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authorAndreas Müller <andreas.mueller@ost.ch>2021-07-22 08:13:09 +0200
committerGitHub <noreply@github.com>2021-07-22 08:13:09 +0200
commit19df8ee38a7240dbb0b1ed93c8f51bf507d491b8 (patch)
treee3a4cf741a20dc5a356de353cca3167996477638 /buch/papers/reedsolomon
parentMerge pull request #43 from tschwall/master (diff)
parentupdate (diff)
downloadSeminarMatrizen-19df8ee38a7240dbb0b1ed93c8f51bf507d491b8.tar.gz
SeminarMatrizen-19df8ee38a7240dbb0b1ed93c8f51bf507d491b8.zip
Merge pull request #44 from JODBaer/Baer
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Diffstat (limited to 'buch/papers/reedsolomon')
-rw-r--r--buch/papers/reedsolomon/RS presentation/images/polynom1 - Kopie.tex33
-rw-r--r--buch/papers/reedsolomon/dtf.tex53
-rw-r--r--buch/papers/reedsolomon/einleitung.tex10
-rw-r--r--buch/papers/reedsolomon/experiments/codiert.txt96
-rw-r--r--buch/papers/reedsolomon/experiments/decodiert.txt96
-rw-r--r--buch/papers/reedsolomon/experiments/empfangen.txt96
-rw-r--r--buch/papers/reedsolomon/experiments/f.m21
-rw-r--r--buch/papers/reedsolomon/experiments/fehler.txt96
-rw-r--r--buch/papers/reedsolomon/experiments/locator.txt96
-rw-r--r--buch/papers/reedsolomon/experiments/plot.tex103
-rw-r--r--buch/papers/reedsolomon/experiments/signal.txt96
-rw-r--r--buch/papers/reedsolomon/experiments/syndrom.txt96
-rw-r--r--buch/papers/reedsolomon/idee.tex78
-rw-r--r--buch/papers/reedsolomon/images/codiert.txt96
-rw-r--r--buch/papers/reedsolomon/images/decodiert.txt96
-rw-r--r--buch/papers/reedsolomon/images/empfangen.txt96
-rw-r--r--buch/papers/reedsolomon/images/fehler.txt96
-rw-r--r--buch/papers/reedsolomon/images/locator.txt96
-rw-r--r--buch/papers/reedsolomon/images/plotfft.tex89
-rw-r--r--buch/papers/reedsolomon/images/polynom2.tex49
-rw-r--r--buch/papers/reedsolomon/images/signal.txt96
-rw-r--r--buch/papers/reedsolomon/images/syndrom.txt96
-rw-r--r--buch/papers/reedsolomon/main.tex2
-rw-r--r--buch/papers/reedsolomon/packages.tex2
-rw-r--r--buch/papers/reedsolomon/teil2.tex40
25 files changed, 1735 insertions, 89 deletions
diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom1 - Kopie.tex b/buch/papers/reedsolomon/RS presentation/images/polynom1 - Kopie.tex
new file mode 100644
index 0000000..038e93e
--- /dev/null
+++ b/buch/papers/reedsolomon/RS presentation/images/polynom1 - Kopie.tex
@@ -0,0 +1,33 @@
+% polynome1
+%-------------------
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\newcommand{\teiler}{40}
+\begin{document}
+
+
+\begin{tikzpicture}[>=latex,thick]
+
+ \begin{axis}[
+ axis lines = left,
+ xlabel = \(x\),
+ ylabel = {\(f(x)\)},
+ ]
+ %Below the red parabola is defined
+ \addplot[
+ color=blue,
+ ]
+ coordinates {
+ (0,23.1)(10,27.5)(20,32)(30,37.8)(40,44.6)(60,61.8)(80,83.8)(100,114)
+ };
+ %Here the blue parabola is defined
+
+ \end{axis}
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/papers/reedsolomon/dtf.tex b/buch/papers/reedsolomon/dtf.tex
index 025be3a..a111527 100644
--- a/buch/papers/reedsolomon/dtf.tex
+++ b/buch/papers/reedsolomon/dtf.tex
@@ -3,28 +3,53 @@
%
% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
%
-\section{Diskrete Fourien Transformation
+\section{Diskrete Fourier Transformation
\label{reedsolomon:section:dtf}}
\rhead{Umwandlung mit DTF}
Um die Polynominterpolation zu umgehen, gehen wir nun über in die Fourientransformation.
Dies wird weder eine erklärung der Forientransorfmation noch ein genauer gebrauch
für den Reed-Solomon-Code. Dieser Abschnitt zeigt nur wie die Fourientransformation auf Fehler reagiert.
