ⇒   subtasks  '''(2)''',  '''(3)''',  and
#  one »Arithmetic Task« with two real-valued computational queries   ⇒   subtask  '''(4)'''.

*However,  most of our exercises are not that easy.  Although MediaWiki also calls an arithmetic task  »quiz«,  answering them is usually much more difficult than in the numerous quiz shows on TV.   Because: 
#  There are no predetermined answers in an arithmetic task,  and moreover:
#  Integrals often have to be solved beforehand,  such as in  [[Aufgaben:Exercise_4.4:_Two-dimensional_Gaussian_PDF|»Exercise 4.4: Two-dimensional Gaussian probabilty density function«.]]

*We recommend:  First print the exercise   ⇒   »printable version«  and solve the exercise  offline  before checking  online.
}}

 

 
===(F)   LNTwww learning videos===

You can access approximately  $30$  learning videos via the link  »Videos«  on the start page.  The realization of a learning video required the following individual steps: 
:Writing the script and texts   ⇒   Creating a set of slides with only slight differences between successive slides   ⇒   Voicing texts and audio editing   ⇒   Combining texts and images into a coherent video stream.
#Clicking on this link brings up a; list of all learning videos,  grouped by textbook.  Some videos appear for multiple books.
#After selecting the desired learning video,  a wiki description page appears with a short content description and user interface.
#From here you can start the video in  »mp4«  and  »ogv«  format.  The browser will search for the appropriate format.
#The videos can be played by many browsers  $($Firefox, Chrome, Safari, ...$)$  as well as smartphones and tablets.
#The bottom link provides all available learning videos in alphabetical order.

Note:   All learning videos are with German language.  English translations are not planned.

{{GraueBox|TEXT=
$\text{Example 4:}$ 
We'll take a look at   [[Analoge_und_digitale_Signale_(Lernvideo)|»Analog and digital signals«]]  as an example.  This provides a two-part video in mp4 and ogv format.
*Each video part can be started by single click and paused by another click.

*The playback speed of the videos can be changed:
** Firefox offers a submenu after right-clicking on the video.
** For Google Chrome you can install e.g. the plugin  »Video Speed Controller«.
}}
 

===(G)   LNTwww applets===

Applets have a similar function as laboratories in mathematical-scientific courses:  Supplementing lecture/exercise with independent work by the student on the topic covered. 

You can access the provided interactive applets via the link of the same name on the home page.  It should be noted:
#Clicking on the link  »'''Applets'''«  a list of all applets appears,  grouped by reference books. 
#We distinguish between the newer  $\text{HTML 5/JavaScript}$  applets  $($in the respective lists above$)$  and the older  $\text{SWF}$  applets  $($below$)$. 
#The SWF applets unfortunately do not work on smartphones and tablets.
#After selecting an HTML 5/JS applet  a wiki description page appears with introductory theory section,  exercises to be solved and sample solutions. 
#At the beginning and end of this wiki description page there are links to the actual applet in German resp. English Language.

{{GraueBox|TEXT=
$\text{Example 5:}$ 
The didactic importance of applets shall be proved by  [[Applets:Eye_Pattern_and_Worst-Case_Error_Probability|»Eye Pattern and Worst-Case Error Probability«]]:
*The  »eye diagram«  is a proven transmission engineering tool,  to capture the influence of  »line dispersion«   ⇒   »intersysmbol interference«  on the quality characteristic  »error probability«  of a digital transmission system.  

*Such applets serve the clarification of more difficult facts,  in this example  »the step-by-step construction of the eye diagram from the symbol sequence«.  The program offers a lot of setting possibilities.  However, not every setting brings the user a relevant learning success and even fewer lead to a so-called "Aha! moment". 

*This is why we guide the user specifically through the program on the basis of the experiment.  He has to solve various tasks:  Predict and evaluate results,  Optimize parameters,  etc.

*A top 10% student has of course the possibility,  to set himself tasks going beyond the execution of experiments with the help of the applet and thus to penetrate very deeply into the presented subject matter.

}}

In addition to these  $\approx\hspace{-0.1cm} 30$  HTML 5/JS based applets  we still offer some of our  $\approx\hspace{-0.1cm}50$  older German-language applets,  which are based on  »Shock Wave Flash«  $\rm (SWF)$.  These were programmed for  »Adobe Flash«. 
#Since the Flashplayer browser plugin is no longer supported for security reasons,  these applets must be opened with the  »projector version«.
#You do not need to install the projector version and it will not be integrated into your browser.  So there are no security concerns in this regard.
#On the corresponding wiki pages you can find the projector version of the flash player and of course the applet itself.
 

===(H)   Glossary===

Due to the fact,  that our e–learning project LNTwww was first conceived in German and the wish for an English version came much later,  in the English version the assignment between  »Formula signs«   and  »Designation«  is not quite easy.   What do for example

#  $f_{\rm T}$,
#  $s_{\rm TP}(t)$, 
#  $e$,
#  $E$?

Here the link  [[LNTwww:Glossary|»Glossary«]]  on the home page below can help with the following alphabetically ordered entries:
::  »Formula sign«   ⇒   »German name«   ⇒   »English name« 

The file is self-explanatory. A few explanations are given under the last menu item  »Some remarks to the Glossary«.

{{GraueBox|TEXT=
$\text{Example 6:}$  In this file you will find the following entries:
:  $f_{\rm T}$   ⇒   Trägerfrequenz   ⇒   carrier frequency 
:  $s_{\rm TP}(t)$   ⇒   äquivalentes Tiefpass–Sendesignal   ⇒   equivalent low-pass transmitted signal 
:  $e= 2.718281828456$...   ⇒   Eulersche Zahl   ⇒   Eulerian number 
:  $ E$   ⇒   $(1)$ Schwellenwert,  $(2)$ Energie   ⇒   $(1)$ threshold value,  $(2)$ energy 

From the context,  the decision for  $(1)$  or  $(2)$  should be easy. }}

 

===(I)    History of LNTwww===

At the  [https://www.ce.cit.tum.de/en/lnt/home/ »Institute for Communications Engineering«]  $\rm (LNT)$  of the  [https://www.tum.de »Technical University of Munich«]  $\rm (TUM)$  two  teaching software packages  $\text{(LNTsim, LNTwin)}$  were realized by  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|»Günter Söder«]] from 1984 to 1996, which were used in our practical courses.  Several other universities have also acquired and used these programs.

At the beginning of the first Internet euphoria,  there were inquiries from students whether we could also provide such simulation and demonstration programs online.  After careful consideration  ("Is the expected big effort worth it?")  Günter Söder began 2001 planning the German-language project  »www.LNTwww.de«.  Co-responsible was his colleague  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|»Klaus Eichin«]].  The project was to be completed by 2011 at the latest,  since both would be retiring this year.

The content was derived from his own teaching materials as well as those of his colleague  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|»Norbert Hanik«]]  $($Associate Professor of Line Transmission Technology$)$.  Other lecture material was also taken into account,  which was produced at the Institute of Communications Engineering under the last four chair holders:
::*Professor [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Hans_Marko_.281962-1993.29|»Hans Marko«]]  $($Head of the LNT from 1962 to 1993$)$,
::*Professor [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|»Joachim Hagenauer«]]  $($Head of the LNT from 1993 to 2006$)$,
::*Professor [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._Ralf_K.C3.B6tter_.282007-2009.29|»Ralf Kötter»]]  $($Head of the LNT from 2007 to 2009$)$,  and
::*Professor [https://www.ce.cit.tum.de/en/lnt/people/professors/kramer/ »Gerhard Kramer«]  $($Head of the LNT since 2010$)$.

