From f12bfc8392b2f09416fb2171a4dd0107ebe16722 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Mon, 26 Jul 2021 14:17:05 +0200 Subject: update some files too --- buch/papers/reedsolomon/Makefile | 50 ++++++++++- buch/papers/reedsolomon/dtf.tex | 29 ++++--- buch/papers/reedsolomon/einleitung.tex | 10 +-- buch/papers/reedsolomon/experiments/plot.tex | 2 +- buch/papers/reedsolomon/figures/plotfft.pdf | Bin 0 -> 60217 bytes buch/papers/reedsolomon/figures/polynom2.pdf | Bin 0 -> 20327 bytes buch/papers/reedsolomon/idee.tex | 13 ++- buch/papers/reedsolomon/images/codiert.txt | 96 --------------------- buch/papers/reedsolomon/images/decodiert.txt | 96 --------------------- buch/papers/reedsolomon/images/empfangen.txt | 96 --------------------- buch/papers/reedsolomon/images/fehler.txt | 96 --------------------- buch/papers/reedsolomon/images/locator.txt | 96 --------------------- buch/papers/reedsolomon/images/plotfft.tex | 89 ------------------- buch/papers/reedsolomon/images/polynom2.tex | 49 ----------- buch/papers/reedsolomon/images/signal.txt | 96 --------------------- buch/papers/reedsolomon/images/syndrom.txt | 96 --------------------- buch/papers/reedsolomon/main.tex | 20 ----- buch/papers/reedsolomon/standalone.tex | 30 +++++++ buch/papers/reedsolomon/standalone/standalone.pdf | Bin 0 -> 1782700 bytes buch/papers/reedsolomon/tikz/codiert.txt | 96 +++++++++++++++++++++ buch/papers/reedsolomon/tikz/decodiert.txt | 96 +++++++++++++++++++++ buch/papers/reedsolomon/tikz/empfangen.txt | 96 +++++++++++++++++++++ buch/papers/reedsolomon/tikz/fehler.txt | 96 +++++++++++++++++++++ buch/papers/reedsolomon/tikz/locator.txt | 96 +++++++++++++++++++++ buch/papers/reedsolomon/tikz/plotfft.tex | 99 ++++++++++++++++++++++ buch/papers/reedsolomon/tikz/polynom2.tex | 57 +++++++++++++ buch/papers/reedsolomon/tikz/signal.txt | 96 +++++++++++++++++++++ buch/papers/reedsolomon/tikz/syndrom.txt | 96 +++++++++++++++++++++ 28 files changed, 939 insertions(+), 853 deletions(-) create mode 100644 buch/papers/reedsolomon/figures/plotfft.pdf create mode 100644 buch/papers/reedsolomon/figures/polynom2.pdf delete mode 100644 buch/papers/reedsolomon/images/codiert.txt delete mode 100644 buch/papers/reedsolomon/images/decodiert.txt delete mode 100644 buch/papers/reedsolomon/images/empfangen.txt delete mode 100644 buch/papers/reedsolomon/images/fehler.txt delete mode 100644 buch/papers/reedsolomon/images/locator.txt delete mode 100644 buch/papers/reedsolomon/images/plotfft.tex delete mode 100644 buch/papers/reedsolomon/images/polynom2.tex delete mode 100644 buch/papers/reedsolomon/images/signal.txt delete mode 100644 buch/papers/reedsolomon/images/syndrom.txt create mode 100644 buch/papers/reedsolomon/standalone.tex create mode 100644 buch/papers/reedsolomon/standalone/standalone.pdf create mode 100644 buch/papers/reedsolomon/tikz/codiert.txt create mode 100644 buch/papers/reedsolomon/tikz/decodiert.txt create mode 100644 buch/papers/reedsolomon/tikz/empfangen.txt create mode 100644 buch/papers/reedsolomon/tikz/fehler.txt create mode 100644 buch/papers/reedsolomon/tikz/locator.txt create mode 100644 buch/papers/reedsolomon/tikz/plotfft.tex create mode 100644 buch/papers/reedsolomon/tikz/polynom2.tex create mode 100644 buch/papers/reedsolomon/tikz/signal.txt create mode 100644 buch/papers/reedsolomon/tikz/syndrom.txt (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/Makefile b/buch/papers/reedsolomon/Makefile index 9c96e88..25fd98b 100644 --- a/buch/papers/reedsolomon/Makefile +++ b/buch/papers/reedsolomon/Makefile @@ -4,6 +4,52 @@ # (c) 2020 Prof Dr Andreas Mueller # -images: - @echo "no images to be created in reedsolomon" +SOURCES := \ + anwendungen.tex \ + codebsp.tex \ + decmitfehler.tex \ + decohnefehler.tex \ + dtf.tex \ + einleitung.tex \ + endlichekoerper.tex \ + hilfstabellen.tex \ + idee.tex \ + main.tex \ + packages.tex \ + rekonstruktion.tex \ + restetabelle1.tex \ + restetabelle2.tex \ + standalone.tex \ + zusammenfassung.tex + +TIKZFIGURES := \ + tikz/polynom2.tex \ + tikz/plotfft.tex + +FIGURES := $(patsubst tikz/%.tex, figures/%.pdf, $(TIKZFIGURES)) + + +all: images standalone + + +.PHONY: images +images: $(FIGURES) + +figures/%.pdf: tikz/%.tex + mkdir -p figures + pdflatex --output-directory=figures $< + +.PHONY: standalone +standalone: standalone.tex $(SOURCES) $(FIGURES) + mkdir -p standalone + cd ../..; \ + pdflatex \ + --halt-on-error \ + --shell-escape \ + --output-directory=papers/reedsolomon/standalone \ + papers/reedsolomon/standalone.tex; + cd standalone; \ + bibtex standalone; \ + makeindex standalone; + diff --git a/buch/papers/reedsolomon/dtf.tex b/buch/papers/reedsolomon/dtf.tex index a111527..62e44cc 100644 --- a/buch/papers/reedsolomon/dtf.tex +++ b/buch/papers/reedsolomon/dtf.tex @@ -14,20 +14,27 @@ wobei sie dann bei späteren Berchnungen ganz nützlich ist. \subsection{Diskrete Fourientransformation Zusamenhang \label{reedsolomon:subsection:dtfzusamenhang}} Die Diskrete Fourientransformation ist definiert als - \[ - \label{ft_discrete} +\begin{equation} \hat{c}_{k} = \frac{1}{N} \sum_{n=0}^{N-1} {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} - \] + \label{reedsolomon:DFT} +\end{equation} + , wenn man nun - \[ - w = e^{-\frac{2\pi j}{N} k} - \] +\begin{equation} + w = + e^{-\frac{2\pi j}{N} k} + \label{reedsolomon:DFT_summand} +\end{equation} + ersetzte, und $N$ konstantbleibt, erhält man - \[ - \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) - \] +\begin{equation} + \hat{c}_{k}= + \frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) + \label{reedsolomon:DFT_polynom} +\end{equation} + was überaust ähnlich zu unserem Polynomidee ist. \subsection{Übertragungsabfolge \label{reedsolomon:subsection:Übertragungsabfolge}} @@ -47,8 +54,8 @@ Das heisst alle information ist in alle Zahlenvorhanden. \begin{figure} \centering \resizebox{0.9\textwidth}{!}{ - %\includegraphics[width=0.5\textwidth]{papers/reedsolomon/images/plot.pdf} - \input{papers/reedsolomon/images/plotfft.tex} + \includegraphics[width=\textwidth]{papers/reedsolomon/figures/plotfft} + %\input{papers/reedsolomon/images/plotfft.tex} } \caption{Übertragungsabfolge \ref{reedsolomon:subsection:Übertragungsabfolge}} \label{fig:sendorder} diff --git a/buch/papers/reedsolomon/einleitung.tex b/buch/papers/reedsolomon/einleitung.tex index 2b1d878..074df05 100644 --- a/buch/papers/reedsolomon/einleitung.tex +++ b/buch/papers/reedsolomon/einleitung.tex @@ -7,13 +7,11 @@ \label{reedsolomon:section:einleitung}} \rhead{Einleitung} Der Reed-Solomon-Code ist entstanden um, -das Problem der Fehler, bei der Datenübertragung, zu lösen. -In diesem Abschnitt wird möglichst verständlich die mathematische Abfolge, Funktion oder Algorithmus erklärt. +das Problem der Fehler bei der Datenübertragung, zu lösen. +In diesem Abschnitt wird möglichst verständlich die mathematische Abfolge, +Funktion oder Algorithmus des Reed-Solomon-Code erklärt. Es wird jedoch nicht auf die technische Umsetzung oder Implementierung eingegangen. -Um beim Datenübertragen Fehler zu erkennen, könnte man die Daten jeweils doppelt senden, -und so jeweilige Fehler zu erkennen. -Doch nur schon um weinige Fehler zu erkennen werden überproportional viele Daten doppelt und dreifach gesendet. -Der Reed-Solomon-Code macht dies auf eine andere, clevere Weise. + diff --git