-wobei sie dann bei späteren Berchnungen ganz nütlich ist.
-
-\subsection{Übertragungsabfolge
-\label{reedsolomon:subsection:Übertragungsabfolge}}
-Das Signal.... sind die Daten, Zahlen welche übertragen werden sollen.
-Das speziell ist das wir 100 Punkte übertragen und von 64 bis 100,
-werden nur Null Punkte übertragen, dies weiss auch unser Empfänger.
-Nun wird das Signal in Abbildung... codiert...
-Somit wird die Information jedes Punktes auf das ganze spektrum von 0 bis 100 übertragen.
-Kommen nuun drei Fehler... hinzu zu diesem codierten Signal sind diese nicht zu erkennen.
-Nach dem Empfangen... und decodieren ... erkennt man die fehlerhafte information in den Punkten 64 bis 100.
-Filtert man nur diese Punkte heraus und Transformiert sie mit Fourier erhält man die stellen an denen die Fehler sich eingeschlichen haben.
+wobei sie dann bei späteren Berchnungen ganz nützlich ist.
\subsection{Diskrete Fourientransformation Zusamenhang
\label{reedsolomon:subsection:dtfzusamenhang}}
Die Diskrete Fourientransformation ist definiert als
-....
+ \[
+ \label{ft_discrete}
+ \hat{c}_{k}
+ = \frac{1}{N} \sum_{n=0}^{N-1}
+ {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn}
+ \]
+, wenn man nun
+ \[
+ w = e^{-\frac{2\pi j}{N} k}
+ \]
+ersetzte, und $N$ konstantbleibt, erhält man
+ \[
+ \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N)
+ \]
+was überaust ähnlich zu unserem Polynomidee ist.
+\subsection{Übertragungsabfolge
+\label{reedsolomon:subsection:Übertragungsabfolge}}
+\begin{enumerate}[1)]
+\item Das Signal hat 64 die Daten, Zahlen welche übertragen werden sollen.
+Dabei zusätzlich nach 16 Fehler abgesichert, macht insgesamt 96 Übertragungszahlen.
+\item Nun wurde mittels der schnellen diskreten Fourientransformation diese 96 codiert.
+Das heisst alle information ist in alle Zahlenvorhanden.
+\item Nun kommen drei Fehler dazu an den Übertragungsstellen 7, 21 und 75.
+\item Dieses wird nun Empfangen und mittels inversen diskreten Fourientransormation, wieder rücktransformiert.
+\item Nun sieht man den Fehler im Decodieren in den Übertragungsstellen 64 bis 96.
+\item Nimmt man nun nur diese Stellen 64 bis 96, auch Syndrom genannt, und Transformiert diese.
+\item Bekommt man die Fehlerstellen im Locator wieder, zwar nichtso genau, dennoch erkkent man wo die Fehler stattgefunden haben.
+\end{enumerate}
+\begin{figure}
+ \centering
+ \resizebox{0.9\textwidth}{!}{
+ %\includegraphics[width=0.5\textwidth]{papers/reedsolomon/images/plot.pdf}
+ \input{papers/reedsolomon/images/plotfft.tex}
+ }
+ \caption{Übertragungsabfolge \ref{reedsolomon:subsection:Übertragungsabfolge}}
+ \label{fig:sendorder}
+\end{figure} \ No newline at end of file
diff --git a/buch/papers/reedsolomon/einleitung.tex b/buch/papers/reedsolomon/einleitung.tex
index 3d40db1..2b1d878 100644
--- a/buch/papers/reedsolomon/einleitung.tex
+++ b/buch/papers/reedsolomon/einleitung.tex
@@ -6,13 +6,13 @@
\section{Einleitung
\label{reedsolomon:section:einleitung}}
\rhead{Einleitung}
-Der Reed-Solomon-Code ist entstaden im ... vom .. um,
-das Problem der Daten Übertragung zu lösen.
-In deiesem Abschnitt wird möglichst verständlich die mathematische Abfolge, Funktion oder Algorithmus erklärt.
+Der Reed-Solomon-Code ist entstanden um,
+das Problem der Fehler, bei der Datenübertragung, zu lösen.
+In diesem Abschnitt wird möglichst verständlich die mathematische Abfolge, Funktion oder Algorithmus erklärt.
Es wird jedoch nicht auf die technische Umsetzung oder Implementierung eingegangen.
-Um beim Daten Übertragen fehler zu erkennen könnte man die Daten jeweils doppelt senden,
+Um beim Datenübertragen Fehler zu erkennen, könnte man die Daten jeweils doppelt senden,
und so jeweilige Fehler zu erkennen.
