Merge pull request #44 from JODBaer/Baer

Baer

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 @@
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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
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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
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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
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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
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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
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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
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+3,6
+4,4
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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
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+94,0.0261327016093158
+95,0.0314909067039411
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.
-
-