Just a few dates about progress of the German-language LNTwww project,  eleven years after the planned completion :
* First of all our own platform had to be developed by students  $($Marin Winkler,  Yven Winter$)$.  The authoring system  »LNTwww«  was based on the http server  »Apache«,  the database  »MySQL«,  the script language  »Perl«  and  »Shock Wave Flash«  $\rm (SWF)$  as a basis for multimedia applications   ⇒   version  »LNTwww.v1«  $($2003$)$.

*Work of the following years was online adaptation of the manuscripts,  input into the database with the rather complicated LNTwww syntax,  creation of the graphs as well as conception and realization of multimedia elements.  After completion of all nine textbooks the desired final state was reached   ⇒   version  »LNTwww.v2«  $($2016$)$.

* At the same time,  it became known that  »SWF«  would not longer be supported by relevant manufacturers.  This fact and the criticism heard from some users about the meanwhile too staid design  $($our authoring system was on the level of 2003$)$  were decisive for a new start based on  »MediaWiki«   ⇒   version  »LNTwww.v3«  $($2021$)$.

Finally,  a few sentences about the English LNTwww version. 
*At the beginning of the Corona pandemic and the associated lockdowns,  the call for  »e-learning«  also became louder and louder at the universities,  even from professors who had previously rather rejected this form of teaching. 

* Suddenly,  funds were also made available,  to provide as many e-learning courses as possible in as short a time as possible.  Our chair  »Gerhard Kramer«  therefore already submitted a corresponding application for working student funds in spring 2020,  which was approved within a few weeks.

*In June 2020 we started the English translation with support of the  »DEEPL«  program  $($free version$)$  and finished it in April 2023.  From the LNT staff were involved:  »Javier Garcia Gomez, Tasnád Kernetzky, Benedikt Leible and Günter Söder«.  Of the students involved,  »Noah Nagi«  and  »Jiwoo Hwang«  deserve special mention.
 

===(J)   Acknowledgement===

The Institute for Communications Engineering would like to thank the many people involved in the creation of  $\rm LNTwww$:

*The persons responsible for the German and/or English LNTwww projects:
::[[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |$\text{Günter Söder}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|$\text{Klaus Eichin}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29| $\text{Tasnád Kernetzky}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29| $\text{Benedikt Leible}$]].

* The  $($former$)$  LNT/LÜT colleagues,  who contributed as co–authors or experts or supervised student work:
::[[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Ronald_B.C3.B6hnke_.28at_LNT_from_2012-2014.29|$\text{Ronald Böhnke}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Joschi_Brauchle_.28at_LNT_from_2007-2015.29|$\text{Joschi Brauchle}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|$\text{Bernhard Göbel}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|$\text{Norbert Hanik}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Thomas_Hindelang_.28at_LNT_from_1994-2000_und_2007-2012.29|$\text{Thomas Hindelang}$]],   [[Biographies_and_Bibliographies/External_Contributors_to_LNTwww#Dr._Gianluigi_Liva|$\text{Gianluigi Liva}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Tobias_Lutz_.28at_LNT_from_2008-2014.29|$\text{Tobias Lutz}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Michael_Mecking_.28at_LNT_from_1997-2012.29|$\text{Michael Mecking}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Markus_Stinner_.28at_LNT_from_2011-2016.29|$\text{Markus Stinner}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Thomas_Stockhammer_.28at_LNT_from_1995-2004.29|$\text{Thomas Stockhammer}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|$\text{Johannes Zangl}$]],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Georg_Zeitler_.28at_LNT_from_2007-2012.29|$\text{Georg Zeitler}$]].

*The more than  »50 students«,  who have worked on subareas,  designed learning videos and applets or implemented the porting to the MediaWiki version within the framework of Engineering practice,  diploma,  bachelor and master theses or within the framework of a working student activity.

*The  [https://www.https://www.ei.tum.de/en/welcome/ »Department of Electrical and Computer Engineering«]  and the  [https://www.tum.de/en/ »Technical University of Munich«]  for funding working students in the years since 2016 within the framework of the  [https://www.ei.tum.de/studium/studienzuschuesse/ »MoliTUM«]  resp.  [https://www.tum.de/en/studies/teaching/awards-and-competitions/ideas-competition »EXIni«]  programs.

        [https://www.ce.cit.tum.de/en/lnt/people/professors/kramer/ $\text{Gerhard Kramer}$]

LNTwww:Imprint for the book "Mobile Communications"

2025-02-03T13:49:10Z

'''Four main chapters with a total of 16 chapters (files) and 121 sections (pages);   47 Exercises   ⇒   Scope:  "$\rm 2L+1E$"      
Development of the German version:   2010–2016.       Development of the English version:   2020/21.      Last corrections:   October 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|'''Benedikt Leible''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:   '''[Hin08]'''<ref name='Hin08'>Hindelang, T.:   Mobile Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2008.</ref>;   '''[Söd10]'''<ref name='Söd10'>Söder, G.: Mobilfunkkanal.   Instructions for the practical course "Simulation digitaler Übertragungssysteme".   LNT/TUM, 2010. </ref>;   '''[Eich11]'''<ref name='Eich11'>Eichin, K.:   Nachrichtensysteme – Kommunikationssysteme (LB).   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2011.</ref>;  '''[Vie17]'''<ref name='Vie17'>Viering, I.:   System Aspects in Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2018</ref>;   '''[Kra18]'''<ref name='Kra18'>Kramer, G.:   Mobile Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2018.</ref>   –   Textbook:   '''[EVB01]'''<ref name='EVB01'>Eberspächer, J.; Vögel, H.J.; Bettstetter, C.: Global System for Mobile Communication. 3. Auflage. Stuttgart: Teubner, 2001.</ref>

*Authors:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29| '''Klaus Eichin''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Thomas_Hindelang_.28at_LNT_from_1994-2000_und_2007-2012.29| '''Thomas Hindelang''']].  

* Participating colleagues in alphabetic order:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|'''Benedikt Leible''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Markus_Stinner_.28at_LNT_from_2011-2016.29|'''Markus Stinner''']],   [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29| '''Johannes Zangl''']]

*Participating students in chronological order:  Martin Winkler '''(2001)''', Yven Winter, Thorsten Kalweit, Slim Lamine, Johannes Schmidt, Hedi Abbes, Thomas Großer, Néjib Kchouk, Khaled Soussi, Alexander Happach, Felix Kristl, Martin Völkl, André Schulz, Noah Nagi, Carolin Mirschina, Jiwoo Hwang '''(2023)'''

$\text{References:}$

LNTwww:Imprint for the book "Examples of Communication Systems"

2025-02-03T13:49:10Z

'''Four main chapters with a total of 17 chapters (files) and 164 sections (pages);   38 Exercises   ⇒   Scope:  "$\rm 3L+1E$"      
Development of the German version:   2002–2017.       Development of the English version:   2022/23.      Last corrections:   March 2023

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|'''Benedikt Leible''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Hin08]'''<ref name='Hin08'>Hindelang, T.:  Mobile Communications. Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2008.</ref>;  '''[Söd10]'''<ref name='Söd10'>Söder, G.: Mobilfunkkanal.   Instructions for the practical course "Simulation digitaler Übertragungssysteme".   LNT/TUM, 2010. </ref>;   '''[Eich11]'''<ref name='Eich11'>Eichin, K.: Nachrichtensysteme – Kommunikationssysteme (LB).  Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2018.</ref>;  '''[Vie17]'''<ref name='Vie17'>Viering, I.:  System Aspects in Communications. Lecture notes. Institute for Communications Engineering. Technical University of Munich, 2017.</ref>; '''[Kra18]'''<ref name='Kra18'>Kramer, G.:   Mobile Communications.   Lecture notes. Institute for Communications Engineering.   Technical University of Munich, 2018.</ref>  –   Textbook:   '''[EVB01]'''<ref name='EVB01'>Eberspächer, J.; Vögel, H.J.; Bettstetter, C.: Global System for Mobile Communication. 3. Auflage. Stuttgart: Teubner, 2001.</ref>