-Doch dies braucht schnell unmengen an Daten, wenn man nach vielen Fehler absichern möchte.
+Doch nur schon um weinige Fehler zu erkennen werden überproportional viele Daten doppelt und dreifach gesendet.
Der Reed-Solomon-Code macht dies auf eine andere, clevere Weise.
diff --git a/buch/papers/reedsolomon/experiments/codiert.txt b/buch/papers/reedsolomon/experiments/codiert.txt
new file mode 100644
index 0000000..4a481d8
--- /dev/null
+++ b/buch/papers/reedsolomon/experiments/codiert.txt
@@ -0,0 +1,96 @@
+0,284
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diff --git a/buch/papers/reedsolomon/experiments/decodiert.txt b/buch/papers/reedsolomon/experiments/decodiert.txt
new file mode 100644
index 0000000..f6221e6
--- /dev/null
+++ b/buch/papers/reedsolomon/experiments/decodiert.txt
@@ -0,0 +1,96 @@
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diff --git a/buch/papers/reedsolomon/experiments/empfangen.txt b/buch/papers/reedsolomon/experiments/empfangen.txt
new file mode 100644
index 0000000..38c13b0
--- /dev/null
+++ b/buch/papers/reedsolomon/experiments/empfangen.txt
@@ -0,0 +1,96 @@
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diff --git a/buch/papers/reedsolomon/experiments/f.m b/buch/papers/reedsolomon/experiments/f.m
index 6bdc741..bf2587c 100644
--- a/buch/papers/reedsolomon/experiments/f.m
+++ b/buch/papers/reedsolomon/experiments/f.m
@@ -1,8 +1,8 @@
-#
-# f.m -- Reed-Solomon-Visualisierung mit FFT
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
+%
+% f.m -- Reed-Solomon-Visualisierung mit FFT
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+
N = 64;
b = 32;
l = N + b;
@@ -51,6 +51,7 @@ syndrom(1:N,1) = zeros(N,1)
plot(abs(syndrom));
xlim([1, l]);
title("Syndrom");
+
pause()
locator = abs(fft(syndrom))
@@ -59,3 +60,13 @@ plot(locator);
xlim([1, l]);
title("Locator");
pause()
+
+
+writematrix([transpose(counter), abs(signal)], 'signal.txt')
+writematrix([transpose(counter), abs(codiert)], 'codiert.txt')
+writematrix([transpose(counter), fehler], 'fehler.txt')
+writematrix([transpose(counter), abs(empfangen)], 'empfangen.txt')
+writematrix([transpose(counter), abs(decodiert)], 'decodiert.txt')
+writematrix([transpose(counter), abs(syndrom)], 'syndrom.txt')
+writematrix([transpose(counter), locator], 'locator.txt')
+
diff --git a/buch/papers/reedsolomon/experiments/fehler.txt b/buch/papers/reedsolomon/experiments/fehler.txt
new file mode 100644
index 0000000..23f1a83
--- /dev/null
+++ b/buch/papers/reedsolomon/experiments/fehler.txt
@@ -0,0 +1,96 @@
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diff --git a/buch/papers/reedsolomon/experiments/locator.txt b/buch/papers/reedsolomon/experiments/locator.txt
new file mode 100644
index 0000000..b28988c
--- /dev/null
+++ b/buch/papers/reedsolomon/experiments/locator.txt
@@ -0,0 +1,96 @@
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diff --git a/buch/papers/reedsolomon/experiments/plot.tex b/buch/papers/reedsolomon/experiments/plot.tex
new file mode 100644
index 0000000..2196c82
--- /dev/null
+++ b/buch/papers/reedsolomon/experiments/plot.tex
@@ -0,0 +1,103 @@
+% polynome1
+%-------------------
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usepackage{pgfplotstable}
+\usepackage{filecontents}
+\usetikzlibrary{arrows,intersections,math}
+\newcommand{\x}{10}
+\newcommand{\y}{-8}
+\begin{document}
+
+\begin{tikzpicture}[]
+
+ %---------------------------------------------------------------
+ %Knote
+ \matrix[draw = none, column sep=20mm, row sep=4mm]{
+ \node(signal) [] {
+ \begin{tikzpicture}
+ \begin{axis}[
+ title = {\Large {Signal}},
+ xlabel={Anzahl Übertragene Zahlen},
+ xtick={0,20,40,64,80,98},]
