From bdc4323142fe07bffe581fc1595e1e0bab0f8f8a Mon Sep 17 00:00:00 2001 From: Philipp Le Date: Sun, 3 May 2020 00:04:47 +0200 Subject: Adding references --- .gitignore | 1 + DCS.bib | 37 ++++ Makefile | 10 +- chapter00/CaesarCipher.jpg | Bin 0 -> 22208 bytes chapter00/OpticalTelegraph.jpg | Bin 0 -> 11480 bytes chapter00/content.tex | 181 ---------------- chapter00/content_ch00.tex | 181 ++++++++++++++++ chapter00/exercise00.tex | 10 + chapter00/preface.tex | 22 +- chapter01/content.tex | 371 --------------------------------- chapter01/content_ch01.tex | 376 +++++++++++++++++++++++++++++++++ chapter01/exercise01.tex | 7 + chapter02/content.tex | 451 ---------------------------------------- chapter02/content_ch02.tex | 456 +++++++++++++++++++++++++++++++++++++++++ common/settings.tex | 41 +++- main/DCS.tex | 10 +- main/chapter00.tex | 11 +- main/chapter01.tex | 4 +- main/chapter02.tex | 2 +- 19 files changed, 1147 insertions(+), 1024 deletions(-) create mode 100644 DCS.bib create mode 100644 chapter00/CaesarCipher.jpg create mode 100644 chapter00/OpticalTelegraph.jpg delete mode 100644 chapter00/content.tex create mode 100644 chapter00/content_ch00.tex create mode 100644 chapter00/exercise00.tex delete mode 100644 chapter01/content.tex create mode 100644 chapter01/content_ch01.tex create mode 100644 chapter01/exercise01.tex delete mode 100644 chapter02/content.tex create mode 100644 chapter02/content_ch02.tex diff --git a/.gitignore b/.gitignore index 567609b..c9c997a 100644 --- a/.gitignore +++ b/.gitignore @@ -1 +1,2 @@ build/ +DCS.bib.bak diff --git a/DCS.bib b/DCS.bib new file mode 100644 index 0000000..9cb3a4d --- /dev/null +++ b/DCS.bib @@ -0,0 +1,37 @@ +% This file was created with JabRef 2.7.1. +% Encoding: UTF8 + +@MISC{Berberich2013, + author = {Hubert Berberich}, + title = {Cipher disc for substitution cipher, manufacturer: Linge, Pleidelsheim + (Germany)}, + howpublished = {Wikimedia}, + year = {2013}, + note = {License: Public Domain, Accessed: 2020-05-02}, + owner = {ple}, + timestamp = {2020.05.02}, + url = {https://en.wikipedia.org/wiki/File:CipherDisk2000.jpg} +} + +@INPROCEEDINGS{friis1946, + author = {Harald Trap Friis}, + title = {A Note on a Simple Transmission Formula}, + booktitle = {IRE Proc.}, + year = {1946}, + pages = {254 -- 256}, + month = {May}, + owner = {ple}, + timestamp = {2019.04.17} +} + +@MISC{WikiSemaphore, + author = {Unknown}, + title = {Chappe's semaphore}, + howpublished = {Wikimedia}, + year = {19th century}, + note = {License: Public Domain, Accessed: 2020-05-02}, + owner = {ple}, + timestamp = {2020.05.02}, + url = {https://en.wikipedia.org/wiki/File:Chappe_semaphore.jpg} +} + diff --git a/Makefile b/Makefile index 34e06e8..581abbc 100644 --- a/Makefile +++ b/Makefile @@ -4,11 +4,15 @@ BUILD_DIR = build LATEXMK = latexmk -pdf -silent -synctex=1 LATEXMK_PVC = $(LATEXMK) -pvc -ALL_TARGETS = $(BUILD_DIR)/chapter00.pdf $(BUILD_DIR)/chapter01.pdf $(BUILD_DIR)/chapter02.pdf -COMMON_DEPS = common/settings.tex common/titlepage.tex common/acronym.tex common/imprint.tex +ALL_CHAPTERS = $(BUILD_DIR)/chapter00.pdf $(BUILD_DIR)/chapter01.pdf $(BUILD_DIR)/chapter02.pdf +COMMON_DEPS = common/settings.tex common/titlepage.tex common/acronym.tex common/imprint.tex DCS.bib -all: $(ALL_TARGETS) +all: chapters complete +.PHONY: chapters +chapters: $(ALL_CHAPTERS) + +.PHONY: complete complete: $(BUILD_DIR)/DCS.pdf clean: diff --git a/chapter00/CaesarCipher.jpg b/chapter00/CaesarCipher.jpg new file mode 100644 index 0000000..ad1dc9a Binary files /dev/null and b/chapter00/CaesarCipher.jpg differ diff --git a/chapter00/OpticalTelegraph.jpg b/chapter00/OpticalTelegraph.jpg new file mode 100644 index 0000000..c3faf86 Binary files /dev/null and b/chapter00/OpticalTelegraph.jpg differ diff --git a/chapter00/content.tex b/chapter00/content.tex deleted file mode 100644 index 3ac57ff..0000000 --- a/chapter00/content.tex +++ /dev/null @@ -1,181 +0,0 @@ -\chapter{Course Organization} - -\section{About the Lecturer} - -\begin{minipage}{0.2\linewidth} - \includegraphics[width=0.8\linewidth]{../chapter00/Philipp.jpg} -\end{minipage} -\hfill -\begin{minipage}{0.75\linewidth} - Philipp Le - - philipp.le@st.ovgu.de -\end{minipage} - -\vspace{1em} - -\underline{Academic CV:} - -\begin{tabular}{p{0.2\linewidth}p{0.7\linewidth}} - 2011 -- 2015 & \makecell[l]{\textbf{B.\,Sc.} in Electrical Engineering and Information Technology,\\ Otto-von-Guericke-University Magdeburg} \\[1.5em] - 2016 & Exchange student, TU Tampere, Finland\\[0.5em] - 2015 -- 2019 & \makecell[l]{\textbf{M.\,Sc.} in Electrical Engineering and Information Technology,\\ Specialization: Communication Technology and Microwave Engineering\\ Otto-von-Guericke-University Magdeburg} \\[0.5em] -\end{tabular} - -\vspace{2em} - -\underline{Current occupation:} - -\begin{quote} - Since 2015: Hardware and Firmware Developer at - - \textbf{metraTec GmbH, Magdeburg} - - located in the Wissenschaftshafen. - - \vspace{1em} - - Main topics: - \begin{itemize} - \item Low power radio devices, IoT - \item Radio localization - \item Microwave engineering - \end{itemize} -\end{quote} - - -\section{Course Overview} - -\textit{Subject to change} - -Facts on the course: -\begin{itemize} - \item Duration: 1 semester (summer semester), 14 weeks - \item E-Learning webpage: \url{https://elearning.ovgu.de/course/view.php?id=7849} - \item Course material is published on the E-Learning webpage: - \begin{itemize} - \item Lecture notes - \item Exercise sheets - \item Supplementary material - \end{itemize} - \item Work load: 150 h (1 h = 45 minutes), yielding \textbf{5 ECTS} - \begin{itemize} - \item 42 h of attendance - \item 108 h autonomous work - \end{itemize} - \item The research report mentioned in the module description is neglected in this semester, due to a lack of time. - \item Written exam (120 minutes) - \item Attending lectures or exercises not mandatory, but recommended. -\end{itemize} - -Dates: -\begin{itemize} - \item Every week: \textbf{Thursdays at 9:15 -- 10:45 (a.m.)} - \begin{itemize} - \item Exception: 2020-05-21 is a public holiday, no lecture! We will meet on 2020-05-19 at 15:15 -- 16:45 instead. - \end{itemize} - \item Additional date if necessary: Tuesdays at 15:15 -- 16:45. Will be explicitly announces on E-Learning. - \item We will use a videoconferencing tool for our meetings. I will announce details in the E-Learning. -\end{itemize} - - -\section{Lecture Details} - -Study organization: -\begin{itemize} - \item You are required to elaborate major parts of the course in \underline{self-study}. - \item The chapters of the lecture notes will be published in the E-Learning on weekly basis. - \item Please go through the lecture notes \underline{on your own}. - \item The lecture will be held in form of a \textbf{consultation} where we will discuss the main topics of the chapter. - \item A pre-requisite is that you have read the lecture notes beforehand. -\end{itemize} - -Remarks on the lecture notes: -\begin{itemize} - \item The lecture notes shall give you a good foundation for your self-studies. It contains the relevant course content. - \item If you find mistakes, don't hesitate to communicate them to me. I'm currently writing the lecture notes in nightshifts, which may lead to mistakes slipping through. ;) -\end{itemize} - - -\section{Exercise Details} - -Goal of the exercises: -\begin{itemize} - \item Strengthening your knowledge that you have gained in the lecture. - \item Bringing your knowledge into action. - \item Learning how to deal with the course material and further sources. - \item Getting used to the nature of questions which will be asked in the exam. However, exam questions will be different. -\end{itemize} - -Exercise in self-study: -\begin{itemize} - \item Exercise sheets will be published in the E-Learning. - \item Solve them individually or \underline{in groups}. - \item You may ask for clarification of an exercise task at any time. - \item \underline{You are required to put reasonable effort in solving the exercise.} This comprises referring to the lecture notes, supplementary material or other sources including the library and the internet. - \item If you get stuck, you can contact me. I will give hints. Please be sure that you have put reasonable effort in solving beforehand. -\end{itemize} - -Optional returning of solutions: -\begin{itemize} - \item You may voluntarily return your solutions to me via E-Learning. This is not mandatory. - \item If you wish to return your solutions, please do so within \underline{two weeks} after their publication. - \item I will skim your solutions. Unfortunately, it is not possible for me to provide you individual corrections, as I'm lecturing besides my full time job. - \item Of course, you may ask questions along with returning the exercises, preferably via e-mail or in the E-Learning. - \item If I see major issues, we will discuss them together. I will integrate this into the \textbf{consultation}. The Tuesday date may be optionally used, if necessary. - \item However, if nobody sends me a solution, we cannot discuss anything. -\end{itemize} - -Please be patient if I don't answer your requests immediately. I'm giving my best to respond as soon as possible. But, I'm having a full time job which is also consuming time. :) - - -\section{Homework} - -Your \underline{weekly} homework comprises: -\begin{itemize} - \item Going through the lecture notes and understand its contents. - \item If necessary, consulting further sources (library, internet) is strongly recommended. - \item Solving the exercises. -\end{itemize} - - -\section{Exam Details} - -\textit{All information are subject to change. There will be an announcement with details on the exam at the end of the course.} - -\begin{itemize} - \item Date: will be announced by the examination office - \item Duration: 120 minutes - \item Information on tools allowed will be announced at the end of the lectures. - \item The exam covers the complete course content (all chapters, all exercise sheets). - \item The exam questions call for different levels of skills: - \begin{itemize} - \item Memorized knowledge - \item Bringing knowledge into practise - \item Creatively deriving solutions from the knowledge - \end{itemize} - \item Grading scale guideline: - \begin{itemize} - \item 1.0 (very good): Exam performance is outstanding and goes beyond the requirements. - \item 2.0 (good): Exam performance fulfils overall requirements. - \item 3.0 (satisfactory): Requirements are fulfilled in general. There are minor deficiencies. - \item 4.0 (pass): Exam performance meets the requirements, but shows major deficiencies. - \item 5.0 (fail): Exam performance is insufficient. - \end{itemize} - \item Individual work. Cheating will result in a 5.0 grade (failed). -\end{itemize} - - -\section{Feedback Poll} - -A feedback poll is required by the universities regulations for quality assurance in the teaching. Furthermore, the poll will help me to improve this course. - -\textbf{So I kindly ask you to participate in the poll.} - -\begin{itemize} - \item The poll will be at the end of the course. - \item There is an online form. - \item The poll is completely anonymous. - \item Participating is voluntary for you. - \item Of course, the poll will not affect the grading. It is anonymous. -\end{itemize} diff --git a/chapter00/content_ch00.tex b/chapter00/content_ch00.tex new file mode 100644 index 0000000..36f2a64 --- /dev/null +++ b/chapter00/content_ch00.tex @@ -0,0 +1,181 @@ +\chapter{Course Organization} + +\section{About the Lecturer} + +\begin{minipage}{0.2\linewidth} + \includegraphics[width=0.8\linewidth]{../chapter00/Philipp.jpg} +\end{minipage} +\hfill +\begin{minipage}{0.75\linewidth} + Philipp Le + + philipp.le@st.ovgu.de +\end{minipage} + +\vspace{1em} + +\underline{Academic CV:} + +\begin{tabular}{p{0.2\linewidth}p{0.7\linewidth}} + 2011 -- 2015 & \makecell[l]{\textbf{B.\,Sc.} in Electrical Engineering and Information Technology,\\ Otto-von-Guericke-University Magdeburg} \\[1.5em] + 2016 & Exchange student, TU Tampere, Finland\\[0.5em] + 2015 -- 2019 & \makecell[l]{\textbf{M.\,Sc.} in Electrical Engineering and Information Technology,\\ Specialization: Communication Technology and Microwave Engineering\\ Otto-von-Guericke-University Magdeburg} \\[0.5em] +\end{tabular} + +\vspace{2em} + +\underline{Current occupation:} + +\begin{quote} + Since 2015: Hardware and Firmware Developer at + + \textbf{metraTec GmbH, Magdeburg} + + located in the Wissenschaftshafen. + + \vspace{1em} + + Main topics: + \begin{itemize} + \item Low power radio devices, IoT + \item Radio localization + \item Microwave engineering + \end{itemize} +\end{quote} + + +\section{Course Overview} + +\textit{Subject to change} + +Facts on the course: +\begin{itemize} + \item Duration: 1 semester (summer semester), 14 weeks + \item E-Learning webpage: \url{https://elearning.ovgu.de/course/view.php?id=7849} + \item Course material is published on the E-Learning webpage: + \begin{itemize} + \item Lecture notes + \item Exercise sheets + \item Supplementary material + \end{itemize} + \item Work load: 150 h (1 h = 45 minutes), yielding \textbf{5 ECTS} + \begin{itemize} + \item 42 h of attendance + \item 108 h autonomous work + \end{itemize} + \item The research report mentioned in the module description is neglected in this semester, due to a lack of time. + \item Written exam (120 minutes) + \item Attending lectures or exercises not mandatory, but recommended. +\end{itemize} + +Dates: +\begin{itemize} + \item Every week: \textbf{Thursdays at 9:15 -- 10:45 (a.m.)