*Authors in alphabetic order:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29| '''Klaus Eichin''']],  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Thomas_Hindelang_.28at_LNT_from_1994-2000_und_2007-2012.29|'''Thomas Hindelang''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']] 

*Participating students in chronological order:      Martin Winkler  '''(2001)''',  Yven Winter,  Franz Kohl,  Thorsten Kalweit,  Thomas Großer,  Franz-Josef Kaupert,  Hichem Kallel,  Khaled Soussi,  Johannes Schmidt,  Sebastian Seitz,  Alexander Happach,  Stefan Müller,  Noah Nagi,  Carolin Mirschina,  Jiwoo Hwang  '''(2022)'''

$\text{References:}$

LNTwww:LNTwww:Imprint for the book "Stochastic Signal Theory"

2025-02-03T13:49:10Z

'''Five main chapters with a total of 28 chapters (files);   93 Exercises   ⇒   Two semester hours per week lecture and one semester hour per week exercises      
Development of the German version:   2011–2015.       Development of the English version:   2021.      Last corrections:   November 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|'''Benedikt Leible''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Söd88]'''<ref name='Söd88'>Söder, G.:  Aufgabensammlung zu &bdquo;Statistische Methoden der Nachrichtentechnik”. Lehrstuhl für Nachrichtentechnik, TU München, 1988.</ref>;  '''[Söd12]'''<ref name='Söd12'>Söder, G.:  Simulationsmethoden in der Nachrichtentechnik.  Anleitung zum gleichnamigen Praktikum. Lehrstuhl für Nachrichtentechnik, TU München, 2012.</ref>;  –  Textbooks:   '''[Söd93]'''<ref name='Söd93'>Söder, G.:  Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin – Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2.</ref>;    '''[Dav87]'''<ref name='Dav87'>Davenport, W. B.: Probability and random processes. An introduction for applied scientists and engineers. New York NY u.a.: McGraw-Hill, 1987. ISBN 0-07-015440-6</ref>;  '''[PP09]'''<ref name='PP09'>Papoulis, A.; Pillai, S. U.: Probability, random variables, and stochastic processes. 4. Aufl. Boston, Mass.: McGraw-Hill, 2009. ISBN 978-0-07122-661-5</ref>

*Author:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|'''Günter Söder''']]

*Participating colleagues in alphabetic order:  [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29| '''Benedikt Leible''']],   [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Thomas_Stockhammer_.28at_LNT_from_1995-2004.29|'''Thomas Stockhammer''']],  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|'''Johannes Zangl''']]

*Participating students in chronological order:      Martin Winkler '''(2001)''', Yven Winter, Reinhold Sixt, Jürgen Veitenhansl, Franz Kohl, Bettina Hirner, Ji Li, Markus Elsberger, Thorsten Kalweit, Thomas Großer, David Jobst, Matthias Niller, Veronika Hofmann, Carolin Mirschina, Noah Nagi, Ji Woo Hwang '''(2022)'''

$\text{List of sources:}$

LNTwww:Imprint for the book "Linear and Time Invariant Systems"

2025-02-03T13:49:10Z

'''Four main chapters with a total of 12 chapters (files) and 93 sections (pages);   90 Exercises   ⇒   Scope:  "$\rm 2L+1E$"      
Development of the German version:   2002–2017.       Development of the English version:   2020/21.      Last corrections:   December 2022

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| $\text{Gerhard Kramer}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|$\text{Tasnád Kernetzky}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|$\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |$\text{Günter Söder}$]]

*Basic materials for the original German version:     Lecture notes of LNT/LÜT:  '''[Eich03]'''<ref name='Eich03'>Eichin, K.:  Nachrichtentechnik I (LB) – Signaldarstellung.  Lecture notes, Chair of Communications Engineering, TU München, 2003. </ref>;  '''[Han15]'''<ref name='Han15'>Hanik, N.:  Nachrichtentechnik 1 (LB): Signaldarstellung.  Lecture notes, Associate Professorship of Line Transmission Technology, TU München, 2015.</ref>;  '''[Söd12]'''<ref name='Söd12'>Söder, G.:   Simulationsmethoden in der Nachrichtentechnik.  Internship notes, Chair of Communications Engineering, TU München, 2012.</ref>   –  Textbooks:   '''[Söd93]'''<ref name='Söd93'>Söder, G.:  Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin, Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2.</ref>;  '''[Mar94]'''<ref name='Mar94'>Marko, H.:  Methoden der Systemtheorie.  3. Auflage. Berlin – Heidelberg: Springer, 1994.</ref>;  '''[HM09]'''<ref name='HM09'>Haykin, S.; Moher, M.: Communication Systems. 5th edition, Hoboken, N.J.: J. Wiley, 2009. ISBN 978-0-47169-790-9.</ref>

*Authors:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |$\text{Günter Söder}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29| $\text{Klaus Eichin}$]],  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|$\text{Norbert Hanik}$]]

* Participating colleagues in alphabetic order:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| $\text{Javier Garcia Gomez}$]],   [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_Übertragungstechnik#Dr.-Ing._Bernhard_G.C3.B6bel_.28at_L.C3.9CT_from_2004-2010.29|$\text{Bernhard Göbel}$]],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|$\text{Benedikt Leible}$]],  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|$\text{Johannes Zangl}$]]

*Participating students in chronological order:      Martin Winkler  '''(2001)''',  Yven Winter,  Franz Kohl,  Thorsten Kalweit,  Markus Elsberger,  Bettina Hirner,  Ji Li,  Thomas Großer,  David Jobst,  Jimmy He,  Xiaohan Liu,  Carolin Mirschina,  Ji Woo Hwang  '''(2021)'''

$\text{References:}$

LNTwww:Imprint for the book "Theory of Stochastic Signals"

2025-02-03T13:49:10Z

'''Five main chapters with a total of 28 chapters (files) and 166 sections (pages);   93 Exercises  ⇒   Scope:  "$\rm 3L+2E$"      
Development of the German version:   2011–2015.       Development of the English version:   2021.      Last corrections:   November 2021

*Project responsibility:  [[Biographies_and_Bibliographies/Chair_holders_of_the_LNT_since_1962#Prof._Dr._sc._techn._Gerhard_Kramer_.28seit_2010.29| '''Gerhard Kramer''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Francisco_Javier_Garc.C3.ADa_G.C3.B3mez_.28at_LNT_from_2016-2021.29| '''Javier Garcia Gomez''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|'''Tasnád Kernetzky''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29|'''Benedikt Leible''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29 |'''Günter Söder''']]

*Basic materials   ⇒   Lecture notes of LNT/LÜT:  '''[Söd88]'''<ref name='Söd88'>Söder, G.:  Aufgabensammlung zu &bdquo;Statistische Methoden der Nachrichtentechnik”. Lecture notes, Chair of Communications Engineering, TU München, 1988.</ref>;  '''[Söd12]'''<ref name='Söd12'>Söder, G.:  Simulationsmethoden in der Nachrichtentechnik.  Internship notes, Chair of Communications Engineering, TU München, 2012</ref>;  –  Textbooks:   '''[Söd93]'''<ref name='Söd93'>Söder, G.:  Modellierung, Simulation und Optimierung von Nachrichtensystemen. Bd. 23. Berlin – Heidelberg: Springer, 1993. ISBN 978-3-54057-215-2.</ref>;    '''[Dav87]'''<ref name='Dav87'>Davenport, W. B.: Probability and random processes. An introduction for applied scientists and engineers. New York NY u.a.: McGraw-Hill, 1987. ISBN 0-07-015440-6</ref>;  '''[PP09]'''<ref name='PP09'>Papoulis, A.; Pillai, S. U.: Probability, random variables, and stochastic processes. 4. Aufl. Boston, Mass.: McGraw-Hill, 2009. ISBN 978-0-07122-661-5</ref>

*Author:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|'''Günter Söder''']]