+ \addplot[blue] table[col sep=comma] {signal.txt};
+ \end{axis}
+ \end{tikzpicture}}; &
+
+ \node(codiert) [] {
+ \begin{tikzpicture}
+ \begin{axis}[title = {\Large {Codiert}}]
+ \addplot[] table[col sep=comma] {codiert.txt};
+ \end{axis}
+ \end{tikzpicture}}; \\
+
+ &\node(fehler) [] {
+ \begin{tikzpicture}
+ \begin{axis}[scale=0.6, title = {\Large {Fehler}}]
+ \addplot[red] table[col sep=comma] {fehler.txt};
+ \end{axis}
+ \end{tikzpicture}};\\
+
+ \node(decodiert) [] {
+ \begin{tikzpicture}
+ \begin{axis}[title = {\Large {Decodiert}}]
+ \addplot[blue] table[col sep=comma] {decodiert.txt};
+ \end{axis}
+ \end{tikzpicture}}; &
+
+ \node(empfangen) [] {
+ \begin{tikzpicture}
+ \begin{axis}[title = {\Large {Empfangen}}]
+ \addplot[] table[col sep=comma] {empfangen.txt};
+ \end{axis}
+ \end{tikzpicture}};\\
+
+ \node(syndrom) [] {
+ \begin{tikzpicture}
+ \begin{axis}[title = {\Large {Syndrom}}]
+ \addplot[blue] table[col sep=comma] {syndrom.txt};
+ \end{axis}
+ \end{tikzpicture}}; &
+
+ \node(locator) [] {
+ \begin{tikzpicture}
+ \begin{axis}[title = {\Large {Locator}}]
+ \addplot[] table[col sep=comma] {locator.txt};
+ \end{axis}
+ \end{tikzpicture}};\\
+ };
+ %-------------------------------------------------------------
+ %FFT & IFFT deskription
+
+ \draw[thin,gray,dashed] (0,12) to (0,-12);
+ \node(IFFT) [scale=0.7] at (0,12.3) {IFFT};
+ \draw[<-](IFFT.south west)--(IFFT.south east);
+ \node(FFT) [scale=0.7, above of=IFFT] {FFT};
+ \draw[->](FFT.north west)--(FFT.north east);
+
+ \draw[thick, ->,] (fehler.west)++(-1,0) +(0.05,0.5) -- +(-0.1,-0.1) -- +(0.1,0.1) -- +(0,-0.5);
+ %Arrows
+ \draw[ultra thick, ->] (signal.east) to (codiert.west);
+ \draw[ultra thick, ->] (codiert.south) to (fehler.north);
+ \draw[ultra thick, ->] (fehler.south) to (empfangen.north);
+ \draw[ultra thick, ->] (empfangen.west) to (decodiert.east);
+ \draw[ultra thick, ->] (syndrom.east) to (locator.west);
+ \draw(decodiert.south east)++(-1.8,1) ellipse (1.3cm and 0.8cm) ++(-1.3,0) coordinate(zoom) ;
+ \draw[ultra thick, ->] (zoom) to[out=180, in=90] (syndrom.north);
+
+ %item
+ \node[circle, draw, fill =lightgray] at (signal.north west)+(1,0) {1};
+ \node[circle, draw, fill =lightgray] at (codiert.north west) {2};
+ \node[circle, draw, fill =lightgray] at (fehler.north west) {3};
+ \node[circle, draw, fill =lightgray] at (empfangen.north west) {4};
+ \node[circle, draw, fill =lightgray] at (decodiert.north west) {5};
+ \node[circle, draw, fill =lightgray] at (syndrom.north west) {6};
+ \node[circle, draw, fill =lightgray] at (locator.north west) {7};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/papers/reedsolomon/experiments/signal.txt b/buch/papers/reedsolomon/experiments/signal.txt
new file mode 100644
index 0000000..c4fa5f8
--- /dev/null
+++ b/buch/papers/reedsolomon/experiments/signal.txt
@@ -0,0 +1,96 @@
+0,6
+1,6
+2,0
+3,6
+4,4
+5,0
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+12,0
+13,6
+14,3
+15,5
+16,7
+17,5
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+19,4
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diff --git a/buch/papers/reedsolomon/experiments/syndrom.txt b/buch/papers/reedsolomon/experiments/syndrom.txt
new file mode 100644
index 0000000..8ca9eed
--- /dev/null
+++ b/buch/papers/reedsolomon/experiments/syndrom.txt
@@ -0,0 +1,96 @@
+0,0
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diff --git a/buch/papers/reedsolomon/idee.tex b/buch/papers/reedsolomon/idee.tex
index 4a7716a..39adbbf 100644
--- a/buch/papers/reedsolomon/idee.tex
+++ b/buch/papers/reedsolomon/idee.tex
@@ -8,51 +8,81 @@
\rhead{Problemstellung}
Das Problem liegt darin Informationen, Zahlen,
zu Übertragen und Fehler zu erkennen.
