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vs/assignment4/main.tex
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412
vs/assignment4/main.tex
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\documentclass[a4paper]{article}
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%\usepackage[singlespacing]{setspace}
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\usepackage[onehalfspacing]{setspace}
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%\usepackage[doublespacing]{setspace}
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\usepackage{geometry} % Required for adjusting page dimensions and margins
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\usepackage{amsmath,amsfonts,stmaryrd,amssymb,mathtools,dsfont} % Math packages
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\usepackage{tabularx}
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\usepackage{colortbl}
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\usepackage{listings}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{enumerate}
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\usepackage{enumitem}
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\usepackage{amsthm}
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\usepackage{subcaption}
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\usepackage{float}
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\usepackage[table,xcdraw]{xcolor}
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\usepackage{tikz-qtree}
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\usepackage{forest}
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\usepackage{changepage,titlesec,fancyhdr} % For styling Header and Titles
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\pagestyle{fancy}
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\renewcommand{\headrulewidth}{0.5pt} % Linienbreite anpassen, falls gewünscht
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\renewcommand{\headrule}{
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\makebox[\textwidth]{\rule{1.0\textwidth}{0.5pt}}
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}
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\usepackage{amsmath}
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\pagestyle{fancy}
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\usepackage{diagbox}
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\usepackage{xfrac}
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\usepackage{pgfplots}
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\usepackage{pgfplotstable}
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\pgfplotsset{compat=1.18}
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\usepackage{enumerate} % Custom item numbers for enumerations
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\usepackage[ruled]{algorithm2e} % Algorithms
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\usepackage[framemethod=tikz]{mdframed} % Allows defining custom boxed/framed environments
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\usepackage{listings} % File listings, with syntax highlighting
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\lstset{
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basicstyle=\ttfamily, % Typeset listings in monospace font
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}
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\usepackage[ddmmyyyy]{datetime}
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\geometry{
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paper=a4paper, % Paper size, change to letterpaper for US letter size
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top=3cm, % Top margin
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bottom=3cm, % Bottom margin
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left=2.5cm, % Left margin
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right=2.5cm, % Right margin
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headheight=25pt, % Header height
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footskip=1.5cm, % Space from the bottom margin to the baseline of the footer
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headsep=1cm, % Space from the top margin to the baseline of the header
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%showframe, % Uncomment to show how the type block is set on the page
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}
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\lhead{\vspace{0.5\baselineskip}Übungsblatt 4}
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\chead{\bfseries{Einführung in Verteilte Systeme\\Sommersemester 2025}}
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\rhead{\vspace{0.5\baselineskip}Werner, 7987847}
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\fancyheadoffset[R]{0cm}
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\begin{document}
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\setcounter{section}{4}
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\subsection{}
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Gegeben sei das folgende Non-Return-to-Zero-kodierte Signal:\\
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\begin{center}
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\begin{tikzpicture}
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\node at (-1.2,0.45) {\textbf{Takt}};
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\begin{axis}[
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axis lines=middle,
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xlabel={t/ns},
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xlabel style={at={(axis description cs:0.97,0.23)}, anchor=north},
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y axis line style={draw=none},
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ylabel={\empty},
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xtick={0,6.25,...,75},
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xticklabels={
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0, , , ,
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25, , , ,
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50, , , ,
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75,
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},
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ytick=\empty,
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width=13cm,
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height=3.3cm,
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ymin=-0.5, ymax=1.5,
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xmin=0, xmax=85,
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enlargelimits=false,
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domain=0:12,
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samples=100,
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]
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\addplot[
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black,
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thick
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] coordinates {
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(0,0)
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(6.25,0)
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(6.25,1)
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(12.5,1)
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(12.5,0)
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(18.75,0)
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(18.75,1)
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(25,1)
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(25,0)
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(31.25,0)
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(31.25,1)
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(37.5,1)
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(37.5,0)
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(43.75,0)
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(43.75,1)
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(50,1)
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(50,0)
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(56.25,0)
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(56.25,1)
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(62.5,1)
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(62.5,0)
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(68.75,0)
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(68.75,1)
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(75,1)
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(75,0)
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};
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\end{axis}
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\end{tikzpicture}
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\end{center}
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\begin{center}
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\begin{tikzpicture}
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\node at (-1.2,0.45) {\textbf{NRZ-Signal}};
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\begin{axis}[
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axis lines=middle,
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xlabel={t/ns},
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xlabel style={at={(axis description cs:0.97,0.23)}, anchor=north},
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y axis line style={draw=none},
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ylabel={\empty},
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xtick={0,6.25,...,75},
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xticklabels={
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0, , , ,
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25, , , ,
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50, , , ,
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75,
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},
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ytick=\empty,
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width=13cm,
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height=3.3cm,
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ymin=-0.5, ymax=1.5,
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xmin=0, xmax=85,
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enlargelimits=false,
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clip=false,
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domain=0:12,
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samples=100,
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]
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\addplot[
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black,
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thick
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] coordinates {
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(-2,1)
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(12.5,1)
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(12.5,0)
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(25,0)
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(25,1)
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(37.5,1)
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(37.5,0)
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(50,0)
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(50,1)
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(62.5,1)
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(75,1)
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};
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\addplot[
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black,
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line width=1.5pt,
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dash pattern=on 1.2pt off 1.8pt
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] coordinates {
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(-2.5,1)
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(-3.5,1)
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};
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\end{axis}
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\end{tikzpicture}
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\end{center}
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\noindent Im Folgenden soll NRZI in der NRZI-M-Kodierung verwendet werden, so dass sich jeweils ein Zustandswechsel bei einem Eins-Bit ergibt.