*Participating colleagues in alphabetic order:  [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Prof._Dr.-Ing._Norbert_Hanik_.28at_LNT_from_1989-1995.2C_at_L.C3.9CT_since_2004.29|'''Norbert Hanik''']],  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29| '''Benedikt Leible''']],   [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Thomas_Stockhammer_.28at_LNT_from_1995-2004.29|'''Thomas Stockhammer''']],  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Johannes_Zangl_.28at_LNT_from_2000-2006.29|'''Johannes Zangl''']]

*Participating students in chronological order:      Martin Winkler '''(2001)''', Yven Winter, Reinhold Sixt, Jürgen Veitenhansl, Franz Kohl, Bettina Hirner, Ji Li, Markus Elsberger, Thorsten Kalweit, Thomas Großer, David Jobst, Matthias Niller, Veronika Hofmann, Carolin Mirschina, Noah Nagi, Ji Woo Hwang '''(2022)'''

$\text{References:}$

Applets:PDF, CDF and Moments of Special Distributions

2025-02-03T13:48:58Z

{{LntAppletLinkEnDe|wdf-vtf_en|wdf-vtf}}

==Applet Description==
 
The applet presents the description forms of two continuous value random variables  $X$  and  $Y\hspace{-0.1cm}$.  For the red random variable  $X$  and the blue random variable  $Y$,  the following basic forms are available for selection:

* Gaussian distribution, uniform distribution, triangular distribution, exponential distribution, Laplace distribution, Rayleigh distribution, Rice distribution, Weibull distribution, Wigner semicircle distribution, Wigner parabolic distribution, Cauchy distribution.

The following data refer to the random variables  $X$. Graphically represented are
* the probability density function  $f_{X}(x)$  (above) and
* the cumulative distribution function  $F_{X}(x)$  (bottom).

In addition, some integral parameters are output, namely
*the linear mean value  $m_X = {\rm E}\big[X \big]$,
*the second order moment  $P_X ={\rm E}\big[X^2 \big] $,
*the variance  $\sigma_X^2 = P_X - m_X^2$,
*the standard deviation  $\sigma_X$,
*the Charlier skewness  $S_X$,
*the kurtosis  $K_X$.

==Definition and Properties of the Presented Descriptive Variables==
 
In this applet we consider only ''(value–)continuous random variables'', i.e. those whose possible numerical values are not countable.
*The range of values of these random variables is thus in general that of the real numbers  $(-\infty \le X \le +\infty)$.
*However, it is possible that the range of values is limited to an interval:  $x_{\rm min} \le X \le +x_{\rm max}$.
 

===Probability density function (PDF)===
For a continuous random variable  $X$  the probabilities that  $X$  takes on quite specific values  $x$  are zero:  ${\rm Pr}(X= x) \equiv 0$.  Therefore, to describe a continuous random variable, we must always refer to the  ''probability density function''  – in short  $\rm PDF$. 

{{BlaueBox|TEXT=
$\text{Definition:}$  The value of the  »'''probability density function'''«  $f_{X}(x)$  at location  $x$  is equal to the probability that the instantaneous value of the random variable  $x$  lies in an  (infinitesimally small)  interval of width  $Δx$  around  $x_\mu$,  divided by  $Δx$:

:$$f_X(x) = \lim_{ {\rm \Delta} x \hspace{0.05cm}\to \hspace{0.05cm} 0} \frac{ {\rm Pr} \big [x - {\rm \Delta} x/2 \le X \le x +{\rm \Delta} x/2 \big ] }{ {\rm \Delta} x}.$$

}}

This extremely important descriptive variable has the following properties:

*For the probability that the random variable  $X$  lies in the range between  $x_{\rm u}$  and  $x_{\rm o} > x_{\rm u}$: 
:$${\rm Pr}(x_{\rm u} \le X \le x_{\rm o}) = \int_{x_{\rm u}}^{x_{\rm o}} f_{X}(x) \ {\rm d}x.$$
*As an important normalization property,  this yields for the area under the PDF with the boundary transitions  $x_{\rm u} → \hspace{0.1cm} – \hspace{0.05cm} ∞$  and  $x_{\rm o} → +∞$:
:$$\int_{-\infty}^{+\infty} f_{X}(x) \ {\rm d}x = 1.$$
 

===Cumulative distribution function (CDF)===

The  ''cumulative distribution function''  – in short  $\rm CDF$  – provides the same information about the random variable  $X$  as the probability density function.

{{BlaueBox|TEXT=
$\text{Definition:}$  The  »'''cumulative distribution function'''«  $F_{X}(x)$  corresponds to the probability that the random variable  $X$  is less than or equal to a real number  $x$: 
:$$F_{X}(x) = {\rm Pr}( X \le x).$$}}

The CDF has the following characteristics:

*The CDF is computable from the probability density function  $f_{X}(x)$  by integration.  It holds:
:$$F_{X}(x) = \int_{-\infty}^{x}f_X(\xi)\,{\rm d}\xi.$$
*Since the PDF is never negative,  $F_{X}(x)$  increases at least weakly monotonically,  and always lies between the following limits:
:$$F_{X}(x → \hspace{0.1cm} – \hspace{0.05cm} ∞) = 0, \hspace{0.5cm}F_{X}(x → +∞) = 1.$$
*Inversely,  the probability density function can be determined from the CDF by differentiation:
:$$f_{X}(x)=\frac{{\rm d} F_{X}(\xi)}{{\rm d}\xi}\Bigg |_{\hspace{0.1cm}x=\xi}.$$
*For the probability that the random variable  $X$  is in the range between  $x_{\rm u}$  and  $x_{\rm o} > x_{\rm u}$  holds:
:$${\rm Pr}(x_{\rm u} \le X \le x_{\rm o}) = F_{X}(x_{\rm o}) - F_{X}(x_{\rm u}).$$
 

===Expected values and moments===
The probability density function provides very extensive information about the random variable under consideration.  Less,  but more compact information is provided by the so-called  "expected values"  and  "moments".

{{BlaueBox|TEXT=
$\text{Definition:}$  The  »'''expected value'''«  with respect to any weighting function  $g(x)$  can be calculated with the PDF  $f_{\rm X}(x)$  in the following way:
:$${\rm E}\big[g (X ) \big] = \int_{-\infty}^{+\infty} g(x)\cdot f_{X}(x) \,{\rm d}x.$$
Substituting into this equation for  $g(x) = x^k$  we get the  »'''moment of $k$-th order'''«:
:$$m_k = {\rm E}\big[X^k \big] = \int_{-\infty}^{+\infty} x^k\cdot f_{X} (x ) \, {\rm d}x.$$}}

From this equation follows.
*with  $k = 1$  for the  ''first order moment''  or the  ''(linear)  mean'':
:$$m_1 = {\rm E}\big[X \big] = \int_{-\infty}^{ \rm +\infty} x\cdot f_{X} (x ) \,{\rm d}x,$$
*with  $k = 2$  for the  ''second order moment''  or the  ''second moment'':
:$$m_2 = {\rm E}\big[X^{\rm 2} \big] = \int_{-\infty}^{ \rm +\infty} x^{ 2}\cdot f_{ X} (x) \,{\rm d}x.$$

In relation to signals,  the following terms are also common:
* $m_1$  indicates the  ''DC component'';    with respect to the random quantity  $X$  in the following we also write  $m_X$.
* $m_2$  corresponds to the ''signal power''  $P_X$   (referred to the unit resistance  $1 \ Ω$ ) .

For example, if  $X$  denotes a voltage, then according to these equations  $m_X$  has the unit  ${\rm V}$  and the power  $P_X$  has the unit  ${\rm V}^2.$ If the power is to be expressed in "watts"  $\rm (W)$, then  $P_X$  must be divided by the resistance value  $R$. 
 