-Beim Reed-Solomon-Code kann man nicht nur Fehler erkenen,
+Beim Reed-Solomon-Code kann man nicht nur Fehler erkennen,
man kann sogar einige Fehler korrigieren.
\rhead{Polynom-Ansatz}
-Eine Idee ist die Daten,
-ein Polynom zu bilden und dieses dann mit bestimmten Punkten überträgt.
+Eine Idee ist aus den Daten
+ein Polynom zu bilden.
+Diese Polynomfunktion bei bestimmten Werten, ausrechnet und diese Punkte dann überträgt.
Nehmen wir als beisbiel die Zahlen \textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5},
welche uns dann das Polynom
\begin{equation}
p(x)
=
-2x^2 + 1x + 5
+\textcolor{blue}{2}x^2 + \textcolor{blue}{1}x + \textcolor{blue}{5}
\label{reedsolomon:equation1}
\end{equation}
ergeben.
-Übertragen werden nun die stellen 1, 2, 3\dots 7 dieses Polynomes.
-Grafisch sieht man dies dann im Abbild //TODO
-Wenn ein Fehler sich in die Übertragung eingeschlichen hatt, muss der Leser/Empfänger erkennen, welches das Richtige Polynom ist.
-Der Leser/Empfänger weiss, mit welchem Grad das Polynom entwickelt wurde.
+Übertragen werden nun die Werte an den stellen 1, 2, 3\dots 7 dieses Polynomes.
+Grafisch sieht man dies dann in Abbildung \ref{fig:polynom},
+mit den Punkten, $p(1),p(2),...,p(7) = (\textcolor{green}{8},
+\textcolor{green}{15}, \textcolor{green}{26},
+\textcolor{green}{41}, \textcolor{green}{60},
+\textcolor{green}{83}, \textcolor{green}{110})$
+Wenn ein Fehler sich in die Übertragung eingeschlichen hatt, muss der Leser/Empfänger diesen erkennen und das Polynom rekonstruieren.
+Der Leser/Empfänger weiss, den Grad des Polynoms und dessen Werte übermittelt wurden.
+
\subsection{Beispiel}
-Für das Beispeil aus der Gleichung \ref{reedsolomon:equation1},
+Für das Beispeil aus der Gleichung \eqref{reedsolomon:equation1},
ist ein Polynome zweiten Grades durch drei Punkte eindeutig bestimmbar.
-Hat es Fehler in der Übertragunge gegeben, kann man diese erkennen,
-da alle Punkte, die korrekt sind, auf dem Polynom liegen müssen.
+Hat es Fehler in der Übertragunge gegeben,(Bei Abbildung \ref{fig:polynom}\textcolor{red}{roten Punkte}) kann man diese erkennen,
+da alle Punkte, die korrekt sind, auf dem Polynom liegen müssen.
+(Bei Abbildung \ref{fig:polynom}\textcolor{green}{grünen Punkte})
Ab wie vielen Fehler ist das Polynom nicht mehr erkennbar beim Übertragen von 7 Punkten?
Bei 2 Fehlern kann man noch eindeutig bestimmen, dass das Polynom mit 4 Punkten,
-gegenüber dem mit 5 Punkten falsch liegt.
-Werden es mehr Fehler kann nur erkennt werden das das Polynom nicht stimmt.
-Das Orginale Polynom kann aber nicht mehr gefunden werden.
-Dabei sollten mehr Übertragungspunkte gegeben werden.
+gegenüber dem mit 5 Punkten falsch liegt.\ref{fig:polynom}
+Werden es mehr Fehler kann nur erkennt werden, dass das Polynom nicht stimmt.
+Das orginale Polynom kann aber nicht mehr gefunden werden.
+Dafür sind mehr übertragene Werte nötig.
+
+\begin{figure}
+ \centering
+ %\includegraphics[width=0.5\textwidth]{papers/reedsolomon/images/polynom2}
+ \input{papers/reedsolomon/images/polynom2.tex}
+ \caption{Polynom $p(x)$ \eqref{reedsolomon:equation1}}
+ \label{fig:polynom}
+\end{figure}
\section{Fehlerbestimmung
\label{reedsolomon:section:Fehlerbestimmmung}}
So wird ein Muster indentifiziert, welches genau vorherbestimmen kann,
wie gross das Polynom sein muss und wie viele Übertragungspunkte gegeben werden müssen.