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\begin{enumerate}[label=\alph*)]
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\item Dekodieren Sie die in dem dargestellten NRZ-Signal übertragenen Bits.\\
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$\Rightarrow$ \textbf{101011}
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\item Bestimmen Sie die Bitrate des dargestellten NRZ-Signals in Mbit/s.\\
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\[\frac{\text{Codelänge (Bitanzahl)}}{\text{Gesamtübermittlungsdauer}}\frac{6\, bit}{75\, ns}=0,08\, bit/ns = 80\, Mbit/s\]
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\item Zeichnen Sie das NRZI-Signal, das der gegebenen Bitfolge entspricht.\\
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\begin{center}
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\begin{tikzpicture}
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\node at (-1.2,0.45) {\textbf{NRZI-Signal}};
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\begin{axis}[
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axis lines=middle,
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xlabel={t/ns},
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xlabel style={at={(axis description cs:0.97,0.23)}, anchor=north},
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y axis line style={draw=none},
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ylabel={\empty},
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xtick={0,6.25,...,75},
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xticklabels={
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0, , , ,
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25, , , ,
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50, , , ,
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75,
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},
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ytick=\empty,
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width=13cm,
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height=3.3cm,
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ymin=-0.5, ymax=1.5,
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xmin=0, xmax=85,
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enlargelimits=false,
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clip=false,
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domain=0:12,
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samples=100,
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]
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\addplot[
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black,
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thick
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] coordinates {
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(-2,1)
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(6.25,1)
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(6.25,0)
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(31.25,0)
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(31.25,1)
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(56.25,1)
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(56.25,0)
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(68.75,0)
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(68.75,1)
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(75,1)
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};
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\addplot[
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black,
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line width=1.5pt,
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dash pattern=on 1.2pt off 1.8pt
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] coordinates {
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(-2.5,1)
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(-3.5,1)
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};
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\end{axis}
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\end{tikzpicture}
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\end{center}
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\item Takt-Rückgewinnung: Kann ein NRZI-Signal alleine verwendet werden, um die Information zu übertragen, oder muss ein Taktsignal zur Synchronisation mitgesendet werden?\\
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Begründen Sie Ihre Antwort in einem Satz. \\\\
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Nein, ohne eine Taktfrequenz oder Taktsignal kann das Signal nicht zuverlässig dekodiert werden. Längere konstante Intervalle bieten nicht genügend Flanken um eine Taktfrequenz zu erschließen.