a/buch/papers/reedsolomon/experiments/plot.tex b/buch/papers/reedsolomon/experiments/plot.tex index 2196c82..4b156bb 100644 --- a/buch/papers/reedsolomon/experiments/plot.tex +++ b/buch/papers/reedsolomon/experiments/plot.tex @@ -90,7 +90,7 @@ \draw[ultra thick, ->] (zoom) to[out=180, in=90] (syndrom.north); %item - \node[circle, draw, fill =lightgray] at (signal.north west)+(1,0) {1}; + \node[circle, draw, fill =lightgray] at (signal.north west) {1}; \node[circle, draw, fill =lightgray] at (codiert.north west) {2}; \node[circle, draw, fill =lightgray] at (fehler.north west) {3}; \node[circle, draw, fill =lightgray] at (empfangen.north west) {4}; diff --git a/buch/papers/reedsolomon/figures/plotfft.pdf b/buch/papers/reedsolomon/figures/plotfft.pdf new file mode 100644 index 0000000..27992c9 Binary files /dev/null and b/buch/papers/reedsolomon/figures/plotfft.pdf differ diff --git a/buch/papers/reedsolomon/figures/polynom2.pdf b/buch/papers/reedsolomon/figures/polynom2.pdf new file mode 100644 index 0000000..ae68385 Binary files /dev/null and b/buch/papers/reedsolomon/figures/polynom2.pdf differ diff --git a/buch/papers/reedsolomon/idee.tex b/buch/papers/reedsolomon/idee.tex index 39adbbf..e18ccd2 100644 --- a/buch/papers/reedsolomon/idee.tex +++ b/buch/papers/reedsolomon/idee.tex @@ -1,15 +1,22 @@ % -% teil1.tex -- Beispiel-File für das Paper +% idee.tex -- Beispiel-File für das Paper % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % \section{Idee \label{reedsolomon:section:idee}} \rhead{Problemstellung} +Um beim Datenübertragen Fehler zu erkennen, könnte man die Daten jeweils doppelt senden, +und so jeweilige Fehler zu erkennen. +Doch nur schon um Fehler zu erkennen werden überproportional viele Daten doppelt und dreifach gesendet. +Der Reed-Solomon-Code macht dies auf eine andere, clevere Weise. Das Problem liegt darin Informationen, Zahlen, zu Übertragen und Fehler zu erkennen. Beim Reed-Solomon-Code kann man nicht nur Fehler erkennen, man kann sogar einige Fehler korrigieren. +Der unterschied des Fehler erkennen und korrigiren, ist das beim Erkennen nur die Frage kommt hat es Fehler oder keine, +beim korrigieren muss man den Fehler erkennun und dann zusätzlich noch den original Wert rekonstruieren. +Auch eine variante wäre es die Daten nach einem Fehler einfach nochmals zu senden, was bei Reed-Solomon-Code-Anwendungen nicht immer sinnvolll ist. \ref(reedsolomon:section:anwendung) \rhead{Polynom-Ansatz} Eine Idee ist aus den Daten @@ -48,8 +55,8 @@ Dafür sind mehr übertragene Werte nötig. \begin{figure} \centering - %\includegraphics[width=0.5\textwidth]{papers/reedsolomon/images/polynom2} - \input{papers/reedsolomon/images/polynom2.tex} + \includegraphics[width=\textwidth]{papers/reedsolomon/figures/polynom2} + %\input{papers/reedsolomon/images/polynom2.tex} \caption{Polynom $p(x)$ \eqref{reedsolomon:equation1}} \label{fig:polynom} \end{figure} diff --git a/buch/papers/reedsolomon/images/codiert.txt b/buch/papers/reedsolomon/images/codiert.txt deleted file mode 100644 index 4a481d8..0000000 --- a/buch/papers/reedsolomon/images/codiert.txt +++ /dev/null @@ -1,96 +0,0 @@ -0,284 -1,131.570790435043 -2,41.9840308053375 -3,12.1189172092243 -4,23.8408857476069 -5,69.1793197789512 -6,24.0186013379153 -7,37.3066577242559 -8,18.2010889773887 -9,12.3214904922455 -10,15.6627133315015 -11,24.5237955316204 -12,32.1114345314062 -13,44.9845039238714 -14,13.5324640263625 -15,10.1736266929292 -16,4.58257569495584 -17,23.217268502288 -18,16.5769107917917 -19,6.89948680823017 -20,4.84567134895776 -21,10.4219666223433 -22,43.6179140616243 -23,35.9073375743642 -24,15.0332963783729 -25,21.7594021268945 -26,23.2496572716993 -27,17.9815599423852 -28,11.3577742151117 -29,38.467599433197 -30,28.3035029562577 -31,9.54321919833388 -32,21.377558326432 -33,17.6292439561917 -34,12.6951848921471 -35,20.0667752354841 -36,22.9097309529208 -37,8.78894645948548 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-86,15.6627133315015 -87,12.3214904922455 -88,18.2010889773887 -89,37.3066577242559 -90,24.0186013379153 -91,69.1793197789512 -92,23.8408857476069 -93,12.1189172092243 -94,41.9840308053375 -95,131.570790435043 diff --git a/buch/papers/reedsolomon/images/decodiert.txt b/buch/papers/reedsolomon/images/decodiert.txt deleted file mode 100644 index f6221e6..0000000 --- a/buch/papers/reedsolomon/images/decodiert.txt +++ /dev/null @@ -1,96 +0,0 @@ -0,6.05208333333333 -1,6.02602539785853 -2,0.0261327016093151 -3,5.98927158561317 -4,4.019445724874 -5,0.0247005083663722 -6,4.97798278395618 -7,1.95246440445439 -8,0.974000110512201 -9,2.00528527696027 -10,1.00071804528155 -11,1.97630907888264 -12,0.0232923747656228 -13,6.01302820392331 -14,3.03567381915226 -15,5.02435590137329 -16,7.00526061008995 -17,5.00739608089369 -18,5.02211514480064 -19,4.02175864806658 -20,1.00236543833726 -21,4.98147315261261 -22,8.97728828610336 -23,8.98481304394618 -24,2.98958333333333 -25,1.98491220960989 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-0,0 -1,0 -2,0 -3,0 -4,0 -5,0 -6,2 -7,0 -8,0 -9,0 -10,0 -11,0 -12,0 -13,0 -14,0 -15,0 -16,0 -17,0 -18,0 -19,0 -20,2 -21,0 -22,0 -23,0 -24,0 -25,0 -26,0 -27,0 -28,0 -29,0 -30,0 -31,0 -32,0 -33,0 -34,0 -35,0 -36,0 -37,0 -38,0 -39,0 -40,0 -41,0 -42,0 -43,0 -44,0 -45,0 -46,0 -47,0 -48,0 -49,0 -50,0 -51,0 -52,0 -53,0 -54,0 -55,0 -56,0 -57,0 -58,0 -59,0 -60,0 -61,0 -62,0 -63,0 -64,0 -65,0 -66,0 -67,0 -68,0 -69,0 -70,0 -71,0 -72,0 -73,0 -74,1 -75,0 -76,0 -77,0 -78,0 -79,0 -80,0 -81,0 -82,0 -83,0 -84,0 -85,0 -86,0 -87,0 -88,0 -89,0 -90,0 -91,0 -92,0 -93,0 -94,0 -95,0 diff --git a/buch/papers/reedsolomon/images/locator.txt b/buch/papers/reedsolomon/images/locator.txt deleted file mode 100644 index b28988c..0000000 --- a/buch/papers/reedsolomon/images/locator.txt +++ /dev/null @@ -1,96 +0,0 @@ -0,0.0301224340567056 -1,0.141653026854885 -2,0.138226631799377 -3,0.0339903276086929 -4,0.310585462557496 -5,0.551427312631385 -6,0.628514858396814 -7,0.51102386251559 -8,0.275861355940449 -9,0.0502396354182268 -10,0.090185502547573 -11,0.110759344849756 -12,0.0684618905063001 -13,0.0362855426992259 -14,0.0697096919781468 -15,0.109288539370248 -16,0.0923187999496653 -17,0.0512198536768088 -18,0.274192386987782 -19,0.51349614953654 -20,0.633154426602466 -21,0.553283743533942 -22,0.307840573214514 -23,0.0341664350328392 -24,0.140270857957 -25,0.138527177682831 -26,0.029637547736156 -27,0.0816962563186052 -28,0.0944383203811073 -29,0.0263932110686261 -30,0.0585881348402056 -31,0.0737117341599984 -32,0.0239973937701886 -33,0.0464215468420038 -34,0.0616218854220964 -35,0.0221963086695009 -36,0.0390764778127646 -37,0.0537637218396934 -38,0.0208333333333332 -39,0.0343107696069045 -40,0.0483441215964552 -41,0.0198077862118806 -42,0.0311207395968725 -43,0.0444955089373458 -44,0.0190533549944159 -45,0.0290049795038723 -46,0.0417536642697558 -47,0.0185261550443084 -48,0.0277059929762261 -49,0.0398606084144816 -50,0.0181978813094817 -51,0.0271098219177584 -52,0.0386836665079729 -53,0.0180518611046889 -54,0.0272138992557141 -55,0.0381891287148314 -56,0.0180809085252469 -57,0.0281418959420061 -58,0.0384596362516637 -59,0.0182864418432272 -60,0.0302250788423173 -61,0.0397874837986351 -62,0.0186786556701694 -63,0.0342489348284216 -64,0.0429932815348666 -65,0.0192777878591759 -66,0.0422808966931999 -67,0.0506815964680563 -68,0.0201167847752226 -69,0.0615048274405271 -70,0.0744953894508454 -71,0.021246054596492 -72,0.142602265816215 -73,0.273502052865436 -74,0.325309673287599 -75,0.272705389655349 -76,0.149074257381345 -77,0.0247199397628712 -78,0.0680137859566976 -79,0.075388270873485 -80,0.0273637831604903 -81,0.0407867704453274 -82,0.0632964886441949 -83,0.0309749128751093 -84,0.0315202035072035 -85,0.0627625211892184 -86,0.0360843918243497 -87,0.02794920551495 -88,0.0677921493367236 -89,0.0437167157553067 -90,0.0270640150996317 -91,0.0783380025231622 -92,0.0561293738314281 -93,0.0278742033265809 -94,0.0981443889498639 -95,0.0794543457386548 diff --git a/buch/papers/reedsolomon/images/plotfft.tex b/buch/papers/reedsolomon/images/plotfft.tex deleted file mode 100644 index 83a89eb..0000000 --- a/buch/papers/reedsolomon/images/plotfft.tex +++ /dev/null @@ -1,89 +0,0 @@ -% -% Plot der Übertrangungsabfolge ins FFT und zurück mit IFFT -% -\begin{tikzpicture}[] - -%--------------------------------------------------------------- - %Knote -\matrix[draw = none, column sep=25mm, row sep=2mm]{ - \node(signal) [] { - \begin{tikzpicture} - \begin{axis} - [title = {\Large {Signal}}, - xlabel={Anzahl Übertragene Zahlen}, - xtick={0,20,40,64,80,98},] - \addplot[blue] table[col sep=comma] {papers/reedsolomon/images/signal.txt}; - \end{axis} - \end{tikzpicture}}; & - - \node(codiert) [] { - \begin{tikzpicture} - \begin{axis}[title = {\Large {Codiert}}] - \addplot[] table[col sep=comma] {papers/reedsolomon/images/codiert.txt}; - \end{axis} - \end{tikzpicture}}; \\ - - &\node(fehler) [] { - \begin{tikzpicture} - \begin{axis}[scale=0.6, title = {\Large {Fehler}}, - xtick={7,21,75}] - \addplot[red] table[col sep=comma] {papers/reedsolomon/images/fehler.txt}; - \end{axis} - \end{tikzpicture}};\\ - - \node(decodiert) [] { - \begin{tikzpicture} - \begin{axis}[title = {\Large {Decodiert}}] - \addplot[blue] table[col sep=comma] {papers/reedsolomon/images/decodiert.txt}; - \end{axis} - \end{tikzpicture}}; & - - \node(empfangen) [] { - \begin{tikzpicture} - \begin{axis}[title = {\Large {Empfangen}}] - \addplot[] table[col sep=comma] {papers/reedsolomon/images/empfangen.txt}; - \end{axis} - \end{tikzpicture}};\\ - - \node(syndrom) [] { - \begin{tikzpicture} - \begin{axis}[title = {\Large {Syndrom}}] - \addplot[blue] table[col sep=comma] {papers/reedsolomon/images/syndrom.txt}; - \end{axis} - \end{tikzpicture}}; & - - \node(locator) [] { - \begin{tikzpicture} - \begin{axis}[title = {\Large {Locator}}] - \addplot[] table[col sep=comma] {papers/reedsolomon/images/locator.txt}; - \end{axis} - \end{tikzpicture}};\\ -}; -%------------------------------------------------------------- - %FFT & IFFT deskription - -\draw[thin,gray,dashed] (0,12) to (0,-12); -\node(IFFT) [scale=0.7] at (0,12.3) {IFFT}; -\draw[<-](IFFT.south west)--(IFFT.south east); -\node(FFT) [scale=0.7, above of=IFFT] {FFT}; -\draw[->](FFT.north west)--(FFT.north east); - -\draw[thick, ->,] (fehler.west)++(-1,0) +(0.05,0.5) -- +(-0.1,-0.1) -- +(0.1,0.1) -- +(0,-0.5); -%Arrows -\draw[ultra thick, ->] (signal.east) to (codiert.west); -\draw[ultra thick, ->] (codiert.south) to (fehler.north); -\draw[ultra thick, ->] (fehler.south) to (empfangen.north); -\draw[ultra thick, ->] (empfangen.west) to (decodiert.east); -\draw[ultra thick, ->] (syndrom.east) to (locator.west); -\draw(decodiert.south east)++(-1.8,1) ellipse (1.3cm and 0.8cm) ++(-1.3,0) coordinate(zoom) ; -\draw[ultra thick, ->] (zoom) to[out=180, in=90] (syndrom.north); - -%item -\node[circle, draw, fill =lightgray] at (signal.north west) {1}; -\node[circle, draw, fill =lightgray] at (codiert.north west) {2}; -\node[circle, draw, fill =lightgray] at (fehler.north west) {3}; -\node[circle, draw, fill =lightgray] at (empfangen.north west) {4}; -\node[circle, draw, fill =lightgray] at (decodiert.north west) {5}; -\node[circle, draw, fill =lightgray] at (syndrom.north west) {6}; -\node[circle, draw, fill =lightgray] at (locator.north west) {7}; -\end{tikzpicture} \ No newline at end of file diff --git a/buch/papers/reedsolomon/images/polynom2.tex b/buch/papers/reedsolomon/images/polynom2.tex deleted file mode 100644 index 288b51c..0000000 --- a/buch/papers/reedsolomon/images/polynom2.tex +++ /dev/null @@ -1,49 +0,0 @@ -% polynome -%------------------- -% Teiler für das Skalieren der Grafik /40 -\newcommand{\teiler}{40} - - -%////////////////////////////////////// - -\begin{tikzpicture}[>=latex,thick] - \draw[color=blue, line width=1.4pt] - plot[domain=0:8, samples=100] - ({\x},{(2*\x^2+1*\x+5)/\teiler}); - - \draw[->] (-0.2,0) -- (8,0) coordinate[label={$x$}]; - \draw[->] (0,-0.2) -- (0,150/\teiler) coordinate[label={right:$p(x)$}]; - - \def\punkt#1{ - \fill[color=green] #1 circle[radius=0.08]; - \draw #1 circle[radius=0.07]; - } - - \def\hellpunkt#1{ - \fill[color=lightgray] #1 circle[radius=0.08]; - \draw #1 circle[radius=0.07]; - } - - \punkt{(1,8/\teiler)} - \hellpunkt{(2,15/\teiler)} - \hellpunkt{(3,26/\teiler)} - \punkt{(4,41/\teiler)} - \punkt{(5,60/\teiler)} - \punkt{(6,83/\teiler)} - \punkt{(7,110/\teiler)} - - \draw[color=gray,line width=1pt,dashed] - plot[domain=0.5:7, samples=100] - ({\x},{(7.832*\x^2-51.5*\x+121.668)/\teiler}); - - \def\erpunkt#1{ - \fill[color=red] #1 circle[radius=0.08]; - \draw #1 circle[radius=0.07]; - } - \erpunkt{(2,50/\teiler)} - \erpunkt{(3,37.66/\teiler)} - - \draw(0,100/\teiler) -- (-0.1,100/\teiler) coordinate[label={left:$100$}]; - \draw(1,0) -- (1,-0.1) coordinate[label={below:$1$}]; -\end{tikzpicture} -%\end{document} diff --git a/buch/papers/reedsolomon/images/signal.txt b/buch/papers/reedsolomon/images/signal.txt deleted file mode 100644 index c4fa5f8..0000000 --- a/buch/papers/reedsolomon/images/signal.txt +++ /dev/null @@ -1,96 +0,0 @@ -0,6 -1,6 -2,0 -3,6 -4,4 -5,0 -6,5 -7,2 -8,1 -9,2 -10,1 -11,2 -12,0 -13,6 -14,3 -15,5 -16,7 -17,5 -18,5 -19,4 -20,1 -21,5 -22,9 -23,9 -24,3 -25,2 -26,6 -27,6 -28,4 -29,2 -30,9 -31,1 -32,1 -33,1 -34,2 -35,6 -36,6 -37,1 -38,9 -39,7 -40,7 -41,1 -42,9 -43,9 -44,10 -45,9 -46,8 -47,5 -48,2 -49,4 -50,1 -51,0 -52,9 -53,3 -54,3 -55,3 -56,5 -57,6 -58,0 -59,8 -60,6 -61,9 -62,3 -63,4 -64,0 -65,0 -66,0 -67,0 -68,0 -69,0 -70,0 -71,0 -72,0 -73,0 -74,0 -75,0 -76,0 -77,0 -78,0 -79,0 -80,0 -81,0 -82,0 -83,0 -84,0 -85,0 -86,0 -87,0 -88,0 -89,0 -90,0 -91,0 -92,0 -93,0 -94,0 -95,0 diff --git a/buch/papers/reedsolomon/images/syndrom.txt b/buch/papers/reedsolomon/images/syndrom.txt deleted file mode 100644 index 8ca9eed..0000000 --- a/buch/papers/reedsolomon/images/syndrom.txt +++ /dev/null @@ -1,96 +0,0 @@ -0,0 -1,0 -2,0 -3,0 -4,0 -5,0 -6,0 -7,0 -8,0 -9,0 -10,0 -11,0 -12,0 -13,0 -14,0 -15,0 -16,0 -17,0 -18,0 -19,0 -20,0 -21,0 -22,0 -23,0 -24,0 -25,0 -26,0 -27,0 -28,0 -29,0 -30,0 -31,0 -32,0 -33,0 -34,0 -35,0 -36,0 -37,0 -38,0 -39,0 -40,0 -41,0 -42,0 -43,0 -44,0 -45,0 -46,0 -47,0 -48,0 -49,0 -50,0 -51,0 -52,0 -53,0 -54,0 -55,0 -56,0 -57,0 -58,0 -59,0 -60,0 -61,0 -62,0 -63,0 -64,0.0275599094902563 -65,0.0115837187254191 -66,0.025877761014238 -67,0.0224618032819697 -68,0.04410594689944 -69,0.0474504002669341 -70,0.0227694695500626 -71,0.0271436638090525 -72,0.0104166666666667 -73,0.0271436638090523 -74,0.0227694695500608 -75,0.0474504002669343 -76,0.0441059468994397 -77,0.0224618032819701 -78,0.0258777610142379 -79,0.0115837187254183 -80,0.027559909490256 -81,0.0245124379481793 -82,0.0499782237195209 -83,0.0401432022864265 -84,0.0232923747656228 -85,0.0237974288564099 -86,0.0143895905726624 -87,0.0271745729691685 -88,0.0275599094902567 -89,0.0515501672184983 -90,0.0358255004834542 -91,0.024700508366373 -92,0.0210194725405171 -93,0.0177592928994296 -94,0.0261327016093158 -95,0.0314909067039411 diff --git a/buch/papers/reedsolomon/main.tex b/buch/papers/reedsolomon/main.tex index e68b947..327d01a 100644 --- a/buch/papers/reedsolomon/main.tex +++ b/buch/papers/reedsolomon/main.tex @@ -8,29 +8,9 @@ \begin{refsection} \chapterauthor{Joshua Bär und Michael Steiner} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} - % Joshua \input{papers/reedsolomon/einleitung.tex} \input{papers/reedsolomon/idee.tex} -%\input{papers/reedsolomon/teil2.tex} \input{papers/reedsolomon/dtf.tex} % Michael diff --git a/buch/papers/reedsolomon/standalone.tex b/buch/papers/reedsolomon/standalone.tex new file mode 100644 index 0000000..c850d1f --- /dev/null +++ b/buch/papers/reedsolomon/standalone.tex @@ -0,0 +1,30 @@ +\documentclass{book} + +\input{common/packages.tex} + +% additional packages used by the individual papers, add a line for +% each paper +\input{papers/common/addpackages.tex} + +% workaround for biblatex bug +\makeatletter +\def\blx@maxline{77} +\makeatother +\addbibresource{chapters/references.bib} + +% Bibresources for each article +\input{papers/common/addbibresources.tex} + +% make sure the last index starts on an odd page +\AtEndDocument{\clearpage\ifodd\value{page}\else\null\clearpage\fi} +\makeindex + +%\pgfplotsset{compat=1.12} +\setlength{\headheight}{15pt} % fix headheight warning +\DeclareGraphicsRule{*}{mps}{*}{} + +\begin{document} + \input{common/macros.tex} + \def\chapterauthor#1{{\large #1}\bigskip\bigskip} + \input{papers/reedsolomon/main.tex} +\end{document} diff --git a/buch/papers/reedsolomon/standalone/standalone.pdf b/buch/papers/reedsolomon/standalone/standalone.pdf new file mode 100644 index 0000000..80af280 Binary files /dev/null and b/buch/papers/reedsolomon/standalone/standalone.pdf differ diff --git a/buch/papers/reedsolomon/tikz/codiert.txt b/buch/papers/reedsolomon/tikz/codiert.txt new file mode 100644 index 0000000..4a481d8 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/codiert.txt @@ -0,0 +1,96 @@ +0,284 +1,131.570790435043 +2,41.9840308053375 +3,12.1189172092243 +4,23.8408857476069 +5,69.1793197789512 +6,24.0186013379153 +7,37.3066577242559 +8,18.2010889773887 +9,12.3214904922455 +10,15.6627133315015 +11,24.5237955316204 +12,32.1114345314062 +13,44.9845039238714 +14,13.5324640263625 +15,10.1736266929292 +16,4.58257569495584 +17,23.217268502288 +18,16.5769107917917 +19,6.89948680823017 +20,4.84567134895776 +21,10.4219666223433 +22,43.6179140616243 +23,35.9073375743642 +24,15.0332963783729 +25,21.7594021268945 +26,23.2496572716993 +27,17.9815599423852 +28,11.3577742151117 +29,38.467599433197 +30,28.3035029562577 +31,9.54321919833388 +32,21.377558326432 +33,17.6292439561917 +34,12.6951848921471 +35,20.0667752354841 +36,22.9097309529208 +37,8.78894645948548 +38,13.360682005498 +39,25.1757616314718 +40,38.0357773686457 +41,18.4633287776253 +42,19.0584505869806 +43,10.8631093309173 +44,12.6147770818983 +45,12.5398140021274 +46,34.901983501949 +47,22.3480442021702 +48,6 +49,22.3480442021702 +50,34.901983501949 +51,12.5398140021274 +52,12.6147770818983 +53,10.8631093309173 +54,19.0584505869806 +55,18.4633287776253 +56,38.0357773686457 +57,25.1757616314718 +58,13.360682005498 +59,8.78894645948548 +60,22.9097309529208 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+75,0.272705389655349 +76,0.149074257381345 +77,0.0247199397628712 +78,0.0680137859566976 +79,0.075388270873485 +80,0.0273637831604903 +81,0.0407867704453274 +82,0.0632964886441949 +83,0.0309749128751093 +84,0.0315202035072035 +85,0.0627625211892184 +86,0.0360843918243497 +87,0.02794920551495 +88,0.0677921493367236 +89,0.0437167157553067 +90,0.0270640150996317 +91,0.0783380025231622 +92,0.0561293738314281 +93,0.0278742033265809 +94,0.0981443889498639 +95,0.0794543457386548 diff --git a/buch/papers/reedsolomon/tikz/plotfft.tex b/buch/papers/reedsolomon/tikz/plotfft.tex new file mode 100644 index 0000000..3036e14 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/plotfft.tex @@ -0,0 +1,99 @@ +% +% Plot der Übertrangungsabfolge ins FFT und zurück mit IFFT +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{pgfplots} +\usepackage{pgfplotstable} +\usepackage{csvsimple} +\usepackage{filecontents} + + +\begin{document} +\begin{tikzpicture}[] + + %--------------------------------------------------------------- + %Knote + \matrix[draw = none, column sep=25mm, row sep=2mm]{ + \node(signal) [] { + \begin{tikzpicture} + \begin{axis} + [title = {\Large {Signal}}, + xtick={0,20,40,64,80,98},] + \addplot[blue] table[col sep=comma] {tikz/signal.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(codiert) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Codiert}}] + \addplot[] table[col sep=comma] {tikz/codiert.txt}; + \end{axis} + \end{tikzpicture}}; \\ + + &\node(fehler) [] { + \begin{tikzpicture} + \begin{axis}[scale=0.6, title = {\Large {Fehler}}, + xtick={7,21,75}] + \addplot[red] table[col sep=comma] {tikz/fehler.txt}; + \end{axis} + \end{tikzpicture}};\\ + + \node(decodiert) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Decodiert}}] + \addplot[blue] table[col sep=comma] {tikz/decodiert.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(empfangen) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Empfangen}}] + \addplot[] table[col sep=comma] {tikz/empfangen.txt}; + \end{axis} + \end{tikzpicture}};\\ + + \node(syndrom) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Syndrom}}] + \addplot[blue] table[col sep=comma] {tikz/syndrom.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(locator) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Locator}}] + \addplot[] table[col sep=comma] {tikz/locator.txt}; + \end{axis} + \end{tikzpicture}};\\ + }; + %------------------------------------------------------------- + %FFT & IFFT deskription + + \draw[thin,gray,dashed] (0,12) to (0,-12); + \node(IFFT) [scale=0.7] at (0,12.3) {IFFT}; + \draw[<-](IFFT.south west)--(IFFT.south east); + \node(FFT) [scale=0.7, above of=IFFT] {FFT}; + \draw[->](FFT.north west)--(FFT.north east); + + \draw[thick, ->,] (fehler.west)++(-1,0) +(0.05,0.5) -- +(-0.1,-0.1) -- +(0.1,0.1) -- +(0,-0.5); + %Arrows + \draw[ultra thick, ->] (signal.east) to (codiert.west); + \draw[ultra thick, ->] (codiert.south) to (fehler.north); + \draw[ultra thick, ->] (fehler.south) to (empfangen.north); + \draw[ultra thick, ->] (empfangen.west) to (decodiert.east); + \draw[ultra thick, ->] (syndrom.east) to (locator.west); + \draw(decodiert.south east)++(-1.8,1) ellipse (1.3cm and 0.8cm) ++(-1.3,0) coordinate(zoom) ; + \draw[ultra thick, ->] (zoom) to[out=180, in=90] (syndrom.north); + + %item + \node[circle, draw, fill =lightgray] at (signal.north west) {1}; + \node[circle, draw, fill =lightgray] at (codiert.north west) {2}; + \node[circle, draw, fill =lightgray] at (fehler.north west) {3}; + \node[circle, draw, fill =lightgray] at (empfangen.north west) {4}; + \node[circle, draw, fill =lightgray] at (decodiert.north west) {5}; + \node[circle, draw, fill =lightgray] at (syndrom.north west) {6}; + \node[circle, draw, fill =lightgray] at (locator.north west) {7}; +\end{tikzpicture} +\end{document} \ No newline at