} + \begin{itemize} + \item Exception: 2020-05-21 is a public holiday, no lecture! We will meet on 2020-05-19 at 15:15 -- 16:45 instead. + \end{itemize} + \item Additional date if necessary: Tuesdays at 15:15 -- 16:45. Will be explicitly announced on E-Learning. + \item We will use a videoconferencing tool for our meetings. I will announce details in the E-Learning. +\end{itemize} + + +\section{Lecture Details} + +Study organization: +\begin{itemize} + \item You are required to elaborate major parts of the course in \underline{self-study}. + \item The chapters of the lecture notes will be published in the E-Learning on weekly basis. + \item Please go through the lecture notes \underline{on your own}. + \item The lecture will be held in form of a \textbf{consultation} where we will discuss the main topics of the chapter. + \item A pre-requisite is that you have read the lecture notes beforehand. +\end{itemize} + +Remarks on the lecture notes: +\begin{itemize} + \item The lecture notes shall give you a good foundation for your self-studies. It contains the relevant course content. + \item If you find mistakes, don't hesitate to communicate them to me. I'm currently writing the lecture notes in nightshifts, which may lead to mistakes slipping through. ;) +\end{itemize} + + +\section{Exercise Details} + +Goal of the exercises: +\begin{itemize} + \item Strengthening your knowledge that you have gained in the lecture. + \item Bringing your knowledge into action. + \item Learning how to deal with the course material and further sources. + \item Getting used to the nature of questions which will be asked in the exam. However, exam questions will be different. +\end{itemize} + +Exercise in self-study: +\begin{itemize} + \item Exercise sheets will be published in the E-Learning. + \item Solve them individually or \underline{in groups}. + \item You may ask for clarification of an exercise task at any time. + \item \underline{You are required to put reasonable effort in solving the exercise.} This comprises referring to the lecture notes, supplementary material or other sources including the library and the internet. + \item If you get stuck, you can contact me. I will give hints. Please be sure that you have put reasonable effort in solving beforehand. +\end{itemize} + +Optional returning of solutions: +\begin{itemize} + \item You may voluntarily return your solutions to me via E-Learning. This is not mandatory. + \item If you wish to return your solutions, please do so within \underline{two weeks} after their publication. + \item I will skim your solutions. Unfortunately, it is not possible for me to provide you individual corrections, as I'm lecturing besides my full time job. + \item Of course, you may ask questions along with returning the exercises, preferably via e-mail or in the E-Learning. + \item If I see major issues, we will discuss them together. I will integrate this into the \textbf{consultation}. The Tuesday date may be optionally used, if necessary. + \item However, if nobody sends me a solution, we cannot discuss anything. +\end{itemize} + +Please be patient if I don't answer your requests immediately. I'm giving my best to respond as soon as possible. But, I'm having a full time job which is also consuming time. :) + + +\section{Homework} + +Your \underline{weekly} homework comprises: +\begin{itemize} + \item Going through the lecture notes and understand its contents. + \item If necessary, consulting further sources (library, internet) is strongly recommended. + \item Solving the exercises. +\end{itemize} + + +\section{Exam Details} + +\textit{All information are subject to change. There will be an announcement with details on the exam at the end of the course.} + +\begin{itemize} + \item Date: will be announced by the examination office + \item Duration: 120 minutes + \item Information on tools allowed will be announced at the end of the lectures. + \item The exam covers the complete course content (all chapters, all exercise sheets). + \item The exam questions call for different levels of skills: + \begin{itemize} + \item Memorized knowledge + \item Bringing knowledge into practise + \item Creatively deriving solutions from the knowledge + \end{itemize} + \item Grading scale guideline: + \begin{itemize} + \item 1.0 (very good): Exam performance is outstanding and goes beyond the requirements. + \item 2.0 (good): Exam performance fulfils overall requirements. + \item 3.0 (satisfactory): Requirements are fulfilled in general. There are minor deficiencies. + \item 4.0 (pass): Exam performance meets the requirements, but shows major deficiencies. + \item 5.0 (fail): Exam performance is insufficient. + \end{itemize} + \item Individual work. Cheating will result in a 5.0 grade (failed). +\end{itemize} + + +\section{Feedback Poll} + +A feedback poll is required by the universities regulations for quality assurance in the teaching. Furthermore, the poll will help me to improve this course. + +\textbf{So I kindly ask you to participate in the poll.} + +\begin{itemize} + \item The poll will be at the end of the course. + \item There is an online form. + \item The poll is completely anonymous. + \item Participating is voluntary for you. + \item Of course, the poll will not affect the grading. It is anonymous. +\end{itemize} diff --git a/chapter00/exercise00.tex b/chapter00/exercise00.tex new file mode 100644 index 0000000..ce5a690 --- /dev/null +++ b/chapter00/exercise00.tex @@ -0,0 +1,10 @@ +\phantomsection +\addcontentsline{toc}{section}{Exercise 0 - Warm Up} +\section*{Exercise 0 -- Warm Up} + +Make up your mind of communication technologies and how they can be used. + +\begin{enumerate} + \item Name 5 communication devices of your every-day life. + \item +\end{enumerate} \ No newline at end of file diff --git a/chapter00/preface.tex b/chapter00/preface.tex index 83e8ae1..4bea33e 100644 --- a/chapter00/preface.tex +++ b/chapter00/preface.tex @@ -2,6 +2,8 @@ \addcontentsline{toc}{chapter}{Preface: Digital Communication -- A Future Technology} \chapter*{Preface: Digital Communication -- A Future Technology} +\begin{refsection} + The current time during the Corona lockdown shows the great significance of telecommunication. It is self-evident to have access to online media and communication platforms. The internet helps to keep in touch with our loved ones. A huge amount of entertainment relies on internet access. The internet is highly integrated in our every-day life. Furthermore, it is a growing economy. There are not only the online services like social media or online warehouses which are going to expand their business in the future. The communication technologies, which the internet is built upon, are a growing and innovative market, too. Besides consumers, the \ac{B2B} market is a huge driver of innovation. It must be expected, that more devices get interconnected. This is a challenge because physical resources are limited. Nevertheless, it is a great possibility for the communication technology to advance. The technologies, which are able to cope with the new requirementes, are digital. This where the course on \emph{Digital Communication Systems} starts. You are warmly welcomed! \addcontentsline{toc}{section}{History of Communications} @@ -22,9 +24,21 @@ Transferring information became more important as societies advanced. \item In the 18th century, semaphore lines had been built. They used visual telegraphy. Semaphores on fixed towers could display a set of symbols, which were relayed along the line. \end{itemize} -\todo{Figure Caesar cipher} - -\todo{Semaphores, beacons, lighthouses} +\begin{minipage}{0.45\linewidth} + \begin{figure}[H] + \centering + \includegraphics[width=\linewidth]{../chapter00/CaesarCipher.jpg} + \caption[Caesar cipher]{Caesar cipher. Each letter is replaced by another one which is a fixed number of letters away from the original one. \licensequote{\cite{Berberich2013}}{Hubert Berberich}{Public Domain}} + \end{figure} +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \begin{figure}[H] + \centering + \includegraphics[width=\linewidth]{../chapter00/OpticalTelegraph.jpg} + \caption[Optical telegraph]{Optical telegraph \cite{WikiSemaphore}} + \end{figure} +\end{minipage} The research of electricity created the foundations for modern communication systems. Electrical telegraphy speeded up telecommunication. At the end of the 19th century, electromagnetic waves have been discovered. James Clerk Maxwell postulated them in 1865. Heinrich Hertz produced the first electromagnetic waves in 1887. The potential had soon been acknowledged by inventors, who developed first radios. The era of analogue radio communication began. The \index{vacuum tube} vacuum tube became an important component in radio electronics. @@ -68,3 +82,5 @@ Understanding digital communication systems is of great importance in many engin \item \textbf{Energy Sector} -- \end{itemize} +\printbibliography[heading=subbibliography] +\end{refsection} diff --git a/chapter01/content.tex b/chapter01/content.tex deleted file mode 100644 index 38706c1..0000000 --- a/chapter01/content.tex +++ /dev/null @@ -1,371 +0,0 @@ -\chapter{Communication Systems} - -\section{What is Communication?} - -\index{communication} -\textbf{Communication} (from Latin \emph{communicare}, ``to share'') is the act of conveying information from one entity to another using mutually understood sign and symbols. - -\index{information} -\textbf{Information} is the knowledge which is being conveyed from the source to the recipient. Information results in increased knowledge at the recipient's side. - -Many research areas concern with communication and information: -\begin{itemize} - \item Information theory: Quantification, storage, communication and information in general - \item Communication studies: Human communication - \item Linguistics: Language as a carrier of information - \item Biosemiotics: Communication in and between living organisms - \item ... -\end{itemize} - -Communication and information are general terms. \textbf{Digital communication} concerns with the technology of conveying information using discrete signals. A \textbf{digital communication system} is a set of components and processes which implement digital communication. The signals carrying the information in a digital communication system are usually electromagnetic waves. - - -\section{Objectives and Distinction from Other Subjects} - -This course will provide an understanding of how a digital communication system can be described. You will learn methods to describe information in their physical form as signals as well as system components. - -The theory of digital communication system is strongly connected to other subjects, for example: -\begin{itemize} - \item Information and coding theory - \item Computer networks - \item Statistics - \item Signals and systems - \item Microwave engineering - \item Electronics -\end{itemize} -There are courses at this university which give you a deeper insight into these subjects. - - -\section{Components of A Communication System} - - -\subsection{Communication Model} - -%\todo{citation} -Claude Shannon and Warren Weaver were engineers at the Bell Telephone Labs, USA. They developed the \index{Shannon-Weaver model} \textbf{Shannon-Weaver Model} (Figure \ref{fig:ch01:shannon_weaver_model}). - -\begin{figure}[H] - \centering - \begin{tikzpicture} - \draw node[draw, block](Source){Information\\ source}; - \draw node[draw, block, right=of Source](STrans){Transducer\\ (sender)}; - \draw node[draw, block, right=of STrans](TX){Transmitter}; - \draw node[draw, block, below right=of TX](Ch){Transmission\\ channel}; - \draw node[draw, block, below left=of Ch](RX){Receiver}; - \draw node[draw, block, left=of RX](RTrans){Transducer\\ (recipient)}; - \draw node[draw, block, left=of RTrans](Sink){Information\\ sink}; - - \draw[-latex] (Source) -- node[midway, align=center, above]{a} (STrans); - \draw[-latex] (STrans) -- node[midway, align=center, above]{b} (TX); - \draw[-latex] (TX) -| node[midway, align=center, above left]{c} (Ch); - \draw[-latex] (Ch) |- node[midway, align=center, above left]{d} (RX); - \draw[-latex] (RX) -- node[midway, align=center, above]{e} (RTrans); - \draw[-latex] (RTrans) -- node[midway, align=center, above]{f} (Sink); - \end{tikzpicture} - \caption{Shannon-Weaver model of communication} - \label{fig:ch01:shannon_weaver_model} -\end{figure} - -\begin{description} - \item[Information source] The information is created here. - \item[Signal a] The original information is represented in physical form by a signal. - \item[Transducer (sender)] The \index{Shannon-Weaver model!transducer} transducer converts the signal from one physical form to another. - \item[Signal b] The signal is in a form which can be processed by the transmitter. - \item[Transmitter] The information is modulated on a carrier, which can be transmitted through the transmission channel. - \item[Signal c] The information is modulated on a carrier and can pass through the transmission channel. - \item[Tansmission channel] The physical system through which the modulated information passes. \index{transmission channels} Transmission channels are noisy and add disturbances to the information. - \item[Signal d] It is basically the Signal c. However, noise and disturbances have been added. - \item[Receiver] The receiver extracts the information from the carrier. Information must be reconstructed from the noisy input signal. - \item[Signal e] The output signal of the receiver. - \item[Transducer (recipient)] The signal must be converted into a physical form which can be processed by the information sink. - \item[Signal f] The signal carries the information in a form which can be used by the information sink. - \item[Information sink] The endpoint of the information. It uses the information to gain knowledge. -\end{description} - -\paragraph{Example: Cell phone} - -\begin{enumerate} - \item The information source is the brain. - \item Electrical impulses and molecules are conveyed by the nerves to the vocal cords (transducer 1). Vocal cords convert the signals to sound. - \item The sound is converted to an electrical signal by a microphone (transducer 2). - \item The electrical pulses are modulated on a radio carrier (transmitter). - \item Radio waves are transmitted over the air (transmission channel). - \item A noisy signal is received. The receivers demodulates the information from the radio carrier. - \item The analogue electrical signal is converted into sound by a speaker (transducer 3). - \item The sound reaches the ear that converts them to electrical pulses (transducer 4). - \item Electrical impulses and molecules are conveyed by the nerves to the brain (information sink). -\end{enumerate} - - -\subsection{Classification of Signals} - -A \index{signal} signal conveys information in a form that can be processed by components of the communication systems. - -\begin{figure}[H] - \centering - \begin{tikzpicture} - \draw node[block](Main){\textbf{Signals carrying}\\ \textbf{information}}; - \draw node[block, below left=of Main](Analogue){Analogue}; - \draw node[block, below right=of Main](Digital){Digital}; - \draw node[block, below left=of Analogue](TimeCont){Time\\ continuous}; - \draw node[block, below right=of Analogue](TimeDis){Time\\ discrete}; - - \draw [-latex] (Main) -- (Analogue); - \draw [-latex] (Main) -- (Digital); - \draw [-latex] (Analogue) -- (TimeCont); - \draw [-latex] (Analogue) -- (TimeDis); - \end{tikzpicture} - \caption{Classification of signals carrying information} - \label{fig:ch01:signals_classif} -\end{figure} - -\paragraph{Analogue signals.