===Central moments===

Of particular importance in statistics in general are the so-called  ''central moments'' from which many characteristics are derived,

{{BlaueBox|TEXT=
$\text{Definition:}$  The  »'''central moments'''«,  in contrast to the conventional moments, are each related to the mean value  $m_1$  in each case. For these, the following applies with  $k = 1, \ 2,$ ...:

:$$\mu_k = {\rm E}\big[(X-m_{\rm 1})^k\big] = \int_{-\infty}^{+\infty} (x-m_{\rm 1})^k\cdot f_x(x) \,\rm d \it x.$$}}

*For mean-free random variables, the central moments  $\mu_k$  coincide with the noncentral moments  $m_k$. 

*The first order central moment is by definition equal to  $\mu_1 = 0$.

* The noncentral moments  $m_k$  and the central moments  $\mu_k$  can be converted directly into each other.  With  $m_0 = 1$  and  $\mu_0 = 1$  it is valid:
:$$\mu_k = \sum\limits_{\kappa= 0}^{k} \left( \begin{array}{*{2}{c}} k \\ \kappa \\ \end{array} \right)\cdot m_\kappa \cdot (-m_1)^{k-\kappa},$$
:$$m_k = \sum\limits_{\kappa= 0}^{k} \left( \begin{array}{*{2}{c}} k \\ \kappa \\ \end{array} \right)\cdot \mu_\kappa \cdot {m_1}^{k-\kappa}.$$
 
===Some Frequently Used Central Moments===

From the last definition the following additional characteristics can be derived:

{{BlaueBox|TEXT=
$\text{Definition:}$  The  »'''variance'''«  of the considered random variable  $X$  is the second order central moment:
:$$\mu_2 = {\rm E}\big[(X-m_{\rm 1})^2\big] = \sigma_X^2.$$
*The variance  $σ_X^2$  corresponds physically to the  "switching power"  and  »'''standard deviation'''«  $σ_X$  gives the "rms value".
*From the linear and the second moment,  the variance can be calculated according to  ''Steiner's theorem''  in the following way:  $\sigma_X^{2} = {\rm E}\big[X^2 \big] - {\rm E}^2\big[X \big].$}}

{{BlaueBox|TEXT=
$\text{Definition:}$  The  »'''Charlier's skewness'''«  $S_X$  of the considered random variable  $X$  denotes the third central moment related to $σ_X^3$.
*For symmetric probability density function,  this parameter   $S_X$  is always zero.
*The larger  $S_X = \mu_3/σ_X^3$  is,  the more asymmetric is the PDF around the mean  $m_X$.
*For example,  for the exponential distribution the (positive) skewness  $S_X =2$, and this is independent of the distribution parameter  $λ$.}}

{{BlaueBox|TEXT=
$\text{Definition:}$  The  »'''kurtosis'''«  of the considered random variable  $X$  is the quotient  $K_X = \mu_4/σ_X^4$    $(\mu_4:$  fourth-order central moment$)$.
*For a Gaussian distributed random variable this always yields the value  $K_X = 3$.
*This parameter can be used, for example, to check whether a given random variable is actually Gaussian or can at least be approximated by a Gaussian distribution. }}

==Compilation of some Continuous–Value Random Variables==
 
The applet considers the following distributions: 

:Gaussian distribution, uniform distribution, triangular distribution, exponential distribution, Laplace distribution, Rayleigh distribution, Rice distribution, Weibull distribution, Wigner semicircle distribution, Wigner parabolic distribution, Cauchy distribution.

Some of these will be described in detail here.

===Gaussian distributed random variables===

[[File:EN_Sto_T_3_5_S2_v2.png |right|frame|Gaussian random variable:  PDF and CDF]]
'''(1)'''    »'''Probability density function'''«   $($axisymmetric around  $m_X)$
:$$f_X(x) = \frac{1}{\sqrt{2\pi}\cdot\sigma_X}\cdot {\rm e}^{-(X-m_X)^2 /(2\sigma_X^2) }.$$
PDF parameters: 
*$m_X$  (mean or DC component),
*$σ_X$  (standard deviation or rms value).

'''(2)'''    »'''Cumulative distribution function'''«   $($point symmetric around  $m_X)$
:$$F_X(x)= \phi(\frac{\it x-m_X}{\sigma_X})\hspace{0.5cm}\rm with\hspace{0.5cm}\rm \phi (\it x\rm ) = \frac{\rm 1}{\sqrt{\rm 2\it \pi}}\int_{-\rm\infty}^{\it x} \rm e^{\it -u^{\rm 2}/\rm 2}\,\, d \it u.$$

$ϕ(x)$:   Gaussian error integral (cannot be calculated analytically, must be taken from tables).

'''(3)'''    »'''Central moments'''«
:$$\mu_{k}=(k- 1)\cdot (k- 3) \ \cdots \ 3\cdot 1\cdot\sigma_X^k\hspace{0.2cm}\rm (if\hspace{0.2cm}\it k\hspace{0.2cm}\rm even).$$
*Charlier's skewness  $S_X = 0$,  since  $\mu_3 = 0$  $($PDF is symmetric about  $m_X)$.
*Kurtosis  $K_X = 3$,  since  $\mu_4 = 3 \cdot \sigma_X^2$  ⇒   $K_X = 3$  results only for the Gaussian PDF.

'''(4)'''    »'''Further remarks'''«
*The naming is due to the important mathematician, physicist and astronomer Carl Friedrich Gauss.
*If  $m_X = 0$  and  $σ_X = 1$, it is often referred to as the  ''normal distribution''.

*The standard deviation can also be determined graphically from the bell-shaped PDF $f_{X}(x)$   (as the distance between the maximum value and the point of inflection).
*Random quantities with Gaussian WDF are realistic models for many physical physical quantities and also of great importance for communications engineering.
*The sum of many small and independent components always leads to the Gaussian PDF   ⇒   Central Limit Theorem of Statistics   ⇒   Basis for noise processes.
*If one applies a Gaussian distributed signal to a linear filter for spectral shaping, the output signal is also Gaussian distributed.

[[File:Gauss_Signal.png|right|frame| Signal and PDF of a Gaussian noise signal]]
{{GraueBox|TEXT=
$\text{Example 1:}$  The graphic shows a section of a stochastic noise signal  $x(t)$  whose instantaneous value can be taken as a continuous random variable  $X$. From the PDF shown on the right, it can be seen that:
* A Gaussian random variable is present.
*Instantaneous values around the mean  $m_X$  occur most frequently.
*If there are no statistical ties between the samples  $x_ν$  of the sequence, such a signal is also called ''"white noise".''}}

===Uniformly distributed random variables===

[[File:Rechteck_WDF_VTF.png|right|frame|Uniform distribution:  PDF and CDF]]
'''(1)'''    »'''Probability density function'''«

*The probability density function (PDF)  $f_{X}(x)$  is in the range from  $x_{\rm min}$  to  $x_{\rm max}$  constant equal to  $1/(x_{\rm max} - x_{\rm min})$  and outside zero.
*At the range limits for  $f_{X}(x)$  only half the value  (mean value between left and right limit value)  is to be set.

'''(2)'''    »'''Cumulative distribution function'''«

*The cumulative distribution function (CDF) increases in the range from  $x_{\rm min}$  to  $x_{\rm max}$  linearly from zero to  $1$. 

'''(3)'''    »''''''«
*Mean and standard deviation have the following values for the uniform distribution:
:$$m_X = \frac{\it x_ {\rm max} \rm + \it x_{\rm min}}{2},\hspace{0.5cm}
\sigma_X^2 = \frac{(\it x_{\rm max} - \it x_{\rm min}\rm )^2}{12}.$$
*For symmetric PDF   ⇒   $x_{\rm min} = -x_{\rm max}$  the mean value  $m_X = 0$  and the variance  $σ_X^2 = x_{\rm max}^2/3.$
*Because of the symmetry around the mean  $m_X$  the Charlier skewness  $S_X = 0$.
*The kurtosis is with   $K_X = 1.8$  significantly smaller than for the Gaussian distribution because of the absence of PDF outliers.

