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+\usetheme{Hannover} +\begin{document} + \author{Joshua Bär und Michael Steiner} + \title{Reed-Solomon-Code} + \subtitle{} + \logo{} + \institute{OST Ostschweizer Fachhochschule} + \date{26.04.2021} + \subject{Mathematisches Seminar} + \setbeamercovered{transparent} + \setbeamertemplate{navigation symbols}{} + \begin{frame}[plain] + \maketitle + \end{frame} + + \begin{frame} + \frametitle{Test} + Ich mag Züge. + \end{frame} +\end{document} \ No newline at end of file diff --git a/buch/papers/reedsolomon/RS presentation/RS.toc b/buch/papers/reedsolomon/RS presentation/RS.toc new file mode 100644 index 0000000..4cd1c86 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.toc @@ -0,0 +1 @@ +\babel@toc {ngerman}{} diff --git a/buch/papers/reedsolomon/RS presentation/Thumbs.db b/buch/papers/reedsolomon/RS presentation/Thumbs.db new file mode 100644 index 0000000..1626e26 Binary files /dev/null and b/buch/papers/reedsolomon/RS presentation/Thumbs.db differ -- cgit v1.2.1 From 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presentation/RS.pdf index 459d7e8..10719b7 100644 Binary files a/buch/papers/reedsolomon/RS presentation/RS.pdf and b/buch/papers/reedsolomon/RS presentation/RS.pdf differ diff --git a/buch/papers/reedsolomon/RS presentation/RS.synctex.gz b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz index fe8adf5..2fe95de 100644 Binary files a/buch/papers/reedsolomon/RS presentation/RS.synctex.gz and b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz differ diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 3d2be8f..fb822da 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -3,7 +3,9 @@ \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage[ngerman]{babel} +\usepackage{tikz} \usetheme{Hannover} + \begin{document} \author{Joshua Bär und Michael Steiner} \title{Reed-Solomon-Code} @@ -17,9 +19,51 @@ \begin{frame}[plain] \maketitle \end{frame} - + \section{Introduction} + \begin{frame} + \frametitle{Idee} + + \end{frame} + \begin{frame} - \frametitle{Test} - Ich mag Züge. + \begin{figure} + \only<1>{ + \includegraphics[width=0.9\linewidth]{images/fig1.pdf} + } + \only<2>{ + \includegraphics[width=0.9\linewidth]{images/fig2.pdf} + } + \only<3>{ + \includegraphics[width=0.9\linewidth]{images/fig3.pdf} + } + \only<4>{ + \includegraphics[width=0.9\linewidth]{images/fig4.pdf} + } + \only<5>{ + \includegraphics[width=0.9\linewidth]{images/fig5.pdf} + } + \only<6>{ + \includegraphics[width=0.9\linewidth]{images/fig6.pdf} + } + \only<7>{ + \includegraphics[width=0.9\linewidth]{images/fig7.pdf} + } + \end{figure} \end{frame} + + \begin{frame} + Übertragen von den Zahlen + \textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5} + als $ p(x) = \textcolor{blue}{2}x^2 + \textcolor{blue}{1}x + \textcolor{blue}{5} $.\newline + Versende $ (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})$ + \only<1>{ + \includegraphics[]{images/polynom1.pdf}} + \only<2>{ + \includegraphics[]{images/polynom2.pdf}} + \end{frame} + + \end{document} \ No newline at end of file diff --git a/buch/papers/reedsolomon/RS presentation/RS.toc b/buch/papers/reedsolomon/RS presentation/RS.toc index 4cd1c86..32e7e8d 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.toc +++ b/buch/papers/reedsolomon/RS presentation/RS.toc @@ -1 +1,2 @@ \babel@toc {ngerman}{} +\beamer@sectionintoc {1}{Introduction}{2}{0}{1} diff --git a/buch/papers/reedsolomon/RS presentation/Thumbs.db b/buch/papers/reedsolomon/RS presentation/Thumbs.db deleted file mode 100644 index 1626e26..0000000 Binary files a/buch/papers/reedsolomon/RS presentation/Thumbs.db and /dev/null differ diff --git a/buch/papers/reedsolomon/RS presentation/images/fig1.pdf b/buch/papers/reedsolomon/RS presentation/images/fig1.pdf index 5cff7fe..abde60c 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\textcolor{green}{83}, \textcolor{green}{110})$ + + + \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]; + } + \punkt{(1,8/\teiler)} + %\punkt{(2,15/\teiler)} + %\punkt{(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},{(0.1958*\x^2-1.2875*\x+3.0417)}); + \def\erpunkt#1{ + \fill[color=red] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + \erpunkt{(2,50/\teiler)} + \erpunkt{(3,0.9414)} + + + \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} -- cgit v1.2.1 From 10f3cdb829c001c341ea31415efb44ff6a2878b8 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Wed, 21 Apr 2021 17:30:50 +0200 Subject: Persentation stand 17:30 --- buch/papers/reedsolomon/RS presentation/RS.aux | 37 +++++- buch/papers/reedsolomon/RS presentation/RS.log | 145 +++++++++++---------- buch/papers/reedsolomon/RS presentation/RS.nav | 26 +++- buch/papers/reedsolomon/RS presentation/RS.out | 4 +- buch/papers/reedsolomon/RS presentation/RS.pdf | Bin 117082 -> 132691 bytes buch/papers/reedsolomon/RS presentation/RS.snm | 1 + .../reedsolomon/RS presentation/RS.synctex.gz | Bin 6763 -> 19501 bytes buch/papers/reedsolomon/RS presentation/RS.tex | 109 ++++++++++++++-- buch/papers/reedsolomon/RS presentation/RS.toc | 4 +- 9 files changed, 235 insertions(+), 91 deletions(-) (limited to 'buch/papers/reedsolomon/RS presentation') diff --git a/buch/papers/reedsolomon/RS presentation/RS.aux 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--git a/buch/papers/reedsolomon/RS presentation/RS.synctex.gz b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz index 2fe95de..96af4cc 100644 Binary files a/buch/papers/reedsolomon/RS presentation/RS.synctex.gz and b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz differ diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index fb822da..9bdf947 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -19,10 +19,15 @@ \begin{frame}[plain] \maketitle \end{frame} - \section{Introduction} + \section{Einführung} \begin{frame} \frametitle{Idee} - + \begin{itemize} + \item Reed-Solomon-Code beschäftigt sich mit der Übertragung von Daten + und deren Fehler Erkennung. + \item Idee Fourier Transformieren und dann senden. + \item Danach Empfangen und Rücktransformieren. + \end{itemize} \end{frame} \begin{frame} @@ -50,20 +55,100 @@ } \end{figure} \end{frame} + \begin{frame} - Übertragen von den Zahlen - \textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5} - als $ p(x) = \textcolor{blue}{2}x^2 + \textcolor{blue}{1}x + \textcolor{blue}{5} $.\newline - Versende $ (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})$ + \uncover<1->{ + Wie ist die Anzahl 0 definiert zum mitgeben? + Indem die Polymereigenschaft genutzt werden. + } + \uncover<2->{ + Wie wird der Fehler lokalisiert? + Indem in einem Endlichen Körper gerechnet wird. + } + + \end{frame} + +\section{Polynom Ansatz} + \begin{frame} + Die Diskrite Fouren Transformation ist so gegeben + \[ + \label{ft_discrete} + \hat{c}_{k} + = \frac{1}{N} \sum_{n=0}^{N-1} + {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} + \]. + + \[ + w = e^{-\frac{2\pi j}{N} k} + \] + Wenn $N$ konstant: + \[ + \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) + \] + \end{frame} + + \begin{frame} + Beispiel 2, 1, 5 Versenden und auf 2 Fehler absichern. + \end{frame} + \begin{frame} + Übertragen von + ${f}_2=$\textcolor{blue}{2}, ${f}_1$\textcolor{blue}{1}, ${f}_0$\textcolor{blue}{5} + als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. + \only<1>{ - \includegraphics[]{images/polynom1.pdf}} + Versende $ (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})$ + \includegraphics[scale = 1.2]{images/polynom1.pdf}} \only<2>{ - \includegraphics[]{images/polynom2.pdf}} + Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, + \textcolor{red}{50}, \textcolor{red}{37}, + \textcolor{green}{ 41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \includegraphics[scale = 1.2]{images/polynom2.pdf} + \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} + \end{frame} + + \begin{frame} + \frametitle{Parameter} + \begin{center} + \begin{tabular}{ c c c } + \hline + "Nutzlast" & Fehler & Versenden \\ + \hline + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + &&\\ + k & t & k+2t Werte eines Polynoms vom Grad k-1 \\ + \hline + \end{tabular} + \end{center} + \end{frame} +\section{Diskrete Fourien Transformation} + \begin{frame} + \[ + \begin{pmatrix} + \hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n + \end{pmatrix} + = + \begin{pmatrix} + w^0 & w^0 & w^0 & \dots &w^0 \\ + w^0 & w^1 &w^2 & \dots &w^n \\ + w^0 & w^2 &w^4 & \dots &w^{2n} \\ + \vdots & \vdots &\vdots &\ddots &\vdots \\ + w^0 & w^{1n}&w^{2n}& \dots &w^{n} \\ + \end{pmatrix} + \begin{pmatrix} + \textcolor{blue}{5} \\ + \textcolor{blue}{1} \\ + \textcolor{blue}{2} \\ + \vdots \\ + 0 \\ + \end{pmatrix} + \] \end{frame} - \end{document} \ No newline at end of file diff --git a/buch/papers/reedsolomon/RS presentation/RS.toc b/buch/papers/reedsolomon/RS presentation/RS.toc index 32e7e8d..ff200c6 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.toc +++ b/buch/papers/reedsolomon/RS presentation/RS.toc @@ -1,2 +1,4 @@ \babel@toc {ngerman}{} -\beamer@sectionintoc {1}{Introduction}{2}{0}{1} +\beamer@sectionintoc {1}{Einführung}{2}{0}{1} +\beamer@sectionintoc {2}{Polynom Ansatz}{12}{0}{2} +\beamer@sectionintoc {3}{Diskrete Fourien Transformation}{17}{0}{3} -- cgit v1.2.1 From 7c0937851938305c2bb760f3cd4c2084c4493217 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Wed, 21 Apr 2021 18:18:22 +0200 Subject: Presentation neu arangiert --- buch/papers/reedsolomon/RS presentation/RS.tex | 186 +++++++++++++------------ 1 file changed, 96 insertions(+), 90 deletions(-) (limited to 'buch/papers/reedsolomon/RS presentation') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 9bdf947..1a1cefd 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -21,12 +21,60 @@ \end{frame} \section{Einführung} \begin{frame} - \frametitle{Idee} + \frametitle{Einführung} \begin{itemize} \item Reed-Solomon-Code beschäftigt sich mit der Übertragung von Daten und deren Fehler Erkennung. - \item Idee Fourier Transformieren und dann senden. - \item Danach Empfangen und Rücktransformieren. + \end{itemize} + \end{frame} +\section{Polynom Ansatz} + \begin{frame} + Beispiel 2, 1, 5 Versenden und auf 2 Fehler absichern. + \end{frame} + \begin{frame} + Übertragen von + ${f}_2=$\textcolor{blue}{2}, ${f}_1$\textcolor{blue}{1}, ${f}_0$\textcolor{blue}{5} + als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. + + \only<1>{ + Versende $ (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})$ + \includegraphics[scale = 1.2]{images/polynom1.pdf}} + \only<2>{ + Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, + \textcolor{red}{50}, \textcolor{red}{37}, + \textcolor{green}{ 41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \includegraphics[scale = 1.2]{images/polynom2.pdf} + \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} + \end{frame} + + \begin{frame} + \frametitle{Parameter} + \begin{center} + \begin{tabular}{ c c c } + \hline + "Nutzlast" & Fehler & Versenden \\ + \hline + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + &&\\ + k & t & k+2t Werte eines Polynoms vom Grad k-1 \\ + \hline + \end{tabular} + \end{center} + + Ausserdem können bis zu 2t Fehler erkannt werden! + \end{frame} +\section{Fourier Transformation} + \begin{frame} + \frametitle{Idee} + \begin{itemize} + \item Idee mit Fourier Transformieren und dann senden. + \item Danach Empfangen und Rücktransformieren. \end{itemize} \end{frame} @@ -56,99 +104,57 @@ \end{figure} \end{frame} - +\section{Diskrete Fourier Transformation} \begin{frame} - \uncover<1->{ - Wie ist die Anzahl 0 definiert zum mitgeben? - Indem die Polymereigenschaft genutzt werden. - } - \uncover<2->{ - Wie wird der Fehler lokalisiert? - Indem in einem Endlichen Körper gerechnet wird. - } - + \frametitle{Diskrete Fourier Transformation} + Die Diskrete Fourier Transformation ist so gegeben: + \[ + \label{ft_discrete} + \hat{c}_{k} + = \frac{1}{N} \sum_{n=0}^{N-1} + {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} + \]. + + \[ + w = e^{-\frac{2\pi j}{N} k} + \] + Wenn $N$ konstant: + \[ + \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) + \] \end{frame} -\section{Polynom Ansatz} - \begin{frame} - Die Diskrite Fouren Transformation ist so gegeben - \[ - \label{ft_discrete} - \hat{c}_{k} - = \frac{1}{N} \sum_{n=0}^{N-1} - {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} - \]. - - \[ - w = e^{-\frac{2\pi j}{N} k} - \] - Wenn $N$ konstant: - \[ - \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) - \] - \end{frame} - - \begin{frame} - Beispiel 2, 1, 5 Versenden und auf 2 Fehler absichern. - \end{frame} - \begin{frame} - Übertragen von - ${f}_2=$\textcolor{blue}{2}, ${f}_1$\textcolor{blue}{1}, ${f}_0$\textcolor{blue}{5} - als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. - \only<1>{ - Versende $ (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})$ - \includegraphics[scale = 1.2]{images/polynom1.pdf}} - \only<2>{ - Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, - \textcolor{red}{50}, \textcolor{red}{37}, - \textcolor{green}{ 41}, \textcolor{green}{60}, - \textcolor{green}{83}, \textcolor{green}{110})$ - \includegraphics[scale = 1.2]{images/polynom2.pdf} - \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} - \end{frame} - - \begin{frame} - \frametitle{Parameter} - \begin{center} - \begin{tabular}{ c c c } - \hline - "Nutzlast" & Fehler & Versenden \\ - \hline - 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ - 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ - 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ - &&\\ - k & t & k+2t Werte eines Polynoms vom Grad k-1 \\ - \hline - \end{tabular} - \end{center} - \end{frame} -\section{Diskrete Fourien Transformation} \begin{frame} + \frametitle{Diskrete Fourier Transformation} \[ - \begin{pmatrix} - \hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n - \end{pmatrix} - = - \begin{pmatrix} - w^0 & w^0 & w^0 & \dots &w^0 \\ - w^0 & w^1 &w^2 & \dots &w^n \\ - w^0 & w^2 &w^4 & \dots &w^{2n} \\ - \vdots & \vdots &\vdots &\ddots &\vdots \\ - w^0 & w^{1n}&w^{2n}& \dots &w^{n} \\ - \end{pmatrix} - \begin{pmatrix} - \textcolor{blue}{5} \\ - \textcolor{blue}{1} \\ - \textcolor{blue}{2} \\ - \vdots \\ - 0 \\ - \end{pmatrix} + \begin{pmatrix} + \hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n + \end{pmatrix} + = + \begin{pmatrix} + w^0 & w^0 & w^0 & \dots &w^0 \\ + w^0 & w^1 &w^2 & \dots &w^n \\ + w^0 & w^2 &w^4 & \dots &w^{2n} \\ + \vdots & \vdots &\vdots &\ddots &\vdots \\ + w^0 & w^{1n}&w^{2n}& \dots &w^{n} \\ + \end{pmatrix} + \begin{pmatrix} + \textcolor{blue}{f_0} \\ + \textcolor{blue}{f_1} \\ + \textcolor{blue}{f_2} \\ + \vdots \\ + 0 \\ + \end{pmatrix} \] \end{frame} - +\section{Probleme und Fragen} + \begin{frame} + \frametitle{Probleme und Fragen} + + Wie wird der Fehler lokalisiert? + \only<2>{ + Indem in einem Endlichen Körper gerechnet wird. + } + \end{frame} \end{document} \ No newline at end of file -- cgit v1.2.1 From 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b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -17,9 +17,715 @@ \begin{frame}[plain] \maketitle \end{frame} - +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Test} Ich mag Züge. \end{frame} + + \begin{frame} + \frametitle{Reed-Solomon in Endlichen Körpern} + + \begin{itemize} + \item Warum Endliche Körper? + + \qquad bessere Laufzeit + + \vspace{10pt} + + \item Nachricht = Nutzdaten + Fehlerkorrekturteil + + \vspace{10pt} + + \item den Fehlerkorrekturteil brauchen wir im Optimalfall nicht + + \vspace{10pt} + + \item Im Fehlerfall sollen wir aus der Nachricht ein Lokatorpolynom berechnen können, welches die Fehlerhaften Stellen beinhaltet + +% Wir sollten im Fehlerfall in der Lage sein, aus der Nachricht ein Lokatorpolynom zu berechnen, welches die Fehlerhaften Stellen beinhaltet + + \end{itemize} + +% TODO + +% erklärung und einführung der endlichen körper, was wollen wir erreichen? + +% wir versenden im endefekt mehr daten als unsere nachricht umfasst, damit die korrektur sichergestellt werden kann + +% sollten wir fehler bekommen, was uns die korrekturstellen mitgeteilt wird, dann ist es unsere aufgabe ein lokatorpolynom zu finden, welches uns verrät, auf welchen zeilen der Fehler aufgetreten ist + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Definition eines Beispiels} + + \begin{itemize} + + \item Endlicher Körper $q = 11$ + + \only<1->{ist eine Primzahl} + + \only<1->{beinhaltet die Zahlen $\mathbb{Z}_{11} = [0,1,2,3,4,5,6,7,8,9,10]$} + + \vspace{10pt} + + \only<1->{\item Nachrichtenblock $n = q-1$} + + wird an den Empfänger gesendet + + \vspace{10pt} + + \only<1->{\item max. Fehler $z = 2$} + + maximale Anzahl von Fehler, die wir noch korrigieren können + + \vspace{10pt} + + \only<1->{\item Nutzlast $k = n -2t = 6$ Zahlen} + + Fehlerstellen $2t = 4$ Zahlen + + \only<1->{Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$} + + \only<1->{als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$} + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \item Ansatz aus den Komplexen Zahlen mit der Fouriertransformation + + \vspace{10pt} + + \item $\mathrm{e}$ existiert nicht in $\mathbb{Z}_{11}$ + + \vspace{10pt} + + \item wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{Z}_{11}$ abdeckt + + $\mathbb{Z}_{11}\setminus\{0\} = [a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9]$ + + \vspace{10pt} + + \item wir wählen $a = 8$ + + $\mathbb{Z}_{11}\setminus\{0\} = [1,8,9,6,4,10,3,2,5,7]$ + + 8 ist eine Primitive Einheitswurzel + + \vspace{10pt} + + \item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$ + + $\Rightarrow$ \qquad können wir auch als Matrix schreiben + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \item Übertragungsvektor $V$ + + \item $V = A \cdot m$ + + \end{itemize} + + \[ + V = \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $V = [5,3,6,5,2,10,2,7,10,4]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Decodierung ohne Fehler} + + \begin{itemize} + \item Der Empfänger erhält den unveränderten Vektor $V = [5,3,6,5,2,10,2,7,10,4]$ + + \vspace{10pt} + + \item Wir suchen die Inverse der Matrix A + + \end{itemize} + + \begin{columns}[t] + \begin{column}{0.50\textwidth} + + Inverse der Fouriertransformation + \vspace{10pt} + \[ + F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt + \] + \vspace{10pt} + \[ + f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega + \] + + \end{column} + \begin{column}{0.50\textwidth} + + Inverse von a + \vspace{10pt} + \[ + 8^{1} \Rightarrow 8^{-1} + \] + + Inverse finden wir über den Eulkidischen Algorithmus + \vspace{10pt} + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus} + + \begin{columns}[t] + \begin{column}{0.50\textwidth} + + Recap aus der Vorlesung: + + Gegeben $a \in \mathbb{F}_p$, finde $b = a^{-1} \in \mathbb{F}_p$ + + \begin{tabular}{rcl} + $a b$ &$\equiv$& $1 \mod p$\\ + $a b$ &$=$& $1 + n p$\\ + $a b - n p$ &$=$& $1$\\ + &&\\ + $\operatorname{ggT}(a,p)$&$=$& $1$\\ + $sa + tp$&$=$& $1$\\ + $b$&$=$&$s$\\ + $n$&$=$&$-t$ + \end{tabular} + + \end{column} + \begin{column}{0.50\textwidth} + + \begin{center} + + \begin{tabular}{| c | c c | c | c c |} + \hline + $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ + \hline + & & & & $1$& $0$\\ + $0$& $8$& $11$& $0$& $0$& $1$\\ + $1$& $11$& $8$& $1$& $1$& $0$\\ + $2$& $8$& $3$& $2$& $-1$& $1$\\ + $3$& $3$& $2$& $1$& $3$& $-2$\\ + $4$& $2$& $1$& $2$& $-4$& $3$\\ + $5$& $1$& $0$& & $11$& $-8$\\ + \hline + \end{tabular} + + \vspace{10pt} + + \begin{tabular}{rcl} + $-4\cdot 8 + 3 \cdot 11$ &$=$& $1$\\ + $7 \cdot 8 + 3 \cdot 11$ &$=$& $1$\\ + $8^{-1}$ &$=$& $7$ + + \end{tabular} + + \end{center} + + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Decodirung mit Inverser Matrix} + + \begin{itemize} + \item $V = [5,3,6,5,2,10,2,7,10,4]$ + + \item $m = 1/10 \cdot A^{-1} \cdot V$ + + \item $m = 10 \cdot A^{-1} \cdot V$ + + \end{itemize} + + \[ + m = \begin{pmatrix} + 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ + 7^0& 7^1& 7^2& 7^3& 7^4& 7^5& 7^6& 7^7& 7^8& 7^9\\ + 7^0& 7^2& 7^4& 7^6& 7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\ + 7^0& 7^3& 7^6& 7^9& 7^{12}& 7^{15}& 7^{18}& 7^{21}& 7^{24}& 7^{27}\\ + 7^0& 7^4& 7^8& 7^{12}& 7^{16}& 7^{20}& 7^{24}& 7^{28}& 7^{32}& 7^{36}\\ + 7^0& 7^5& 7^{10}& 7^{15}& 7^{20}& 7^{25}& 7^{30}& 7^{35}& 7^{40}& 7^{45}\\ + 7^0& 7^6& 7^{12}& 7^{18}& 7^{24}& 7^{30}& 7^{36}& 7^{42}& 7^{48}& 7^{54}\\ + 7^0& 7^7& 7^{14}& 7^{21}& 7^{28}& 7^{35}& 7^{42}& 7^{49}& 7^{56}& 7^{63}\\ + 7^0& 7^8& 7^{16}& 7^{24}& 7^{32}& 7^{40}& 7^{48}& 7^{56}& 7^{64}& 7^{72}\\ + 7^0& 7^9& 7^{18}& 7^{27}& 7^{36}& 7^{45}& 7^{54}& 7^{63}& 7^{72}& 7^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $m = [0,0,0,0,4,7,2,5,8,1]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Decodierung mit Fehler - Ansatz} + + \begin{itemize} + \item Gesendet: $V = [5,3,6,5,2,10,2,7,10,4]$ + + \item Empfangen: $W = [5,3,6,8,2,10,2,7,1,4]$ + + \item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerstellen}5,4,5,7,6,7]$ + \end{itemize} + + Wie finden wir die Fehler? + + \begin{itemize} + \item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ + + \item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ + + \item $e(X) = r(X) - m(X)$ + \end{itemize} + + \begin{center} + + \begin{tabular}{c c c c c c c c c c c} + \hline + $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ + \hline + $r(a^{i})$& $5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$\\ + $m(a^{i})$& $5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$\\ + $e(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\ + \hline + \end{tabular} + + \end{center} + + \begin{itemize} + \item Alle Stellen, die nicht Null sind, sind Fehler + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + + \vspace{10pt} + + \item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$ + + \vspace{10pt} + + \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + + \vspace{10pt} + + \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + + \vspace{10pt} + + \item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat + + \vspace{10pt} + + $\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + + \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + + \vspace{10pt} + + \item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$ + + \vspace{10pt} + + \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + + \vspace{10pt} + + \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + + \vspace{10pt} + + \item $\operatorname{kgV}$ gibt uns eine Liste von aller Nullstellen, die wir in $e$ und $d$ zerlegen können + + \vspace{10pt} + + $\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot $ + + \qquad \qquad \qquad \qquad $(X-a^7)(X-a^8)(X-a^9) \cdot q(X)$ + + $= d(X) \cdot e(X)$ + + \vspace{10pt} + + \item Lokatorpolynom $d(X) = (X-a^3)(X-a^8)$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{kennen wir $e$?} + + \begin{itemize} + + \item $e$ ist unbekannt auf der Empfängerseite + + \vspace{10pt} + + \item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt? + + \vspace{10pt} + + \item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ + + In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat + + \vspace{10pt} + + \item daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$ + + \vspace{10pt} + + \item $f(X) = X^{10} - 1 = X^{10} + 10$ + + \vspace{10pt} + + \item jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus (nochmal)} + + $\operatorname{ggT}(f(X),e(X))$ hat den Grad 8 + + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + X^{10}& & & & & & &+& 10& & & & &:&5X^9&+&7X^8&+& 4X^7&+&10X^6&+&p(X)&=&9X&+&5\\ + X^{10}&+& 8X^9&+& 3X^8&+&2X^7&+& p(X)& & & & & & & & & & & & & & & & \\ \cline{1-9} + && 3X^9&+& 8X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ + && 3X^9&+& 2X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ \cline{3-9} + & & & &6X^8&+&0X^7&+&p(X)& & & & & & & & & & & & \\ + \end{array} + \] + + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + 5X^9&+& 7X^8&+& 4X^7&+& 10X^6&+& p(X)& & & & &:&6X^8&+&0X^7& & & & & & &=&10X&+&3\\ + 5X^9&+& 0X^8&+& p(X)& & & & & & & & & & & & & & & & & & & & \\ \cline{1-5} + && 7X^8&+& p(X)& & & & & & & & & & & & & & & & \\ + \end{array} + \] + + \vspace{10pt} + + $\operatorname{ggT}(f(X),e(X)) = 6X^8$ + + \vspace{10pt} + + $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen + + \end{frame} + +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Erweiterte Euklidische Algorithmus} + + \begin{center} + + \begin{tabular}{| c | c | c c |} + \hline + $k$ & $q_i$ & $e_i$ & $f_i$\\ + \hline + & & $0$& $1$\\ + $0$& $9X + 5$& $1$& $0$\\ + $1$& $10X + 3$& $9X+5$& $1$\\ + $2$& & $2X^2 + 0X + 5$& $10X + 3$\\ + \hline + \end{tabular} + + \end{center} + + \vspace{10pt} + + \begin{tabular}{ll} + Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\ + Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\ + Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$ + \end{tabular} + + \vspace{10pt} + + \begin{center} + $a^i = 5 \qquad \Rightarrow \qquad i = 3$ + + $a^i = 6 \qquad \Rightarrow \qquad i = 8$ + \end{center} + + $D = [3,8]$ + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \begin{itemize} + + \item $W = [5,3,6,8,2,10,2,7,1,4]$ + + \item $D = [3,8]$ + + \end{itemize} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 8 \\ 2 \\ 10 \\ 2 \\ 7 \\ 1 \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item Fehlerstellen entfernen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item Nullstellen entfernen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \item Matrix in eine Quadratische Form bringen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \item Matrix Invertieren + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 1& 1& 1& 1& 1& 1\\ + 1& 8& 9& 6& 4& 10\\ + 1& 9& 4& 3& 5& 1\\ + 1& 4& 5& 9& 3& 1\\ + 1& 10& 1& 10& 1& 10\\ + 1& 3& 9& 5& 4& 1\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \begin{center} + $\Downarrow$ + \end{center} + \[ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + \] + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $m = [4,7,2,5,8,1]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- \end{document} \ No newline at end of file -- cgit v1.2.1 From 8473571bc77425cd198b4bba515a3f5fe10c8cd2 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Wed, 21 Apr 2021 22:53:49 +0200 Subject: Style verbessert --- buch/papers/reedsolomon/RS presentation/RS.tex | 17 +++++++++++------ 1 file changed, 11 insertions(+), 6 deletions(-) (limited to 'buch/papers/reedsolomon/RS presentation') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 1a1cefd..65f8431 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -64,12 +64,16 @@ &&\\ k & t & k+2t Werte eines Polynoms vom Grad k-1 \\ \hline + &&\\ + &&\\ + &Ausserdem können bis zu 2t Fehler erkannt werden!\\ \end{tabular} \end{center} - Ausserdem können bis zu 2t Fehler erkannt werden! + + \end{frame} -\section{Fourier Transformation} +\section{Diskrete Fourier Transformation} \begin{frame} \frametitle{Idee} \begin{itemize} @@ -104,7 +108,7 @@ \end{figure} \end{frame} -\section{Diskrete Fourier Transformation} + \begin{frame} \frametitle{Diskrete Fourier Transformation} Die Diskrete Fourier Transformation ist so gegeben: @@ -134,10 +138,10 @@ = \begin{pmatrix} w^0 & w^0 & w^0 & \dots &w^0 \\ - w^0 & w^1 &w^2 & \dots &w^n \\ - w^0 & w^2 &w^4 & \dots &w^{2n} \\ + w^0 & w^1 &w^2 & \dots &w^N \\ + w^0 & w^2 &w^4 & \dots &w^{2N} \\ \vdots & \vdots &\vdots &\ddots &\vdots \\ - w^0 & w^{1n}&w^{2n}& \dots &w^{n} \\ + w^0 & w^{1(N-1)}&w^{2(N-1)}& \dots &w^{(N-1)(N-1)} \\ \end{pmatrix} \begin{pmatrix} \textcolor{blue}{f_0} \\ @@ -154,6 +158,7 @@ Wie wird der Fehler lokalisiert? \only<2>{ + \newline Indem in einem Endlichen Körper gerechnet wird. } \end{frame} -- cgit v1.2.1 From 38d0c69842308be5f096375ff070c5233b395c4c Mon Sep 17 00:00:00 2001 From: JODBaer Date: Thu, 22 Apr 2021 16:01:46 +0200 Subject: kleine korrekturen --- buch/papers/reedsolomon/RS presentation/RS.tex | 45 +++++++++++++++----------- 1 file changed, 26 insertions(+), 19 deletions(-) (limited to 'buch/papers/reedsolomon/RS presentation') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index eecd66b..618121c 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -19,14 +19,18 @@ \begin{frame}[plain] \maketitle \end{frame} - \section{Einführung} +%------------------------------------------------------------------------------- +\section{Einführung} \begin{frame} \frametitle{Einführung} \begin{itemize} \item Reed-Solomon-Code beschäftigt sich mit der Übertragung von Daten und deren Fehler Erkennung. + \item Wird verwendet in: + \only<2>{CD, QR-Codes, Voyager-Sonde, etc.