} - -\index{signal!analogue signal} -\index{signal!value-continuous} -Analogue signals are represented by values out of a continuous range (\emph{value-continuous}). The range can be limited. However, each real value in this range can be taken. - -Examples: -\begin{itemize} - \item Acoustic signals (speech, sound) - \item Electric signals (voltage, current) - \item Light signals (microscope, photograph) -\end{itemize} - -\index{signal!time-continuous} -\index{signal!time-discrete} -Analogue signals can be time-continuous or time-discrete. \emph{Time-continuity} means that the signal is defined at any real point of time. A \emph{time-discrete} signal is only defined at certain time instances. The number of time instances can be unlimited. However, the signal is not defined between two time points. - -\begin{figure}[H] - \centering - \begin{adjustbox}{scale=0.8} - \begin{tikzpicture} - \draw node[draw, block](Continuous){Value-continuous,\\ time-continuous\\ signal}; - \draw node[draw, block, right=3cm of Continuous](Sampled){Value-continuous,\\ time-discrete\\ signal}; - \draw node[draw, block, right=3cm of Sampled](Digital){Value-discrete,\\ time-discrete\\ signal}; - - \draw [-latex] (Continuous) -- node[midway, align=center, above]{Sampling} (Sampled); - \draw [-latex] (Sampled) -- node[midway, align=center, above]{Quantization} (Digital); - - \draw[decorate, decoration={brace, amplitude=3mm, mirror}] ([yshift=-5mm] Continuous.south west) -- ([yshift=-5mm] Sampled.south east) node[midway, below, yshift=-3mm]{\textbf{Analogue}}; - \draw[decorate, decoration={brace, amplitude=3mm, mirror}] ([yshift=-5mm] Digital.south west) -- ([yshift=-5mm] Digital.south east) node[midway, below, yshift=-3mm]{\textbf{Digital}}; - \end{tikzpicture} - \end{adjustbox} - \caption{Conversion from analogue to digital signals} - \label{fig:ch01:signals_sampling} -\end{figure} - -\begin{figure}[H] - \centering - \includegraphics{../chapter01/Signal_Analogue.jpg} - \caption[An analogue, value-continuous, time-continuous signal]{An analogue, value-continuous, time-continuous signal. Both time and value can be any real number.} - \label{fig:ch01:Signal_Analogue} -\end{figure} - -\begin{figure}[H] - \centering - \includegraphics{../chapter01/Signal_TimeDiscr.jpg} - \caption[An analogue, value-continuous, but time-discrete signal]{An analogue, value-continuous, but time-discrete signal. Only certain time points are valid, but the values can be any real number.} - \label{fig:ch01:Signal_TimeDiscr} -\end{figure} - -\paragraph{Digital signals.} - -\index{signal!digital signal} -\index{signal!value-discrete} -Digital signals are both time-discrete and value-discrete. \emph{Value-discrete} means that they can take only one state out of a limited set of states. - -\begin{figure}[H] - \centering - \includegraphics{../chapter01/Signal_Digital.jpg} - \caption[A digital, value-discrete, time-discrete signal]{A digital, value-discrete, time-discrete signal. Only certain time points and a limited set of values are valid.} - \label{fig:ch01:Signal_Digital} -\end{figure} - -Examples: -\begin{itemize} - \item Text letters - \item Morse code - \item Coded data -\end{itemize} - -\textit{Remark:} In fact, the physical form of a digital signal is again an analogue signal. A binary signal can take the discrete states ``high'' and ``low''. If the signal is on a wire, its states are represented by voltage levels, for example \SI{0}{V} and \SI{3.3}{V}. However, if processed by a digital system, the physical representation is of minor importance. Only the discrete, logical states are considered. - -\index{signal!binary signal} -A special kind of digital signal is the \textbf{binary signal}. It has two discrete states. - - -\subsection{Transmission Channels} - -\index{transmission channel} -Digital communication systems employ electromagnetic waves to convey information. Therefore, only transmission channels transporting electromagnetic waves are considered. - -\begin{figure}[H] - \centering - \begin{tikzpicture} - \draw node[block](Main){\textbf{Transmission}\\ \textbf{channels}}; - \draw node[block, below left=of Main](Wired){Wired\\ channels}; - \draw node[block, below right=of Main](Wireless){Wireless\\ channels}; - - \draw [-latex] (Main) -- (Wired); - \draw [-latex] (Main) -- (Wireless); - \end{tikzpicture} - \caption{Classification of transmission channels} - \label{fig:ch01:trans_ch_classif} -\end{figure} - -\paragraph{Wired Channels.} - -The electromagnetic wave propagates along a transmission line. - -Examples of transmission lines: -\begin{itemize} - \item Cables - \begin{itemize} - \item Two wire, twisted-pair \index{twisted-pair cable} - \item Coaxial cable \index{coaxial cable} - \end{itemize} - \item Waveguides \index{waveguide} - \item Planar lines (on printed circuit boards or integrated circuits) - \begin{itemize} - \item Microstrip \index{microstrip} - \item Coplanar waveguide \index{coplanar waveguide} - \end{itemize} - \item Glass fibre (light is an electromagnetic wave, too) -\end{itemize} - -\paragraph{Wireless Channels.} - -The electromagnetic wave is not bound to a transmission line. It propagates through the space. A medium is not necessary. Electromagnetic wave can also travel through vacuum. - - -\section{Computer Networks} - - -This course focuses on the technologies which convey information between endpoints, using electromagnetic waves. The information, being conveyed, are called \textbf{data}. The handling of the data is a subject of computer science, especially \emph{computer networks} \index{computer network}. Since data processing is a part of digital communication systems, too, this digression shall give an overview about the employed concepts. - - -\subsection{Protocols} - -Modern communication systems convey information world-wide. These communication links are established over myriads of devices, which form a network. The biggest computer network is the internet. - -These devices mainly operate automatically without human interaction. Therefore, they are required to follow certain rules, which are called \textbf{communication protocols} \index{communication protocol}. Protocols define -\begin{itemize} - \item the structure and semantics of data, - \item synchronization of communication, and - \item possible error recovery methods. -\end{itemize} - -Protocols are standardized and must be implemented in every device, which interacts with other devices. Important standardization organizations are: -\begin{itemize} - \item The non-profit organization \textbf{\acf{IETF}} issues standards concerning the internet. The standards are called \emph{Request For Comment} (RFC) and are available for everyone for free. Example standards: \ac{IP}, \ac{HTTP} - \item The \textbf{\acf{IEEE}} has standards committees which develop and publish standards. With respect to the internet, the IEEE\,802 LAN/MAN Standards Committee is the most important one. Example standards: IEEE\,802.11 (Wifi) - \item The \textbf{\acf{ETSI}} is an independent, non-profit standardization organization. It is recognized by the European Council and officially responsible for standardization of information and communication technologies in Europe. Example standards: 3G (cell phone system), 4G (cell phone system), TETRA (professional mobile radio system) -\end{itemize} - - -\subsection{\acs{OSI} Model} - -There are many task which a digital communication systems must accomplish. -\begin{itemize} - \item An application processes user input and displays data to the user. - \item The application data must be reliably transferred over a network with many nodes. - \item The network is shared with other users and applications. - \item The network consists of many links using different physical transmission channels, for example, wired and wireless. -\end{itemize} -For each task, there are communication protocols to solve it. Communication protocols are grouped by the task which they fulfil. There is an increasing level of abstraction from the physical link to the application data. The \index{OSI model} \textbf{\acs{OSI} Model} (Figure \ref{fig:ch01:osi_model}) defines a layer structure for classifying communication protocols, which regards the level of abstraction. - -\begin{figure}[H] - \centering - \begin{tikzpicture}[ - layer/.style={ - rectangle, - minimum height=1cm, - minimum width=8cm - } - ] - \draw node[draw, layer](L7){Application layer}; - \draw node[draw, layer, below=0 of L7](L6){Presentation layer}; - \draw node[draw, layer, below=0 of L6](L5){Session layer}; - \draw node[draw, layer, below=0 of L5](L4){Transport layer}; - \draw node[draw, layer, below=0 of L4](L3){Network layer}; - \draw node[draw, layer, below=0 of L3](L2){Data link layer}; - \draw node[draw, layer, below=0 of L2](L1){Physical layer}; - - \node [anchor=east, align=right] at([xshift=-5mm] L7.west) {Layer 7}; - \node [anchor=east, align=right] at([xshift=-5mm] L6.west) {Layer 6}; - \node [anchor=east, align=right] at([xshift=-5mm] L5.west) {Layer 5}; - \node [anchor=east, align=right] at([xshift=-5mm] L4.west) {Layer 4}; - \node [anchor=east, align=right] at([xshift=-5mm] L3.west) {Layer 3}; - \node [anchor=east, align=right] at([xshift=-5mm] L2.west) {Layer 2}; - \node [anchor=east, align=right] at([xshift=-5mm] L1.west) {Layer 1}; - - \filldraw[fill=gray!60, draw=none] ([xshift=10mm, yshift=3mm] L7.north east) -- node[midway, above, anchor=south, align=center]{Level of\\ abstraction} ([xshift=15mm, yshift=3mm] L7.north east) -- ([xshift=12.5mm, yshift=-3mm] L1.south east); - \end{tikzpicture} - \caption[OSI Model with seven layers]{\ac{OSI} Model with seven layers} - \label{fig:ch01:osi_model} -\end{figure} - -\begin{table}[H] - \caption[Description of the layers of the OSI Model]{Description of the layers of the \ac{OSI} Model (Figure \ref{fig:ch01:osi_model}). The protocol data unit is the information } - \begin{tabular}{|l|l|p{0.5\linewidth}|} - \hline - Layer & PDU & Function \\ - \hline - \hline - 7: Application & Data & Processing user inputs, displaying data, providing services \\ - \hline - 6: Presentation & Data & Translation between network service and application (encryption, compression, etc.) \\ - \hline - 5: Session & Data & Managing sessions (retaining the communication state across multiple contacts) \\ - \hline - 4: Transport & Datagram, Segment & Reliable communication (segmentation, multiplexing, data loss detection) \\ - \hline - 3: Network & Packet & Data transfer across multiple nodes (addressing, routing, traffic control) \\ - \hline - 2: Data link & Frame & Transmission between two devices (medium access, flow control) \\ - \hline - 1: Physical & Symbol & Transmission over a physical medium \\ - \hline - \end{tabular} -\end{table} - -Each protocol has a standardized interface exposed to the upper layer, called \index{service access point} \textbf{\ac{SAP}}. They allow an upper layer protocol to execute functions of the lower layer protocol. These functions are, for example: -\begin{itemize} - \item Sending or receiving data - \item Control operations - \item Network registration and de-registration -\end{itemize} - -Protocol layers add own information to the data received from the upper layer. This additional information is required to provide the protocol's functionality. For example, the \acf{IP} needs to add the source and destination address, so that the packet can be routed to the correct endpoint. One can imagine this like data which is written on a letter, which is put into an envelope, which itself is put into another envelope, and so on. - -\begin{figure}[H] - \centering - \includegraphics{../chapter01/Frame_Wrapping.jpg} - \caption{Principle of adding more information in each protocol layer} - \label{fig:ch01:frame_construction} -\end{figure} - -Communication protocols may be exchanged in one layer without affecting the functionality of the other layers. For example, \ac{HTTP} operates on \acs{TCP}/\acs{IP}. But the \acf{IP} works on multiple physical links like Ethernet (IEEE\,802.3), Wifi (IEEE\,802.11) or 4G. The transmission media can even change along the communication path. Information travelling through the internet experience lots of \index{media change} \textbf{media changes}. - - -\begin{figure}[H] - \centering - \caption{Media change on the internet. } - \label{fig:ch01:media_changes} -\end{figure} - -This course on digital communication systems mainly considers the physical layer (layer 1) and the data link layer (layer 2). This physical layer converts the information to physical signals which then leave the device to be transmitted over a physical transmission channel. Networks, which are enabled by protocols of layer 3 and above, are outside the scope of this course. - - -\subsection{Network Topologies} - -\begin{itemize} - \item \textbf{Ring} - \item \textbf{Star} - \item \textbf{Tree} - \item \textbf{Chain} - \item \textbf{Bus} - \item \textbf{Mesh}, special form \emph{Full Mesh} -\end{itemize} - diff --git a/chapter01/content_ch01.tex b/chapter01/content_ch01.tex new file mode 100644 index 0000000..aeadf28 --- /dev/null +++ b/chapter01/content_ch01.tex @@ -0,0 +1,376 @@ +\chapter{Communication Systems} + +\begin{refsection} + +\section{What is Communication?