-Durch ein klein wenig Überlegung ist klar das die anzahl Zahlen (Daten, ab hier verwenden wir das Wort Nutzlast),
-die dan Entschlüsselt werden sollen den Grad des Polynoms minus 1 ergeben.
+Um zu bestimmen wie viel Fehler erkennt und korriegiert werden können.
+Die Anzahl Zahlen (Daten, ab hier verwenden wir das Wort Nutzlast),
+die Entschlüsselt werden sollen, brauchen die gleiche Anzahl an Polynomgraden, beginnend bei Grad 0. ( \( k-1 \) )
Für die Anzahl an Übertragungspunkte, muss bestimmt werden wieviel Fehler erkennt und korrigiert werden sollen.
-Mit Hilfe der Tabelle.... sieht man das es bei $$t$$ Fehlern und $$k$$ Nutzlast,
-für das Übertragen $$k+2t$$ Punkte gegben werden müssen.
+Mit Hilfe der Tabelle, sieht man das es bei $t$ Fehlern und $k$ Nutzlast Zahlen,
+$k+2t$ Punkte übertragen werden müssen.
-Ein toller Nebeneffekt ist das dadurch auch $$2t$$ Fehler erkannt werden.
-um zurück auf unser Beispiel zu kommen,
-können von den 7 Übertragungspunkten bis zu $$2t = 2*2 = 4 $$ Punkten falsch liegen
-und es wird kein eindeutiges Polynom 2ten Grades erkannt, und somit die Nutzlast Daten als fehlerhaft deklariert.
+\begin{center}
+ \begin{tabular}{ c c c }
+ \hline
+ Nutzlas & Fehler & Übertragen \\
+ \hline
+ 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\
+ 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\
+ 3 & 3 & 9 Werte eines Polynoms vom Grad 2 \\
+ \hline
+ $k$ & $t$ & $k+2t$ Werte eines Polynoms vom Grad $k-1$ \\
+ \hline
+ \end{tabular}
+\end{center}
-Ein Polynom durch Punkt mit Polynom Interpolation zu rekonstruieren ist schwierig und Fehleranfällig.
+Ein toller Nebeneffekt ist das dadurch auch $2t$ Fehler erkannt werden.
+Um zurück auf unser Beispiel zu kommen,
+können von den 7 Übertragungspunkten bis zu $2t = 2\cdot2 = 4 $ Punkten falsch liegen
+und es wird kein eindeutiges Polynom zweiten Grades erkannt, und somit die Nutzlast Daten als fehlerhaft deklariert.
+Um aus den Übertragenen Zahlen wieder die Nutzlastzahlen zu bekommen könnte man eine Polynominterpolation anwenden,
+doch die Punkte mit Polynominterpolation zu einem Polynom zu rekonstruieren ist schwierig und Fehleranfällig.
diff --git a/buch/papers/reedsolomon/images/codiert.txt b/buch/papers/reedsolomon/images/codiert.txt
new file mode 100644
index 0000000..4a481d8
--- /dev/null
+++ b/buch/papers/reedsolomon/images/codiert.txt
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diff --git a/buch/papers/reedsolomon/images/decodiert.txt b/buch/papers/reedsolomon/images/decodiert.txt
new file mode 100644
index 0000000..f6221e6
--- /dev/null
+++ b/buch/papers/reedsolomon/images/decodiert.txt
@@ -0,0 +1,96 @@
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diff --git a/buch/papers/reedsolomon/images/empfangen.txt b/buch/papers/reedsolomon/images/empfangen.txt
new file mode 100644
index 0000000..38c13b0
--- /dev/null
+++ b/buch/papers/reedsolomon/images/empfangen.txt
@@ -0,0 +1,96 @@
+0,284
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diff --git a/buch/papers/reedsolomon/images/fehler.txt b/buch/papers/reedsolomon/images/fehler.txt
new file mode 100644
index 0000000..23f1a83
--- /dev/null
+++ b/buch/papers/reedsolomon/images/fehler.txt
@@ -0,0 +1,96 @@
+0,0
+1,0
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diff --git a/buch/papers/reedsolomon/images/locator.txt b/buch/papers/reedsolomon/images/locator.txt
new file mode 100644
index 0000000..b28988c
--- /dev/null
+++ b/buch/papers/reedsolomon/images/locator.txt
@@ -0,0 +1,96 @@
+0,0.0301224340567056