} \end{itemize} \end{frame} +%------------------------------------------------------------------------------- \section{Polynom Ansatz} \begin{frame} Beispiel 2, 1, 5 Versenden und auf 2 Fehler absichern. @@ -50,7 +54,7 @@ \includegraphics[scale = 1.2]{images/polynom2.pdf} \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} \end{frame} - +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Parameter} \begin{center} @@ -59,20 +63,24 @@ "Nutzlast" & Fehler & Versenden \\ \hline 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ - 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ - 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ +\only<2->{3}& +\only<2->{2}& +\only<2->{7 Werte eines Polynoms vom Grad 2} \\ &&\\ - k & t & k+2t Werte eines Polynoms vom Grad k-1 \\ +\only<3->{k} & +\only<3->{t} & +\only<3->{k+2t Werte eines Polynoms vom Grad k-1} \\ \hline &&\\ &&\\ - &Ausserdem können bis zu 2t Fehler erkannt werden!\\ + \multicolumn{3}{l} { + \only<4>{Ausserdem können bis zu 2t Fehler erkannt werden!} + } \end{tabular} - \end{center} - - - + \end{center} \end{frame} +%------------------------------------------------------------------------------- \section{Diskrete Fourier Transformation} \begin{frame} \frametitle{Idee} @@ -81,7 +89,7 @@ \item Danach Empfangen und Rücktransformieren. \end{itemize} \end{frame} - +%------------------------------------------------------------------------------- \begin{frame} \begin{figure} \only<1>{ @@ -107,8 +115,7 @@ } \end{figure} \end{frame} - - +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Diskrete Fourier Transformation} Die Diskrete Fourier Transformation ist so gegeben: @@ -117,8 +124,8 @@ \hat{c}_{k} = \frac{1}{N} \sum_{n=0}^{N-1} {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} - \]. - + \] + Ersetzten als: \[ w = e^{-\frac{2\pi j}{N} k} \] @@ -128,14 +135,14 @@ \] \end{frame} - +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Diskrete Fourier Transformation} \[ \begin{pmatrix} \hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n \end{pmatrix} - = + = \frac{1}{N} \begin{pmatrix} w^0 & w^0 & w^0 & \dots &w^0 \\ w^0 & w^1 &w^2 & \dots &w^N \\ @@ -152,7 +159,7 @@ \end{pmatrix} \] \end{frame} - +%------------------------------------------------------------------------------- \section{Probleme und Fragen} \begin{frame} \frametitle{Probleme und Fragen} @@ -163,7 +170,7 @@ Indem in einem Endlichen Körper gerechnet wird. } \end{frame} - +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Reed-Solomon in Endlichen Körpern} -- cgit v1.2.1 From 9ce4fb55792c297989d1c001a621793303f31689 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Thu, 22 Apr 2021 22:13:29 +0200 Subject: Verbesserungen und anmerkungen umgesetzt --- buch/papers/reedsolomon/RS presentation/RS.tex | 56 ++++++++++++++------------ 1 file changed, 31 insertions(+), 25 deletions(-) (limited to 'buch/papers/reedsolomon/RS presentation') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 618121c..9811cf6 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -22,36 +22,38 @@ %------------------------------------------------------------------------------- \section{Einführung} \begin{frame} - \frametitle{Einführung} + \frametitle{Reed-Solomon-Code:} \begin{itemize} - \item Reed-Solomon-Code beschäftigt sich mit der Übertragung von Daten - und deren Fehler Erkennung. - \item Wird verwendet in: - \only<2>{CD, QR-Codes, Voyager-Sonde, etc.} + \item \only<1>{Für Übertragung von Daten} + \item \only<2->{Ermöglicht Korrektur von Übertragungsfehler} + \item \only<3->{Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc.} \end{itemize} \end{frame} %------------------------------------------------------------------------------- \section{Polynom Ansatz} \begin{frame} - Beispiel 2, 1, 5 Versenden und auf 2 Fehler absichern. + \begin{itemize} + \item Beispiel $2, 1, 5$ versenden und auf 2 Fehler absichern + \end{itemize} \end{frame} \begin{frame} Übertragen von - ${f}_2=$\textcolor{blue}{2}, ${f}_1$\textcolor{blue}{1}, ${f}_0$\textcolor{blue}{5} + ${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. \only<1>{ - Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, + Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, \textcolor{green}{15}, \textcolor{green}{26}, - \textcolor{green}{ 41}, \textcolor{green}{60}, + \textcolor{green}{41}, \textcolor{green}{60}, \textcolor{green}{83}, \textcolor{green}{110})$ \includegraphics[scale = 1.2]{images/polynom1.pdf}} \only<2>{ - Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, + Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, \textcolor{red}{50}, \textcolor{red}{37}, - \textcolor{green}{ 41}, \textcolor{green}{60}, + \textcolor{green}{41}, \textcolor{green}{60}, \textcolor{green}{83}, \textcolor{green}{110})$ \includegraphics[scale = 1.2]{images/polynom2.pdf} + \newline \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} \end{frame} %------------------------------------------------------------------------------- @@ -60,22 +62,22 @@ \begin{center} \begin{tabular}{ c c c } \hline - "Nutzlast" & Fehler & Versenden \\ + ``Nutzlas´´ & Fehler & Versenden \\ \hline 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ \only<2->{3}& -\only<2->{2}& -\only<2->{7 Werte eines Polynoms vom Grad 2} \\ +\only<2->{3}& +\only<3->{9 Werte eines Polynoms vom Grad 2} \\ &&\\ -\only<3->{k} & -\only<3->{t} & -\only<3->{k+2t Werte eines Polynoms vom Grad k-1} \\ +\only<4->{$k$} & +\only<4->{$t$} & +\only<4->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ \hline &&\\ &&\\ \multicolumn{3}{l} { - \only<4>{Ausserdem können bis zu 2t Fehler erkannt werden!} + \only<4>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} } \end{tabular} \end{center} @@ -85,8 +87,9 @@ \begin{frame} \frametitle{Idee} \begin{itemize} - \item Idee mit Fourier Transformieren und dann senden. - \item Danach Empfangen und Rücktransformieren. + \item Fourier-transformieren + \item Übertragung + \item Rücktransformieren \end{itemize} \end{frame} %------------------------------------------------------------------------------- @@ -118,14 +121,16 @@ %------------------------------------------------------------------------------- \begin{frame} \frametitle{Diskrete Fourier Transformation} - Die Diskrete Fourier Transformation ist so gegeben: + \begin{itemize} + \item Diskrete Fourier-Transformation gegeben durch: + \[ \label{ft_discrete} \hat{c}_{k} = \frac{1}{N} \sum_{n=0}^{N-1} {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} \] - Ersetzten als: + \item Ersetzte \[ w = e^{-\frac{2\pi j}{N} k} \] @@ -133,6 +138,7 @@ \[ \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) \] + \end{itemize} \end{frame} %------------------------------------------------------------------------------- @@ -145,8 +151,8 @@ = \frac{1}{N} \begin{pmatrix} w^0 & w^0 & w^0 & \dots &w^0 \\ - w^0 & w^1 &w^2 & \dots &w^N \\ - w^0 & w^2 &w^4 & \dots &w^{2N} \\ + w^0 & w^1 &w^2 & \dots &w^{N-1} \\ + w^0 & w^2 &w^4 & \dots &w^{2(N-1)} \\ \vdots & \vdots &\vdots &\ddots &\vdots \\ w^0 & w^{1(N-1)}&w^{2(N-1)}& \dots &w^{(N-1)(N-1)} \\ \end{pmatrix} @@ -167,7 +173,7 @@ Wie wird der Fehler lokalisiert? \only<2>{ \newline - Indem in einem Endlichen Körper gerechnet wird. + Indem in einem endlichen Körper gerechnet wird. } \end{frame} %------------------------------------------------------------------------------- -- cgit v1.2.1 From 5bca0960f8c9635375d2ca53c93d2bc5a2e37c10 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Thu, 22 Apr 2021 22:59:07 +0200 Subject: Animation verbessert --- buch/papers/reedsolomon/RS presentation/RS.tex | 37 ++++++++++++++------------ 1 file changed, 20 insertions(+), 17 deletions(-) (limited to 'buch/papers/reedsolomon/RS presentation') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 9811cf6..732cee5 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -24,9 +24,9 @@ \begin{frame} \frametitle{Reed-Solomon-Code:} \begin{itemize} - \item \only<1>{Für Übertragung von Daten} - \item \only<2->{Ermöglicht Korrektur von Übertragungsfehler} - \item \only<3->{Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc.} + \visible<1->{\item Für Übertragung von Daten} + \visible<2->{\item Ermöglicht Korrektur von Übertragungsfehler} + \visible<3->{\item Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc.} \end{itemize} \end{frame} %------------------------------------------------------------------------------- @@ -37,6 +37,7 @@ \end{itemize} \end{frame} \begin{frame} + \frametitle{Beispiel} Übertragen von ${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. @@ -66,18 +67,18 @@ \hline 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ -\only<2->{3}& -\only<2->{3}& -\only<3->{9 Werte eines Polynoms vom Grad 2} \\ +\visible<2->{3}& +\visible<2->{3}& +\visible<3->{9 Werte eines Polynoms vom Grad 2} \\ &&\\ -\only<4->{$k$} & -\only<4->{$t$} & -\only<4->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ +\visible<4->{$k$} & +\visible<4->{$t$} & +\visible<4->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ \hline &&\\ &&\\ \multicolumn{3}{l} { - \only<4>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} + \visible<4>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} } \end{tabular} \end{center} @@ -123,21 +124,23 @@ \frametitle{Diskrete Fourier Transformation} \begin{itemize} \item Diskrete Fourier-Transformation gegeben durch: - + \visible<1->{ \[ \label{ft_discrete} \hat{c}_{k} = \frac{1}{N} \sum_{n=0}^{N-1} {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} - \] + \]} + \visible<2->{ \item Ersetzte \[ w = e^{-\frac{2\pi j}{N} k} - \] - Wenn $N$ konstant: + \]} + \visible<3->{ + \item Wenn $N$ konstant: \[ \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) - \] + \]} \end{itemize} \end{frame} @@ -166,12 +169,12 @@ \] \end{frame} %------------------------------------------------------------------------------- -\section{Probleme und Fragen} + \begin{frame} \frametitle{Probleme und Fragen} Wie wird der Fehler lokalisiert? - \only<2>{ + \visible<2>{ \newline Indem in einem endlichen Körper gerechnet wird. } -- cgit v1.2.1 From 967ff1f33d3faaa1e344ff687aff6c07cde29b77 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Thu, 22 Apr 2021 23:33:02 +0200 Subject: Update RS.tex --- buch/papers/reedsolomon/RS presentation/RS.tex | 288 ++++++++++++++----------- 1 file changed, 165 insertions(+), 123 deletions(-) (limited to 'buch/papers/reedsolomon/RS presentation') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 732cee5..61324f7 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -15,6 +15,7 @@ \date{26.04.2021} \subject{Mathematisches Seminar} \setbeamercovered{transparent} + %\setbeamercovered{invisible} \setbeamertemplate{navigation symbols}{} \begin{frame}[plain] \maketitle @@ -83,7 +84,11 @@ \end{tabular} \end{center} \end{frame} +<<<<<<< Updated upstream %------------------------------------------------------------------------------- +======= + +>>>>>>> Stashed changes \section{Diskrete Fourier Transformation} \begin{frame} \frametitle{Idee} @@ -179,26 +184,38 @@ Indem in einem endlichen Körper gerechnet wird. } \end{frame} +<<<<<<< Updated upstream %------------------------------------------------------------------------------- +======= + +\section{Reed-Solomon in Endlichen Körpern} + +>>>>>>> Stashed changes \begin{frame} \frametitle{Reed-Solomon in Endlichen Körpern} \begin{itemize} - \item Warum Endliche Körper? + \onslide<1->{\item Warum endliche Körper?} - \qquad bessere Laufzeit + \onslide<1->{\qquad konkrete Zahlen $\rightarrow$ keine Rundungsfehler} - \vspace{10pt} + \onslide<1->{\qquad digitale Fehlerkorrektur} - \item Nachricht = Nutzdaten + Fehlerkorrekturteil + \onslide<1->{\qquad bessere Laufzeit} \vspace{10pt} - \item den Fehlerkorrekturteil brauchen wir im Optimalfall nicht + \onslide<1->{\item Nachricht = Nutzdaten + Fehlerkorrekturteil} \vspace{10pt} - \item Im Fehlerfall sollen wir aus der Nachricht ein Lokatorpolynom berechnen können, welches die Fehlerhaften Stellen beinhaltet + \onslide<1->{\item aus Fehlerkorrekturteil die Fehlerstellen finden} + + \onslide<1->{\qquad $\Rightarrow$ gesucht ist ein Lokatorpolynom} + +% \vspace{10pt} + +% \onslide<1->{\item Im Fehlerfall sollen wir aus der Nachricht ein Lokatorpolynom berechnen können, welches die fehlerhaften Stellen beinhaltet} % Wir sollten im Fehlerfall in der Lage sein, aus der Nachricht ein Lokatorpolynom zu berechnen, welches die Fehlerhaften Stellen beinhaltet @@ -212,35 +229,35 @@ % sollten wir fehler bekommen, was uns die korrekturstellen mitgeteilt wird, dann ist es unsere aufgabe ein lokatorpolynom zu finden, welches uns verrät, auf welchen zeilen der Fehler aufgetreten ist \end{frame} -%------------------------------------------------------------------------------- +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Definition eines Beispiels} \begin{itemize} - \item Endlicher Körper $q = 11$ + \only<1->{\item endlicher Körper $q = 11$} \only<1->{ist eine Primzahl} - \only<1->{beinhaltet die Zahlen $\mathbb{Z}_{11} = [0,1,2,3,4,5,6,7,8,9,10]$} + \only<1->{beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$} \vspace{10pt} - \only<1->{\item Nachrichtenblock $n = q-1$} + \only<1->{\item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen - wird an den Empfänger gesendet + $n = q - 1 = 10$ Zahlen} \vspace{10pt} - \only<1->{\item max. Fehler $z = 2$} + \only<1->{\item Max.