end of file diff --git a/buch/papers/reedsolomon/tikz/polynom2.tex b/buch/papers/reedsolomon/tikz/polynom2.tex new file mode 100644 index 0000000..456e067 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/polynom2.tex @@ -0,0 +1,57 @@ +% polynome +%------------------- + +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{pgfplots} + + +\begin{document} +% Teiler für das Skalieren der Grafik /40 +\newcommand{\teiler}{40} + + +%////////////////////////////////////// + +\begin{tikzpicture}[>=latex,thick] + \draw[color=blue, line width=1.4pt] + plot[domain=0:8, samples=100] + ({\x},{(2*\x^2+1*\x+5)/\teiler}); + + \draw[->] (-0.2,0) -- (8,0) coordinate[label={$x$}]; + \draw[->] (0,-0.2) -- (0,150/\teiler) coordinate[label={right:$p(x)$}]; + + \def\punkt#1{ + \fill[color=green] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + + \def\hellpunkt#1{ + \fill[color=lightgray] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + + \punkt{(1,8/\teiler)} + \hellpunkt{(2,15/\teiler)} + \hellpunkt{(3,26/\teiler)} + \punkt{(4,41/\teiler)} + \punkt{(5,60/\teiler)} + \punkt{(6,83/\teiler)} + \punkt{(7,110/\teiler)} + + \draw[color=gray,line width=1pt,dashed] + plot[domain=0.5:7, samples=100] + ({\x},{(7.832*\x^2-51.5*\x+121.668)/\teiler}); + + \def\erpunkt#1{ + \fill[color=red] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + \erpunkt{(2,50/\teiler)} + \erpunkt{(3,37.66/\teiler)} + + \draw(0,100/\teiler) -- (-0.1,100/\teiler) coordinate[label={left:$100$}]; + \draw(1,0) -- (1,-0.1) coordinate[label={below:$1$}]; +\end{tikzpicture} +\end{document} diff --git a/buch/papers/reedsolomon/tikz/signal.txt b/buch/papers/reedsolomon/tikz/signal.txt new file mode 100644 index 0000000..c4fa5f8 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/signal.txt @@ -0,0 +1,96 @@ +0,6 +1,6 +2,0 +3,6 +4,4 +5,0 +6,5 +7,2 +8,1 +9,2 +10,1 +11,2 +12,0 +13,6 +14,3 +15,5 +16,7 +17,5 +18,5 +19,4 +20,1 +21,5 +22,9 +23,9 +24,3 +25,2 +26,6 +27,6 +28,4 +29,2 +30,9 +31,1 +32,1 +33,1 +34,2 +35,6 +36,6 +37,1 +38,9 +39,7 +40,7 +41,1 +42,9 +43,9 +44,10 +45,9 +46,8 +47,5 +48,2 +49,4 +50,1 +51,0 +52,9 +53,3 +54,3 +55,3 +56,5 +57,6 +58,0 +59,8 +60,6 +61,9 +62,3 +63,4 +64,0 +65,0 +66,0 +67,0 +68,0 +69,0 +70,0 +71,0 +72,0 +73,0 +74,0 +75,0 +76,0 +77,0 +78,0 +79,0 +80,0 +81,0 +82,0 +83,0 +84,0 +85,0 +86,0 +87,0 +88,0 +89,0 +90,0 +91,0 +92,0 +93,0 +94,0 +95,0 diff --git a/buch/papers/reedsolomon/tikz/syndrom.txt b/buch/papers/reedsolomon/tikz/syndrom.txt new file mode 100644 index 0000000..8ca9eed --- /dev/null +++ b/buch/papers/reedsolomon/tikz/syndrom.txt @@ -0,0 +1,96 @@ +0,0 +1,0 +2,0 +3,0 +4,0 +5,0 +6,0 +7,0 +8,0 +9,0 +10,0 +11,0 +12,0 +13,0 +14,0 +15,0 +16,0 +17,0 +18,0 +19,0 +20,0 +21,0 +22,0 +23,0 +24,0 +25,0 +26,0 +27,0 +28,0 +29,0 +30,0 +31,0 +32,0 +33,0 +34,0 +35,0 +36,0 +37,0 +38,0 +39,0 +40,0 +41,0 +42,0 +43,0 +44,0 +45,0 +46,0 +47,0 +48,0 +49,0 +50,0 +51,0 +52,0 +53,0 +54,0 +55,0 +56,0 +57,0 +58,0 +59,0 +60,0 +61,0 +62,0 +63,0 +64,0.0275599094902563 +65,0.0115837187254191 +66,0.025877761014238 +67,0.0224618032819697 +68,0.04410594689944 +69,0.0474504002669341 +70,0.0227694695500626 +71,0.0271436638090525 +72,0.0104166666666667 +73,0.0271436638090523 +74,0.0227694695500608 +75,0.0474504002669343 +76,0.0441059468994397 +77,0.0224618032819701 +78,0.0258777610142379 +79,0.0115837187254183 +80,0.027559909490256 +81,0.0245124379481793 +82,0.0499782237195209 +83,0.0401432022864265 +84,0.0232923747656228 +85,0.0237974288564099 +86,0.0143895905726624 +87,0.0271745729691685 +88,0.0275599094902567 +89,0.0515501672184983 +90,0.0358255004834542 +91,0.024700508366373 +92,0.0210194725405171 +93,0.0177592928994296 +94,0.0261327016093158 +95,0.0314909067039411 -- cgit v1.2.1 From 88c208363cf560043f87c2c83fa251177e74cd1b Mon Sep 17 00:00:00 2001 From: JODBaer Date: Tue, 27 Jul 2021 13:20:05 +0200 Subject: save --- buch/papers/reedsolomon/dtf.tex | 2 +- buch/papers/reedsolomon/idee.tex | 18 +++++------ buch/papers/reedsolomon/tikz/plotfft.tex | 55 +++++++++++++++----------------- 3 files changed, 35 insertions(+), 40 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/dtf.tex b/buch/papers/reedsolomon/dtf.tex index 62e44cc..ffe98f8 100644 --- a/buch/papers/reedsolomon/dtf.tex +++ b/buch/papers/reedsolomon/dtf.tex @@ -53,7 +53,7 @@ Das heisst alle information ist in alle Zahlenvorhanden. \begin{figure} \centering - \resizebox{0.9\textwidth}{!}{ + \resizebox{\textwidth}{!}{ \includegraphics[width=\textwidth]{papers/reedsolomon/figures/plotfft} %\input{papers/reedsolomon/images/plotfft.tex} } diff --git a/buch/papers/reedsolomon/idee.tex b/buch/papers/reedsolomon/idee.tex index e18ccd2..519e642 100644 --- a/buch/papers/reedsolomon/idee.tex +++ b/buch/papers/reedsolomon/idee.tex @@ -22,7 +22,7 @@ Auch eine variante wäre es die Daten nach einem Fehler einfach nochmals zu send Eine Idee ist aus den Daten ein Polynom zu bilden. Diese Polynomfunktion bei bestimmten Werten, ausrechnet und diese Punkte dann überträgt. -Nehmen wir als beisbiel die Zahlen \textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5}, +Nehmen wir als Beispiel die Zahlen \textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5}, welche uns dann das Polynom \begin{equation} p(x) @@ -31,21 +31,21 @@ p(x) \label{reedsolomon:equation1} \end{equation} ergeben. -Übertragen werden nun die Werte an den stellen 1, 2, 3\dots 7 dieses Polynomes. +Übertragen werden nun die Werte dieses Polynomes an den Stellen 1, 2, 3\dots 7 dieses Polynomes. Grafisch sieht man dies dann in Abbildung \ref{fig:polynom}, -mit den Punkten, $p(1),p(2),...,p(7) = (\textcolor{green}{8}, -\textcolor{green}{15}, \textcolor{green}{26}, -\textcolor{green}{41}, \textcolor{green}{60}, -\textcolor{green}{83}, \textcolor{green}{110})$ -Wenn ein Fehler sich in die Übertragung eingeschlichen hatt, muss der Leser/Empfänger diesen erkennen und das Polynom rekonstruieren. +mit den Punkten, $p(1),p(2),...,p(7) = (\textcolor{darkgreen}{8}, +\textcolor{darkgreen}{15}, \textcolor{darkgreen}{26}, +\textcolor{darkgreen}{41}, \textcolor{darkgreen}{60}, +\textcolor{darkgreen}{83}, \textcolor{darkgreen}{110})$ +Wenn ein Fehler sich in die Übertragung eingeschlichen hat, muss der Leser/Empfänger diesen erkennen und das Polynom rekonstruieren. Der Leser/Empfänger weiss, den Grad des Polynoms und dessen Werte übermittelt wurden. \subsection{Beispiel} -Für das Beispeil aus der Gleichung \eqref{reedsolomon:equation1}, +Für das Beispiel aus der Gleichung \eqref{reedsolomon:equation1}, ist ein Polynome zweiten Grades durch drei Punkte eindeutig bestimmbar. Hat es Fehler in der Übertragunge gegeben,(Bei Abbildung \ref{fig:polynom}\textcolor{red}{roten Punkte}) kann man diese erkennen, da alle Punkte, die korrekt sind, auf dem Polynom liegen müssen. -(Bei Abbildung \ref{fig:polynom}\textcolor{green}{grünen Punkte}) +(Bei Abbildung \ref{fig:polynom}\textcolor{darkgreen}{grünen Punkte}) Ab wie vielen Fehler ist das Polynom nicht mehr erkennbar beim Übertragen von 7 Punkten? Bei 2 Fehlern kann man noch eindeutig bestimmen, dass das Polynom mit 4 Punkten, gegenüber dem mit 5 Punkten falsch liegt.