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diff --git a/buch/papers/reedsolomon/images/plotfft.tex b/buch/papers/reedsolomon/images/plotfft.tex
new file mode 100644
index 0000000..83a89eb
--- /dev/null
+++ b/buch/papers/reedsolomon/images/plotfft.tex
@@ -0,0 +1,89 @@
+%
+% Plot der Übertrangungsabfolge ins FFT und zurück mit IFFT
+%
+\begin{tikzpicture}[]
+
+%---------------------------------------------------------------
+ %Knote
+\matrix[draw = none, column sep=25mm, row sep=2mm]{
+ \node(signal) [] {
+ \begin{tikzpicture}
+ \begin{axis}
+ [title = {\Large {Signal}},
+ xlabel={Anzahl Übertragene Zahlen},
+ xtick={0,20,40,64,80,98},]
+ \addplot[blue] table[col sep=comma] {papers/reedsolomon/images/signal.txt};
+ \end{axis}
+ \end{tikzpicture}}; &
+
+ \node(codiert) [] {
+ \begin{tikzpicture}
+ \begin{axis}[title = {\Large {Codiert}}]
+ \addplot[] table[col sep=comma] {papers/reedsolomon/images/codiert.txt};
+ \end{axis}
+ \end{tikzpicture}}; \\
+
+ &\node(fehler) [] {
+ \begin{tikzpicture}
+ \begin{axis}[scale=0.6, title = {\Large {Fehler}},
+ xtick={7,21,75}]
+ \addplot[red] table[col sep=comma] {papers/reedsolomon/images/fehler.txt};
+ \end{axis}
+ \end{tikzpicture}};\\
+
+ \node(decodiert) [] {
+ \begin{tikzpicture}
+ \begin{axis}[title = {\Large {Decodiert}}]
+ \addplot[blue] table[col sep=comma] {papers/reedsolomon/images/decodiert.txt};
+ \end{axis}
+ \end{tikzpicture}}; &
+
+ \node(empfangen) [] {
+ \begin{tikzpicture}
+ \begin{axis}[title = {\Large {Empfangen}}]
+ \addplot[] table[col sep=comma] {papers/reedsolomon/images/empfangen.txt};
+ \end{axis}
+ \end{tikzpicture}};\\
+
+ \node(syndrom) [] {
+ \begin{tikzpicture}
+ \begin{axis}[title = {\Large {Syndrom}}]
+ \addplot[blue] table[col sep=comma] {papers/reedsolomon/images/syndrom.txt};
+ \end{axis}
+ \end{tikzpicture}}; &
+
+ \node(locator) [] {
+ \begin{tikzpicture}
+ \begin{axis}[title = {\Large {Locator}}]
+ \addplot[] table[col sep=comma] {papers/reedsolomon/images/locator.txt};
+ \end{axis}
+ \end{tikzpicture}};\\
+};
+%-------------------------------------------------------------
+ %FFT & IFFT deskription
+
+\draw[thin,gray,dashed] (0,12) to (0,-12);
+\node(IFFT) [scale=0.7] at (0,12.3) {IFFT};
+\draw[<-](IFFT.south west)--(IFFT.south east);
+\node(FFT) [scale=0.7, above of=IFFT] {FFT};
+\draw[->](FFT.north west)--(FFT.north east);
+
+\draw[thick, ->,] (fehler.west)++(-1,0) +(0.05,0.5) -- +(-0.1,-0.1) -- +(0.1,0.1) -- +(0,-0.5);
+%Arrows
+\draw[ultra thick, ->] (signal.east) to (codiert.west);
+\draw[ultra thick, ->] (codiert.south) to (fehler.north);
+\draw[ultra thick, ->] (fehler.south) to (empfangen.north);
+\draw[ultra thick, ->] (empfangen.west) to (decodiert.east);
+\draw[ultra thick, ->] (syndrom.east) to (locator.west);
+\draw(decodiert.south east)++(-1.8,1) ellipse (1.3cm and 0.8cm) ++(-1.3,0) coordinate(zoom) ;
+\draw[ultra thick, ->] (zoom) to[out=180, in=90] (syndrom.north);
+
+%item
+\node[circle, draw, fill =lightgray] at (signal.north west) {1};
+\node[circle, draw, fill =lightgray] at (codiert.north west) {2};
+\node[circle, draw, fill =lightgray] at (fehler.north west) {3};
+\node[circle, draw, fill =lightgray] at (empfangen.north west) {4};
+\node[circle, draw, fill =lightgray] at (decodiert.north west) {5};
+\node[circle, draw, fill =lightgray] at (syndrom.north west) {6};
+\node[circle, draw, fill =lightgray] at (locator.north west) {7};
+\end{tikzpicture} \ No newline at end of file
diff --git a/buch/papers/reedsolomon/images/polynom2.tex b/buch/papers/reedsolomon/images/polynom2.tex
new file mode 100644
index 0000000..288b51c
--- /dev/null