~Fehler $z = 2$ - maximale Anzahl von Fehler, die wir noch korrigieren können + maximale Anzahl von Fehler, die wir noch korrigieren können} \vspace{10pt} \only<1->{\item Nutzlast $k = n -2t = 6$ Zahlen} - Fehlerstellen $2t = 4$ Zahlen + \only<1->{Fehlerkorrkturstellen $2t = 4$ Zahlen} \only<1->{Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$} @@ -250,52 +267,54 @@ \end{frame} %------------------------------------------------------------------------------- +\section{Codierung eines Beispiels} \begin{frame} \frametitle{Codierung} \begin{itemize} - \item Ansatz aus den Komplexen Zahlen mit der Fouriertransformation + \only<1->{\item Ansatz aus den komplexen Zahlen mit der diskreten Fouriertransformation} \vspace{10pt} - \item $\mathrm{e}$ existiert nicht in $\mathbb{Z}_{11}$ + \only<1->{\item Eulersche Zahl $\mathrm{e}$ existiert nicht in $\mathbb{F}_{11}$} \vspace{10pt} - \item wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{Z}_{11}$ abdeckt + \only<1->{\item Wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken - $\mathbb{Z}_{11}\setminus\{0\} = [a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9]$ + $\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}$} \vspace{10pt} - \item wir wählen $a = 8$ + \only<1->{\item Wir wählen $a = 8$} - $\mathbb{Z}_{11}\setminus\{0\} = [1,8,9,6,4,10,3,2,5,7]$ + \only<1->{$\mathbb{Z}_{11}\setminus\{0\} = \{1,8,9,6,4,10,3,2,5,7\}$} - 8 ist eine Primitive Einheitswurzel + \only<1->{$8$ ist eine primitive Einheitswurzel} \vspace{10pt} - \item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$ + \only<1->{\item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$} - $\Rightarrow$ \qquad können wir auch als Matrix schreiben + \only<1->{$\Rightarrow$ \qquad können wir auch als Matrix schreiben} \end{itemize} \end{frame} -%------------------------------------------------------------------------------- +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Codierung} \begin{itemize} - \item Übertragungsvektor $V$ + \only<1->{\item Übertragungsvektor $v$} - \item $V = A \cdot m$ + \only<1->{\item $v = A \cdot m$} \end{itemize} \[ - V = \begin{pmatrix} + \only<1->{ + v = \begin{pmatrix} 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ @@ -311,29 +330,34 @@ \begin{pmatrix} 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{pmatrix} + } \] - + \only<1->{ \begin{itemize} - \item $V = [5,3,6,5,2,10,2,7,10,4]$ + \item $v = [5,3,6,5,2,10,2,7,10,4]$ \end{itemize} - + } \end{frame} %------------------------------------------------------------------------------- +\section{Decodierung ohne Fehler} \begin{frame} \frametitle{Decodierung ohne Fehler} \begin{itemize} - \item Der Empfänger erhält den unveränderten Vektor $V = [5,3,6,5,2,10,2,7,10,4]$ + \only<1->{\item Der Empfänger erhält den unveränderten Vektor + $v = [5,3,6,5,2,10,2,7,10,4]$} \vspace{10pt} - \item Wir suchen die Inverse der Matrix A + \only<1->{\item Wir suchen die Inverse der Matrix $A$} + + \vspace{10pt} \end{itemize} \begin{columns}[t] \begin{column}{0.50\textwidth} - + \only<1->{ Inverse der Fouriertransformation \vspace{10pt} \[ @@ -341,25 +365,26 @@ \] \vspace{10pt} \[ - f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega + \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega \] - + } \end{column} \begin{column}{0.50\textwidth} - - Inverse von a + \only<1->{ + Inverse von $a$} \vspace{10pt} + \only<1->{ \[ 8^{1} \Rightarrow 8^{-1} \] - - Inverse finden wir über den Eulkidischen Algorithmus + } + \only<1->{Inverse finden wir über den Eulkidischen Algorithmus} \vspace{10pt} \end{column} \end{columns} \end{frame} -%------------------------------------------------------------------------------- +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Der Euklidische Algorithmus} @@ -385,8 +410,8 @@ \begin{column}{0.50\textwidth} \begin{center} - - \begin{tabular}{| c | c c | c | c c |} + \only<1->{ + \begin{tabular}{| c | c c | c | r r |} \hline $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ \hline @@ -395,17 +420,17 @@ $1$& $11$& $8$& $1$& $1$& $0$\\ $2$& $8$& $3$& $2$& $-1$& $1$\\ $3$& $3$& $2$& $1$& $3$& $-2$\\ - $4$& $2$& $1$& $2$& $-4$& $3$\\ + $4$& $2$& $1$& $2$& \textcolor<3->{blue}{$-4$}& \textcolor<3->{red}{$3$}\\ $5$& $1$& $0$& & $11$& $-8$\\ \hline \end{tabular} - + } \vspace{10pt} \begin{tabular}{rcl} - $-4\cdot 8 + 3 \cdot 11$ &$=$& $1$\\ - $7 \cdot 8 + 3 \cdot 11$ &$=$& $1$\\ - $8^{-1}$ &$=$& $7$ + \only<1->{$\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$}\\ + \only<1->{$7 \cdot 8 + 3 \cdot 11$ &$=$& $1$}\\ + \only<1->{$8^{-1}$ &$=$& $7$} \end{tabular} @@ -417,17 +442,17 @@ \end{frame} %------------------------------------------------------------------------------- \begin{frame} - \frametitle{Decodirung mit Inverser Matrix} + \frametitle{Decodierung mit Inverser Matrix} \begin{itemize} - \item $V = [5,3,6,5,2,10,2,7,10,4]$ + \only<1->{\item $v = [5,3,6,5,2,10,2,7,10,4]$} - \item $m = 1/10 \cdot A^{-1} \cdot V$ + \only<1->{\item $m = 1/10 \cdot A^{-1} \cdot v$} - \item $m = 10 \cdot A^{-1} \cdot V$ + \only<1->{\item $m = 10 \cdot A^{-1} \cdot v$} \end{itemize} - + \only<1->{ \[ m = \begin{pmatrix} 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ @@ -446,85 +471,95 @@ 5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\ \end{pmatrix} \] - + } + \only<1->{ \begin{itemize} \item $m = [0,0,0,0,4,7,2,5,8,1]$ \end{itemize} - + } \end{frame} %------------------------------------------------------------------------------- +\section{Decodierung mit Fehler} \begin{frame} \frametitle{Decodierung mit Fehler - Ansatz} \begin{itemize} - \item Gesendet: $V = [5,3,6,5,2,10,2,7,10,4]$ + \only<1->{\item Gesendet: $v = [5,3,6,5,2,10,2,7,10,4]$} - \item Empfangen: $W = [5,3,6,8,2,10,2,7,1,4]$ + \only<1->{\item Empfangen: $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$} + + \only<1->{\item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]$} - \item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerstellen}5,4,5,7,6,7]$ \end{itemize} - Wie finden wir die Fehler? + \only<1->{Wie finden wir die Fehler?} + \only<1->{ \begin{itemize} \item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ \item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ + %\only<7->{\item $e(X) = r(X) - m(X)$} + \item $e(X) = r(X) - m(X)$ + \end{itemize} - + } + \begin{center} - + \only<1->{ \begin{tabular}{c c c c c c c c c c c} \hline $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ \hline - $r(a^{i})$& $5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$\\ - $m(a^{i})$& $5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$\\ - $e(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\ + $r(a^{i})$& \only<1->{$5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$}\\ + $m(a^{i})$& \only<1->{$5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$}\\ + $e(a^{i})$& \only<1->{$0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$}\\ \hline \end{tabular} - + } \end{center} - + + \only<1->{ \begin{itemize} \item Alle Stellen, die nicht Null sind, sind Fehler \end{itemize} - + } + \end{frame} -%------------------------------------------------------------------------------- +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Nullstellen des Fehlerpolynoms finden} \begin{itemize} - \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + \only<1->{\item Satz von Fermat: $f(X) = X^{q-1}-1=0$} \vspace{10pt} - \item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$ + \only<1->{\item $f(X) = X^{10}-1 = 0$ \qquad für $X \in \{1,2,3,4,5,6,7,8,9,10\}$} \vspace{10pt} - \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + \only<1->{\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ - \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$} \vspace{10pt} - \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + \only<1->{\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ - \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$} \vspace{10pt} - \item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat + \only<1->{\item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat} \vspace{10pt} - $\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + \only<1->{$\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ - \qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$ + \qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$} \end{itemize} @@ -574,33 +609,33 @@ \end{frame} %------------------------------------------------------------------------------- \begin{frame} - \frametitle{kennen wir $e$?} + \frametitle{Kennen wir $e(X)$?} \begin{itemize} - \item $e$ ist unbekannt auf der Empfängerseite + \only<1->{\item $e(X)$ ist unbekannt auf der Empfängerseite} \vspace{10pt} - \item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt? + \only<1->{\item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt?} \vspace{10pt} - \item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ + \only<1->{\item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ - In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat + In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat} \vspace{10pt} - \item daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$ + \only<1->{\item Daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$} \vspace{10pt} - \item $f(X) = X^{10} - 1 = X^{10} + 10$ + \only<1->{\item $f(X) = X^{10} - 1 = X^{10} + 10$} \vspace{10pt} - \item jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen + \only<1->{\item Jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen} \end{itemize} \end{frame} @@ -608,8 +643,8 @@ \begin{frame} \frametitle{Der Euklidische Algorithmus (nochmal)} - $\operatorname{ggT}(f(X),e(X))$ hat den Grad 8 - + \only<1->{$\operatorname{ggT}(f(X),e(X))$ hat den Grad $8$} + \only<1->{ \[ \arraycolsep=1.4pt \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} @@ -620,7 +655,8 @@ & & & &6X^8&+&0X^7&+&p(X)& & & & & & & & & & & & \\ \end{array} \] - + } + \only<1->{ \[ \arraycolsep=1.4pt \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} @@ -629,14 +665,14 @@ && 7X^8&+& p(X)& & & & & & & & & & & & & & & & \\ \end{array} \] - + } \vspace{10pt} - $\operatorname{ggT}(f(X),e(X)) = 6X^8$ + \only<1->{$\operatorname{ggT}(f(X),e(X)) = 6X^8$} \vspace{10pt} - $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen + \only<1->{ $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen } \end{frame} @@ -653,7 +689,7 @@ & & $0$& $1$\\ $0$& $9X + 5$& $1$& $0$\\ $1$& $10X + 3$& $9X+5$& $1$\\ - $2$& & $2X^2 + 0X + 5$& $10X + 3$\\ + $2$& & \textcolor<2->{blue}{$2X^2 + 0X + 5$}& $10X + 3$\\ \hline \end{tabular} @@ -662,49 +698,54 @@ \vspace{10pt} \begin{tabular}{ll} - Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\ - Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\ - Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$ + \only<1->{Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\} + \only<1->{Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\} + \only<1->{Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$} \end{tabular} \vspace{10pt} - + \only<1->{ \begin{center} $a^i = 5 \qquad \Rightarrow \qquad i = 3$ $a^i = 6 \qquad \Rightarrow \qquad i = 8$ \end{center} - - $D = [3,8]$ + } + \only<1->{$d(X) = (X-a^3)(X-a^8)$} \end{frame} -%------------------------------------------------------------------------------- +%------------------------------------------------------------------------------- +\section{Nachricht Rekonstruieren} \begin{frame} \frametitle{Rekonstruktion der Nachricht} \begin{itemize} - \item $W = [5,3,6,8,2,10,2,7,1,4]$ + \only<1->{\item $w = [5,3,6,8,2,10,2,7,1,4]$} - \item $D = [3,8]$ + \only<1->{\item $d(X) = (X-\textcolor<4->{red}{a^3})(X-\textcolor<4->{red}{a^8})$} \end{itemize} - + \only<1->{ \[ + \textcolor{gray}{ \begin{pmatrix} - 5 \\ 3 \\ 6 \\ 8 \\ 2 \\ 10 \\ 2 \\ 7 \\ 1 \\ 4 \\ + a^0 \\ a^1 \\ a^2 \\ \textcolor<4->{red}{a^3} \\ a^4 \\ a^5 \\ a^6 \\ a^7 \\ \textcolor<4->{red}{a^8} \\ a^9 \\ + \end{pmatrix}} + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ \textcolor<4->{red}{8} \\ 2 \\ 10 \\ 2 \\ 7 \\ \textcolor<4->{red}{1} \\ 4 \\ \end{pmatrix} = \begin{pmatrix} 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ - 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + \textcolor<4->{red}{8^0}& \textcolor<4->{red}{8^3}& \textcolor<4->{red}{8^6}& \textcolor<4->{red}{8^9}& \textcolor<4->{red}{8^{12}}& \textcolor<4->{red}{8^{15}}& \textcolor<4->{red}{8^{18}}& \textcolor<4->{red}{8^{21}}& \textcolor<4->{red}{8^{24}}& \textcolor<4->{red}{8^{27}}\\ 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ - 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + \textcolor<4->{red}{8^0}& \textcolor<4->{red}{8^8}& \textcolor<4->{red}{8^{16}}& \textcolor<4->{red}{8^{24}}& \textcolor<4->{red}{8^{32}}& \textcolor<4->{red}{8^{40}}& \textcolor<4->{red}{8^{48}}& \textcolor<4->{red}{8^{56}}& \textcolor<4->{red}{8^{64}}& \textcolor<4->{red}{8^{72}}\\ 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ \end{pmatrix} \cdot @@ -712,13 +753,14 @@ m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ \end{pmatrix} \] - + } + \only<1->{ \begin{itemize} \item Fehlerstellen entfernen \end{itemize} - + } \end{frame} -%------------------------------------------------------------------------------- +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Rekonstruktion der Nachricht} @@ -728,25 +770,25 @@ \end{pmatrix} = \begin{pmatrix} - 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ - 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ - 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ - 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ - 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ - 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ - 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ - 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor<3->{green}{8^0}& \textcolor<3->{green}{8^0}& \textcolor<3->{green}{8^0}& \textcolor<3->{green}{8^0}\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor<3->{green}{8^6}& \textcolor<3->{green}{8^7}& \textcolor<3->{green}{8^8}& \textcolor<3->{green}{8^9}\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor<3->{green}{8^{12}}& \textcolor<3->{green}{8^{14}}& \textcolor<3->{green}{8^{16}}& \textcolor<3->{green}{8^{18}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor<3->{green}{8^{24}}& \textcolor<3->{green}{8^{28}}& \textcolor<3->{green}{8^{32}}& \textcolor<3->{green}{8^{36}}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor<3->{green}{8^{30}}& \textcolor<3->{green}{8^{35}}& \textcolor<3->{green}{8^{40}}& \textcolor<3->{green}{8^{45}}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor<3->{green}{8^{36}}& \textcolor<3->{green}{8^{42}}& \textcolor<3->{green}{8^{48}}& \textcolor<3->{green}{8^{54}}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor<3->{green}{8^{42}}& \textcolor<3->{green}{8^{49}}& \textcolor<3->{green}{8^{56}}& \textcolor<3->{green}{8^{63}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor<3->{green}{8^{54}}& \textcolor<3->{green}{8^{63}}& \textcolor<3->{green}{8^{72}}& \textcolor<3->{green}{8^{81}}\\ \end{pmatrix} \cdot \begin{pmatrix} - m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor<2->{green}{m_6} \\ \textcolor<2->{green}{m_7} \\ \textcolor<2->{green}{m_8} \\ \textcolor<2->{green}{m_9} \\ \end{pmatrix} \] - + \only<1->{ \begin{itemize} \item Nullstellen entfernen \end{itemize} - + } \end{frame} %------------------------------------------------------------------------------- \begin{frame} @@ -754,7 +796,7 @@ \[ \begin{pmatrix} - 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor<2->{red}{7} \\ \textcolor<2->{red}{4} \\ \end{pmatrix} = \begin{pmatrix} @@ -764,8 +806,8 @@ 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ - 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}\\ - 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}\\ + \textcolor<2->{red}{8^0}& \textcolor<2->{red}{8^7}& \textcolor<2->{red}{8^{14}}& \textcolor<2->{red}{8^{21}}& \textcolor<2->{red}{8^{28}}& \textcolor<2->{red}{8^{35}}\\ + \textcolor<2->{red}{8^0}& \textcolor<2->{red}{8^9}& \textcolor<2->{red}{8^{18}}& \textcolor<2->{red}{8^{27}}& \textcolor<2->{red}{8^{36}}& \textcolor<2->{red}{8^{45}}\\ \end{pmatrix} \cdot \begin{pmatrix} @@ -774,11 +816,11 @@ \] \vspace{5pt} - + \only<1->{ \begin{itemize} \item Matrix in eine Quadratische Form bringen \end{itemize} - + } \end{frame} %------------------------------------------------------------------------------- \begin{frame} -- cgit v1.2.1 From 8c6a8e56c125c238dc64c21d1269fcdc7542c5cd Mon Sep 17 00:00:00 2001 From: JODBaer Date: Thu, 22 Apr 2021 23:45:32 +0200 Subject: =?UTF-8?q?merge=20lines=20gel=C3=B6scht?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- buch/papers/reedsolomon/RS presentation/RS.tex | 9 +++------ 1 file changed, 3 insertions(+), 6 deletions(-) (limited to 'buch/papers/reedsolomon/RS presentation') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 61324f7..943f2da 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -84,11 +84,9 @@ \end{tabular} \end{center} \end{frame} -<<<<<<< Updated upstream + %------------------------------------------------------------------------------- -======= ->>>>>>> Stashed changes \section{Diskrete Fourier Transformation} \begin{frame} \frametitle{Idee} @@ -184,13 +182,12 @@ Indem in einem endlichen Körper gerechnet wird. } \end{frame} -<<<<<<< Updated upstream + %------------------------------------------------------------------------------- -======= + \section{Reed-Solomon in Endlichen Körpern} ->>>>>>> Stashed changes \begin{frame} \frametitle{Reed-Solomon in Endlichen Körpern} -- cgit v1.2.1 From 179ea16b001b6640e9b720d53ffc06f3e2389ff2 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Fri, 23 Apr 2021 00:30:36 +0200 Subject: appostroph verbessert --- buch/papers/reedsolomon/RS presentation/RS.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'buch/papers/reedsolomon/RS presentation') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 943f2da..d09d77d 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -64,7 +64,7 @@ \begin{center} \begin{tabular}{ c c c } \hline - ``Nutzlas´´ & Fehler & Versenden \\ + ``Nutzlast'' & Fehler & Versenden \\ \hline 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ -- cgit v1.2.1 From ded210e33924d4c078e5a0d899c0585d7f987565 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Fri, 23 Apr 2021 12:58:40 +0200 Subject: Folien Verbesserungen animation --- buch/papers/reedsolomon/RS presentation/RS.aux | 167 +++++++++++----- buch/papers/reedsolomon/RS presentation/RS.log | 212 ++++++++++++++++----- buch/papers/reedsolomon/RS presentation/RS.nav | 117 ++++++++---- buch/papers/reedsolomon/RS presentation/RS.out | 9 +- buch/papers/reedsolomon/RS presentation/RS.pdf | Bin 135643 -> 207741 bytes 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[2][]{Outline0.3}{Diskrete\040Fourier\040Transformation}{}% 3 +\BOOKMARK [2][]{Outline0.4}{Reed-Solomon in Endlichen Körpern}{}% 4 +\BOOKMARK [2][]{Outline0.5}{Codierung\040eines\040Beispiels}{}% 5 +\BOOKMARK [2][]{Outline0.6}{Decodierung\040ohne\040Fehler}{}% 6 +\BOOKMARK [2][]{Outline0.7}{Decodierung\040mit\040Fehler}{}% 7 +\BOOKMARK [2][]{Outline0.8}{Nachricht\040Rekonstruieren}{}% 8 diff --git a/buch/papers/reedsolomon/RS presentation/RS.pdf b/buch/papers/reedsolomon/RS presentation/RS.pdf index 913bc42..d9d6693 100644 Binary files a/buch/papers/reedsolomon/RS presentation/RS.pdf and b/buch/papers/reedsolomon/RS presentation/RS.pdf differ diff --git a/buch/papers/reedsolomon/RS presentation/RS.snm b/buch/papers/reedsolomon/RS presentation/RS.snm index 6607ea8..86859c9 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.snm +++ b/buch/papers/reedsolomon/RS presentation/RS.snm @@ -1 +1 @@ -\beamer@slide {ft_discrete}{15} +\beamer@slide {ft_discrete}{21} diff --git a/buch/papers/reedsolomon/RS presentation/RS.synctex.gz b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz index 001b5c8..04bd239 100644 Binary files a/buch/papers/reedsolomon/RS presentation/RS.synctex.gz and b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz differ diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index d09d77d..7b2c4da 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -43,18 +43,18 @@ ${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. - \only<1>{ - Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, - \textcolor{green}{15}, \textcolor{green}{26}, - \textcolor{green}{41}, \textcolor{green}{60}, - \textcolor{green}{83}, \textcolor{green}{110})$ - \includegraphics[scale = 1.2]{images/polynom1.pdf}} - \only<2>{ - Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, - \textcolor{red}{50}, \textcolor{red}{37}, - \textcolor{green}{41}, \textcolor{green}{60}, - \textcolor{green}{83}, \textcolor{green}{110})$ - \includegraphics[scale = 1.2]{images/polynom2.pdf} + + Versende $ (p(1),p(2),\dots,p(7))$ + \visible<2->{ = (\textcolor{green}{8},} + \only<2>{\textcolor{green}{15},} + \only<3>{\textcolor{red}{50},} + \only<2>{\textcolor{green}{26},} + \only<3>{\textcolor{red}{37},} + \visible<2->{\textcolor{green}{41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})} + \only<2>{\includegraphics[scale = 1.2]{images/polynom1.pdf}} + \only<3>{\includegraphics[scale = 1.2]{images/polynom2.pdf}} + \visible<3>{ \newline \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} \end{frame} diff --git a/buch/papers/reedsolomon/RS presentation/RS.toc b/buch/papers/reedsolomon/RS presentation/RS.toc index 44c06ab..095b5e6 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.toc +++ b/buch/papers/reedsolomon/RS presentation/RS.toc @@ -1,6 +1,9 @@ \babel@toc {ngerman}{} \beamer@sectionintoc {1}{Einführung}{2}{0}{1} -\beamer@sectionintoc {2}{Polynom Ansatz}{3}{0}{2} -\beamer@sectionintoc {3}{Fourier Transformation}{7}{0}{3} -\beamer@sectionintoc {4}{Diskrete Fourier Transformation}{15}{0}{4} -\beamer@sectionintoc {5}{Probleme und Fragen}{17}{0}{5} +\beamer@sectionintoc {2}{Polynom Ansatz}{5}{0}{2} +\beamer@sectionintoc {3}{Diskrete Fourier Transformation}{13}{0}{3} +\beamer@sectionintoc {4}{Reed-Solomon in Endlichen Körpern}{27}{0}{4} +\beamer@sectionintoc {5}{Codierung eines Beispiels}{29}{0}{5} +\beamer@sectionintoc {6}{Decodierung ohne Fehler}{31}{0}{6} +\beamer@sectionintoc {7}{Decodierung mit Fehler}{36}{0}{7} +\beamer@sectionintoc {8}{Nachricht Rekonstruieren}{43}{0}{8} -- cgit v1.2.1 From 0a80be4477602e2d909e5eda40dae485ec6acd56 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Fri, 23 Apr 2021 13:02:38 +0200 Subject: Read me erstellt --- buch/papers/reedsolomon/RS presentation/README.txt | 1 + 1 file changed, 1 insertion(+) create mode 100644 buch/papers/reedsolomon/RS presentation/README.txt (limited to 'buch/papers/reedsolomon/RS presentation') diff --git a/buch/papers/reedsolomon/RS presentation/README.txt b/buch/papers/reedsolomon/RS presentation/README.txt new file mode 100644 index 0000000..4d0620f --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/README.txt @@ -0,0 +1 @@ +Dies ist die Presentation des Reed-Solomon-Code \ No newline at end of file -- cgit v1.2.1 From d1b6d92a02d9c44b3860b73d5660c5c6863de0df Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Fri, 23 Apr 2021 21:19:34 +0200 Subject: handout added --- buch/papers/reedsolomon/RS presentation/RS.tex | 290 +++---- .../reedsolomon/RS presentation/RS_handout.tex | 921 +++++++++++++++++++++ 2 files changed, 1069 insertions(+), 142 deletions(-) create mode 100644 buch/papers/reedsolomon/RS presentation/RS_handout.tex (limited to 'buch/papers/reedsolomon/RS presentation') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 943f2da..c215e66 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -14,8 +14,8 @@ \institute{OST Ostschweizer Fachhochschule} \date{26.04.2021} \subject{Mathematisches Seminar} - \setbeamercovered{transparent} - %\setbeamercovered{invisible} + %\setbeamercovered{transparent} + \setbeamercovered{invisible} \setbeamertemplate{navigation symbols}{} \begin{frame}[plain] \maketitle @@ -64,22 +64,22 @@ \begin{center} \begin{tabular}{ c c c } \hline - ``Nutzlas´´ & Fehler & Versenden \\ + Nutzlas & Fehler & Versenden \\ \hline 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ -\visible<2->{3}& -\visible<2->{3}& -\visible<3->{9 Werte eines Polynoms vom Grad 2} \\ +\visible<1->{3}& +\visible<1->{3}& +\visible<1->{9 Werte eines Polynoms vom Grad 2} \\ &&\\ -\visible<4->{$k$} & -\visible<4->{$t$} & -\visible<4->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ +\visible<1->{$k$} & +\visible<1->{$t$} & +\visible<1->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ \hline &&\\ &&\\ \multicolumn{3}{l} { - \visible<4>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} + \visible<1>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} } \end{tabular} \end{center} @@ -194,21 +194,21 @@ \begin{itemize} \onslide<1->{\item Warum endliche Körper?} - \onslide<1->{\qquad konkrete Zahlen $\rightarrow$ keine Rundungsfehler} + \onslide<2->{\qquad konkrete Zahlen $\rightarrow$ keine Rundungsfehler} - \onslide<1->{\qquad digitale Fehlerkorrektur} + \onslide<3->{\qquad digitale Fehlerkorrektur} - \onslide<1->{\qquad bessere Laufzeit} + %\onslide<4->{\qquad bessere Laufzeit} \vspace{10pt} - \onslide<1->{\item Nachricht = Nutzdaten + Fehlerkorrekturteil} + \onslide<4->{\item Nachricht = Nutzdaten + Fehlerkorrekturteil} \vspace{10pt} - \onslide<1->{\item aus Fehlerkorrekturteil die Fehlerstellen finden} + \onslide<5->{\item aus Fehlerkorrekturteil die Fehlerstellen finden} - \onslide<1->{\qquad $\Rightarrow$ gesucht ist ein Lokatorpolynom} + \onslide<6->{\qquad $\Rightarrow$ gesucht ist ein Lokatorpolynom} % \vspace{10pt} @@ -232,33 +232,33 @@ \begin{itemize} - \only<1->{\item endlicher Körper $q = 11$} + \onslide<1->{\item endlicher Körper $q = 11$} - \only<1->{ist eine Primzahl} + \onslide<2->{ist eine Primzahl} - \only<1->{beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$} + \onslide<3->{beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$} \vspace{10pt} - \only<1->{\item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen + \onslide<4->{\item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen} - $n = q - 1 = 10$ Zahlen} + \onslide<5->{$n = q - 1 = 10$ Zahlen} \vspace{10pt} - \only<1->{\item Max.~Fehler $z = 2$ + \onslide<6->{\item Max.