} + +\index{communication} +\textbf{Communication} (from Latin \emph{communicare}, ``to share'') is the act of conveying information from one entity to another using mutually understood sign and symbols. + +\index{information} +\textbf{Information} is the knowledge which is being conveyed from the source to the recipient. Information results in increased knowledge at the recipient's side. + +Many research areas concern with communication and information: +\begin{itemize} + \item Information theory: Quantification, storage, communication and information in general + \item Communication studies: Human communication + \item Linguistics: Language as a carrier of information + \item Biosemiotics: Communication in and between living organisms + \item ... +\end{itemize} + +Communication and information are general terms. \textbf{Digital communication} concerns with the technology of conveying information using discrete signals. A \textbf{digital communication system} is a set of components and processes which implement digital communication. The signals carrying the information in a digital communication system are usually electromagnetic waves. + + +\section{Objectives and Distinction from Other Subjects} + +This course will provide an understanding of how a digital communication system can be described. You will learn methods to describe information in their physical form as signals as well as system components. + +The theory of digital communication system is strongly connected to other subjects, for example: +\begin{itemize} + \item Information and coding theory + \item Computer networks + \item Statistics + \item Signals and systems + \item Microwave engineering + \item Electronics +\end{itemize} +There are courses at this university which give you a deeper insight into these subjects. + + +\section{Components of A Communication System} + + +\subsection{Communication Model} + +%\todo{citation} +Claude Shannon and Warren Weaver were engineers at the Bell Telephone Labs, USA. They developed the \index{Shannon-Weaver model} \textbf{Shannon-Weaver Model} (Figure \ref{fig:ch01:shannon_weaver_model}). + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \draw node[draw, block](Source){Information\\ source}; + \draw node[draw, block, right=of Source](STrans){Transducer\\ (sender)}; + \draw node[draw, block, right=of STrans](TX){Transmitter}; + \draw node[draw, block, below right=of TX](Ch){Transmission\\ channel}; + \draw node[draw, block, below left=of Ch](RX){Receiver}; + \draw node[draw, block, left=of RX](RTrans){Transducer\\ (recipient)}; + \draw node[draw, block, left=of RTrans](Sink){Information\\ sink}; + + \draw[-latex] (Source) -- node[midway, align=center, above]{a} (STrans); + \draw[-latex] (STrans) -- node[midway, align=center, above]{b} (TX); + \draw[-latex] (TX) -| node[midway, align=center, above left]{c} (Ch); + \draw[-latex] (Ch) |- node[midway, align=center, above left]{d} (RX); + \draw[-latex] (RX) -- node[midway, align=center, above]{e} (RTrans); + \draw[-latex] (RTrans) -- node[midway, align=center, above]{f} (Sink); + \end{tikzpicture} + \caption{Shannon-Weaver model of communication} + \label{fig:ch01:shannon_weaver_model} +\end{figure} + +\begin{description} + \item[Information source] The information is created here. + \item[Signal a] The original information is represented in physical form by a signal. + \item[Transducer (sender)] The \index{Shannon-Weaver model!transducer} transducer converts the signal from one physical form to another. + \item[Signal b] The signal is in a form which can be processed by the transmitter. + \item[Transmitter] The information is modulated on a carrier, which can be transmitted through the transmission channel. + \item[Signal c] The information is modulated on a carrier and can pass through the transmission channel. + \item[Tansmission channel] The physical system through which the modulated information passes. \index{transmission channels} Transmission channels are noisy and add disturbances to the information. + \item[Signal d] It is basically the Signal c. However, noise and disturbances have been added. + \item[Receiver] The receiver extracts the information from the carrier. Information must be reconstructed from the noisy input signal. + \item[Signal e] The output signal of the receiver. + \item[Transducer (recipient)] The signal must be converted into a physical form which can be processed by the information sink. + \item[Signal f] The signal carries the information in a form which can be used by the information sink. + \item[Information sink] The endpoint of the information. It uses the information to gain knowledge. +\end{description} + +\paragraph{Example: Cell phone} + +\begin{enumerate} + \item The information source is the brain. + \item Electrical impulses and molecules are conveyed by the nerves to the vocal cords (transducer 1). Vocal cords convert the signals to sound. + \item The sound is converted to an electrical signal by a microphone (transducer 2). + \item The electrical pulses are modulated on a radio carrier (transmitter). + \item Radio waves are transmitted over the air (transmission channel). + \item A noisy signal is received. The receivers demodulates the information from the radio carrier. + \item The analogue electrical signal is converted into sound by a speaker (transducer 3). + \item The sound reaches the ear that converts them to electrical pulses (transducer 4). + \item Electrical impulses and molecules are conveyed by the nerves to the brain (information sink). +\end{enumerate} + + +\subsection{Classification of Signals} + +A \index{signal} signal conveys information in a form that can be processed by components of the communication systems. + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \draw node[block](Main){\textbf{Signals carrying}\\ \textbf{information}}; + \draw node[block, below left=of Main](Analogue){Analogue}; + \draw node[block, below right=of Main](Digital){Digital}; + \draw node[block, below left=of Analogue](TimeCont){Time\\ continuous}; + \draw node[block, below right=of Analogue](TimeDis){Time\\ discrete}; + + \draw [-latex] (Main) -- (Analogue); + \draw [-latex] (Main) -- (Digital); + \draw [-latex] (Analogue) -- (TimeCont); + \draw [-latex] (Analogue) -- (TimeDis); + \end{tikzpicture} + \caption{Classification of signals carrying information} + \label{fig:ch01:signals_classif} +\end{figure} + +\paragraph{Analogue signals.} + +\index{signal!analogue signal} +\index{signal!value-continuous} +Analogue signals are represented by values out of a continuous range (\emph{value-continuous}). The range can be limited. However, each real value in this range can be taken. + +Examples: +\begin{itemize} + \item Acoustic signals (speech, sound) + \item Electric signals (voltage, current) + \item Light signals (microscope, photograph) +\end{itemize} + +\index{signal!time-continuous} +\index{signal!time-discrete} +Analogue signals can be time-continuous or time-discrete. \emph{Time-continuity} means that the signal is defined at any real point of time. A \emph{time-discrete} signal is only defined at certain time instances. The number of time instances can be unlimited. However, the signal is not defined between two time points. + +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=0.8} + \begin{tikzpicture} + \draw node[draw, block](Continuous){Value-continuous,\\ time-continuous\\ signal}; + \draw node[draw, block, right=3cm of Continuous](Sampled){Value-continuous,\\ time-discrete\\ signal}; + \draw node[draw, block, right=3cm of Sampled](Digital){Value-discrete,\\ time-discrete\\ signal}; + + \draw [-latex] (Continuous) -- node[midway, align=center, above]{Sampling} (Sampled); + \draw [-latex] (Sampled) -- node[midway, align=center, above]{Quantization} (Digital); + + \draw[decorate, decoration={brace, amplitude=3mm, mirror}] ([yshift=-5mm] Continuous.south west) -- ([yshift=-5mm] Sampled.south east) node[midway, below, yshift=-3mm]{\textbf{Analogue}}; + \draw[decorate, decoration={brace, amplitude=3mm, mirror}] ([yshift=-5mm] Digital.south west) -- ([yshift=-5mm] Digital.south east) node[midway, below, yshift=-3mm]{\textbf{Digital}}; + \end{tikzpicture} + \end{adjustbox} + \caption{Conversion from analogue to digital signals} + \label{fig:ch01:signals_sampling} +\end{figure} + +\begin{figure}[H] + \centering + \includegraphics{../chapter01/Signal_Analogue.jpg} + \caption[An analogue, value-continuous, time-continuous signal]{An analogue, value-continuous, time-continuous signal. Both time and value can be any real number.} + \label{fig:ch01:Signal_Analogue} +\end{figure} + +\begin{figure}[H] + \centering + \includegraphics{../chapter01/Signal_TimeDiscr.jpg} + \caption[An analogue, value-continuous, but time-discrete signal]{An analogue, value-continuous, but time-discrete signal. Only certain time points are valid, but the values can be any real number.} + \label{fig:ch01:Signal_TimeDiscr} +\end{figure} + +\paragraph{Digital signals.} + +\index{signal!digital signal} +\index{signal!value-discrete} +Digital signals are both time-discrete and value-discrete. \emph{Value-discrete} means that they can take only one state out of a limited set of states. + +\begin{figure}[H] + \centering + \includegraphics{../chapter01/Signal_Digital.jpg} + \caption[A digital, value-discrete, time-discrete signal]{A digital, value-discrete, time-discrete signal. Only certain time points and a limited set of values are valid.} + \label{fig:ch01:Signal_Digital} +\end{figure} + +Examples: +\begin{itemize} + \item Text letters + \item Morse code + \item Coded data +\end{itemize} + +\textit{Remark:} In fact, the physical form of a digital signal is again an analogue signal. A binary signal can take the discrete states ``high'' and ``low''. If the signal is on a wire, its states are represented by voltage levels, for example \SI{0}{V} and \SI{3.3}{V}. However, if processed by a digital system, the physical representation is of minor importance. Only the discrete, logical states are considered. + +\index{signal!binary signal} +A special kind of digital signal is the \textbf{binary signal}. It has two discrete states. + + +\subsection{Transmission Channels} + +\index{transmission channel} +Digital communication systems employ electromagnetic waves to convey information. Therefore, only transmission channels transporting electromagnetic waves are considered. + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \draw node[block](Main){\textbf{Transmission}\\ \textbf{channels}}; + \draw node[block, below left=of Main](Wired){Wired\\ channels}; + \draw node[block, below right=of Main](Wireless){Wireless\\ channels}; + + \draw [-latex] (Main) -- (Wired); + \draw [-latex] (Main) -- (Wireless); + \end{tikzpicture} + \caption{Classification of transmission channels} + \label{fig:ch01:trans_ch_classif} +\end{figure} + +\paragraph{Wired Channels.} + +The electromagnetic wave propagates along a transmission line. + +Examples of transmission lines: +\begin{itemize} + \item Cables + \begin{itemize} + \item Two wire, twisted-pair \index{twisted-pair cable} + \item Coaxial cable \index{coaxial cable} + \end{itemize} + \item Waveguides \index{waveguide} + \item Planar lines (on printed circuit boards or integrated circuits) + \begin{itemize} + \item Microstrip \index{microstrip} + \item Coplanar waveguide \index{coplanar waveguide} + \end{itemize} + \item Glass fibre (light is an electromagnetic wave, too) +\end{itemize} + +\paragraph{Wireless Channels.} + +The electromagnetic wave is not bound to a transmission line. It propagates through the space. A medium is not necessary. Electromagnetic wave can also travel through vacuum. + + +\section{Computer Networks} + + +This course focuses on the technologies which convey information between endpoints, using electromagnetic waves. The information, being conveyed, are called \textbf{data}. The handling of the data is a subject of computer science, especially \emph{computer networks} \index{computer network}. Since data processing is a part of digital communication systems, too, this digression shall give an overview about the employed concepts. + + +\subsection{Protocols} + +Modern communication systems convey information world-wide. These communication links are established over myriads of devices, which form a network. The biggest computer network is the internet. + +These devices mainly operate automatically without human interaction. Therefore, they are required to follow certain rules, which are called \textbf{communication protocols} \index{communication protocol}. Protocols define +\begin{itemize} + \item the structure and semantics of data, + \item synchronization of communication, and + \item possible error recovery methods. +\end{itemize} + +Protocols are standardized and must be implemented in every device, which interacts with other devices. Important standardization organizations are: +\begin{itemize} + \item The non-profit organization \textbf{\acf{IETF}} issues standards concerning the internet. The standards are called \emph{Request For Comment} (RFC) and are available for everyone for free. Example standards: \ac{IP}, \ac{HTTP} + \item The \textbf{\acf{IEEE}} has standards committees which develop and publish standards. With respect to the internet, the IEEE\,802 LAN/MAN Standards Committee is the most important one. Example standards: IEEE\,802.11 (Wifi) + \item The \textbf{\acf{ETSI}} is an independent, non-profit standardization organization. It is recognized by the European Council and officially responsible for standardization of information and communication technologies in Europe. Example standards: 3G (cell phone system), 4G (cell phone system), TETRA (professional mobile radio system) +\end{itemize} + + +\subsection{\acs{OSI} Model} + +There are many task which a digital communication systems must accomplish. +\begin{itemize} + \item An application processes user input and displays data to the user. + \item The application data must be reliably transferred over a network with many nodes. + \item The network is shared with other users and applications. + \item The network consists of many links using different physical transmission channels, for example, wired and wireless. +\end{itemize} +For each task, there are communication protocols to solve it. Communication protocols are grouped by the task which they fulfil. There is an increasing level of abstraction from the physical link to the application data. The \index{OSI model} \textbf{\acs{OSI} Model} (Figure \ref{fig:ch01:osi_model}) defines a layer structure for classifying communication protocols, which regards the level of abstraction. + +\begin{figure}[H] + \centering + \begin{tikzpicture}[ + layer/.style={ + rectangle, + minimum height=1cm, + minimum width=8cm + } + ] + \draw node[draw, layer](L7){Application layer}; + \draw node[draw, layer, below=0 of L7](L6){Presentation layer}; + \draw node[draw, layer, below=0 of L6](L5){Session layer}; + \draw node[draw, layer, below=0 of L5](L4){Transport layer}; + \draw