\ref{fig:polynom} diff --git a/buch/papers/reedsolomon/tikz/plotfft.tex b/buch/papers/reedsolomon/tikz/plotfft.tex index 3036e14..db141a8 100644 --- a/buch/papers/reedsolomon/tikz/plotfft.tex +++ b/buch/papers/reedsolomon/tikz/plotfft.tex @@ -15,30 +15,27 @@ %--------------------------------------------------------------- %Knote - \matrix[draw = none, column sep=25mm, row sep=2mm]{ + \matrix(m) [draw = none, column sep=25mm, row sep=2mm]{ + \node(signal) [] { \begin{tikzpicture} \begin{axis} [title = {\Large {Signal}}, - xtick={0,20,40,64,80,98},] - \addplot[blue] table[col sep=comma] {tikz/signal.txt}; + xtick={0,20,40,64,80,98}] + \addplot[black] table[col sep=comma] {tikz/signal.txt}; \end{axis} \end{tikzpicture}}; & \node(codiert) [] { - \begin{tikzpicture} - \begin{axis}[title = {\Large {Codiert}}] - \addplot[] table[col sep=comma] {tikz/codiert.txt}; + \begin{tikzpicture}[] + \begin{axis}[ title = {\Large {Codiert \space + \space Fehler}}, + xtick={0,40,60,100}, axis y line*=left] + \addplot[green] table[col sep=comma] {tikz/codiert.txt}; \end{axis} - \end{tikzpicture}}; \\ - - &\node(fehler) [] { - \begin{tikzpicture} - \begin{axis}[scale=0.6, title = {\Large {Fehler}}, - xtick={7,21,75}] - \addplot[red] table[col sep=comma] {tikz/fehler.txt}; + \begin{axis}[xtick={7,21,75}, axis y line*=right] + \addplot[red] table[col sep=comma] {tikz/fehler.txt}; \end{axis} - \end{tikzpicture}};\\ + \end{tikzpicture}}; \\ \node(decodiert) [] { \begin{tikzpicture} @@ -50,7 +47,7 @@ \node(empfangen) [] { \begin{tikzpicture} \begin{axis}[title = {\Large {Empfangen}}] - \addplot[] table[col sep=comma] {tikz/empfangen.txt}; + \addplot[green] table[col sep=comma] {tikz/empfangen.txt}; \end{axis} \end{tikzpicture}};\\ @@ -71,26 +68,24 @@ %------------------------------------------------------------- %FFT & IFFT deskription - \draw[thin,gray,dashed] (0,12) to (0,-12); - \node(IFFT) [scale=0.7] at (0,12.3) {IFFT}; - \draw[<-](IFFT.south west)--(IFFT.south east); - \node(FFT) [scale=0.7, above of=IFFT] {FFT}; - \draw[->](FFT.north west)--(FFT.north east); + \draw[thin,gray,dashed] (0,9) to (0,-9); + \node(IFFT) [scale=0.8] at (0,9.3) {IFFT}; + \draw[stealth-](IFFT.south west)--(IFFT.south east); + \node(FFT) [scale=0.8, above of=IFFT] {FFT}; + \draw[-stealth](FFT.north west)--(FFT.north east); - \draw[thick, ->,] (fehler.west)++(-1,0) +(0.05,0.5) -- +(-0.1,-0.1) -- +(0.1,0.1) -- +(0,-0.5); + \draw[thick, ->,] (codiert)++(-1,0) +(0.05,0.5) -- +(-0.1,-0.1) -- +(0.1,0.1) -- +(0,-0.5); %Arrows - \draw[ultra thick, ->] (signal.east) to (codiert.west); - \draw[ultra thick, ->] (codiert.south) to (fehler.north); - \draw[ultra thick, ->] (fehler.south) to (empfangen.north); - \draw[ultra thick, ->] (empfangen.west) to (decodiert.east); - \draw[ultra thick, ->] (syndrom.east) to (locator.west); - \draw(decodiert.south east)++(-1.8,1) ellipse (1.3cm and 0.8cm) ++(-1.3,0) coordinate(zoom) ; - \draw[ultra thick, ->] (zoom) to[out=180, in=90] (syndrom.north); + \draw[thick, ->] (signal.east) to (codiert.west); + \draw[thick, ->] (codiert.south) to (empfangen.north); + \draw[thick, ->] (empfangen.west) to (decodiert.east); + \draw[thick, ->] (syndrom.east) to (locator.west); + \draw[thick](decodiert.south east)++(-1.8,1) ellipse (1.3cm and 0.8cm) ++(-1.3,0) coordinate(zoom) ; + \draw[thick, ->] (zoom) to[out=180, in=90] (syndrom.north); %item \node[circle, draw, fill =lightgray] at (signal.north west) {1}; - \node[circle, draw, fill =lightgray] at (codiert.north west) {2}; - \node[circle, draw, fill =lightgray] at (fehler.north west) {3}; + \node[circle, draw, fill =lightgray] at (codiert.north west) {2+3}; \node[circle, draw, fill =lightgray] at (empfangen.north west) {4}; \node[circle, draw, fill =lightgray] at (decodiert.north west) {5}; \node[circle, draw, fill =lightgray] at (syndrom.north west) {6}; -- cgit v1.2.1 From c3c7a6320004974ba56eb98305b5ac9fa13d4a52 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Tue, 27 Jul 2021 17:10:19 +0200 Subject: save --- buch/papers/reedsolomon/dtf.tex | 20 +++-- buch/papers/reedsolomon/experiments/codiert.txt | 96 ---------------------- buch/papers/reedsolomon/experiments/decodiert.txt | 96 ---------------------- buch/papers/reedsolomon/experiments/empfangen.txt | 96 ---------------------- buch/papers/reedsolomon/experiments/fehler.txt | 96 ---------------------- buch/papers/reedsolomon/experiments/locator.txt | 96 ---------------------- buch/papers/reedsolomon/experiments/signal.txt | 96 ---------------------- buch/papers/reedsolomon/experiments/syndrom.txt | 96 ---------------------- buch/papers/reedsolomon/figures/plotfft.pdf | Bin 60217 -> 59617 bytes buch/papers/reedsolomon/figures/polynom2.pdf | Bin 20327 -> 20327 bytes buch/papers/reedsolomon/standalone/standalone.pdf | Bin 1782700 -> 1828186 bytes buch/papers/reedsolomon/tikz/plotfft.tex | 6 +- 12 files changed, 15 insertions(+), 683 deletions(-) delete mode 100644 buch/papers/reedsolomon/experiments/codiert.txt delete mode 100644 buch/papers/reedsolomon/experiments/decodiert.txt delete mode 100644 buch/papers/reedsolomon/experiments/empfangen.txt delete mode 100644 buch/papers/reedsolomon/experiments/fehler.txt delete mode 100644 buch/papers/reedsolomon/experiments/locator.txt delete mode 100644 buch/papers/reedsolomon/experiments/signal.txt delete mode 100644 buch/papers/reedsolomon/experiments/syndrom.txt (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/dtf.tex b/buch/papers/reedsolomon/dtf.tex index ffe98f8..73d0d12 100644 --- a/buch/papers/reedsolomon/dtf.tex +++ b/buch/papers/reedsolomon/dtf.tex @@ -7,21 +7,21 @@ \label{reedsolomon:section:dtf}} \rhead{Umwandlung mit DTF} Um die Polynominterpolation zu umgehen, gehen wir nun über in die Fourientransformation. -Dies wird weder eine erklärung der Forientransorfmation noch ein genauer gebrauch -für den Reed-Solomon-Code. Dieser Abschnitt zeigt nur wie die Fourientransformation auf Fehler reagiert. +Dies wird weder eine erklärung der Forientransorfmation noch ein genauer gebrauchfür den Reed-Solomon-Code. +Dieser Abschnitt zeigt nur wie die Fourientransformation auf Fehler reagiert. wobei sie dann bei späteren Berchnungen ganz nützlich ist. -\subsection{Diskrete Fourientransformation Zusamenhang +\subsection{Diskrete Fourietransformation Zusamenhang \label{reedsolomon:subsection:dtfzusamenhang}} -Die Diskrete Fourientransformation ist definiert als +Die Diskrete Fourietransformation ist definiert als \begin{equation} \hat{c}_{k} = \frac{1}{N} \sum_{n=0}^{N-1} {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} - \label{reedsolomon:DFT} + ,\label{reedsolomon:DFT} \end{equation} -, wenn man nun +wenn man nun \begin{equation} w = e^{-\frac{2\pi j}{N} k} @@ -38,8 +38,12 @@ ersetzte, und $N$ konstantbleibt, erhält man was überaust ähnlich zu unserem Polynomidee ist. \subsection{Übertragungsabfolge \label{reedsolomon:subsection:Übertragungsabfolge}} - -\begin{enumerate}[1)] +Der Auftrag ist nun 64 Daten zu übertragen und nach 16 Fehler abzusicheren, +16 Fehler erkennen und rekonstruieren. +Dieser Auftrag soll mittels Fouriertransformation bewerkstelligt werden. +In der Abbildung \ref{reedsolomon:subsection:Übertragungsabfolge} sieht man dies Schritt für schritt, +und hier werden die einzelne Schritte erklärt. +\begin{enumerate}[(1)] \item Das Signal hat 64 die Daten, Zahlen welche übertragen werden sollen. Dabei zusätzlich nach 16 Fehler abgesichert, macht insgesamt 96 Übertragungszahlen. \item Nun wurde mittels der schnellen diskreten Fourientransformation diese 96 codiert. diff --git a/buch/papers/reedsolomon/experiments/codiert.txt