~Fehler $t = 2$} - maximale Anzahl von Fehler, die wir noch korrigieren können} + \onslide<7->{maximale Anzahl von Fehler, die wir noch korrigieren können} \vspace{10pt} - \only<1->{\item Nutzlast $k = n -2t = 6$ Zahlen} + \onslide<8->{\item Nutzlast $k = n -2t = 6$ Zahlen} - \only<1->{Fehlerkorrkturstellen $2t = 4$ Zahlen} + \onslide<9->{Fehlerkorrkturstellen $2t = 4$ Zahlen} - \only<1->{Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$} + \onslide<10->{Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$} - \only<1->{als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$} + \onslide<11->{als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$} \end{itemize} @@ -269,31 +269,31 @@ \frametitle{Codierung} \begin{itemize} - \only<1->{\item Ansatz aus den komplexen Zahlen mit der diskreten Fouriertransformation} + \onslide<1->{\item Ansatz aus den komplexen Zahlen mit der diskreten Fouriertransformation} \vspace{10pt} - \only<1->{\item Eulersche Zahl $\mathrm{e}$ existiert nicht in $\mathbb{F}_{11}$} + \onslide<2->{\item Eulersche Zahl $\mathrm{e}$ existiert nicht in $\mathbb{F}_{11}$} \vspace{10pt} - \only<1->{\item Wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken + \onslide<3->{\item Wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken} - $\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}$} + \onslide<4->{$\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}$} \vspace{10pt} - \only<1->{\item Wir wählen $a = 8$} + \onslide<5->{\item Wir wählen $a = 8$} - \only<1->{$\mathbb{Z}_{11}\setminus\{0\} = \{1,8,9,6,4,10,3,2,5,7\}$} + \onslide<6->{$\mathbb{Z}_{11}\setminus\{0\} = \{1,8,9,6,4,10,3,2,5,7\}$} - \only<1->{$8$ ist eine primitive Einheitswurzel} + \onslide<7->{$8$ ist eine primitive Einheitswurzel} \vspace{10pt} - \only<1->{\item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$} + \onslide<8->{\item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$} - \only<1->{$\Rightarrow$ \qquad können wir auch als Matrix schreiben} + \onslide<9->{$\Rightarrow$ \qquad können wir auch als Matrix schreiben} \end{itemize} @@ -303,14 +303,14 @@ \frametitle{Codierung} \begin{itemize} - \only<1->{\item Übertragungsvektor $v$} + \onslide<1->{\item Übertragungsvektor $v$} - \only<1->{\item $v = A \cdot m$} + \onslide<2->{\item $v = A \cdot m$} \end{itemize} \[ - \only<1->{ + \onslide<3->{ v = \begin{pmatrix} 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ @@ -329,11 +329,11 @@ \end{pmatrix} } \] - \only<1->{ + \begin{itemize} - \item $v = [5,3,6,5,2,10,2,7,10,4]$ + \onslide<4->{\item $v = [5,3,6,5,2,10,2,7,10,4]$} \end{itemize} - } + \end{frame} %------------------------------------------------------------------------------- \section{Decodierung ohne Fehler} @@ -341,41 +341,44 @@ \frametitle{Decodierung ohne Fehler} \begin{itemize} - \only<1->{\item Der Empfänger erhält den unveränderten Vektor - $v = [5,3,6,5,2,10,2,7,10,4]$} + \onslide<1->{\item Der Empfänger erhält den unveränderten Vektor $v = [5,3,6,5,2,10,2,7,10,4]$} \vspace{10pt} - \only<1->{\item Wir suchen die Inverse der Matrix $A$} + \onslide<2->{\item Wir suchen die Inverse der Matrix $A$} \vspace{10pt} \end{itemize} \begin{columns}[t] - \begin{column}{0.50\textwidth} - \only<1->{ - Inverse der Fouriertransformation + \begin{column}{0.55\textwidth} + \onslide<3->{ Inverse der Fouriertransformation} \vspace{10pt} + \onslide<4->{ \[ F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt \] + } \vspace{10pt} + \onslide<5->{ \[ \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega \] } \end{column} - \begin{column}{0.50\textwidth} - \only<1->{ - Inverse von $a$} + \begin{column}{0.45\textwidth} + \onslide<6->{Inverse von $a$} + \vspace{10pt} - \only<1->{ + + \onslide<7->{ \[ 8^{1} \Rightarrow 8^{-1} \] } - \only<1->{Inverse finden wir über den Eulkidischen Algorithmus} + + \onslide<8->{Inverse finden wir über den Eulkidischen Algorithmus} \vspace{10pt} \end{column} \end{columns} @@ -407,7 +410,7 @@ \begin{column}{0.50\textwidth} \begin{center} - \only<1->{ + \onslide<1->{ \begin{tabular}{| c | c c | c | r r |} \hline $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ @@ -417,17 +420,18 @@ $1$& $11$& $8$& $1$& $1$& $0$\\ $2$& $8$& $3$& $2$& $-1$& $1$\\ $3$& $3$& $2$& $1$& $3$& $-2$\\ - $4$& $2$& $1$& $2$& \textcolor<3->{blue}{$-4$}& \textcolor<3->{red}{$3$}\\ + $4$& $2$& $1$& $2$& \textcolor<2->{blue}{$-4$}& \textcolor<2->{red}{$3$}\\ $5$& $1$& $0$& & $11$& $-8$\\ \hline \end{tabular} } + \vspace{10pt} \begin{tabular}{rcl} - \only<1->{$\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$}\\ - \only<1->{$7 \cdot 8 + 3 \cdot 11$ &$=$& $1$}\\ - \only<1->{$8^{-1}$ &$=$& $7$} + \onslide<3->{$\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$}\\ + \onslide<4->{$7 \cdot 8 + 3 \cdot 11$ &$=$& $1$}\\ + \onslide<5->{$8^{-1}$ &$=$& $7$} \end{tabular} @@ -442,16 +446,16 @@ \frametitle{Decodierung mit Inverser Matrix} \begin{itemize} - \only<1->{\item $v = [5,3,6,5,2,10,2,7,10,4]$} + \onslide<1->{\item $v = [5,3,6,5,2,10,2,7,10,4]$} - \only<1->{\item $m = 1/10 \cdot A^{-1} \cdot v$} + \onslide<2->{\item $m = 1/10 \cdot A^{-1} \cdot v$} - \only<1->{\item $m = 10 \cdot A^{-1} \cdot v$} + \onslide<3->{\item $m = 10 \cdot A^{-1} \cdot v$} \end{itemize} - \only<1->{ + \onslide<4->{ \[ - m = \begin{pmatrix} + m = 10 \cdot \begin{pmatrix} 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ 7^0& 7^1& 7^2& 7^3& 7^4& 7^5& 7^6& 7^7& 7^8& 7^9\\ 7^0& 7^2& 7^4& 7^6& 7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\ @@ -469,11 +473,11 @@ \end{pmatrix} \] } - \only<1->{ + \begin{itemize} - \item $m = [0,0,0,0,4,7,2,5,8,1]$ + \onslide<5->{\item $m = [0,0,0,0,4,7,2,5,8,1]$} \end{itemize} - } + \end{frame} %------------------------------------------------------------------------------- \section{Decodierung mit Fehler} @@ -481,48 +485,46 @@ \frametitle{Decodierung mit Fehler - Ansatz} \begin{itemize} - \only<1->{\item Gesendet: $v = [5,3,6,5,2,10,2,7,10,4]$} + \onslide<1->{\item Gesendet: $v = [5,3,6,5,2,10,2,7,10,4]$} - \only<1->{\item Empfangen: $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$} + \onslide<2->{\item Empfangen: $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$} - \only<1->{\item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]$} + \onslide<3->{\item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]$} \end{itemize} - \only<1->{Wie finden wir die Fehler?} + \onslide<4->{Wie finden wir die Fehler?} - \only<1->{ \begin{itemize} - \item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ + \onslide<5->{\item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$} - \item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ + \onslide<6->{\item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$} %\only<7->{\item $e(X) = r(X) - m(X)$} - \item $e(X) = r(X) - m(X)$ + \onslide<7->{\item $e(X) = r(X) - m(X)$} \end{itemize} - } \begin{center} - \only<1->{ + \onslide<8->{ \begin{tabular}{c c c c c c c c c c c} \hline $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ \hline - $r(a^{i})$& \only<1->{$5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$}\\ - $m(a^{i})$& \only<1->{$5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$}\\ - $e(a^{i})$& \only<1->{$0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$}\\ + $r(a^{i})$& \onslide<9->{$5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$}\\ + $m(a^{i})$& \onslide<10->{$5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$}\\ + $e(a^{i})$& \onslide<11->{$0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$}\\ \hline \end{tabular} } \end{center} - \only<1->{ + \begin{itemize} - \item Alle Stellen, die nicht Null sind, sind Fehler + \onslide<12->{\item Alle Stellen, die nicht Null sind, sind Fehler} \end{itemize} - } + \end{frame} %------------------------------------------------------------------------------- @@ -530,31 +532,31 @@ \frametitle{Nullstellen des Fehlerpolynoms finden} \begin{itemize} - \only<1->{\item Satz von Fermat: $f(X) = X^{q-1}-1=0$} + \onslide<1->{\item Satz von Fermat: $f(X) = X^{q-1}-1=0$} \vspace{10pt} - \only<1->{\item $f(X) = X^{10}-1 = 0$ \qquad für $X \in \{1,2,3,4,5,6,7,8,9,10\}$} + \onslide<2->{\item $f(X) = X^{10}-1 = 0$ \qquad für $X \in \{1,2,3,4,5,6,7,8,9,10\}$} \vspace{10pt} - \only<1->{\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + \onslide<3->{\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$} \vspace{10pt} - \only<1->{\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + \onslide<4->{\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$} \vspace{10pt} - \only<1->{\item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat} + \onslide<5->{\item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat} \vspace{10pt} - \only<1->{$\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + \onslide<6->{$\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ \qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$} @@ -567,39 +569,39 @@ \begin{itemize} - \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + \onslide<1->{\item Satz von Fermat: $f(X) = X^{q-1}-1=0$} \vspace{10pt} - \item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$ + \onslide<1->{\item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$} \vspace{10pt} - \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + \onslide<1->{\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ - \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$} \vspace{10pt} - \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + \onslide<1->{\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ - \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$} \vspace{10pt} - \item $\operatorname{kgV}$ gibt uns eine Liste von aller Nullstellen, die wir in $e$ und $d$ zerlegen können + \onslide<1->{\item $\operatorname{kgV}$ gibt uns eine Liste von aller Nullstellen, die wir in $e$ und $d$ zerlegen können} \vspace{10pt} - $\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot $ + \onslide<2->{$\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot $ - \qquad \qquad \qquad \qquad $(X-a^7)(X-a^8)(X-a^9) \cdot q(X)$ + \qquad \qquad \qquad \qquad $(X-a^7)(X-a^8)(X-a^9) \cdot q(X)$} - $= d(X) \cdot e(X)$ + \onslide<3->{$= d(X) \cdot e(X)$} \vspace{10pt} - \item Lokatorpolynom $d(X) = (X-a^3)(X-a^8)$ + \onslide<4->{\item Lokatorpolynom $d(X) = (X-a^3)(X-a^8)$} \end{itemize} @@ -610,29 +612,29 @@ \begin{itemize} - \only<1->{\item $e(X)$ ist unbekannt auf der Empfängerseite} + \onslide<1->{\item $e(X)$ ist unbekannt auf der Empfängerseite} \vspace{10pt} - \only<1->{\item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt?} + \onslide<2->{\item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt?} \vspace{10pt} - \only<1->{\item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ + \onslide<3->{\item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat} \vspace{10pt} - \only<1->{\item Daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$} + \onslide<4->{\item Daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$} \vspace{10pt} - \only<1->{\item $f(X) = X^{10} - 1 = X^{10} + 10$} + \onslide<5->{\item $f(X) = X^{10} - 1 = X^{10} + 10$} \vspace{10pt} - \only<1->{\item Jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen} + \onslide<6->{\item Jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen} \end{itemize} \end{frame} @@ -640,8 +642,8 @@ \begin{frame} \frametitle{Der Euklidische Algorithmus (nochmal)} - \only<1->{$\operatorname{ggT}(f(X),e(X))$ hat den Grad $8$} - \only<1->{ + \onslide<1->{$\operatorname{ggT}(f(X),e(X))$ hat den Grad $8$} + \onslide<2->{ \[ \arraycolsep=1.4pt \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} @@ -653,7 +655,7 @@ \end{array} \] } - \only<1->{ + \onslide<3->{ \[ \arraycolsep=1.4pt \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} @@ -665,11 +667,11 @@ } \vspace{10pt} - \only<1->{$\operatorname{ggT}(f(X),e(X)) = 6X^8$} + \onslide<4->{$\operatorname{ggT}(f(X),e(X)) = 6X^8$} \vspace{10pt} - \only<1->{ $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen } + \onslide<5->{ $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen } \end{frame} @@ -695,20 +697,22 @@ \vspace{10pt} \begin{tabular}{ll} - \only<1->{Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\} - \only<1->{Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\} - \only<1->{Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$} + \onslide<3->{Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\} + \onslide<4->{Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\} + \onslide<5->{Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$} \end{tabular} \vspace{10pt} - \only<1->{ + + \onslide<6->{ \begin{center} $a^i = 5 \qquad \Rightarrow \qquad i = 3$ $a^i = 6 \qquad \Rightarrow \qquad i = 8$ \end{center} - } - \only<1->{$d(X) = (X-a^3)(X-a^8)$} + } + + \onslide<7->{$d(X) = (X-a^3)(X-a^8)$} \end{frame} %------------------------------------------------------------------------------- @@ -718,12 +722,12 @@ \begin{itemize} - \only<1->{\item $w = [5,3,6,8,2,10,2,7,1,4]$} + \onslide<1->{\item $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$} - \only<1->{\item $d(X) = (X-\textcolor<4->{red}{a^3})(X-\textcolor<4->{red}{a^8})$} + \onslide<2->{\item $d(X) = (X-\textcolor<4->{red}{a^3})(X-\textcolor<4->{red}{a^8})$} \end{itemize} - \only<1->{ + \onslide<3->{ \[ \textcolor{gray}{ \begin{pmatrix} @@ -751,11 +755,11 @@ \end{pmatrix} \] } - \only<1->{ + \begin{itemize} - \item Fehlerstellen entfernen + \onslide<5->{\item Fehlerstellen entfernen} \end{itemize} - } + \end{frame} %------------------------------------------------------------------------------- \begin{frame} @@ -767,25 +771,25 @@ \end{pmatrix} = \begin{pmatrix} - 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor<3->{green}{8^0}& \textcolor<3->{green}{8^0}& \textcolor<3->{green}{8^0}& \textcolor<3->{green}{8^0}\\ - 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor<3->{green}{8^6}& \textcolor<3->{green}{8^7}& \textcolor<3->{green}{8^8}& \textcolor<3->{green}{8^9}\\ - 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor<3->{green}{8^{12}}& \textcolor<3->{green}{8^{14}}& \textcolor<3->{green}{8^{16}}& \textcolor<3->{green}{8^{18}}\\ - 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor<3->{green}{8^{24}}& \textcolor<3->{green}{8^{28}}& \textcolor<3->{green}{8^{32}}& \textcolor<3->{green}{8^{36}}\\ - 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor<3->{green}{8^{30}}& \textcolor<3->{green}{8^{35}}& \textcolor<3->{green}{8^{40}}& \textcolor<3->{green}{8^{45}}\\ - 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor<3->{green}{8^{36}}& \textcolor<3->{green}{8^{42}}& \textcolor<3->{green}{8^{48}}& \textcolor<3->{green}{8^{54}}\\ - 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor<3->{green}{8^{42}}& \textcolor<3->{green}{8^{49}}& \textcolor<3->{green}{8^{56}}& \textcolor<3->{green}{8^{63}}\\ - 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor<3->{green}{8^{54}}& \textcolor<3->{green}{8^{63}}& \textcolor<3->{green}{8^{72}}& \textcolor<3->{green}{8^{81}}\\ + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor<4->{green}{8^0}& \textcolor<4->{green}{8^0}& \textcolor<4->{green}{8^0}& \textcolor<4->{green}{8^0}\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor<4->{green}{8^6}& \textcolor<4->{green}{8^7}& \textcolor<4->{green}{8^8}& \textcolor<4->{green}{8^9}\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor<4->{green}{8^{12}}& \textcolor<4->{green}{8^{14}}& \textcolor<4->{green}{8^{16}}& \textcolor<4->{green}{8^{18}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor<4->{green}{8^{24}}& \textcolor<4->{green}{8^{28}}& \textcolor<4->{green}{8^{32}}& \textcolor<4->{green}{8^{36}}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor<4->{green}{8^{30}}& \textcolor<4->{green}{8^{35}}& \textcolor<4->{green}{8^{40}}& \textcolor<4->{green}{8^{45}}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor<4->{green}{8^{36}}& \textcolor<4->{green}{8^{42}}& \textcolor<4->{green}{8^{48}}& \textcolor<4->{green}{8^{54}}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor<4->{green}{8^{42}}& \textcolor<4->{green}{8^{49}}& \textcolor<4->{green}{8^{56}}& \textcolor<4->{green}{8^{63}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor<4->{green}{8^{54}}& \textcolor<4->{green}{8^{63}}& \textcolor<4->{green}{8^{72}}& \textcolor<4->{green}{8^{81}}\\ \end{pmatrix} \cdot \begin{pmatrix} m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor<2->{green}{m_6} \\ \textcolor<2->{green}{m_7} \\ \textcolor<2->{green}{m_8} \\ \textcolor<2->{green}{m_9} \\ \end{pmatrix} \] - \only<1->{ + \begin{itemize} - \item Nullstellen entfernen + \onslide<3->{\item Nullstellen entfernen} \end{itemize} - } + \end{frame} %------------------------------------------------------------------------------- \begin{frame} @@ -793,7 +797,7 @@ \[ \begin{pmatrix} - 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor<2->{red}{7} \\ \textcolor<2->{red}{4} \\ + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor<3->{red}{7} \\ \textcolor<3->{red}{4} \\ \end{pmatrix} = \begin{pmatrix} @@ -803,8 +807,8 @@ 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ - \textcolor<2->{red}{8^0}& \textcolor<2->{red}{8^7}& \textcolor<2->{red}{8^{14}}& \textcolor<2->{red}{8^{21}}& \textcolor<2->{red}{8^{28}}& \textcolor<2->{red}{8^{35}}\\ - \textcolor<2->{red}{8^0}& \textcolor<2->{red}{8^9}& \textcolor<2->{red}{8^{18}}& \textcolor<2->{red}{8^{27}}& \textcolor<2->{red}{8^{36}}& \textcolor<2->{red}{8^{45}}\\ + \textcolor<3->{red}{8^0}& \textcolor<3->{red}{8^7}& \textcolor<3->{red}{8^{14}}& \textcolor<3->{red}{8^{21}}& \textcolor<3->{red}{8^{28}}& \textcolor<3->{red}{8^{35}}\\ + \textcolor<3->{red}{8^0}& \textcolor<3->{red}{8^9}& \textcolor<3->{red}{8^{18}}& \textcolor<3->{red}{8^{27}}& \textcolor<3->{red}{8^{36}}& \textcolor<3->{red}{8^{45}}\\ \end{pmatrix} \cdot \begin{pmatrix} @@ -813,11 +817,11 @@ \] \vspace{5pt} - \only<1->{ + \begin{itemize} - \item Matrix in eine Quadratische Form bringen + \onslide<2->{\item Matrix in eine Quadratische Form bringen} \end{itemize} - } + \end{frame} %------------------------------------------------------------------------------- \begin{frame} @@ -845,7 +849,7 @@ \vspace{5pt} \begin{itemize} - \item Matrix Invertieren + \onslide<2->{\item Matrix Invertieren} \end{itemize} \end{frame} @@ -873,9 +877,10 @@ \] \begin{center} - $\Downarrow$ + \onslide<2->{$\Downarrow$} \end{center} \[ + \onslide<3->{ \begin{pmatrix} m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \end{pmatrix} @@ -892,6 +897,7 @@ \begin{pmatrix} 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \end{pmatrix} + } \] \end{frame} @@ -919,7 +925,7 @@ \] \begin{itemize} - \item $m = [4,7,2,5,8,1]$ + \onslide<2->{\item $m = [4,7,2,5,8,1]$} \end{itemize} \end{frame} diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.tex b/buch/papers/reedsolomon/RS presentation/RS_handout.tex new file mode 100644 index 0000000..863b3a2 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.tex @@ -0,0 +1,921 @@ +\documentclass[11pt,aspectratio=169]{beamer} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[ngerman]{babel} +\usepackage{tikz} +\usetheme{Hannover} + +\begin{document} + \author{Joshua Bär und Michael Steiner} + \title{Reed-Solomon-Code} + \subtitle{} + \logo{} + \institute{OST Ostschweizer Fachhochschule} + \date{26.04.2021} + \subject{Mathematisches Seminar} + %\setbeamercovered{transparent} + \setbeamercovered{invisible} + \setbeamertemplate{navigation symbols}{} + \begin{frame}[plain] + \maketitle + \end{frame} +%------------------------------------------------------------------------------- +\section{Einführung} + \begin{frame} + \frametitle{Reed-Solomon-Code:} + \begin{itemize} + \visible<1->{\item Für Übertragung von Daten} + \visible<2->{\item Ermöglicht Korrektur von Übertragungsfehler} + \visible<3->{\item Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc.} + \end{itemize} + \end{frame} +%------------------------------------------------------------------------------- +\section{Polynom Ansatz} + \begin{frame} + \begin{itemize} + \item Beispiel $2, 1, 5$ versenden und auf 2 Fehler absichern + \end{itemize} + \end{frame} + \begin{frame} + \frametitle{Beispiel} + Übertragen von + ${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ + als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. + + \only<1>{ + Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, + \textcolor{green}{15}, \textcolor{green}{26}, + \textcolor{green}{41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \includegraphics[scale = 1.2]{images/polynom1.pdf}} + \only<2>{ + Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, + \textcolor{red}{50}, \textcolor{red}{37}, + \textcolor{green}{41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \includegraphics[scale = 1.2]{images/polynom2.pdf} + \newline + \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Parameter} + \begin{center} + \begin{tabular}{ c c c } + \hline + Nutzlas & Fehler & Versenden \\ + \hline + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ +\visible<1->{3}& +\visible<1->{3}& +\visible<1->{9 Werte eines Polynoms vom Grad 2} \\ + &&\\ +\visible<1->{$k$} & +\visible<1->{$t$} & +\visible<1->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ + \hline + &&\\ + &&\\ + \multicolumn{3}{l} { + \visible<1>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} + } + \end{tabular} + \end{center} + \end{frame} + +%------------------------------------------------------------------------------- + +\section{Diskrete Fourier Transformation} + \begin{frame} + \frametitle{Idee} + \begin{itemize} + \item Fourier-transformieren + \item Übertragung + \item Rücktransformieren + \end{itemize} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \begin{figure} + \only<1>{ + \includegraphics[width=0.9\linewidth]{images/fig1.pdf} + } + \only<2>{ + \includegraphics[width=0.9\linewidth]{images/fig2.pdf} + } + \only<3>{ + \includegraphics[width=0.9\linewidth]{images/fig3.pdf} + } + \only<4>{ + \includegraphics[width=0.9\linewidth]{images/fig4.pdf} + } + \only<5>{ + \includegraphics[width=0.9\linewidth]{images/fig5.pdf} + } + \only<6>{ + \includegraphics[width=0.9\linewidth]{images/fig6.pdf} + } + \only<7>{ + \includegraphics[width=0.9\linewidth]{images/fig7.pdf} + } + \end{figure} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Diskrete Fourier Transformation} + \begin{itemize} + \item Diskrete Fourier-Transformation gegeben durch: + \visible<1->{ + \[ + \label{ft_discrete} + \hat{c}_{k} + = \frac{1}{N} \sum_{n=0}^{N-1} + {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} + \]} + \visible<2->{ + \item Ersetzte + \[ + w = e^{-\frac{2\pi j}{N} k} + \]} + \visible<3->{ + \item Wenn $N$ konstant: + \[ + \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) + \]} + \end{itemize} + \end{frame} + +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Diskrete Fourier Transformation} + \[ + \begin{pmatrix} + \hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n + \end{pmatrix} + = \frac{1}{N} + \begin{pmatrix} + w^0 & w^0 & w^0 & \dots &w^0 \\ + w^0 & w^1 &w^2 & \dots &w^{N-1} \\ + w^0 & w^2 &w^4 & \dots &w^{2(N-1)} \\ + \vdots & \vdots &\vdots &\ddots &\vdots \\ + w^0 & w^{1(N-1)}&w^{2(N-1)}& \dots &w^{(N-1)(N-1)} \\ + \end{pmatrix} + \begin{pmatrix} + \textcolor{blue}{f_0} \\ + \textcolor{blue}{f_1} \\ + \textcolor{blue}{f_2} \\ + \vdots \\ + 0 \\ + \end{pmatrix} + \] + \end{frame} +%------------------------------------------------------------------------------- + + \begin{frame} + \frametitle{Probleme und Fragen} + + Wie wird der Fehler lokalisiert? + \visible<2>{ + \newline + Indem in einem endlichen Körper gerechnet wird. + } + \end{frame} + +%------------------------------------------------------------------------------- + + +\section{Reed-Solomon in Endlichen Körpern} + + \begin{frame} + \frametitle{Reed-Solomon in Endlichen Körpern} + + \begin{itemize} + \item Warum endliche Körper? + + \qquad konkrete Zahlen $\rightarrow$ keine Rundungsfehler + + \qquad digitale Fehlerkorrektur + + %\onslide<4->{\qquad bessere Laufzeit} + + \vspace{10pt} + + \item Nachricht = Nutzdaten + Fehlerkorrekturteil + + \vspace{10pt} + + \item aus Fehlerkorrekturteil die Fehlerstellen finden + + \qquad $\Rightarrow$ gesucht ist ein Lokatorpolynom + +% \vspace{10pt} + +% \onslide<1->{\item Im Fehlerfall sollen wir aus der Nachricht ein Lokatorpolynom berechnen können, welches die fehlerhaften Stellen beinhaltet} + +% Wir sollten im Fehlerfall in der Lage sein, aus der Nachricht ein Lokatorpolynom zu berechnen, welches die Fehlerhaften Stellen beinhaltet + + \end{itemize} + +% TODO + +% erklärung und einführung der endlichen körper, was wollen wir erreichen? + +% wir versenden im endefekt mehr daten als unsere nachricht umfasst, damit die korrektur sichergestellt werden kann + +% sollten wir fehler bekommen, was uns die korrekturstellen mitgeteilt wird, dann ist es unsere aufgabe ein lokatorpolynom zu finden, welches uns verrät, auf welchen zeilen der Fehler aufgetreten ist + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Definition eines Beispiels} + + \begin{itemize} + + \item endlicher Körper $q = 11$ + + ist eine Primzahl + + beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$ + + \vspace{10pt} + + \item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen + + $n = q - 1 = 10$ Zahlen + + \vspace{10pt} + + \item Max.~Fehler $t = 2$ + + maximale Anzahl von Fehler, die wir noch korrigieren können + + \vspace{10pt} + + \item Nutzlast $k = n -2t = 6$ Zahlen + + Fehlerkorrkturstellen $2t = 4$ Zahlen + + Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$ + + als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Codierung eines Beispiels} + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \item Ansatz aus den komplexen Zahlen mit der diskreten Fouriertransformation + + \vspace{10pt} + + \item Eulersche Zahl $\mathrm{e}$ existiert nicht in $\mathbb{F}_{11}$ + + \vspace{10pt} + + \item Wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken + + $\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}$ + + \vspace{10pt} + + \item Wir wählen $a = 8$ + + $\mathbb{Z}_{11}\setminus\{0\} = \{1,8,9,6,4,10,3,2,5,7\}$ + + $8$ ist eine primitive Einheitswurzel + + \vspace{10pt} + + \item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$ + + $\Rightarrow$ \qquad können wir auch als Matrix schreiben + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \item Übertragungsvektor $v$ + + \item $v = A \cdot m$ + + \end{itemize} + + \[ + v = \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $v = [5,3,6,5,2,10,2,7,10,4]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Decodierung ohne Fehler} + \begin{frame} + \frametitle{Decodierung ohne Fehler} + + \begin{itemize} + \item Der Empfänger erhält den unveränderten Vektor $v = [5,3,6,5,2,10,2,7,10,4]$ + + \vspace{10pt} + + \item Wir suchen die Inverse der Matrix $A$ + + \vspace{10pt} + + \end{itemize} + + \begin{columns}[t] + \begin{column}{0.55\textwidth} + Inverse der Fouriertransformation + \vspace{10pt} + + \[ + F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt + \] + + \vspace{10pt} + + \[ + \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega + \] + + \end{column} + \begin{column}{0.45\textwidth} + Inverse von $a$ + + \vspace{10pt} + + \[ + 8^{1} \Rightarrow 8^{-1} + \] + + Inverse finden wir über den Eulkidischen Algorithmus + \vspace{10pt} + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus} + + \begin{columns}[t] + \begin{column}{0.50\textwidth} + + Recap aus der Vorlesung: + + Gegeben $a \in \mathbb{F}_p$, finde $b = a^{-1} \in \mathbb{F}_p$ + + \begin{tabular}{rcl} + $a b$ &$\equiv$& $1 \mod p$\\ + $a b$ &$=$& $1 + n p$\\ + $a b - n p$ &$=$& $1$\\ + &&\\ + $\operatorname{ggT}(a,p)$&$=$& $1$\\ + $sa + tp$&$=$& $1$\\ + $b$&$=$&$s$\\ + $n$&$=$&$-t$ + \end{tabular} + + \end{column} + \begin{column}{0.50\textwidth} + + \begin{center} + + \begin{tabular}{| c | c c | c | r r |} + \hline + $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ + \hline + & & & & $1$& $0$\\ + $0$& $8$& $11$& $0$& $0$& $1$\\ + $1$& $11$& $8$& $1$& $1$& $0$\\ + $2$& $8$& $3$& $2$& $-1$& $1$\\ + $3$& $3$& $2$& $1$& $3$& $-2$\\ + $4$& $2$& $1$& $2$& \textcolor{blue}{$-4$}& \textcolor{red}{$3$}\\ + $5$& $1$& $0$& & $11$& $-8$\\ + \hline + \end{tabular} + + + \vspace{10pt} + + \begin{tabular}{rcl} + $\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$\\ + $7 \cdot 8 + 3 \cdot 11$ &$=$& $1$\\ + $8^{-1}$ &$=$& $7$ + + \end{tabular} + + \end{center} + + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Decodierung mit Inverser Matrix} + + \begin{itemize} + \item $v = [5,3,6,5,2,10,2,7,10,4]$ + + \item $m = 1/10 \cdot A^{-1} \cdot v$ + + \item $m = 10 \cdot A^{-1} \cdot v$ + + \end{itemize} + + \[ + m = 10 \cdot \begin{pmatrix} + 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ + 7^0& 7^1& 7^2& 7^3& 7^4& 7^5& 7^6& 7^7& 7^8& 7^9\\ + 7^0& 7^2& 7^4& 7^6& 7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\ + 7^0& 7^3& 7^6& 7^9& 7^{12}& 7^{15}& 7^{18}& 7^{21}& 7^{24}& 7^{27}\\ + 7^0& 7^4& 7^8& 7^{12}& 7^{16}& 7^{20}& 7^{24}& 7^{28}& 7^{32}& 7^{36}\\ + 7^0& 7^5& 7^{10}& 7^{15}& 7^{20}& 7^{25}& 7^{30}& 7^{35}& 7^{40}& 7^{45}\\ + 7^0& 7^6& 7^{12}& 7^{18}& 7^{24}& 7^{30}& 7^{36}& 7^{42}& 7^{48}& 7^{54}\\ + 7^0& 7^7& 7^{14}& 7^{21}& 7^{28}& 7^{35}& 7^{42}& 7^{49}& 7^{56}& 7^{63}\\ + 7^0& 7^8& 7^{16}& 7^{24}& 7^{32}& 7^{40}& 7^{48}& 7^{56}& 7^{64}& 7^{72}\\ + 7^0& 7^9& 7^{18}& 7^{27}& 7^{36}& 7^{45}& 7^{54}& 7^{63}& 7^{72}& 7^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $m = [0,0,0,0,4,7,2,5,8,1]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Decodierung mit Fehler} + \begin{frame} + \frametitle{Decodierung mit Fehler - Ansatz} + + \begin{itemize} + \item Gesendet: $v = [5,3,6,5,2,10,2,7,10,4]$ + + \item Empfangen: $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$ + + \item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]$ + + \end{itemize} + + Wie finden wir die Fehler? + + \begin{itemize} + \item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ + + \item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ + + \item $e(X) = r(X) - m(X)$ + + \end{itemize} + + \begin{center} + + \begin{tabular}{c c c c c c c c c c c} + \hline + $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ + \hline + $r(a^{i})$& $5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$\\ + $m(a^{i})$& $5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$\\ + $e(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\ + \hline + \end{tabular} + + \end{center} + + \begin{itemize} + \item Alle Stellen, die nicht Null sind, sind Fehler + \end{itemize} + + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + + \vspace{10pt} + + \item $f(X) = X^{10}-1 = 0$ \qquad für $X \in \{1,2,3,4,5,6,7,8,9,10\}$ + + \vspace{10pt} + + \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + + \vspace{10pt} + + \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + + \vspace{10pt} + + \item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat + + \vspace{10pt} + + $\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + + \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + + \vspace{10pt} + + \item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$ + + \vspace{10pt} + + \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + + \vspace{10pt} + + \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + + \vspace{10pt} + + \item $\operatorname{kgV}$ gibt uns eine Liste von aller Nullstellen, die wir in $e$ und $d$ zerlegen können + + \vspace{10pt} + + $\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot $ + + \qquad \qquad \qquad \qquad $(X-a^7)(X-a^8)(X-a^9) \cdot q(X)$ + + $= d(X) \cdot e(X)$ + + \vspace{10pt} + + \item Lokatorpolynom $d(X) = (X-a^3)(X-a^8)$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Kennen wir $e(X)$?