node[draw, layer, below=0 of L4](L3){Network layer}; + \draw node[draw, layer, below=0 of L3](L2){Data link layer}; + \draw node[draw, layer, below=0 of L2](L1){Physical layer}; + + \node [anchor=east, align=right] at([xshift=-5mm] L7.west) {Layer 7}; + \node [anchor=east, align=right] at([xshift=-5mm] L6.west) {Layer 6}; + \node [anchor=east, align=right] at([xshift=-5mm] L5.west) {Layer 5}; + \node [anchor=east, align=right] at([xshift=-5mm] L4.west) {Layer 4}; + \node [anchor=east, align=right] at([xshift=-5mm] L3.west) {Layer 3}; + \node [anchor=east, align=right] at([xshift=-5mm] L2.west) {Layer 2}; + \node [anchor=east, align=right] at([xshift=-5mm] L1.west) {Layer 1}; + + \filldraw[fill=gray!60, draw=none] ([xshift=10mm, yshift=3mm] L7.north east) -- node[midway, above, anchor=south, align=center]{Level of\\ abstraction} ([xshift=15mm, yshift=3mm] L7.north east) -- ([xshift=12.5mm, yshift=-3mm] L1.south east); + \end{tikzpicture} + \caption[OSI Model with seven layers]{\ac{OSI} Model with seven layers} + \label{fig:ch01:osi_model} +\end{figure} + +\begin{table}[H] + \caption[Description of the layers of the OSI Model]{Description of the layers of the \ac{OSI} Model (Figure \ref{fig:ch01:osi_model}). The protocol data unit is the information } + \begin{tabular}{|l|l|p{0.5\linewidth}|} + \hline + Layer & PDU & Function \\ + \hline + \hline + 7: Application & Data & Processing user inputs, displaying data, providing services \\ + \hline + 6: Presentation & Data & Translation between network service and application (encryption, compression, etc.) \\ + \hline + 5: Session & Data & Managing sessions (retaining the communication state across multiple contacts) \\ + \hline + 4: Transport & Datagram, Segment & Reliable communication (segmentation, multiplexing, data loss detection) \\ + \hline + 3: Network & Packet & Data transfer across multiple nodes (addressing, routing, traffic control) \\ + \hline + 2: Data link & Frame & Transmission between two devices (medium access, flow control) \\ + \hline + 1: Physical & Symbol & Transmission over a physical medium \\ + \hline + \end{tabular} +\end{table} + +Each protocol has a standardized interface exposed to the upper layer, called \index{service access point} \textbf{\ac{SAP}}. They allow an upper layer protocol to execute functions of the lower layer protocol. These functions are, for example: +\begin{itemize} + \item Sending or receiving data + \item Control operations + \item Network registration and de-registration +\end{itemize} + +Protocol layers add own information to the data received from the upper layer. This additional information is required to provide the protocol's functionality. For example, the \acf{IP} needs to add the source and destination address, so that the packet can be routed to the correct endpoint. One can imagine this like data which is written on a letter, which is put into an envelope, which itself is put into another envelope, and so on. + +\begin{figure}[H] + \centering + \includegraphics{../chapter01/Frame_Wrapping.jpg} + \caption{Principle of adding more information in each protocol layer} + \label{fig:ch01:frame_construction} +\end{figure} + +Communication protocols may be exchanged in one layer without affecting the functionality of the other layers. For example, \ac{HTTP} operates on \acs{TCP}/\acs{IP}. But the \acf{IP} works on multiple physical links like Ethernet (IEEE\,802.3), Wifi (IEEE\,802.11) or 4G. The transmission media can even change along the communication path. Information travelling through the internet experience lots of \index{media change} \textbf{media changes}. + + +\begin{figure}[H] + \centering + \caption{Media change on the internet. } + \label{fig:ch01:media_changes} +\end{figure} + +This course on digital communication systems mainly considers the physical layer (layer 1) and the data link layer (layer 2). This physical layer converts the information to physical signals which then leave the device to be transmitted over a physical transmission channel. Networks, which are enabled by protocols of layer 3 and above, are outside the scope of this course. + + +\subsection{Network Topologies} + +\begin{itemize} + \item \textbf{Ring} + \item \textbf{Star} + \item \textbf{Tree} + \item \textbf{Chain} + \item \textbf{Bus} + \item \textbf{Mesh}, special form \emph{Full Mesh} +\end{itemize} + +\printbibliography[heading=subbibliography] +\end{refsection} + diff --git a/chapter01/exercise01.tex b/chapter01/exercise01.tex new file mode 100644 index 0000000..9b939e9 --- /dev/null +++ b/chapter01/exercise01.tex @@ -0,0 +1,7 @@ +\phantomsection +\addcontentsline{toc}{section}{Exercise 1} +\section*{Exercise 1} + +\begin{enumerate} + \item What is the difference between a \emph{digital communication system} and a \emph{service}? To which OSI layers are they associated? +\end{enumerate} \ No newline at end of file diff --git a/chapter02/content.tex b/chapter02/content.tex deleted file mode 100644 index f22fc80..0000000 --- a/chapter02/content.tex +++ /dev/null @@ -1,451 +0,0 @@ -\chapter{Signals and Systems} - -All signals considered in this chapter are \index{signal!deterministic signal} \textbf{deterministic}, i.e., its values are predictable at any time. Especially, the values can be calculated by a mathematical equation. In contrast, \emph{random} signals are not predictable. Its values are subject to a random process, which must be modelled stochastically. - -\index{signal!time-continuous} -\begin{figure}[H] - \centering - \begin{tikzpicture} - \draw node[block](Signals){\textbf{Signal}\\ \textbf{(deterministic)}}; - \draw node[block, below left=of Signals](Periodic){Periodic}; - \draw node[block, below right=of Signals](NonPeriodic){Non-periodic}; - \draw node[block, below left=of Periodic](Mono){Mono-chromatic}; - \draw node[block, below right=of Periodic](Multi){Multi-frequent}; - - \draw [-latex] (Signals) -- (Periodic); - \draw [-latex] (Signals) -- (NonPeriodic); - \draw [-latex] (Periodic) -- (Mono); - \draw [-latex] (Periodic) -- (Multi); - \end{tikzpicture} - \caption{Classification of time-continuous signals} - \label{fig:ch02:timecont_signals_classif} -\end{figure} - -\section{Mono-Chromatic Signals} - -\paragraph{Representation by A Real-Valued Function.} - -The mono-chromatic signal $x_{mc}(t)$ is defined by: -\begin{equation} - x_{mc}(t) = \hat{X} \cdot \cos\left(\omega_0 t - \varphi_0\right) - \label{eq:ch02:mono_chrom_eq} -\end{equation} -where - -\begin{tabular}{ll} - $\hat{X}$ & is the \index{amplitude} \textbf{amplitude} of the signal, \\ - $\omega_0$ & is the \index{angular frequency} \textbf{angular frequency} of the signal, \\ - $\varphi_0$ & is the \index{phase} \textbf{phase} of the signal, \\ - $t \in \mathbb{R}$ & is the real-value time variable and continuously defined. -\end{tabular} - -In fact, the sine function $\sin()$ is mono-chromatic, too. However, it can be derived from \eqref{eq:ch02:mono_chrom_eq} with $\varphi_0 = \SI{90}{\degree}$. - -\begin{equation*} - x_{sin}(t) = \hat{X} \cdot \sin\left(\omega_0 t\right) = \cos\left(\omega_0 t - \SI{90}{\degree}\right) -\end{equation*} - -The angular frequency is connected to the \index{frequency} \textbf{frequency}. -\begin{equation} - \omega_0 = 2 \pi f_0 -\end{equation} - -\begin{attention} - You must not confuse the terms \emph{frequency} and \emph{angular frequency}! -\end{attention} - -The inverse of the frequency is the \index{period} \textbf{period} $T_0$. It is the time interval at which the signal repeats. -\begin{equation} - T_0 = \frac{1}{f_0} = \frac{2 \pi}{\omega_0} -\end{equation} - -Be aware of the units. The period $T_0$ is defined in seconds \si{s}. The frequency $f_0$ is the inverse of seconds, which is Hertz \si{Hz}. The angular frequency $\omega_0$ is the inverse of seconds, too. However, it is never given in Hertz, only in \si{rad/s} or, more commonly, \si{1/s}. - -\begin{table}[H] - \centering - \caption{Units} - \begin{tabular}{|l|l|} - \hline - Period $T_0$ & \si{s} \\ - \hline - Frequency $f_0$ & \si{Hz} \\ - \hline - Angular frequency $\omega_0$ & \si{1/s} \; (never Hertz!) \\ - \hline - \end{tabular} -\end{table} - -The actual unit of the signal is derived from its amplitude $\hat{X}$ which can be any physical measure. - -\paragraph{Representation by A Complex-Valued Phasor.} - -A graphical view on the creation of a cosine signal is depicted in Figure \ref{fig:ch02:cos_creation}. - -\begin{figure}[H] - \caption{Imagine, there is a pointer (red) with one side fixed to a point. Now, it begins rotating counter-clockwise with an angular frequency of $\omega_0$ (blue). The arrow of the pointer draws a circle (left side). Each angle of the pointer is related to a time instance (green). The blue pointer is the current position at time instance $t$. Its vertical value is projected into the time plot, forming the cosine wave (orange).} - \label{fig:ch02:cos_creation} -\end{figure} - -You may now some relations: -\begin{itemize} - \item A full rotation of the pointer takes exactly one period $T_0$. - \item The orange cosine curve can be horizontally shifted by redefining the original angle of the pointer at $T_0$. This offset angle is the phase $\varphi_0$. - \item The length of the pointer and the radius of the circle is the amplitude $\hat{X}$. -\end{itemize} - -A mono-chromatic signal can be described by its three parameters -\begin{itemize} - \item Amplitude $\hat{X}$ - \item Phase $\varphi_0$ - \item Frequency $\omega_0$ -\end{itemize} - -When a signal passes through a \ac{LTI} system, the amplitude, the phase or both may change. However, the frequency never changes. Thus, the frequency $\omega_0$ is assumed to be constant and neglected. Consequently, the parameters -\begin{itemize} - \item amplitude $\hat{X}$ and - \item phase $\varphi_0$ -\end{itemize} -remain. Both are absorbed by the complex-valued \index{phasor} \textbf{phasor} $\underline{X}$, which uniquely describes a mono-chromatic signal. -\begin{equation} - \underline{X} = \hat{X} \cdot e^{-j \varphi_0} -\end{equation} - -The phasor $\underline{X} \in \mathbb{C}$ is a complex number, which is mostly represented in polar coordinates (see Figure \ref{fig:ch02:cmplxplane_phasor}). - -\begin{figure}[H] - \centering - \begin{tikzpicture} - \draw[->] (-3.2,0) -- (3.2,0) node[below, align=left]{$\Re$}; - \draw[->] (0,-3.2) -- (0,3.2) node[left, align=right]{$\Im$}; - \draw[->, thick] (0, 0) -- (-40:3) node[right, align=left]{Complex phasor $\underline{X}$\\ (position at $t = 0$)}; - \draw (0:1.5) arc(0:-40:1.5) node[midway, right, align=left]{Phase $\varphi_0$}; - - \draw[->, dashed] (-50:1) arc(-50:30:1) node[right, align=left]{$\omega_0$}; - \end{tikzpicture} - \caption{Phasor in the complex plane} - \label{fig:ch02:cmplxplane_phasor} -\end{figure} - -Figure \ref{fig:ch02:cmplxplane_phasor} depicts the phasor in the complex plane. Figure \ref{fig:ch02:cos_creation} shows a complex plane, too. Please note that both complex planes are rotated by \SI{90}{\degree} with respect to each other. - -\begin{fact} - The phasor of a signal is a signal parameter, constant and \underline{not} time-dependent. -\end{fact} - -The current position of the pointer $\underline{x}(t)$ in the complex plane is obtained by rotating it. It makes a full rotation each $T_0$ periods. Therefore, it rotates at an angular frequency of $\omega_0$. The rotation is a multiplication by $e^{j \omega t}$ in the complex plane. $\underline{x}(t) \in \mathbb{C}$ is a complex value, too. -\begin{equation} - \underline{x_{mc}}(t) = \underline{X} \cdot e^{j \omega t} = \hat{X} \cdot e^{-j \varphi_0} \cdot e^{j \omega t} -\end{equation} - -\todo{Proof} - -The real-valued function can be obtained by extracting the real part of the complex-valued current value. -\begin{equation} - x_{mc}(t) = \Re\left\{\underline{x_{mc}}(t)\right\} -\end{equation} - -% Exercise: Is a sine wave with DC bias mono-chromatic -> no - -\section{Periodic Signals and Fourier Series} - -Periodic signals $x_p(t)$ comprises a class of signals which indefinitely repeat at constant time intervals $T_0$. -\begin{equation} - x_p(t + n T_0) = x_p(t) \qquad \forall \; n \in \mathbb{Z}, \quad \mathbb{Z} = \left\{..., -2, -1, 0, 1, 2, ...\right\} -\end{equation} - -Mono-chromatic signals are a special kind of periodic signals. Multi-frequent signals are composed a limited or unlimited number of mono-chromatic signals, which superimpose. Multi-frequent signals are periodic signals in general. - -\begin{fact} - Each periodic signal can be decomposed into a superposition of mono-chromatic signals. -\end{fact} - -The inverse of the period $T_0$ is $f_0$, which is the \textbf{base frequency}. This is the frequency at the periodic pattern repeats. Again, frequency and angular frequency $\omega_0 = 2 \pi f_0$ must be distinguished. - -The periodic signal can now be decomposed in cosine and sine functions with integer multiples of the base frequency $f_0$ or base angular frequency $\omega_0$, respectively. They are called \index{harmonics} \textbf{harmonics}. -\begin{equation} - \begin{split} - x_p(t) &= \sum\limits_{n=0}^{\infty} a_n \cos\left(n \omega_0 t\right) + \sum\limits_{m=0}^{\infty} b_m \sin\left(m \omega_0 t\right) \qquad \forall \; n, m \in \mathbb{N} = \left\{0, 1, 2, ...\right\} \\ - &= a_0 + \sum\limits_{n=1}^{\infty} a_n \cos\left(n \omega_0 t\right) + \sum\limits_{m=1}^{\infty} b_m \sin\left(m \omega_0 t\right) \\ - \end{split} - \label{eq:ch02:fourier_series} -\end{equation} - -What happened to $n = 0$ and $m = 0$? $\cos(0) = 1$ and $\sin(0) = 0$. That's it. - -Comparing to the mono-chromatic signals, what happened to the phase $\varphi_0$? The phase $\varphi_0$ is a characteristic of mono-chromatic signals. It is completely absorbed by the coefficients $a_n$ and $b_n$ of the cosine and sine functions. - -\subsection{Orthogonality} -\index{orthogonality} -The cosine and sine functions are orthogonal to each other. In geometry, two vectors $\vect{A}$ and $\vect{B}$ are said to be orthogonal, if the angle between them is \SI{90}{\degree}. In this case, their inner product is zero. -\begin{equation} - \langle \vect{A}, \vect{B} \rangle = 0 -\end{equation} - -More generally, two functions $f(x)$ and $g(x)$ are orthogonal if their \index{inner product} \textbf{inner product} $\langle f, g \rangle$ is zero. -\begin{equation} - 0 \stackrel{!