'''(4)'''    »'''Further remarks'''«

*For modeling transmission systems, uniformly distributed random variables are the exception. An example of an actual (nearly) uniformly distributed random variable is the phase in circularly symmetric interference, such as occurs in  ''quadrature amplitude modulation''  (QAM) schemes.

*The importance of uniformly distributed random variables for information and communication technology lies rather in the fact that, from the point of view of information theory, this PDF form represents an optimum with respect to differential entropy under the constraint of "peak limitation".

*In ''image processing & encoding'', the uniform distribution is often used instead of the actual distribution of the original image, which is usually much more complicated, because the difference in information content between a ''natural image'' and the model based on the uniform distribution is relatively small.

*In the simulation of intelligence systems, one often uses "pseudo-random generators" based on the uniform distribution (which are relatively easy to realize), from which other distributions  (Gaussian distribution, exponential distribution, etc.)  can be easily derived.

===Exponentially distributed random variables===

'''(1)'''    »'''Probability distribution function'''«
[[File:Exponential_WDF_VTF.png|right|frame|Exponential distribution:  PDF and CDF]]

An exponentially distributed random variable  $X$  can only take on non–negative values. For  $x>0$  the PDF has the following shape:
:$$f_X(x)=\it \lambda_X\cdot\rm e^{\it -\lambda_X \hspace{0.05cm}\cdot \hspace{0.03cm} x}.$$
*The larger the distribution parameter  $λ_X$,  the steeper the drop.
*By definition,  $f_{X}(0) = λ_X/2$, which is the average of the left-hand limit  $(0)$  and the right-hand limit  $(\lambda_X)$.

'''(2)'''    »'''Cumulative distribution function'''«

Distribution function PDF, we obtain for  $x > 0$:
:$$F_{X}(x)=1-\rm e^{\it -\lambda_X\hspace{0.05cm}\cdot \hspace{0.03cm} x}.$$

'''(3)'''    »'''Moments and central moments'''«

*The  ''moments''  of the (one-sided) exponential distribution are generally equal to:
:$$m_k = \int_{-\infty}^{+\infty} x^k \cdot f_{X}(x) \,\,{\rm d} x = \frac{k!}{\lambda_X^k}.$$
*From this and from Steiner's theorem we get for mean and standard deviation:
:$$m_X = m_1=\frac{1}{\lambda_X},\hspace{0.6cm}\sigma_X^2={m_2-m_1^2}={\frac{2}{\lambda_X^2}-\frac{1}{\lambda_X^2}}=\frac{1}{\lambda_X^2}.$$
*The PDF is clearly asymmetric here. For the Charlier skewness  $S_X = 2$.
*The kurtosis with   $K_X = 9$  is clearly larger than for the Gaussian distribution, because the PDF foothills extend much further.

'''(4)'''    »'''Further remarks'''«

*The exponential distribution has great importance for reliability studies; in this context, the term "lifetime distribution" is also commonly used.
*In these applications, the random variable is often the time  $t$, that elapses before a component fails.
*Furthermore, it should be noted that the exponential distribution is closely related to the Laplace distribution.

===Laplace distributed random variables===

[[File:Laplace_WDF_VTF.png|right|frame|Laplace distribution:  PDF and CDF]]
'''(1)'''    »'''Probability density function'''«

As can be seen from the graph, the Laplace distribution is a "two-sided exponential distribution":

:$$f_{X}(x)=\frac{\lambda_X} {2}\cdot{\rm e}^ { - \lambda_X \hspace{0.05cm} \cdot \hspace{0.05cm} \vert \hspace{0.05cm} x \hspace{0.05cm} \vert}.$$

* The maximum value here is  $\lambda_X/2$.
*The tangent at  $x=0$  intersects the abscissa at  $1/\lambda_X$, as in the exponential distribution.

'''(2)'''    »'''Cumulative distribution function'''«

:$$F_{X}(x) = {\rm Pr}\big [X \le x \big ] = \int_{-\infty}^{x} f_{X}(\xi) \,\,{\rm d}\xi $$
:$$\Rightarrow \hspace{0.5cm} F_{X}(x) = 0.5 + 0.5 \cdot {\rm sign}(x) \cdot \big [ 1 - {\rm e}^ { - \lambda_X \hspace{0.05cm} \cdot \hspace{0.05cm} \vert \hspace{0.05cm} x \hspace{0.05cm} \vert}\big ] $$
:$$\Rightarrow \hspace{0.5cm} F_{X}(-\infty) = 0, \hspace{0.5cm}F_{X}(0) = 0.5, \hspace{0.5cm} F_{X}(+\infty) = 1.$$

'''(3)'''    »'''Moments and central moments'''«

* For odd  $k$,  the Laplace distribution always gives  $m_k= 0$ due to symmetry. Among others:  Linear mean  $m_X =m_1 = 0$.

* For even  $k$  the moments of Laplace distribution and exponential distribution agree:  $m_k = {k!}/{\lambda^k}$.

* For the variance  $(=$ second order central moment $=$ second order moment$)$  holds:  $\sigma_X^2 = {2}/{\lambda_X^2}$   ⇒   twice as large as for the exponential distribution.

*For the Charlier skewness,  $S_X = 0$ is obtained here due to the symmetric PDF.

*The kurtosis is  $K_X = 6$,  significantly larger than for the Gaussian distribution, but smaller than for the exponential distribution.

'''(4)'''    »'''Further remarks'''«

*The instantaneous values of speech and music signals are Laplace distributed with good approximation. See learning video  [[Wahrscheinlichkeit_und_WDF_(Lernvideo)|"Wahrscheinlichkeit und Wahrscheinlichkeitsdichtefunktion"]],  part 2.
*By adding a Dirac delta function at  $x=0$,  speech pauses can also be modeled.
 
===Brief description of other distributions===
 
$\text{(A) Rayleigh distribution}$     [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading|$\text{More detailed description}$]]

*Probability density function:
:$$f_X(x) =
\left\{ \begin{array}{c} x/\lambda_X^2 \cdot {\rm e}^{- x^2/(2 \hspace{0.05cm}\cdot\hspace{0.05cm} \lambda_X^2)} \\
0 \end{array} \right.\hspace{0.15cm}
\begin{array}{*{1}c} {\rm for}\hspace{0.1cm} x\hspace{-0.05cm} \ge \hspace{-0.05cm}0,
\\ {\rm for}\hspace{0.1cm} x \hspace{-0.05cm}<\hspace{-0.05cm} 0. \\ \end{array}.$$
*Application:     Modeling of the cellular channel (non-frequency selective fading, attenuation, diffraction, and refraction effects only, no line-of-sight).

$\text{(B) Rice distribution}$     [[Mobile_Communications/Non-Frequency-Selective_Fading_With_Direct_Component|$\text{More detailed description}$]]

*Probability density function  $(\rm I_0$  denotes the modified zero-order Bessel function$)$:
:$$f_X(x) = \frac{x}{\lambda_X^2} \cdot {\rm exp} \big [ -\frac{x^2 + C_X^2}{2\cdot \lambda_X^2}\big ] \cdot {\rm I}_0 \left [ \frac{x \cdot C_X}{\lambda_X^2} \right ]\hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k}}{k! \cdot \Gamma (k+1)}
\hspace{0.05cm}.$$
*Application:     Cellular channel modeling (non-frequency selective fading, attenuation, diffraction, and refraction effects only, with line-of-sight).

$\text{(C) Weibull distribution}$     [https://en.wikipedia.org/wiki/Weibull_distribution $\text{More detailed description}$]

*Probability density function:
:$$f_X(x) = \lambda_X \cdot k_X \cdot (\lambda_X \cdot x)^{k_X-1} \cdot {\rm e}^{(\lambda_X \cdot x)^{k_X}}
\hspace{0.05cm}.$$

*Application:     PDF with adjustable skewness $S_X$; exponential distribution  $(k_X = 1)$  and Rayleigh distribution  $(k_X = 2)$  included as special cases.