} + + \begin{itemize} + + \item $e(X)$ ist unbekannt auf der Empfängerseite + + \vspace{10pt} + + \item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt? + + \vspace{10pt} + + \item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ + + In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat + + \vspace{10pt} + + \item Daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$ + + \vspace{10pt} + + \item $f(X) = X^{10} - 1 = X^{10} + 10$ + + \vspace{10pt} + + \item Jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus (nochmal)} + + $\operatorname{ggT}(f(X),e(X))$ hat den Grad $8$ + + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + X^{10}& & & & & & &+& 10& & & & &:&5X^9&+&7X^8&+& 4X^7&+&10X^6&+&p(X)&=&9X&+&5\\ + X^{10}&+& 8X^9&+& 3X^8&+&2X^7&+& p(X)& & & & & & & & & & & & & & & & \\ \cline{1-9} + && 3X^9&+& 8X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ + && 3X^9&+& 2X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ \cline{3-9} + & & & &6X^8&+&0X^7&+&p(X)& & & & & & & & & & & & \\ + \end{array} + \] + + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + 5X^9&+& 7X^8&+& 4X^7&+& 10X^6&+& p(X)& & & & &:&6X^8&+&0X^7& & & & & & &=&10X&+&3\\ + 5X^9&+& 0X^8&+& p(X)& & & & & & & & & & & & & & & & & & & & \\ \cline{1-5} + && 7X^8&+& p(X)& & & & & & & & & & & & & & & & \\ + \end{array} + \] + + \vspace{10pt} + + $\operatorname{ggT}(f(X),e(X)) = 6X^8$ + + \vspace{10pt} + + $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen + + \end{frame} + +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Erweiterte Euklidische Algorithmus} + + \begin{center} + + \begin{tabular}{| c | c | c c |} + \hline + $k$ & $q_i$ & $e_i$ & $f_i$\\ + \hline + & & $0$& $1$\\ + $0$& $9X + 5$& $1$& $0$\\ + $1$& $10X + 3$& $9X+5$& $1$\\ + $2$& & \textcolor{blue}{$2X^2 + 0X + 5$}& $10X + 3$\\ + \hline + \end{tabular} + + \end{center} + + \vspace{10pt} + + \begin{tabular}{ll} + Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\ + Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\ + Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$ + \end{tabular} + + \vspace{10pt} + + \begin{center} + $a^i = 5 \qquad \Rightarrow \qquad i = 3$ + + $a^i = 6 \qquad \Rightarrow \qquad i = 8$ + \end{center} + + + $d(X) = (X-a^3)(X-a^8)$ + + \end{frame} +%------------------------------------------------------------------------------- +\section{Nachricht Rekonstruieren} + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \begin{itemize} + + \item $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$ + + \item $d(X) = (X-\textcolor{red}{a^3})(X-\textcolor{red}{a^8})$ + + \end{itemize} + + \[ + \textcolor{gray}{ + \begin{pmatrix} + a^0 \\ a^1 \\ a^2 \\ \textcolor{red}{a^3} \\ a^4 \\ a^5 \\ a^6 \\ a^7 \\ \textcolor{red}{a^8} \\ a^9 \\ + \end{pmatrix}} + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ \textcolor{red}{8} \\ 2 \\ 10 \\ 2 \\ 7 \\ \textcolor{red}{1} \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^3}& \textcolor{red}{8^6}& \textcolor{red}{8^9}& \textcolor{red}{8^{12}}& \textcolor{red}{8^{15}}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{27}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^8}& \textcolor{red}{8^{16}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{32}}& \textcolor{red}{8^{40}}& \textcolor{red}{8^{48}}& \textcolor{red}{8^{56}}& \textcolor{red}{8^{64}}& \textcolor{red}{8^{72}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item Fehlerstellen entfernen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor{green}{8^0}& \textcolor{green}{8^0}& \textcolor{green}{8^0}& \textcolor{green}{8^0}\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor{green}{8^6}& \textcolor{green}{8^7}& \textcolor{green}{8^8}& \textcolor{green}{8^9}\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor{green}{8^{12}}& \textcolor{green}{8^{14}}& \textcolor{green}{8^{16}}& \textcolor{green}{8^{18}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor{green}{8^{24}}& \textcolor{green}{8^{28}}& \textcolor{green}{8^{32}}& \textcolor{green}{8^{36}}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor{green}{8^{30}}& \textcolor{green}{8^{35}}& \textcolor{green}{8^{40}}& \textcolor{green}{8^{45}}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor{green}{8^{36}}& \textcolor{green}{8^{42}}& \textcolor{green}{8^{48}}& \textcolor{green}{8^{54}}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor{green}{8^{42}}& \textcolor{green}{8^{49}}& \textcolor{green}{8^{56}}& \textcolor{green}{8^{63}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor{green}{8^{54}}& \textcolor{green}{8^{63}}& \textcolor{green}{8^{72}}& \textcolor{green}{8^{81}}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor{green}{m_6} \\ \textcolor{green}{m_7} \\ \textcolor{green}{m_8} \\ \textcolor{green}{m_9} \\ + \end{pmatrix} + \] + + \begin{itemize} + \item Nullstellen entfernen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor{red}{7} \\ \textcolor{red}{4} \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^7}& \textcolor{red}{8^{14}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{28}}& \textcolor{red}{8^{35}}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^9}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{27}}& \textcolor{red}{8^{36}}& \textcolor{red}{8^{45}}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \item Matrix in eine Quadratische Form bringen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \item Matrix Invertieren + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 1& 1& 1& 1& 1& 1\\ + 1& 8& 9& 6& 4& 10\\ + 1& 9& 4& 3& 5& 1\\ + 1& 4& 5& 9& 3& 1\\ + 1& 10& 1& 10& 1& 10\\ + 1& 3& 9& 5& 4& 1\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \begin{center} + $\Downarrow$ + \end{center} + \[ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + \] + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $m = [4,7,2,5,8,1]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + +\end{document} -- cgit v1.2.1 From df810d1315cfb1c4b876d5145846d6ea70753141 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Sat, 24 Apr 2021 15:27:05 +0200 Subject: Handout animation deleted --- .../reedsolomon/RS presentation/RS_handout.aux | 143 +++ 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a/buch/papers/reedsolomon/RS presentation/RS_handout.out b/buch/papers/reedsolomon/RS presentation/RS_handout.out new file mode 100644 index 0000000..364319e --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.out @@ -0,0 +1,8 @@ +\BOOKMARK [2][]{Outline0.1}{Einführung}{}% 1 +\BOOKMARK [2][]{Outline0.2}{Polynom\040Ansatz}{}% 2 +\BOOKMARK [2][]{Outline0.3}{Diskrete\040Fourier\040Transformation}{}% 3 +\BOOKMARK [2][]{Outline0.4}{Reed-Solomon in Endlichen Körpern}{}% 4 +\BOOKMARK [2][]{Outline0.5}{Codierung\040eines\040Beispiels}{}% 5 +\BOOKMARK [2][]{Outline0.6}{Decodierung\040ohne\040Fehler}{}% 6 +\BOOKMARK [2][]{Outline0.7}{Decodierung\040mit\040Fehler}{}% 7 +\BOOKMARK [2][]{Outline0.8}{Nachricht\040Rekonstruieren}{}% 8 diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.pdf b/buch/papers/reedsolomon/RS presentation/RS_handout.pdf new file mode 100644 index 0000000..382049d Binary files /dev/null and b/buch/papers/reedsolomon/RS presentation/RS_handout.pdf differ diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.snm b/buch/papers/reedsolomon/RS presentation/RS_handout.snm new file mode 100644 index 0000000..1796304 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.snm @@ -0,0 +1 @@ +\beamer@slide {ft_discrete}{13} diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.synctex.gz b/buch/papers/reedsolomon/RS presentation/RS_handout.synctex.gz new file mode 100644 index 0000000..c28a28a Binary files /dev/null and b/buch/papers/reedsolomon/RS presentation/RS_handout.synctex.gz differ diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.tex b/buch/papers/reedsolomon/RS presentation/RS_handout.tex index 863b3a2..1cbb6ef 100644 --- a/buch/papers/reedsolomon/RS presentation/RS_handout.tex +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.tex @@ -25,38 +25,29 @@ \begin{frame} \frametitle{Reed-Solomon-Code:} \begin{itemize} - \visible<1->{\item Für Übertragung von Daten} - \visible<2->{\item Ermöglicht Korrektur von Übertragungsfehler} - \visible<3->{\item Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc.} + \item Für Übertragung von Daten + \item Ermöglicht Korrektur von Übertragungsfehler + \item Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc. \end{itemize} \end{frame} %------------------------------------------------------------------------------- \section{Polynom Ansatz} \begin{frame} \begin{itemize} - \item Beispiel $2, 1, 5$ versenden und auf 2 Fehler absichern + \item $2, 1, 5$ versenden und auf 2 Fehler absichern \end{itemize} - \end{frame} - \begin{frame} \frametitle{Beispiel} Übertragen von ${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. - - \only<1>{ - Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, - \textcolor{green}{15}, \textcolor{green}{26}, - \textcolor{green}{41}, \textcolor{green}{60}, - \textcolor{green}{83}, \textcolor{green}{110})$ - \includegraphics[scale = 1.2]{images/polynom1.pdf}} - \only<2>{ - Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, - \textcolor{red}{50}, \textcolor{red}{37}, - \textcolor{green}{41}, \textcolor{green}{60}, - \textcolor{green}{83}, \textcolor{green}{110})$ - \includegraphics[scale = 1.2]{images/polynom2.pdf} - \newline - \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} + \newline + Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, + \textcolor{red}{50}, \textcolor{red}{37}, + \textcolor{green}{41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \includegraphics[scale = 1.2]{images/polynom2.pdf} + \newline + \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern. \end{frame} %------------------------------------------------------------------------------- \begin{frame} @@ -68,18 +59,14 @@ \hline 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ -\visible<1->{3}& -\visible<1->{3}& -\visible<1->{9 Werte eines Polynoms vom Grad 2} \\ + 3& 3& 9 Werte eines Polynoms vom Grad 2 \\ &&\\ -\visible<1->{$k$} & -\visible<1->{$t$} & -\visible<1->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ + $k$ & $t$ & $k+2t$ Werte eines Polynoms vom Grad $k-1$ \\ \hline &&\\ &&\\ \multicolumn{3}{l} { - \visible<1>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} + Ausserdem können bis zu $2t$ Fehler erkannt werden! } \end{tabular} \end{center} @@ -127,23 +114,23 @@ \frametitle{Diskrete Fourier Transformation} \begin{itemize} \item Diskrete Fourier-Transformation gegeben durch: - \visible<1->{ + \[ \label{ft_discrete} \hat{c}_{k} = \frac{1}{N} \sum_{n=0}^{N-1} {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} - \]} - \visible<2->{ + \] + \item Ersetzte \[ w = e^{-\frac{2\pi j}{N} k} - \]} - \visible<3->{ + \] + \item Wenn $N$ konstant: \[ \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) - \]} + \] \end{itemize} \end{frame} @@ -177,10 +164,9 @@ \frametitle{Probleme und Fragen} Wie wird der Fehler lokalisiert? - \visible<2>{ \newline Indem in einem endlichen Körper gerechnet wird. - } + \end{frame} %------------------------------------------------------------------------------- diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.toc b/buch/papers/reedsolomon/RS presentation/RS_handout.toc new file mode 100644 index 0000000..ce1bdc2 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.toc @@ -0,0 +1,9 @@ +\babel@toc {ngerman}{} +\beamer@sectionintoc {1}{Einführung}{2}{0}{1} +\beamer@sectionintoc {2}{Polynom Ansatz}{3}{0}{2} +\beamer@sectionintoc {3}{Diskrete Fourier Transformation}{5}{0}{3} +\beamer@sectionintoc {4}{Reed-Solomon in Endlichen Körpern}{16}{0}{4} +\beamer@sectionintoc {5}{Codierung eines Beispiels}{18}{0}{5} +\beamer@sectionintoc {6}{Decodierung ohne Fehler}{20}{0}{6} +\beamer@sectionintoc {7}{Decodierung mit Fehler}{23}{0}{7} +\beamer@sectionintoc {8}{Nachricht Rekonstruieren}{29}{0}{8} -- cgit v1.2.1