}{=} \langle f, g \rangle_w = \int\limits_{a}^{b} f(x) g(x) w(x) \, \mathrm{d} x -\end{equation} -$w(x)$ is a non-negative weight function, which is $w(x) = 1$ in simple cases like this one. - -Now, you can prove that the cosine and sine functions are orthogonal to each other. -\begin{equation} - \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \cos\left(n \omega_0 t\right) \sin\left(m \omega_0 t\right) \, \mathrm{d} t = 0 \qquad \forall n, m \in \mathbb{Z} - \label{eq:ch02:orth_rel_cos_sin} -\end{equation} - -Furthermore, the sine and cosine functions with \underline{different} indices are orthogonal to each other. -\begin{equation} - \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \cos\left(n \omega_0 t\right) \cos\left(p \omega_0 t\right) \, \mathrm{d} t = \frac{\pi}{\omega_0} \cdot \delta_{np} \qquad \forall \; n, p \in \mathbb{N} - \label{eq:ch02:orth_rel_cos} -\end{equation} -\begin{equation} - \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \sin\left(m \omega_0 t\right) \sin\left(q \omega_0 t\right) \, \mathrm{d} t = \frac{\pi}{\omega_0} \cdot \delta_{mq} \qquad \forall \; m, q \in \mathbb{N} - \label{eq:ch02:orth_rel_sin} -\end{equation} -with the Kronecker delta -\begin{equation} - \delta_{uv} = \begin{cases} - 1 & \qquad \text{if } u = v, \\ - 0 & \qquad \text{if } u \neq v - \end{cases} - \label{eq:ch02:kronecker_delta} -\end{equation} - -The \index{orthogonality relations} \textbf{orthogonality relations} \eqref{eq:ch02:orth_rel_cos_sin}, \eqref{eq:ch02:orth_rel_cos} and \eqref{eq:ch02:orth_rel_sin} point out: -\begin{itemize} - \item Cosine functions are orthogonal if their indices are different. I.e., $n \neq p$ in \eqref{eq:ch02:orth_rel_cos}. - \item Sine functions are orthogonal if their indices are different. I.e., $m \neq q$ in \eqref{eq:ch02:orth_rel_sin}. - \item Cosine and sine function are orthogonal independent of their indices. - \item The indices are the integer multiples of the base frequency $\omega_0$ (harmonics). -\end{itemize} - -\subsection{Extraction of The Coefficients} - -The orthogonality relations are useful to extract the coefficients $a_n$ and $b_n$ in \eqref{eq:ch02:fourier_series}. Given is the input signal $\tilde{x}_p(t)$ whose coefficient shall be determined. Following assumptions can be derived from the properties of a periodic signal: -\begin{itemize} - \item $\tilde{x}_p(t)$ is composed of mono-chromatic cosine and sine functions. - \item All cosine and sine functions have integer multiples of the base frequency. - \item Each cosine and sine function has a different weight -- the coefficient. -\end{itemize} - -Using the orthogonality relations, the coefficients $\tilde{a}_n$ and $\tilde{b}_n$ can be obtained by: -\begin{subequations} - \begin{align} - \tilde{a}_n &= \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \tilde{x}_p(t) \cdot \cos\left(n \omega_0 t\right) \, \mathrm{d} t \label{eq_ch02_fourier_series_coeff_an} \\ - \tilde{b}_m &= \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \tilde{x}_p(t) \cdot \sin\left(n \omega_0 t\right) \, \mathrm{d} t \label{eq_ch02_fourier_series_coeff_bm} - \end{align} -\end{subequations} - -\begin{proof}{Parameter Extraction for $\tilde{a}_n$} - Given is a periodic function $\tilde{x}_p(t)$, which can be decomposed into: - \begin{equation} - \tilde{x}_p(t) = \sum\limits_{p=0}^{\infty} \tilde{a}_p \cos\left(p \omega_0 t\right) + \sum\limits_{q=0}^{\infty} \tilde{b}_q \sin\left(q \omega_0 t\right) - \label{eq_ch02_proof_per_sig_example} - \end{equation} - The coefficient $\tilde{a}_n$ is of interest. - - Inserting \eqref{eq_ch02_proof_per_sig_example} into \eqref{eq_ch02_fourier_series_coeff_an}, yields - \begin{equation} - \tilde{a}_n = \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \left(\sum\limits_{p=0}^{\infty} \tilde{a}_p \cos\left(p \omega_0 t\right) + \sum\limits_{q=0}^{\infty} \tilde{b}_q \sin\left(q \omega_0 t\right)\right) \cdot \cos\left(n \omega_0 t\right) \, \mathrm{d} t - \end{equation} - Due to the orthogonality relations, \underline{all products containing a sine function} and \underline{all products containing a cosine function with the index $n \neq p$} become zero. Furthermore, following must be true: $n = p$ - - \begin{equation} - \tilde{a}_n = \tilde{a}_p \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \cos\left(p \omega_0 t\right) \cdot \cos\left(n \omega_0 t\right) \, \mathrm{d} t \qquad \text{if } \; n = p - \end{equation} - - Using \eqref{eq:ch02:orth_rel_cos}, the integral resolves to: - \begin{equation} - \tilde{a}_n = \tilde{a}_p \frac{\omega_0}{\pi} \frac{\pi}{\omega_0} \qquad \text{if } \; n = p - \end{equation} - - In the end, it could be proven that $\tilde{a}_n = \tilde{a}_p$ for $n = p$. - - The proof is analogous for the coefficient $b_n$. -\end{proof} - -$\cos\left(n \omega_0 t\right)$ can be seen as a ``test function'', which is used to extract the component with the index $n$. The proof points out: -\begin{itemize} - \item All sine components are erased by $\cos\left(n \omega_0 t\right)$, due to the orthogonality relations. - \item All cosine function with index $p \neq n$ are erased by $\cos\left(n \omega_0 t\right)$, due to the orthogonality relations. -\end{itemize} -For $b_m$, $\sin\left(m \omega_0 t\right)$ is analogous. - -\begin{excursus}{Illustration of The ``Test Function''} - For illustration of the ``test functions'', image you have a radio and want to hear a specific station. You tune to the frequency on which the station is broadcasting. All other signals are filtered out, you don't want to hear them. Actually, the radio does not employ orthogonality in this case. However, this illustration might help to understand the meaning of $\cos\left(n \omega_0 t\right)$ and $\sin\left(m \omega_0 t\right)$ \underline{in connection} with the orthogonality relations. -\end{excursus} - -A special case is the coefficient $\tilde{a}_0$. -\begin{equation} - \tilde{a}_0 = \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \tilde{x}_p(t) \, \mathrm{d} t -\end{equation} -$\cos\left(n \omega_0 t\right)$ is $1$ for $n = 0$. $\tilde{a}_0$ is the \index{DC offset} \textbf{\ac{DC} offset} of the signal. The above formula is known as the calculation of the signal mean in electrical engineering. - -The composition of a series of mono-chromatic signals as shown in \eqref{eq:ch02:fourier_series} is called \index{Fourier series} \textbf{Fourier series}. -\begin{equation*} - x_p(t) = \sum\limits_{n=1}^{\infty} a_n \cos\left(n \omega_0 t\right) + \sum\limits_{m=1}^{\infty} b_m \sin\left(m \omega_0 t\right) -\end{equation*} - -\subsection{Complex-Valued Fourier Series} - -A complex-valued, periodic signal $\underline{x_p}(t)$ can be decomposed into complex-valued mono-chromatic signals. The coefficients $\underline{c}_n$ are phasors. -\begin{equation} - \underline{x_p}(t) = \sum\limits_{n = -\infty}^{\infty} \underline{c}_n \cdot e^{j n \omega_0 t} \qquad \forall \; n \in \mathbb{Z} - \label{eq:ch02:fourier_series_cmplx} -\end{equation} - -The coefficients $\underline{\tilde{c}}_n$ of an input signal $\underline{\tilde{x}_p}(t)$ can be determined by: -\begin{equation} - \underline{\tilde{c}}_n = \frac{\omega_0}{2 \pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \underline{\tilde{x}_p}(t) \cdot e^{-j n \omega_0 t} \, \mathrm{d} t - \label{eq_ch02_fourier_series_coeff_cn} -\end{equation} - -It is based on the orthogonality relation: -\begin{equation} - \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} e^{j n \omega_0 t} e^{-j p \omega_0 t} \, \mathrm{d} t = \frac{2 \pi}{\omega_0} \cdot \delta_{np} \qquad \forall \; n, p \in \mathbb{Z} - \label{eq:ch02:orth_rel_exp} -\end{equation} - -\subsection{Amplitude and Phase Spectra} - -\section{Non-Periodic Signals and Fourier Transform} - -\subsection{Derivation of The Fourier Transform} - -Non-periodic signals have no repeating pattern. Consequently, there is no period $T_0$. Mathematically, the period is indefinite $T_0 \rightarrow \infty$. - -A non-periodic signal $\underline{x_{np}}(t)$ cannot be simply decomposed by a Fourier series \eqref{eq:ch02:fourier_series_cmplx}. -\begin{equation} - \begin{split} - \underline{x_{np}}(t) &= \lim\limits_{T_0 \rightarrow \infty} \sum\limits_{n = -\infty}^{\infty} \underline{c}_n \cdot e^{j n \omega_0 t} \\ - &= \lim\limits_{T_0 \rightarrow \infty} \sum\limits_{n = -\infty}^{\infty} \underline{c}_n \cdot e^{j \frac{2 \pi n}{T_0} t} - \end{split} - \label{eq:ch02:sig_np_fourier_series} -\end{equation} - -The coefficient $\underline{c}_n$ is defined by \eqref{eq_ch02_fourier_series_coeff_cn}: -\begin{equation*} - \begin{split} - \underline{c}_n &= \frac{\omega_0}{2 \pi} \int\limits_{t = -\frac{T_0}{2}}^{\frac{T_0}{2}} \underline{x_{np}}(t) \cdot e^{-j n \omega_0 t} \, \mathrm{d} t \\ - &= \frac{1}{T_0} \int\limits_{t = -\frac{T_0}{2}}^{\frac{T_0}{2}} \underline{x_{np}}(t) \cdot e^{-j n \omega_0 t} \, \mathrm{d} t - \end{split} - \label{eq:ch02:sig_np_cn} -\end{equation*} - -In this case where $T_0 \rightarrow \infty$, $n \omega_0$ is substituted by the frequency variable $\omega$. -\begin{equation} - \omega = n \omega_0 - \label{eq:ch02:omega_subst} -\end{equation} - -Inserting \eqref{eq:ch02:sig_np_cn} into \eqref{eq:ch02:sig_np_fourier_series} while considering \eqref{eq:ch02:omega_subst}, yields: -\begin{equation} - \underline{x_{np}}(t) = \lim\limits_{T_0 \rightarrow \infty} \sum\limits_{n = -\infty}^{\infty} \frac{1}{T_0} \left( \int\limits_{t' = -\frac{T_0}{2}}^{\frac{T_0}{2}} \underline{x_{np}}(t') \cdot e^{-j \omega t'} \, \mathrm{d} t' \right) \cdot e^{j \omega t} -\end{equation} -Remember, that $n$ is still in the sum, since it has been absorbed by $\omega = n \omega_0$. - -The outer sum is a Rieman sum. $\frac{1}{T_0}$ is substituted by $\frac{\Delta \omega}{2 \pi}$. With $T_0 \rightarrow \infty$, it can be rewritten as an integral. -\begin{equation} - \underline{x_{np}}(t) = \underbrace{\frac{1}{2 \pi} \int\limits_{\omega = -\infty}^{\infty} \underbrace{\left( \int\limits_{t' = -\infty}^{\infty} \underline{x_{np}}(t') \cdot e^{-j \omega t'} \, \mathrm{d} t' \right)}_{\text{Fourier transform}} \cdot e^{j \omega t} \, \mathrm{d} \omega}_{\text{Inverse Fourier transform}} -\end{equation} - -The inner integral is the \index{Fourier transform} \textbf{Fourier transform}. - -\begin{definition}{Fourier Transform} - The \index{Fourier transform} \textbf{Fourier transform} of the function $\underline{x}(t)$ is: - \begin{equation} - \underline{X}(j \omega) = \mathcal{F} \left\{\underline{x}(t)\right\} = \int\limits_{t = -\infty}^{\infty} \underline{x}(t) \cdot e^{-j \omega t} \, \mathrm{d} t - \end{equation} - - The \index{inverse Fourier transform} \textbf{inverse Fourier transform} is: - \begin{equation} - \underline{x}(t) = \mathcal{F}^{-1} \left\{\underline{X}(j \omega)\right\} = \frac{1}{2 \pi} \int\limits_{\omega = -\infty}^{\infty} \underline{X}(j \omega) \cdot e^{+j \omega t} \, \mathrm{d} \omega - \end{equation} -\end{definition} - -\subsection{Amplitude and Phase Spectra} - -\subsection{Time Domain and Frequency Domain} - -\section{Properties of The Fourier Transform} - -\subsection{Energy Signals and Power Signals} - -Besides the classification of signals into periodic and non-periodic, signals can be divided into \index{energy signals} \textbf{energy signals} and \index{power signals} \textbf{power signals}. - -\begin{definition}{Energy and Power Signals} - \begin{itemize} - \item \textbf{Energy signals} have a finite, positive signal energy $0 < E < \infty$, but their average power is zero $P = 0$. - \item \textbf{Power signals} have a finite, positive average signal power $0 < P < \infty$, but their signal energy is indefinite $E = \infty$. - \end{itemize} -\end{definition} - -The \index{average signal power} \textbf{average signal power} $P$ is a measure for the amount of energy transferred per unit time and defined by: -\begin{equation} - P = \lim\limits_{T \rightarrow \infty} \frac{1}{T} \int\limits_{-\frac{T}{2}}^{\frac{T}{2}} \left|x(t)\right|^2 \; \mathrm{d} t -\end{equation} -The signal power is connected to the \ac{RMS} value, which is often used in electrical engineering. -\begin{equation} - \hat{x}_{RMS} = \lim\limits_{T \rightarrow \infty} \sqrt{ \frac{1}{T} \int\limits_{-\frac{T}{2}}^{\frac{T}{2}} \left|x(t)\right|^2 \; \mathrm{d} t} -\end{equation} - -The \index{signal energy} \textbf{signal energy} $E$ is: -\begin{equation} - E = \int\limits_{-\infty}^{\infty} \left|x(t)\right|^2 \; \mathrm{d} t -\end{equation} - -The property of power signals, which have an indefinite signal energy, is a problem for the Fourier transform. The transform would yield an indefinite value. Thus: -\begin{fact} - Every energy signal has a Fourier transform. -\end{fact} - -Only some power signals have a Fourier transform. There are distributions which are power signals, but have a Fourier transform, too. Especially, all \emph{tempered distributions} have a Fourier transform. - -\subsection{Dirac Delta Function} - -An important distribution is the \index{Dirac delta function} \textbf{Dirac delta function} $\delta(t)$. The Dirac delta function is zero everywhere except at its origin, where it is an indefinitely narrow, indefinitely high pulse. -\begin{equation} - \delta(t) = \begin{cases} - +\infty & \qquad \text{if } t = 0, \\ - 0 & \qquad \text{if } t \neq 0 - \end{cases} - \label{eq:ch02:dirac_delta} -\end{equation} -It is constrained by -\begin{equation} - \int\limits_{-\infty}^{\infty} \delta(t) \; \mathrm{d} t = 1 -\end{equation} - -\begin{attention} - The Dirac delta function $\delta(t)$ must not be confused with the Kronecker delta \eqref{eq:ch02:kronecker_delta}. The Dirac delta function operates in continuous space $t \in \mathbb{R}$. The Kronecker delta $\delta_n$ (here one-dimensional) operates in discrete space $n \in \mathbb{Z}$. -\end{attention} - -A special feature of the function is called \index{Dirac measure} \textbf{Dirac measure}. -\begin{equation} - \int\limits_{-\infty}^{\infty} f(t) \delta(t) \; \mathrm{d} t = f(0) - \label{eq:ch02:dirac_measure} -\end{equation} - -Using the Dirac measure, the Fourier transform can be calculated: -\begin{equation} - \mathcal{F} \left\{\delta(t)\right\} = \int\limits_{-\infty}^{\infty} \delta(t) \cdot e^{-j \omega t} \; \mathrm{d} t = 1 -\end{equation} -The Fourier transform of the Dirac delta function is the frequency-independent constant $1$. - -\subsection{Operations 1: Linearity} - -\subsection{Operations 2: Differentiation and Integration} - -\subsection{Operations 3: Multiplication} - -\subsection{Operations 4: Time Shift} - -\subsection{Duality} - -\section{\acs{LTI} Systems} - -\subsection{Transfer Function and Impulse Response} - -\subsection{Convolution} - -\subsection{Poles and Zeroes} diff --git a/chapter02/content_ch02.tex b/chapter02/content_ch02.tex new file mode 100644 index 0000000..8a9ec88 --- /dev/null +++ b/chapter02/content_ch02.tex @@ -0,0 +1,456 @@ +\chapter{Signals and Systems} + +\begin{refsection} + +All signals considered in this chapter are \index{signal!deterministic signal} \textbf{deterministic}, i.e., its values are predictable at any time. Especially, the values can be calculated by a mathematical equation. In contrast, \emph{random} signals are not predictable. Its values are subject to a random process, which must be modelled stochastically. + +\index{signal!time-continuous} +\begin{figure}[H] + \centering + \begin{tikzpicture} + \draw node[block](Signals){\textbf{Signal}\\ \textbf{(deterministic)}}; + \draw node[block, below left=of Signals](Periodic){Periodic}; + \draw node[block, below right=of Signals](NonPeriodic){Non-periodic}; + \draw node[block, below left=of Periodic](Mono){Mono-chromatic}; + \draw node[block, below right=of Periodic](Multi){Multi-frequent}; + + \draw [-latex] (Signals) -- (Periodic); + \draw [-latex] (Signals) -- (NonPeriodic); + \draw [-latex] (Periodic) -- (Mono); + \draw [-latex] (Periodic) -- (Multi); + \end{tikzpicture} + \caption{Classification of time-continuous signals} + \label{fig:ch02:timecont_signals_classif} +\end{figure} + +\section{Mono-Chromatic Signals} + +\paragraph{Representation by A Real-Valued Function.