+++ b/buch/papers/reedsolomon/images/polynom2.tex
@@ -0,0 +1,49 @@
+% polynome
+%-------------------
+% Teiler für das Skalieren der Grafik /40
+\newcommand{\teiler}{40}
+
+
+%//////////////////////////////////////
+
+\begin{tikzpicture}[>=latex,thick]
+ \draw[color=blue, line width=1.4pt]
+ plot[domain=0:8, samples=100]
+ ({\x},{(2*\x^2+1*\x+5)/\teiler});
+
+ \draw[->] (-0.2,0) -- (8,0) coordinate[label={$x$}];
+ \draw[->] (0,-0.2) -- (0,150/\teiler) coordinate[label={right:$p(x)$}];
+
+ \def\punkt#1{
+ \fill[color=green] #1 circle[radius=0.08];
+ \draw #1 circle[radius=0.07];
+ }
+
+ \def\hellpunkt#1{
+ \fill[color=lightgray] #1 circle[radius=0.08];
+ \draw #1 circle[radius=0.07];
+ }
+
+ \punkt{(1,8/\teiler)}
+ \hellpunkt{(2,15/\teiler)}
+ \hellpunkt{(3,26/\teiler)}
+ \punkt{(4,41/\teiler)}
+ \punkt{(5,60/\teiler)}
+ \punkt{(6,83/\teiler)}
+ \punkt{(7,110/\teiler)}
+
+ \draw[color=gray,line width=1pt,dashed]
+ plot[domain=0.5:7, samples=100]
+ ({\x},{(7.832*\x^2-51.5*\x+121.668)/\teiler});
+
+ \def\erpunkt#1{
+ \fill[color=red] #1 circle[radius=0.08];
+ \draw #1 circle[radius=0.07];
+ }
+ \erpunkt{(2,50/\teiler)}
+ \erpunkt{(3,37.66/\teiler)}
+
+ \draw(0,100/\teiler) -- (-0.1,100/\teiler) coordinate[label={left:$100$}];
+ \draw(1,0) -- (1,-0.1) coordinate[label={below:$1$}];
+\end{tikzpicture}
+%\end{document}
diff --git a/buch/papers/reedsolomon/images/signal.txt b/buch/papers/reedsolomon/images/signal.txt
new file mode 100644
index 0000000..c4fa5f8
--- /dev/null
+++ b/buch/papers/reedsolomon/images/signal.txt
@@ -0,0 +1,96 @@
+0,6
+1,6
+2,0
+3,6
+4,4
+5,0
+6,5
+7,2
+8,1
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+10,1
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+14,3
+15,5
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+24,3
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+26,6
+27,6
+28,4
+29,2
+30,9
+31,1
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+33,1
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diff --git a/buch/papers/reedsolomon/images/syndrom.txt b/buch/papers/reedsolomon/images/syndrom.txt
new file mode 100644
index 0000000..8ca9eed
--- /dev/null
+++ b/buch/papers/reedsolomon/images/syndrom.txt
@@ -0,0 +1,96 @@
+0,0
+1,0
+2,0
+3,0
+4,0
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diff --git a/buch/papers/reedsolomon/main.tex b/buch/papers/reedsolomon/main.tex
index a400508..e68b947 100644
--- a/buch/papers/reedsolomon/main.tex
+++ b/buch/papers/reedsolomon/main.tex
@@ -30,7 +30,7 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren
% Joshua
\input{papers/reedsolomon/einleitung.tex}
\input{papers/reedsolomon/idee.tex}
-\input{papers/reedsolomon/teil2.tex}
+%\input{papers/reedsolomon/teil2.tex}
\input{papers/reedsolomon/dtf.tex}
% Michael
diff --git a/buch/papers/reedsolomon/packages.tex b/buch/papers/reedsolomon/packages.tex
index 3643731..b84e228 100644
--- a/buch/papers/reedsolomon/packages.tex
+++ b/buch/papers/reedsolomon/packages.tex
@@ -8,3 +8,5 @@
% following example
%\usepackage{packagename}
+\usepackage{pgfplots}
+\usepackage{filecontents}
diff --git a/buch/papers/reedsolomon/teil2.tex b/buch/papers/reedsolomon/teil2.tex
deleted file mode 100644
index b2adc9f..0000000
--- a/buch/papers/reedsolomon/teil2.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil2.tex -- Beispiel-File für teil2
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 2
-\label{reedsolomon:section:teil2}}
-\rhead{Teil 2}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{reedsolomon:subsection:bonorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
-
-