$\text{(D) Wigner semicircle distribution}$     [https://en.wikipedia.org/wiki/Wigner_semicircle_distribution $\text{More detailed description}$]

*Probability density function:
:$$f_X(x) =
\left\{ \begin{array}{c} 2/(\pi \cdot {R_X}^2) \cdot \sqrt{{R_X}^2 - (x- m_X)^2} \\
0 \end{array} \right.\hspace{0.15cm}
\begin{array}{*{1}c} {\rm for}\hspace{0.1cm} |x- m_X|\hspace{-0.05cm} \le \hspace{-0.05cm}R_X,
\\ {\rm for}\hspace{0.1cm} |x- m_X| \hspace{-0.05cm} > \hspace{-0.05cm} R_X \\ \end{array}.$$
*Application:     PDF of Chebyshev nodes   ⇒   zeros of Chebyshev polynomials from numerics.

$\text{(E) Wigner parabolic distribution}$

*Probability density function:
:$$f_X(x) =
\left\{ \begin{array}{c} 3/(4 \cdot {R_X}^3) \cdot \big ({R_X}^2 - (x- m_X)^2\big ) \\
0 \end{array} \right.\hspace{0.15cm}
\begin{array}{*{1}c} {\rm for}\hspace{0.1cm} |x|\hspace{-0.05cm} \le \hspace{-0.05cm}R_X,
\\ {\rm for}\hspace{0.1cm} |x| \hspace{-0.05cm} > \hspace{-0.05cm} R_X \\ \end{array}.$$
*Application:     PDF of eigenvalues of symmetric random matrices whose dimension approaches infinity.

$\text{(F) Cauchy distribution}$     [[Theory_of_Stochastic_Signals/Further_Distributions#Cauchy_PDF|$\text{More detailed description}$]]

*Probability density function and distribution function:
:$$f_{X}(x)=\frac{1}{\pi}\cdot\frac{\lambda_X}{\lambda_X^2+x^2}, \hspace{2cm} F_{X}(x)={\rm 1}/{2}+{\rm arctan}({x}/{\lambda_X}).$$
*In the Cauchy distribution, all moments  $m_k$  for even  $k$  have an infinitely large value, independent of the parameter  $λ_X$.
*Thus, this distribution also has an infinitely large variance:  $\sigma_X^2 \to \infty$.
*Due to symmetry, for odd  $k$  all moments  $m_k = 0$, if one assumes the "Cauchy Principal Value" as in the program:  $m_X = 0, \ S_X = 0$.
*Example:     The quotient of two Gaussian mean-free random variables is Cauchy distributed. For practical applications the Cauchy distribution has less meaning.

==Exercises==
 
*First, select the number  $(1,\ 2, \text{...} \ )$  of the task to be processed.  The number  "$0$"  corresponds to a  "Reset":  Same setting as at program start.
*A task description is displayed.  The parameter values are adjusted.  Solution after pressing  "Show Solution".
*In the following  $\text{Red}$  stands for the random variable  $X$  and  $\text{Blue}$  for  $Y$.

{{BlueBox|TEXT=
'''(1)'''  Select  $\text{red: Gaussian PDF}\ (m_X = 1, \ \sigma_X = 0.4)$  and  $\text{blue: Rectangular PDF}\ (y_{\rm min} = -2, \ y_{\rm max} = +3)$.  Interpret the  $\rm PDF$  graph.}}

* $\text{Gaussian PDF}$:  The  $\rm PDF$ maximum is equal to  $f_{X}(x = m_X) = \sqrt{1/(2\pi \cdot \sigma_X^2)} = 0.9974 \approx 1$.
* $\text{Rectangular PDF}$:  All  $\rm PDF$ values are equal  $0.2$  in the range  $-2 < y < +3$.  At the edges  $f_Y(-2) = f_Y(+3)= 0.1$  (half value) holds.

{{BlueBox|TEXT=
'''(2)'''  Same setting as for  $(1)$.  What are the probabilities  ${\rm Pr}(X = 0)$,   ${\rm Pr}(0.5 \le X \le 1.5)$,   ${\rm Pr}(Y = 0)$   and  ${\rm Pr}(0.5 \le Y \le 1.5)$ .}}

* ${\rm Pr}(X = 0)={\rm Pr}(Y = 0) \equiv 0$   ⇒   Probability of a discrete random variable to take exactly a certain value.
* The other two probabilities can be obtained by integration over the PDF in the range  $+0.5\ \text{...} \ +\hspace{-0.1cm}1.5$.
* Or:  ${\rm Pr}(0.5 \le X \le 1.5)= F_X(1.5) - F_X(0.5) = 0.8944-0.1056 = 0.7888$. Correspondingly:  ${\rm Pr}(0.5 \le Y \le 1.5)= 0.7-0.5=0.2$.

{{BlueBox|TEXT=
'''(3)'''  Same settings as before.  How must the standard deviation  $\sigma_X$  be changed so that with the same mean  $m_X$  it holds for the second order moment:  $P_X=2$ ?}}

* According to Steiner's theorem:  $P_X=m_X^2 + \sigma_X^2$   ⇒   $\sigma_X^2 = P_X-m_X^2 = 2 - 1^2 = 1 $   ⇒   $\sigma_X = 1$.

{{BlueBox|TEXT=
'''(4)'''  Same settings as before:  How must the parameters  $y_{\rm min}$  and  $y_{\rm max}$  of the rectangular PDF be changed to yield  $m_Y = 0$  and  $\sigma_Y^2 = 0.75$?}}

* Starting from the previous setting  $(y_{\rm min} = -2, \ y_{\rm max} = +3)$  we change  $y_{\rm max}$ until  $\sigma_Y^2 = 0.75$  occurs   ⇒   $y_{\rm max} = 1$.
* The width of the rectangle is now  $3$.  The desired mean   $m_Y = 0$  is obtained by shifting:  $y_{\rm min} = -1.5, \ y_{\rm max} = +1.5$.
* You could also consider that for a mean-free random variable  $(y_{\rm min} = -y_{\rm max})$  the following equation holds:   $\sigma_Y^2 = y_{\rm max}^2/3$.

{{BlueBox|TEXT=
'''(5)'''  For which of the adjustable distributions is the Charlier skewness  $S \ne 0$ ? }}

* The Charlier's skewness denotes the third central moment related to  $σ_X^3$   ⇒  $S_X = \mu_3/σ_X^3$  $($valid for the random variable  $X)$.
* If the PDF  $f_X(x)$  is symmetric around the mean  $m_X$  then the parameter  $S_X$  is always zero.
* Exponential distribution:  $S_X =2$;  Rayleigh distribution:  $S_X =0.631$   $($both independent of  $λ_X)$;   Rice distribution:  $S_X >0$  $($dependent of  $C_X, \ λ_X)$.
* With the Weibull distribution, the Charlier skewness  $S_X$  can be zero, positive or negative,  depending on the PDF parameter  $k_X$.
*  Weibull distribution,  $\lambda_X=0.4$:  With  $k_X = 1.5$  ⇒   PDF is curved to the left  $(S_X > 0)$;   $k_X = 7$  ⇒   PDF is curved to the right  $(S_X < 0)$.

{{BlueBox|TEXT=
'''(6)'''  Select  $\text{Red: Gaussian PDF}\ (m_X = 1, \ \sigma_X = 0.4)$  and  $\text{Blue: Gaussian PDF}\ (m_X = 0, \ \sigma_X = 1)$.  What is the kurtosis in each case?}}

* For each Gaussian distribution the kurtosis has the same value:   $K_X = K_Y =3$.  Therefore,  $K-3$  is called "excess".
*This parameter can be used to check whether a given random variable can be approximated by a Gaussian distribution.