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\end{enumerate}
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\clearpage
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\subsection{}
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\begin{enumerate}[label=\alph*)]
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\item Dekodieren Sie das dargestellte Manchester-kodierte Signal.\\
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(Es gilt, analog zur Vorlesung, folgende Definition: Ein $0 \to 1$-Übergang sei eine logische Null, $1 \to 0$ eine logische Eins.)\\
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\begin{center}
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\begin{tikzpicture}
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\node at (-1.2,0.45) {\textbf{Signal}};
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\begin{axis}[
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axis lines=middle,
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xlabel={$t$},
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xlabel style={at={(axis description cs:0.97,0.23)}, anchor=north},
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y axis line style={draw=none},
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ylabel={\empty},
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xtick={\empty},
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ytick=\empty,
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width=13cm,
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height=3.3cm,
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ymin=-0.5, ymax=1.5,
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xmin=0, xmax=13,
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enlargelimits=false,
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domain=0:12,
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samples=100,
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]
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\addplot[
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black,
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thick
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] coordinates {
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(0,1)
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(1,1)
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(1,0)
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(2,0)
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(2,1)
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(3,1)
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(3,0)
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(4,0)
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(4,1)
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(5,1)
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(5,0)
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(6,0)
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(7,0)
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(7,1)
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(8,1)
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(8,0)
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(9,0)
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(9,1)
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(11,1)
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(11,0)
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(12,0)
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};
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\addplot[purple] coordinates {(2,1.5) (2,-0.5)};
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\addplot[purple] coordinates {(4,1.5) (4,-0.5)};
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\addplot[purple] coordinates {(6,1.5) (6,-0.5)};
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\addplot[purple] coordinates {(8,1.5) (8,-0.5)};
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\addplot[purple] coordinates {(10,1.5) (10,-0.5)};
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\addplot[purple] coordinates {(12,1.5) (12,-0.5)};
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\end{axis}
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\end{tikzpicture}
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\end{center}
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\vspace*{-1cm}
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\begin{center}
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\begin{tikzpicture}
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\node at (-1.2,0.45) {\textbf{Signal}};
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\begin{axis}[
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axis lines=middle,
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xlabel={$t$},
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xlabel style={at={(axis description cs:0.97,0.23)}, anchor=north},
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y axis line style={draw=none},
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ylabel={\empty},
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xtick={\empty},
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ytick=\empty,
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width=13cm,
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height=3.3cm,
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ymin=-0.5, ymax=1.5,
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xmin=0, xmax=13,
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enlargelimits=false,
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domain=0:12,
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samples=100,
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]
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\addplot[
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black,
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thick
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] coordinates {
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(0,0)
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(1,0)
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(1,1)
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(2,1)
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(2,1)
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(3,1)
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(3,0)
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(4,0)
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(4,1)
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(5,1)
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(5,0)
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(6,0)
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(6,1)
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(7,1)
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(7,0)
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(8,0)
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(9,0)
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(9,1)
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(10,1)
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(10,0)
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(11,0)
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(11,1)
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(12,1)
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};
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\addplot[purple] coordinates {(2,1.5) (2,-0.5)};
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\addplot[purple] coordinates {(4,1.5) (4,-0.5)};
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\addplot[purple] coordinates {(6,1.5) (6,-0.5)};
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\addplot[purple] coordinates {(8,1.5) (8,-0.5)};
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\addplot[purple] coordinates {(10,1.5) (10,-0.5)};
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\addplot[purple] coordinates {(12,1.5) (12,-0.5)};
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\end{axis}
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\end{tikzpicture}
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\end{center}
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Markieren Sie in der Abbildung die Grenzen zwischen übertragenen Bits mit vertikalen Strichen.\\
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\item Nun soll die Bitfolge 011100 übertragen werden. Zeichnen Sie den Verlauf des Manchesterkodierten Signals.
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\item Begründen Sie, ob sich dieses Manchesterkodierte Signal eindeutig bestimmen lässt.\\
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\begin{center}
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\begin{tikzpicture}
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\node at (-1.2,0.45) {\textbf{Signal}};Takt
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\begin{axis}[
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axis lines=middle,
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xlabel={$t$},
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xlabel style={at={(axis description cs:0.97,0.23)}, anchor=north},
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y axis line style={draw=none},
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ylabel={\empty},
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xtick={\empty},
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ytick=\empty,
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width=13cm,
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height=3.3cm,
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ymin=-0.5, ymax=1.5,
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xmin=0, xmax=13,
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enlargelimits=false,
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domain=0:12,
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samples=100,
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]
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\addplot[
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black,
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thick
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] coordinates {
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(0,0)
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(1,0)
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(1,1)
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(3,1)
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(3,0)
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(5,0)
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(5,1)
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(7,1)
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(7,0)
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(9,0)
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(9,1)
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(11,1)
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(11,0)
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(12,0)
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};
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\end{axis}
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\end{tikzpicture}
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\end{center}
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Ja, aus der Vorlesung wissen wir, dass Signale mit der Manchester-Kodierung eine Taktrückgewinnung ermöglichen. Durch die spezielle Kodierung gibt es pro Takt mindestens eine Flanke welche Orientierung und Synchronisierung bietet.
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\item Takt-Rückgewinnung: Kann das Manchester-Signal alleine verwendet werden, um die Information zu übertragen, oder muss ein Taktsignal zur Synchronisation mitgesendet werden?\\
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Begründen Sie Ihre Antwort in einem Satz\\\bigskip
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Ja, wie schon in der Teilaufgabe c) erwähnt, bietet die Manchester-Kodierung mindestens eine Flanke pro Takt, also für jedee Bitstelle, zur Eindeutigen Bestimmung des Signals. Somit erfolgt auch die Takt-Rückgewinnung, welche üblicherweise im 2. Schritt zur Bestimmung des Signals genutzt wird.
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\end{enumerate}
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\end{document}
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