} + +The mono-chromatic signal $x_{mc}(t)$ is defined by: +\begin{equation} + x_{mc}(t) = \hat{X} \cdot \cos\left(\omega_0 t - \varphi_0\right) + \label{eq:ch02:mono_chrom_eq} +\end{equation} +where + +\begin{tabular}{ll} + $\hat{X}$ & is the \index{amplitude} \textbf{amplitude} of the signal, \\ + $\omega_0$ & is the \index{angular frequency} \textbf{angular frequency} of the signal, \\ + $\varphi_0$ & is the \index{phase} \textbf{phase} of the signal, \\ + $t \in \mathbb{R}$ & is the real-value time variable and continuously defined. +\end{tabular} + +In fact, the sine function $\sin()$ is mono-chromatic, too. However, it can be derived from \eqref{eq:ch02:mono_chrom_eq} with $\varphi_0 = \SI{90}{\degree}$. + +\begin{equation*} + x_{sin}(t) = \hat{X} \cdot \sin\left(\omega_0 t\right) = \cos\left(\omega_0 t - \SI{90}{\degree}\right) +\end{equation*} + +The angular frequency is connected to the \index{frequency} \textbf{frequency}. +\begin{equation} + \omega_0 = 2 \pi f_0 +\end{equation} + +\begin{attention} + You must not confuse the terms \emph{frequency} and \emph{angular frequency}! +\end{attention} + +The inverse of the frequency is the \index{period} \textbf{period} $T_0$. It is the time interval at which the signal repeats. +\begin{equation} + T_0 = \frac{1}{f_0} = \frac{2 \pi}{\omega_0} +\end{equation} + +Be aware of the units. The period $T_0$ is defined in seconds \si{s}. The frequency $f_0$ is the inverse of seconds, which is Hertz \si{Hz}. The angular frequency $\omega_0$ is the inverse of seconds, too. However, it is never given in Hertz, only in \si{rad/s} or, more commonly, \si{1/s}. + +\begin{table}[H] + \centering + \caption{Units} + \begin{tabular}{|l|l|} + \hline + Period $T_0$ & \si{s} \\ + \hline + Frequency $f_0$ & \si{Hz} \\ + \hline + Angular frequency $\omega_0$ & \si{1/s} \; (never Hertz!) \\ + \hline + \end{tabular} +\end{table} + +The actual unit of the signal is derived from its amplitude $\hat{X}$ which can be any physical measure. + +\paragraph{Representation by A Complex-Valued Phasor.} + +A graphical view on the creation of a cosine signal is depicted in Figure \ref{fig:ch02:cos_creation}. + +\begin{figure}[H] + \caption{Imagine, there is a pointer (red) with one side fixed to a point. Now, it begins rotating counter-clockwise with an angular frequency of $\omega_0$ (blue). The arrow of the pointer draws a circle (left side). Each angle of the pointer is related to a time instance (green). The blue pointer is the current position at time instance $t$. Its vertical value is projected into the time plot, forming the cosine wave (orange).} + \label{fig:ch02:cos_creation} +\end{figure} + +You may now some relations: +\begin{itemize} + \item A full rotation of the pointer takes exactly one period $T_0$. + \item The orange cosine curve can be horizontally shifted by redefining the original angle of the pointer at $T_0$. This offset angle is the phase $\varphi_0$. + \item The length of the pointer and the radius of the circle is the amplitude $\hat{X}$. +\end{itemize} + +A mono-chromatic signal can be described by its three parameters +\begin{itemize} + \item Amplitude $\hat{X}$ + \item Phase $\varphi_0$ + \item Frequency $\omega_0$ +\end{itemize} + +When a signal passes through a \ac{LTI} system, the amplitude, the phase or both may change. However, the frequency never changes. Thus, the frequency $\omega_0$ is assumed to be constant and neglected. Consequently, the parameters +\begin{itemize} + \item amplitude $\hat{X}$ and + \item phase $\varphi_0$ +\end{itemize} +remain. Both are absorbed by the complex-valued \index{phasor} \textbf{phasor} $\underline{X}$, which uniquely describes a mono-chromatic signal. +\begin{equation} + \underline{X} = \hat{X} \cdot e^{-j \varphi_0} +\end{equation} + +The phasor $\underline{X} \in \mathbb{C}$ is a complex number, which is mostly represented in polar coordinates (see Figure \ref{fig:ch02:cmplxplane_phasor}). + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \draw[->] (-3.2,0) -- (3.2,0) node[below, align=left]{$\Re$}; + \draw[->] (0,-3.2) -- (0,3.2) node[left, align=right]{$\Im$}; + \draw[->, thick] (0, 0) -- (-40:3) node[right, align=left]{Complex phasor $\underline{X}$\\ (position at $t = 0$)}; + \draw (0:1.5) arc(0:-40:1.5) node[midway, right, align=left]{Phase $\varphi_0$}; + + \draw[->, dashed] (-50:1) arc(-50:30:1) node[right, align=left]{$\omega_0$}; + \end{tikzpicture} + \caption{Phasor in the complex plane} + \label{fig:ch02:cmplxplane_phasor} +\end{figure} + +Figure \ref{fig:ch02:cmplxplane_phasor} depicts the phasor in the complex plane. Figure \ref{fig:ch02:cos_creation} shows a complex plane, too. Please note that both complex planes are rotated by \SI{90}{\degree} with respect to each other. + +\begin{fact} + The phasor of a signal is a signal parameter, constant and \underline{not} time-dependent. +\end{fact} + +The current position of the pointer $\underline{x}(t)$ in the complex plane is obtained by rotating it. It makes a full rotation each $T_0$ periods. Therefore, it rotates at an angular frequency of $\omega_0$. The rotation is a multiplication by $e^{j \omega t}$ in the complex plane. $\underline{x}(t) \in \mathbb{C}$ is a complex value, too. +\begin{equation} + \underline{x_{mc}}(t) = \underline{X} \cdot e^{j \omega t} = \hat{X} \cdot e^{-j \varphi_0} \cdot e^{j \omega t} +\end{equation} + +\todo{Proof} + +The real-valued function can be obtained by extracting the real part of the complex-valued current value. +\begin{equation} + x_{mc}(t) = \Re\left\{\underline{x_{mc}}(t)\right\} +\end{equation} + +% Exercise: Is a sine wave with DC bias mono-chromatic -> no + +\section{Periodic Signals and Fourier Series} + +Periodic signals $x_p(t)$ comprises a class of signals which indefinitely repeat at constant time intervals $T_0$. +\begin{equation} + x_p(t + n T_0) = x_p(t) \qquad \forall \; n \in \mathbb{Z}, \quad \mathbb{Z} = \left\{..., -2, -1, 0, 1, 2, ...\right\} +\end{equation} + +Mono-chromatic signals are a special kind of periodic signals. Multi-frequent signals are composed a limited or unlimited number of mono-chromatic signals, which superimpose. Multi-frequent signals are periodic signals in general. + +\begin{fact} + Each periodic signal can be decomposed into a superposition of mono-chromatic signals. +\end{fact} + +The inverse of the period $T_0$ is $f_0$, which is the \textbf{base frequency}. This is the frequency at the periodic pattern repeats. Again, frequency and angular frequency $\omega_0 = 2 \pi f_0$ must be distinguished. + +The periodic signal can now be decomposed in cosine and sine functions with integer multiples of the base frequency $f_0$ or base angular frequency $\omega_0$, respectively. They are called \index{harmonics} \textbf{harmonics}. +\begin{equation} + \begin{split} + x_p(t) &= \sum\limits_{n=0}^{\infty} a_n \cos\left(n \omega_0 t\right) + \sum\limits_{m=0}^{\infty} b_m \sin\left(m \omega_0 t\right) \qquad \forall \; n, m \in \mathbb{N} = \left\{0, 1, 2, ...\right\} \\ + &= a_0 + \sum\limits_{n=1}^{\infty} a_n \cos\left(n \omega_0 t\right) + \sum\limits_{m=1}^{\infty} b_m \sin\left(m \omega_0 t\right) \\ + \end{split} + \label{eq:ch02:fourier_series} +\end{equation} + +What happened to $n = 0$ and $m = 0$? $\cos(0) = 1$ and $\sin(0) = 0$. That's it. + +Comparing to the mono-chromatic signals, what happened to the phase $\varphi_0$? The phase $\varphi_0$ is a characteristic of mono-chromatic signals. It is completely absorbed by the coefficients $a_n$ and $b_n$ of the cosine and sine functions. + +\subsection{Orthogonality} +\index{orthogonality} +The cosine and sine functions are orthogonal to each other. In geometry, two vectors $\vect{A}$ and $\vect{B}$ are said to be orthogonal, if the angle between them is \SI{90}{\degree}. In this case, their inner product is zero. +\begin{equation} + \langle \vect{A}, \vect{B} \rangle = 0 +\end{equation} + +More generally, two functions $f(x)$ and $g(x)$ are orthogonal if their \index{inner product} \textbf{inner product} $\langle f, g \rangle$ is zero. +\begin{equation} + 0 \stackrel{!}{=} \langle f, g \rangle_w = \int\limits_{a}^{b} f(x) g(x) w(x) \, \mathrm{d} x +\end{equation} +$w(x)$ is a non-negative weight function, which is $w(x) = 1$ in simple cases like this one. + +Now, you can prove that the cosine and sine functions are orthogonal to each other. +\begin{equation} + \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \cos\left(n \omega_0 t\right) \sin\left(m \omega_0 t\right) \, \mathrm{d} t = 0 \qquad \forall n, m \in \mathbb{Z} + \label{eq:ch02:orth_rel_cos_sin} +\end{equation} + +Furthermore, the sine and cosine functions with \underline{different} indices are orthogonal to each other. +\begin{equation} + \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \cos\left(n \omega_0 t\right) \cos\left(p \omega_0 t\right) \, \mathrm{d} t = \frac{\pi}{\omega_0} \cdot \delta_{np} \qquad \forall \; n, p \in \mathbb{N} + \label{eq:ch02:orth_rel_cos} +\end{equation} +\begin{equation} + \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \sin\left(m \omega_0 t\right) \sin\left(q \omega_0 t\right) \, \mathrm{d} t = \frac{\pi}{\omega_0} \cdot \delta_{mq} \qquad \forall \; m, q \in \mathbb{N} + \label{eq:ch02:orth_rel_sin} +\end{equation} +with the Kronecker delta +\begin{equation} + \delta_{uv} = \begin{cases} + 1 & \qquad \text{if } u = v, \\ + 0 & \qquad \text{if } u \neq v + \end{cases} + \label{eq:ch02:kronecker_delta} +\end{equation} + +The \index{orthogonality relations} \textbf{orthogonality relations} \eqref{eq:ch02:orth_rel_cos_sin}, \eqref{eq:ch02:orth_rel_cos} and \eqref{eq:ch02:orth_rel_sin} point out: +\begin{itemize} + \item Cosine functions are orthogonal if their indices are different. I.e., $n \neq p$ in \eqref{eq:ch02:orth_rel_cos}. + \item Sine functions are orthogonal if their indices are different. I.e., $m \neq q$ in \eqref{eq:ch02:orth_rel_sin}. + \item Cosine and sine function are orthogonal independent of their indices. + \item The indices are the integer multiples of the base frequency $\omega_0$ (harmonics). +\end{itemize} + +\subsection{Extraction of The Coefficients} + +The orthogonality relations are useful to extract the coefficients $a_n$ and $b_n$ in \eqref{eq:ch02:fourier_series}. Given is the input signal $\tilde{x}_p(t)$ whose coefficient shall be determined. Following assumptions can be derived from the properties of a periodic signal: +\begin{itemize} + \item $\tilde{x}_p(t)$ is composed of mono-chromatic cosine and sine functions. + \item All cosine and sine functions have integer multiples of the base frequency. + \item Each cosine and sine function has a different weight -- the coefficient. +\end{itemize} + +Using the orthogonality relations, the coefficients $\tilde{a}_n$ and $\tilde{b}_n$ can be obtained by: +\begin{subequations} + \begin{align} + \tilde{a}_n &= \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \tilde{x}_p(t) \cdot \cos\left(n \omega_0 t\right) \, \mathrm{d} t \label{eq_ch02_fourier_series_coeff_an} \\ + \tilde{b}_m &= \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \tilde{x}_p(t) \cdot \sin\left(n \omega_0 t\right) \, \mathrm{d} t \label{eq_ch02_fourier_series_coeff_bm} + \end{align} +\end{subequations} + +\begin{proof}{Parameter Extraction for $\tilde{a}_n$} + Given is a periodic function $\tilde{x}_p(t)$, which can be decomposed into: + \begin{equation} + \tilde{x}_p(t) = \sum\limits_{p=0}^{\infty} \tilde{a}_p \cos\left(p \omega_0 t\right) + \sum\limits_{q=0}^{\infty} \tilde{b}_q \sin\left(q \omega_0 t\right) + \label{eq_ch02_proof_per_sig_example} + \end{equation} + The coefficient $\tilde{a}_n$ is of interest. + + Inserting \eqref{eq_ch02_proof_per_sig_example} into \eqref{eq_ch02_fourier_series_coeff_an}, yields + \begin{equation} + \tilde{a}_n = \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \left(\sum\limits_{p=0}^{\infty} \tilde{a}_p \cos\left(p \omega_0 t\right) + \sum\limits_{q=0}^{\infty} \tilde{b}_q \sin\left(q \omega_0 t\right)\right) \cdot \cos\left(n \omega_0 t\right) \, \mathrm{d} t + \end{equation} + Due to the orthogonality relations, \underline{all products containing a sine function} and \underline{all products containing a cosine function with the index $n \neq p$} become zero. Furthermore, following must be true: $n = p$ + + \begin{equation} + \tilde{a}_n = \tilde{a}_p \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \cos\left(p \omega_0 t\right) \cdot \cos\left(n \omega_0 t\right) \, \mathrm{d} t \qquad \text{if } \; n = p + \end{equation} + + Using \eqref{eq:ch02:orth_rel_cos}, the integral resolves to: + \begin{equation} + \tilde{a}_n = \tilde{a}_p \frac{\omega_0}{\pi} \frac{\pi}{\omega_0} \qquad \text{if } \; n = p + \end{equation} + + In the end, it could be proven that $\tilde{a}_n = \tilde{a}_p$ for $n = p$. + + The proof is analogous for the coefficient $b_n$. +\end{proof} + +$\cos\left(n \omega_0 t\right)$ can be seen as a ``test function'', which is used to extract the component with the index $n$. The proof points out: +\begin{itemize} + \item All sine components are erased by $\cos\left(n \omega_0 t\right)$, due to the orthogonality relations. + \item All cosine function with index $p \neq n$ are erased by $\cos\left(n \omega_0 t\right)$, due to the orthogonality relations. +\end{itemize} +For $b_m$, $\sin\left(m \omega_0 t\right)$ is analogous. + +\begin{excursus}{Illustration of The ``Test Function''} + For illustration of the ``test functions'', image you have a radio and want to hear a specific station. You tune to the frequency on which the station is broadcasting. All other signals are filtered out, you don't want to hear them. Actually, the radio does not employ orthogonality in this case. However, this illustration might help to understand the meaning of $\cos\left(n \omega_0 t\right)$ and $\sin\left(m \omega_0 t\right)$ \underline{in connection} with the orthogonality relations. +\end{excursus} + +A special case is the coefficient $\tilde{a}_0$. +\begin{equation} + \tilde{a}_0 = \frac{\omega_0}{\pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \tilde{x}_p(t) \, \mathrm{d} t +\end{equation} +$\cos\left(n \omega_0 t\right)$ is $1$ for $n = 0$. $\tilde{a}_0$ is the \index{DC offset} \textbf{\ac{DC} offset} of the signal. The above formula is known as the calculation of the signal mean in electrical engineering. + +The composition of a series of mono-chromatic signals as shown in \eqref{eq:ch02:fourier_series} is called \index{Fourier series} \textbf{Fourier series}. +\begin{equation*} + x_p(t) = \sum\limits_{n=1}^{\infty} a_n \cos\left(n \omega_0 t\right) + \sum\limits_{m=1}^{\infty} b_m \sin\left(m \omega_0 t\right) +\end{equation*} + +\subsection{Complex-Valued Fourier Series} + +A complex-valued, periodic signal $\underline{x_p}(t)$ can be decomposed into complex-valued mono-chromatic signals. The coefficients $\underline{c}_n$ are phasors. +\begin{equation} + \underline{x_p}(t) = \sum\limits_{n = -\infty}^{\infty} \underline{c}_n \cdot e^{j n \omega_0 t} \qquad \forall \; n \in \mathbb{Z} + \label{eq:ch02:fourier_series_cmplx} +\end{equation} + +The coefficients $\underline{\tilde{c}}_n$ of an input signal $\underline{\tilde{x}_p}(t)$ can be determined by: +\begin{equation} + \underline{\tilde{c}}_n = \frac{\omega_0}{2 \pi} \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \underline{\tilde{x}_p}(t) \cdot e^{-j n \omega_0 t} \, \mathrm{d} t + \label{eq_ch02_fourier_series_coeff_cn} +\end{equation} + +It is based on the orthogonality relation: +\begin{equation} + \int\limits_{-\frac{T_0}{2}}^{\frac{T_0}{2}} e^{j n \omega_0 t} e^{-j p \omega_0 t} \, \mathrm{d} t = \frac{2 \pi}{\omega_0} \cdot \delta_{np} \qquad \forall \; n, p \in \mathbb{Z} + \label{eq:ch02:orth_rel_exp} +\end{equation} + +\subsection{Amplitude and Phase Spectra} + +\section{Non-Periodic Signals and Fourier Transform} + +\subsection{Derivation of The Fourier Transform} + +Non-periodic signals have no repeating pattern. Consequently, there is no period $T_0$. Mathematically, the period is indefinite $T_0 \rightarrow \infty$. + +A non-periodic signal $\underline{x_{np}}(t)$ cannot be simply decomposed by a Fourier series \eqref{eq:ch02:fourier_series_cmplx}. +\begin{equation} + \begin{split} + \underline{x_{np}}(t) &= \lim\limits_{T_0 \rightarrow \infty} \sum\limits_{n = -\infty}^{\infty} \underline{c}_n \cdot e^{j n \omega_0 t} \\ + &= \lim\limits_{T_0 \rightarrow \infty} \sum\limits_{n = -\infty}^{\infty} \underline{c}_n \cdot e^{j \frac{2 \pi n}{T_0} t} + \end{split} + \label{eq:ch02:sig_np_fourier_series} +\end{equation} + +The coefficient $\underline{c}_n$ is defined by \eqref{eq_ch02_fourier_series_coeff_cn}: +\begin{equation*} + \begin{split} + \underline{c}_n &= \frac{\omega_0}{2 \pi} \int\limits_{t = -\frac{T_0}{2}}^{\frac{T_0}{2}} \underline{x_{np}}(t) \cdot e^{-j n \omega_0 t} \, \mathrm{d} t \\ + &= \frac{1}{T_0} \int\limits_{t = -\frac{T_0}{2}}^{\frac{T_0}{2}} \underline{x_{np}}(t) \cdot e^{-j n \omega_0 t} \, \mathrm{d} t + \end{split} + \label{eq:ch02:sig_np_cn} +\end{equation*} + +In this case where $T_0 \rightarrow \infty$, $n \omega_0$ is substituted by the frequency variable $\omega$. +\begin{equation} + \omega = n \omega_0 + \label{eq:ch02:omega_subst} +\end{equation} + +Inserting \eqref{eq:ch02:sig_np_cn} into \eqref{eq:ch02:sig_np_fourier_series} while considering \eqref{eq:ch02:omega_subst}, yields: +\begin{equation} + \underline{x_{np}}(t) = \lim\limits_{T_0 \rightarrow \infty} \sum\limits_{n = -\infty}^{\infty} \frac{1}{T_0} \left( \int\limits_{t' = -\frac{T_0}{2}}^{\frac{T_0}{2}} \underline{x_{np}}(t') \cdot e^{-j \omega t'} \, \mathrm{d} t' \right) \cdot e^{j \omega t} +\end{equation} +Remember, that $n$ is still in the sum, since it has been absorbed by $\omega = n \omega_0$. + +The outer sum is a Rieman sum. $\frac{1}{T_0}$ is substituted by $\frac{\Delta \omega}{2 \pi}$. With $T_0 \rightarrow \infty$, it can be rewritten as an integral. +\begin{equation} + \underline{x_{np}}(t) = \underbrace{\frac{1}{2 \pi} \int\limits_{\omega = -\infty}^{\infty} \underbrace{\left( \int\limits_{t' = -\infty}^{\infty} \underline{x_{np}}(t') \cdot e^{-j \omega t'} \, \mathrm{d} t' \right)}_{\text{Fourier transform}} \cdot e^{j \omega t} \, \mathrm{d} \omega}_{\text{Inverse Fourier transform}} +\end{equation} + +The inner integral is the \index{Fourier transform} \textbf{Fourier transform}. + +\begin{definition}{Fourier Transform} + The \index{Fourier transform} \textbf{Fourier transform} of the function $\underline{x}(t)$ is: + \begin{equation} + \underline{X}(j \omega) = \mathcal{F} \left\{\underline{x}(t)\right\} = \int\limits_{t = -\infty}^{\infty} \underline{x}(t) \cdot e^{-j \omega t} \, \mathrm{d} t + \end{equation} + + The \index{inverse Fourier transform} \textbf{inverse Fourier transform} is: + \begin{equation} + \underline{x}(t) = \mathcal{F}^{-1} \left\{\underline{X}(j \omega)\right\} = \frac{1}{2 \pi} \int\limits_{\omega = -\infty}^{\infty} \underline{X}(j \omega) \cdot e^{+j \omega t} \, \mathrm{d} \omega + \end{equation} +\end{definition} + +\subsection{Amplitude and Phase Spectra} + +\subsection{Time Domain and Frequency Domain} + +\section{Properties of The Fourier Transform} + +\subsection{Energy Signals and Power Signals} + +Besides the classification of signals into periodic and non-periodic, signals can be divided into \index{energy signals} \textbf{energy signals} and \index{power signals} \textbf{power signals}. + +\begin{definition}{Energy and Power Signals} + \begin{itemize} + \item \textbf{Energy signals} have a finite, positive signal energy $0 < E < \infty$, but their average power is zero $P = 0$. + \item \textbf{Power signals} have a finite, positive average signal power $0 < P < \infty$, but their signal energy is indefinite $E = \infty$. + \end{itemize} +\end{definition} + +The \index{average signal power} \textbf{average signal power} $P$ is a measure for the amount of energy transferred per unit time and defined by: +\begin{equation} + P = \lim\limits_{T \rightarrow \infty} \frac{1}{T} \int\limits_{-\frac{T}{2}}^{\frac{T}{2}} \left|x(t)\right|^2 \; \mathrm{d} t +\end{equation} +The signal power is connected to the \ac{RMS} value, which is often used in electrical engineering. +\begin{equation} + \hat{x}_{RMS} = \lim\limits_{T \rightarrow \infty} \sqrt{ \frac{1}{T} \int\limits_{-\frac{T}{2}}^{\frac{T}{2}} \left|x(t)\right|^2 \; \mathrm{d} t} +\end{equation} + +The \index{signal energy} \textbf{signal energy} $E$ is: +\begin{equation} + E = \int\limits_{-\infty}^{\infty} \left|x(t)\right|^2 \; \mathrm{d} t +\end{equation} + +The property of power signals, which have an indefinite signal energy, is a problem for the Fourier transform. The transform would yield an indefinite value. Thus: +\begin{fact} + Every energy signal has a Fourier transform. +\end{fact} + +Only some power signals have a Fourier transform. There are distributions which are power signals, but have a Fourier transform, too. Especially, all \emph{tempered distributions} have a Fourier transform. + +\subsection{Dirac Delta Function} + +An important distribution is the \index{Dirac delta function} \textbf{Dirac delta function} $\delta(t)$. The Dirac delta function is zero everywhere except at its origin, where it is an indefinitely narrow, indefinitely high pulse. +\begin{equation} + \delta(t) = \begin{cases} + +\infty & \qquad \text{if } t = 0, \\ + 0 & \qquad \text{if } t \neq 0 + \end{cases} + \label{eq:ch02:dirac_delta} +\end{equation} +It is constrained by +\begin{equation} + \int\limits_{-\infty}^{\infty} \delta(t) \; \mathrm{d} t = 1 +\end{equation} + +\begin{attention} + The Dirac delta function $\delta(t)$ must not be confused with the Kronecker delta \eqref{eq:ch02:kronecker_delta}. The Dirac delta function operates in continuous space $t \in \mathbb{R}$. The Kronecker delta $\delta_n$ (here one-dimensional) operates in discrete space $n \in \mathbb{Z}$. +\end{attention} + +A special feature of the function is called \index{Dirac measure} \textbf{Dirac measure}. +\begin{equation} + \int\limits_{-\infty}^{\infty} f(t) \delta(t) \; \mathrm{d} t = f(0) + \label{eq:ch02:dirac_measure} +\end{equation} + +Using the Dirac measure, the Fourier transform can be calculated: +\begin{equation} + \mathcal{F} \left\{\delta(t)\right\} = \int\limits_{-\infty}^{\infty} \delta(t) \cdot e^{-j \omega t} \; \mathrm{d} t = 1 +\end{equation} +The Fourier transform of the Dirac delta function is the frequency-independent constant $1$. + +\subsection{Operations 1: Linearity} + +\subsection{Operations 2: Differentiation and Integration} + +\subsection{Operations 3: Multiplication} + +\subsection{Operations 4: Time Shift} + +\subsection{Duality} + +\section{\acs{LTI} Systems} + +\subsection{Transfer Function and Impulse Response} + +\subsection{Convolution} + +\subsection{Poles and Zeroes} + +\printbibliography[heading=subbibliography] +\end{refsection} diff --git a/common/settings.tex b/common/settings.tex index 0e73eb1..e98b536 100644 --- a/common/settings.tex +++ b/common/settings.tex @@ -1,7 +1,7 @@ \documentclass[% a4paper,% A4 Papier twoside,% einseitig (linker und rechter Seitenrand sind gleich groß) - bibliography=totocnumbered,% Literaturverzeichnis nummeriert mit ins + %bibliography=totocnumbered,% Literaturverzeichnis nummeriert mit ins % Inhaltsverzeichnis einfügen numbers=noenddot,% hinter der Gliederungsnummer soll kein Punkt gesetzt werden (siehe Duden) parskip=half,% europäischer Satz mit Abstand zwischen Absätzen @@ -177,8 +177,38 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Bibliography -%\usepackage{bibgerm} -%\usepackage[numbers]{natbib} +%\usepackage[square, numbers, sectionbib]{natbib} +%\usepackage{chapterbib} +%\def\thebibfile{../DCS} +%\def\thebibstyle{unsrtnat} +% +%\makeatletter +%\renewenvironment{thebibliography}[1] +%{\section*{\bibname}% <-- this line was changed from \chapter* to \section* +% \@mkboth{\MakeUppercase\bibname}{\MakeUppercase\bibname}% +% \list{\@biblabel{\@arabic\c@enumiv}}% +% {\settowidth\labelwidth{\@biblabel{#1}}% +% \leftmargin\labelwidth +% \advance\leftmargin\labelsep +% \@openbib@code +% \usecounter{enumiv}% +% \let\p@enumiv\@empty +% \renewcommand\theenumiv{\@arabic\c@enumiv}}% +% \sloppy +% \clubpenalty4000 +% \@clubpenalty \clubpenalty +% \widowpenalty4000% +% \sfcode`\.\@m} +%{\def\@noitemerr +% {\@latex@warning{Empty `thebibliography' environment}}% +% \endlist} +%\makeatother + +%\usepackage[square, numbers, sectionbib]{natbib} +%\usepackage[backend=bibtex,style=unsrtnat]{biblatex} +\usepackage[backend=biber,sorting=none]{biblatex} +\addbibresource{../DCS.bib} +%\bibliography{../DCS} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Directories @@ -233,6 +263,11 @@ hidelinks]{hyperref} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Own commands + +\newcommand{\licensequote}[3]{\textit{Reproduced from #1. Copyright by #2. License: #3.}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Own environments diff --git a/main/DCS.tex b/main/DCS.tex index 3986e61..595d615 100644 --- a/main/DCS.tex +++ b/main/DCS.tex @@ -48,13 +48,13 @@ \setcounter{chapter}{-1} -\input{../chapter00/content.tex} +\input{../chapter00/content_ch00.tex} \clearpage -\input{../chapter01/content.tex} +\input{../chapter01/content_ch01.tex} \clearpage -\input{../chapter02/content.tex} +\input{../chapter02/content_ch02.tex} \clearpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -86,8 +86,8 @@ \newpage % Print default index -\phantomsection -\addcontentsline{toc}{chapter}{Index} +\phantomsection +\addcontentsline{toc}{chapter}{Index} \printindex \newpage diff --git a/main/chapter00.tex b/main/chapter00.tex index c96a1b6..a452bbf 100644 --- a/main/chapter00.tex +++ b/main/chapter00.tex @@ -33,13 +33,16 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Content -\clearpage \input{../chapter00/preface.tex} +\clearpage \setcounter{chapter}{-1} +\input{../chapter00/content_ch00.tex} +\clearpage + +\input{../chapter00/exercise00.tex} \clearpage -\input{../chapter00/content.tex} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Appendix @@ -74,8 +77,8 @@ \newpage % Print default index -\phantomsection -\addcontentsline{toc}{chapter}{Index} +\phantomsection +\addcontentsline{toc}{chapter}{Index} \printindex \newpage diff --git a/main/chapter01.tex b/main/chapter01.tex index 2e63445..33f8ba2 100644 --- a/main/chapter01.tex +++ b/main/chapter01.tex @@ -35,8 +35,8 @@ \setcounter{chapter}{0} +\input{../chapter01/content_ch01.tex} \clearpage -\input{../chapter01/content.tex} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Appendix @@ -72,7 +72,7 @@ % Print default index \phantomsection -\addcontentsline{toc}{chapter}{Index} +\addcontentsline{toc}{chapter}{Index} \printindex \newpage diff --git a/main/chapter02.tex b/main/chapter02.tex index b7b11c7..213ad45 100644 --- a/main/chapter02.tex +++ b/main/chapter02.tex @@ -35,8 +35,8 @@ \setcounter{chapter}{1} +\input{../chapter02/content_ch02.tex} \clearpage -\input{../chapter02/content.tex} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Appendix -- cgit v1.1