{{BlueBox|TEXT=
'''(7)'''  For which distributions does a significantly smaller kurtosis value result than  $K=3$?  And for which distributions does a significantly larger one?}}

* $K<3$  always results when the PDF values are more concentrated around the mean than in the Gaussian distribution.
* This is true, for example, for the uniform distribution  $(K=1.8)$  and for the triangular distribution  $(K=2.4)$.
* $K>3$,  if the PDF offshoots are more pronounced than for the Gaussian distribution.  Example:  Exponential PDF  $(K=9)$.

{{BlueBox|TEXT=
'''(8)'''  Select  $\text{Red: Exponential PDF}\ (\lambda_X = 1)$  and  $\text{Blue: Laplace PDF}\ (\lambda_Y = 1)$.  Interpret the differences.}}

* The Laplace distribution is symmetric around its mean  $(S_Y=0, \ m_Y=0)$  unlike the exponential distribution  $(S_X=2, \ m_X=1)$.
* The even moments  $m_2, \ m_4, \ \text{...}$  are equal,  for example:  $P_X=P_Y=2$.  But not the variances:  $\sigma_X^2 =1, \ \sigma_Y^2 =2$.
* The probabilities  ${\rm Pr}(|X| < 2) = F_X(2) = 0.864$  and  ${\rm Pr}(|Y| < 2) = F_Y(2) - F_Y(-2)= 0.932 - 0.068 = 0.864$  are equal.
* In the Laplace PDF, the values are more tightly concentrated around the mean than in the exponential PDF:  $K_Y =6 < K_X = 9$.

{{BlueBox|TEXT=
'''(9)'''  Select  $\text{Red: Rice PDF}\ (\lambda_X = 1, \ C_X = 1)$  and  $\text{Blue: Rayleigh PDF}\ (\lambda_Y = 1)$.  Interpret the differences.}}

*  With  $C_X = 0$  the Rice PDF transitions to the Rayleigh PDF.  A larger  $C_X$  improves the performance, e.g., in mobile communications.
*  Both, in  "Rayleigh"  and  "Rice"  the abscissa is the magnitude  $A$  of the received signal.  Favorably, if  ${\rm Pr}(A \le A_0)$  is small  $(A_0$  given$)$.
*  For  $C_X \ne 0$  and equal  $\lambda$  the Rice CDF is below the Rayleigh CDF   ⇒   smaller  ${\rm Pr}(A \le A_0)$  for all  $A_0$.

{{BlueBox|TEXT=
'''(10)'''  Select  $\text{Red: Rice PDF}\ (\lambda_X = 0.6, \ C_X = 2)$.  By which distribution  $F_Y(y)$  can this Rice distribution be well approximated? }}

*  The kurtosis   $K_X = 2.9539 \approx 3$  indicates the Gaussian distribution.   Favorable parameters:  $m_Y = 2.1 > C_X, \ \ \sigma_Y = \lambda_X = 0.6$.
*  The larger tht quotient  $C_X/\lambda_X$  is, the better the Rice PDF is approximated by a Gaussian PDF.
*  For large   $C_X/\lambda_X$  the Rice PDF has no more similarity with the Rayleigh PDF.

{{BlueBox|TEXT=
'''(11)'''  Select  $\text{Red: Weibull PDF}\ (\lambda_X = 1, \ k_X = 1)$  and  $\text{Blue: Weibull PDF}\ (\lambda_Y = 1, \ k_Y = 2)$. Interpret the results. }}

*  The Weibull PDF  $f_X(x)$  is identical to the exponential PDF and  $f_Y(y)$  to the Rayleigh PDF.
*  However, after best fit, the parameters  $\lambda_{\rm Weibull} = 1$  and  $\lambda_{\rm Rayleigh} = 0.7$ differ.
*  Moreover, it holds  $f_X(x = 0) \to \infty$  for  $k_X < 1$.  However, this does not have the affect of infinite moments.

{{BlueBox|TEXT=
'''(12)'''  Select  $\text{Red: Weibull PDF}\ (\lambda_X = 1, \ k_X = 1.6)$  and   $\text{Blue: Weibull PDF}\ (\lambda_Y = 1, \ k_Y = 5.6)$.  Interpret the Charlier skewness. }}

*  One observes:   For the PDF parameter  $k < k_*$  the Charlier skewness is positive and for  $k > k_*$  negative.  It is approximately  $k_* = 3.6$.

{{BlueBox|TEXT=
'''(13)'''  Select  $\text{Red: Semicircle PDF}\ (m_X = 0, \ R_X = 1)$  and  $\text{Blue: Parabolic PDF}\ (m_Y = 0, \ R_Y = 1)$.  Vary the parameter  $R$  in each case. }}

*  The PDF in each case is mean-free and symmetric  $(S_X = S_Y =0)$  with  $\sigma_X^2 = 0.25, \ K_X = 2$  respectively,  $\sigma_Y^2 = 0.2, \ K_Y \approx 2.2$.

==Applet Manual==
 
[[File:Bildschirm_WDF_VTF_neu.png|right|600px|frame|Screenshot of the German version]]
    '''(A)'''     Selection of the distribution  $f_X(x)$  (red curves and output values)

    '''(B)'''     Parameter input for the "red distribution" via slider

    '''(C)'''     Selection of the distribution  $f_Y(y)$  (blue curves and output values)

    '''(D)'''     Parameter input for the "red distribution" via slider

    '''(E)'''     Graphic area for the probability density function (PDF)

    '''(F)'''     Graphic area for the distribution function (CDF)

    '''(G)'''     Numerical output for the "red distribution"

    '''(H)'''     Numerical output for the "blue distribution"

    '''( I )'''     Input of  $x_*$  and  $y_*$  abscissa values for the numerics outputs

    '''(J)'''     Experiment execution area:   task selection

    '''(K)'''     Experiment execution area:   task description

    '''( L)'''     Experiment execution area:   sample solution
 

'''Selection options for''' for  $\rm A$  and  $\rm C$:  

Gaussian distribution,   uniform distribution,   triangular distribution,   exponential distribution,   Laplace distribution,   Rayleigh distribution,  Rice distribution,   Weibull distribution,   Wigner semicircle distribution,   Wigner parabolic distribution,   Cauchy distribution.

The following »'''integral parameters'''« are output  $($with respect to $X)$:  

Linear mean value  $m_X = {\rm E}\big[X \big]$,   second order moment  $P_X ={\rm E}\big[X^2 \big] $,   variance  $\sigma_X^2 = P_X - m_X^2$,   standard deviation  $\sigma_X$,  Charlier's skewness  $S_X$,   kurtosis  $K_X$.

'''In all applets top right''':    Changeable graphical interface design   ⇒   '''Theme''':
* Dark:   black background  (recommended by the authors).
* Bright:   white background  (recommended for beamers and printouts)
* Deuteranopia:   for users with pronounced green–visual impairment
* Protanopia:   for users with pronounced red–visual impairment
 
==About the Authors==
 
This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].
*The first version was created in 2005 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29|»Bettina Hirner«]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29| »Günter Söder« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29| »Klaus Eichin« ]]).

*In 2019 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Matthias_Niller_.28Ingenieurspraxis_Math_2019.29|»Matthias Niller«]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Benedikt_Leible_.28at_L.C3.9CT_since_2017.29| »Benedikt Leible« ]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29| »Tasnád Kernetzky« ]] ).

*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|»Carolin Mirschina«]]  in the context of a working student activity. 

*The conversion of this applet was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ $\text{Studienzuschüsse}$]  (TUM Department of Electrical and Computer Engineering).  We thank.

==Once again: Open Applet in new Tab==

{{LntAppletLinkEnDe|wdf-vtf_en|wdf-vtf}}