b/buch/papers/reedsolomon/experiments/codiert.txt deleted file mode 100644 index 4a481d8..0000000 --- a/buch/papers/reedsolomon/experiments/codiert.txt +++ /dev/null @@ -1,96 +0,0 @@ -0,284 -1,131.570790435043 -2,41.9840308053375 -3,12.1189172092243 -4,23.8408857476069 -5,69.1793197789512 -6,24.0186013379153 -7,37.3066577242559 -8,18.2010889773887 -9,12.3214904922455 -10,15.6627133315015 -11,24.5237955316204 -12,32.1114345314062 -13,44.9845039238714 -14,13.5324640263625 -15,10.1736266929292 -16,4.58257569495584 -17,23.217268502288 -18,16.5769107917917 -19,6.89948680823017 -20,4.84567134895776 -21,10.4219666223433 -22,43.6179140616243 -23,35.9073375743642 -24,15.0332963783729 -25,21.7594021268945 -26,23.2496572716993 -27,17.9815599423852 -28,11.3577742151117 -29,38.467599433197 -30,28.3035029562577 -31,9.54321919833388 -32,21.377558326432 -33,17.6292439561917 -34,12.6951848921471 -35,20.0667752354841 -36,22.9097309529208 -37,8.78894645948548 -38,13.360682005498 -39,25.1757616314718 -40,38.0357773686457 -41,18.4633287776253 -42,19.0584505869806 -43,10.8631093309173 -44,12.6147770818983 -45,12.5398140021274 -46,34.901983501949 -47,22.3480442021702 -48,6 -49,22.3480442021702 -50,34.901983501949 -51,12.5398140021274 -52,12.6147770818983 -53,10.8631093309173 -54,19.0584505869806 -55,18.4633287776253 -56,38.0357773686457 -57,25.1757616314718 -58,13.360682005498 -59,8.78894645948548 -60,22.9097309529208 -61,20.0667752354841 -62,12.6951848921471 -63,17.6292439561917 -64,21.377558326432 -65,9.54321919833388 -66,28.3035029562577 -67,38.467599433197 -68,11.3577742151117 -69,17.9815599423852 -70,23.2496572716993 -71,21.7594021268945 -72,15.0332963783729 -73,35.9073375743642 -74,43.6179140616243 -75,10.4219666223433 -76,4.84567134895776 -77,6.89948680823017 -78,16.5769107917917 -79,23.217268502288 -80,4.58257569495584 -81,10.1736266929292 -82,13.5324640263625 -83,44.9845039238714 -84,32.1114345314062 -85,24.5237955316204 -86,15.6627133315015 -87,12.3214904922455 -88,18.2010889773887 -89,37.3066577242559 -90,24.0186013379153 -91,69.1793197789512 -92,23.8408857476069 -93,12.1189172092243 -94,41.9840308053375 -95,131.570790435043 diff --git a/buch/papers/reedsolomon/experiments/decodiert.txt b/buch/papers/reedsolomon/experiments/decodiert.txt deleted file mode 100644 index f6221e6..0000000 --- a/buch/papers/reedsolomon/experiments/decodiert.txt +++ /dev/null @@ -1,96 +0,0 @@ -0,6.05208333333333 -1,6.02602539785853 -2,0.0261327016093151 -3,5.98927158561317 -4,4.019445724874 -5,0.0247005083663722 -6,4.97798278395618 -7,1.95246440445439 -8,0.974000110512201 -9,2.00528527696027 -10,1.00071804528155 -11,1.97630907888264 -12,0.0232923747656228 -13,6.01302820392331 -14,3.03567381915226 -15,5.02435590137329 -16,7.00526061008995 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a/buch/papers/reedsolomon/experiments/fehler.txt b/buch/papers/reedsolomon/experiments/fehler.txt deleted file mode 100644 index 23f1a83..0000000 --- a/buch/papers/reedsolomon/experiments/fehler.txt +++ /dev/null @@ -1,96 +0,0 @@ -0,0 -1,0 -2,0 -3,0 -4,0 -5,0 -6,2 -7,0 -8,0 -9,0 -10,0 -11,0 -12,0 -13,0 -14,0 -15,0 -16,0 -17,0 -18,0 -19,0 -20,2 -21,0 -22,0 -23,0 -24,0 -25,0 -26,0 -27,0 -28,0 -29,0 -30,0 -31,0 -32,0 -33,0 -34,0 -35,0 -36,0 -37,0 -38,0 -39,0 -40,0 -41,0 -42,0 -43,0 -44,0 -45,0 -46,0 -47,0 -48,0 -49,0 -50,0 -51,0 -52,0 -53,0 -54,0 -55,0 -56,0 -57,0 -58,0 -59,0 -60,0 -61,0 -62,0 -63,0 -64,0 -65,0 -66,0 -67,0 -68,0 -69,0 -70,0 -71,0 -72,0 -73,0 -74,1 -75,0 -76,0 -77,0 -78,0 -79,0 -80,0 -81,0 -82,0 -83,0 -84,0 -85,0 -86,0 -87,0 -88,0 -89,0 -90,0 -91,0 -92,0 -93,0 -94,0 -95,0 diff --git a/buch/papers/reedsolomon/experiments/locator.txt b/buch/papers/reedsolomon/experiments/locator.txt deleted file mode 100644 index b28988c..0000000 --- a/buch/papers/reedsolomon/experiments/locator.txt +++ /dev/null @@ -1,96 +0,0 @@ -0,0.0301224340567056 -1,0.141653026854885 -2,0.138226631799377 -3,0.0339903276086929 -4,0.310585462557496 -5,0.551427312631385 -6,0.628514858396814 -7,0.51102386251559 -8,0.275861355940449 -9,0.0502396354182268 -10,0.090185502547573 -11,0.110759344849756 -12,0.0684618905063001 -13,0.0362855426992259 -14,0.0697096919781468 -15,0.109288539370248 -16,0.0923187999496653 -17,0.0512198536768088 -18,0.274192386987782 -19,0.51349614953654 -20,0.633154426602466 -21,0.553283743533942 -22,0.307840573214514 -23,0.0341664350328392 -24,0.140270857957 -25,0.138527177682831 -26,0.029637547736156 -27,0.0816962563186052 -28,0.0944383203811073 -29,0.0263932110686261 -30,0.0585881348402056 -31,0.0737117341599984 -32,0.0239973937701886 -33,0.0464215468420038 -34,0.0616218854220964 -35,0.0221963086695009 -36,0.0390764778127646 -37,0.0537637218396934 -38,0.0208333333333332 -39,0.0343107696069045 -40,0.0483441215964552 -41,0.0198077862118806 -42,0.0311207395968725 -43,0.0444955089373458 -44,0.0190533549944159 -45,0.0290049795038723 -46,0.0417536642697558 -47,0.0185261550443084 -48,0.0277059929762261 -49,0.0398606084144816 -50,0.0181978813094817 -51,0.0271098219177584 -52,0.0386836665079729 -53,0.0180518611046889 -54,0.0272138992557141 -55,0.0381891287148314 -56,0.0180809085252469 -57,0.0281418959420061 -58,0.0384596362516637 -59,0.0182864418432272 -60,0.0302250788423173 -61,0.0397874837986351 -62,0.0186786556701694 -63,0.0342489348284216 -64,0.0429932815348666 -65,0.0192777878591759 -66,0.0422808966931999 -67,0.0506815964680563 -68,0.0201167847752226 -69,0.0615048274405271 -70,0.0744953894508454 -71,0.021246054596492 -72,0.142602265816215 -73,0.273502052865436 -74,0.325309673287599 -75,0.272705389655349 -76,0.149074257381345 -77,0.0247199397628712 -78,0.0680137859566976 -79,0.075388270873485 -80,0.0273637831604903 -81,0.0407867704453274 -82,0.0632964886441949 -83,0.0309749128751093 -84,0.0315202035072035 -85,0.0627625211892184 -86,0.0360843918243497 -87,0.02794920551495 -88,0.0677921493367236 -89,0.0437167157553067 -90,0.0270640150996317 -91,0.0783380025231622 -92,0.0561293738314281 -93,0.0278742033265809 -94,0.0981443889498639 -95,0.0794543457386548 diff --git a/buch/papers/reedsolomon/experiments/signal.txt b/buch/papers/reedsolomon/experiments/signal.txt deleted file mode 100644 index c4fa5f8..0000000 --- a/buch/papers/reedsolomon/experiments/signal.txt +++ /dev/null @@ -1,96 +0,0 @@ -0,6 -1,6 -2,0 -3,6 -4,4 -5,0 -6,5 -7,2 -8,1 -9,2 -10,1 -11,2 -12,0 -13,6 -14,3 -15,5 -16,7 -17,5 -18,5 -19,4 -20,1 -21,5 -22,9 -23,9 -24,3 -25,2 -26,6 -27,6 -28,4 -29,2 -30,9 -31,1 -32,1 -33,1 -34,2 -35,6 -36,6 -37,1 -38,9 -39,7 -40,7 -41,1 -42,9 -43,9 -44,10 -45,9 -46,8 -47,5 -48,2 -49,4 -50,1 -51,0 -52,9 -53,3 -54,3 -55,3 -56,5 -57,6 -58,0 -59,8 -60,6 -61,9 -62,3 -63,4 -64,0 -65,0 -66,0 -67,0 -68,0 -69,0 -70,0 -71,0 -72,0 -73,0 -74,0 -75,0 -76,0 -77,0 -78,0 -79,0 -80,0 -81,0 -82,0 -83,0 -84,0 -85,0 -86,0 -87,0 -88,0 -89,0 -90,0 -91,0 -92,0 -93,0 -94,0 -95,0 diff --git a/buch/papers/reedsolomon/experiments/syndrom.txt b/buch/papers/reedsolomon/experiments/syndrom.txt deleted file mode 100644 index 8ca9eed..0000000 --- a/buch/papers/reedsolomon/experiments/syndrom.txt +++ /dev/null @@ -1,96 +0,0 @@ -0,0 -1,0 -2,0 -3,0 -4,0 -5,0 -6,0 -7,0 -8,0 -9,0 -10,0 -11,0 -12,0 -13,0 -14,0 -15,0 -16,0 -17,0 -18,0 -19,0 -20,0 -21,0 -22,0 -23,0 -24,0 -25,0 -26,0 -27,0 -28,0 -29,0 -30,0 -31,0 -32,0 -33,0 -34,0 -35,0 -36,0 -37,0 -38,0 -39,0 -40,0 -41,0 -42,0 -43,0 -44,0 -45,0 -46,0 -47,0 -48,0 -49,0 -50,0 -51,0 -52,0 -53,0 -54,0 -55,0 -56,0 -57,0 -58,0 -59,0 -60,0 -61,0 -62,0 -63,0 -64,0.0275599094902563 -65,0.0115837187254191 -66,0.025877761014238 -67,0.0224618032819697 -68,0.04410594689944 -69,0.0474504002669341 -70,0.0227694695500626 -71,0.0271436638090525 -72,0.0104166666666667 -73,0.0271436638090523 -74,0.0227694695500608 -75,0.0474504002669343 -76,0.0441059468994397 -77,0.0224618032819701 -78,0.0258777610142379 -79,0.0115837187254183 -80,0.027559909490256 -81,0.0245124379481793 -82,0.0499782237195209 -83,0.0401432022864265 -84,0.0232923747656228 -85,0.0237974288564099 -86,0.0143895905726624 -87,0.0271745729691685 -88,0.0275599094902567 -89,0.0515501672184983 -90,0.0358255004834542 -91,0.024700508366373 -92,0.0210194725405171 -93,0.0177592928994296 -94,0.0261327016093158 -95,0.0314909067039411 diff --git a/buch/papers/reedsolomon/figures/plotfft.pdf b/buch/papers/reedsolomon/figures/plotfft.pdf index 27992c9..c5e21e3 100644 Binary files a/buch/papers/reedsolomon/figures/plotfft.pdf and b/buch/papers/reedsolomon/figures/plotfft.pdf differ diff --git a/buch/papers/reedsolomon/figures/polynom2.pdf b/buch/papers/reedsolomon/figures/polynom2.pdf index ae68385..dd6cdd3 100644 Binary files a/buch/papers/reedsolomon/figures/polynom2.pdf and b/buch/papers/reedsolomon/figures/polynom2.pdf differ diff --git a/buch/papers/reedsolomon/standalone/standalone.pdf b/buch/papers/reedsolomon/standalone/standalone.pdf index 80af280..a984f35 100644 Binary files a/buch/papers/reedsolomon/standalone/standalone.pdf and b/buch/papers/reedsolomon/standalone/standalone.pdf differ diff --git a/buch/papers/reedsolomon/tikz/plotfft.tex b/buch/papers/reedsolomon/tikz/plotfft.tex index db141a8..14af683 100644 --- a/buch/papers/reedsolomon/tikz/plotfft.tex +++ b/buch/papers/reedsolomon/tikz/plotfft.tex @@ -22,7 +22,7 @@ \begin{axis} [title = {\Large {Signal}}, xtick={0,20,40,64,80,98}] - \addplot[black] table[col sep=comma] {tikz/signal.txt}; + \addplot[blue] table[col sep=comma] {tikz/signal.txt}; \end{axis} \end{tikzpicture}}; & @@ -54,14 +54,14 @@ \node(syndrom) [] { \begin{tikzpicture} \begin{axis}[title = {\Large {Syndrom}}] - \addplot[blue] table[col sep=comma] {tikz/syndrom.txt}; + \addplot[black] table[col sep=comma] {tikz/syndrom.txt}; \end{axis} \end{tikzpicture}}; & \node(locator) [] { \begin{tikzpicture} \begin{axis}[title = {\Large {Locator}}] - \addplot[] table[col sep=comma] {tikz/locator.txt}; + \addplot[gray] table[col sep=comma] {tikz/locator.txt}; \end{axis} \end{tikzpicture}};\\ }; -- cgit v1.2.1 From 5daff6cc906d9abb2a913569588a0666b4d53b4a Mon Sep 17 00:00:00 2001 From: JODBaer Date: Wed, 28 Jul 2021 17:52:37 +0200 Subject: rewrite some texts --- buch/papers/reedsolomon/dtf.tex | 42 ++++++++----- buch/papers/reedsolomon/figures/polynom2.pdf | Bin 20327 -> 20317 bytes buch/papers/reedsolomon/idee.tex | 73 +++++++++++++--------- buch/papers/reedsolomon/packages.tex | 2 + buch/papers/reedsolomon/standalone/standalone.pdf | Bin 1828186 -> 1835615 bytes buch/papers/reedsolomon/tikz/polynom2.tex | 11 ++-- 6 files changed, 79 insertions(+), 49 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/dtf.tex b/buch/papers/reedsolomon/dtf.tex index 73d0d12..e9aacfb 100644 --- a/buch/papers/reedsolomon/dtf.tex +++ b/buch/papers/reedsolomon/dtf.tex @@ -3,57 +3,65 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Diskrete Fourier Transformation +\section{Übertragung mit hilfe der Diskrete Fourier Transformation \label{reedsolomon:section:dtf}} \rhead{Umwandlung mit DTF} Um die Polynominterpolation zu umgehen, gehen wir nun über in die Fourientransformation. -Dies wird weder eine erklärung der Forientransorfmation noch ein genauer gebrauchfür den Reed-Solomon-Code. +Dies wird weder eine Erklärung der Forientransorfmation, noch ein genauer gebrauch für den Reed-Solomon-Code. Dieser Abschnitt zeigt nur wie die Fourientransformation auf Fehler reagiert. wobei sie dann bei späteren Berchnungen ganz nützlich ist. \subsection{Diskrete Fourietransformation Zusamenhang \label{reedsolomon:subsection:dtfzusamenhang}} -Die Diskrete Fourietransformation ist definiert als +Mit hilfe der Fourietransformation werden die \textcolor{blue}{blauen Datenpunkte} transformiert, +zu den \textcolor{darkgreen}{grünen Übertragungspunkten}. +Durch eine Rücktransformation könnnen die \textcolor{blue}{blauen Datenpunkte} wieder rekonstruiert werden. +Nun zur definition der Diskrete Fourietransformation, diese ist definiert als \begin{equation} \hat{c}_{k} = \frac{1}{N} \sum_{n=0}^{N-1} {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} ,\label{reedsolomon:DFT} \end{equation} - wenn man nun \begin{equation} w = e^{-\frac{2\pi j}{N} k} \label{reedsolomon:DFT_summand} \end{equation} - ersetzte, und $N$ konstantbleibt, erhält man \begin{equation} \hat{c}_{k}= \frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) \label{reedsolomon:DFT_polynom} \end{equation} - was überaust ähnlich zu unserem Polynomidee ist. -\subsection{Übertragungsabfolge + +\subsection{Beispiel \label{reedsolomon:subsection:Übertragungsabfolge}} -Der Auftrag ist nun 64 Daten zu übertragen und nach 16 Fehler abzusicheren, -16 Fehler erkennen und rekonstruieren. +Der Auftrag ist nun 64 Daten zu übertragen und nach 32 Fehler abzusicheren, +16 Fehler erkennen und rekonstruieren. + Dieser Auftrag soll mittels Fouriertransformation bewerkstelligt werden. In der Abbildung \ref{reedsolomon:subsection:Übertragungsabfolge} sieht man dies Schritt für schritt, -und hier werden die einzelne Schritte erklärt. +und hier werden die einzelne Schritte erklärt: \begin{enumerate}[(1)] \item Das Signal hat 64 die Daten, Zahlen welche übertragen werden sollen. Dabei zusätzlich nach 16 Fehler abgesichert, macht insgesamt 96 Übertragungszahlen. -\item Nun wurde mittels der schnellen diskreten Fourientransformation diese 96 codiert. -Das heisst alle information ist in alle Zahlenvorhanden. -\item Nun kommen drei Fehler dazu an den Übertragungsstellen 7, 21 und 75. -\item Dieses wird nun Empfangen und mittels inversen diskreten Fourientransormation, wieder rücktransformiert. -\item Nun sieht man den Fehler im Decodieren in den Übertragungsstellen 64 bis 96. -\item Nimmt man nun nur diese Stellen 64 bis 96, auch Syndrom genannt, und Transformiert diese. -\item Bekommt man die Fehlerstellen im Locator wieder, zwar nichtso genau, dennoch erkkent man wo die Fehler stattgefunden haben. +(siehe Abschnitt \externaldocument{papers/reedsolomon/idee}\ref{reedsolomon:section:Fehlerkorrekturstellen}) +Die 32 Fehlerkorrekturstellen werden als Null Übertragen +\item Nun wurde mittels der diskreten Fourientransformation diese 96 codiert. +Das heisst alle Informationen ist in alle Zahlenvorhanden. (Auch die Fehlerkorrekturstellen Null) +\item Nun kommen drei Fehler dazu an den Übertragungsstellen 7, 21 und 75.(die Skala ist Rechts) +Die Fehler können auf den ganzen 96 Übertragungswerten liegen, wie die 75 zeigt. +\item Dieses wird nun Empfangen und mittels inversen diskreten Fourientransormation, wieder rücktransformiert.(Iklusive der Fehler) +\item Nun sieht man den Fehler im Decodieren in den Übertragungsstellen 64 bis 96, da es dort nicht mehr Null ist. +\item Nimmt man nun nur diese Stellen 64 bis 96, dies definieren wir als Syndrom, und transformiert nur dieses Syndrom. +\item Bekommt man die Fehlerstellen wieder, zwar nichtso genau, dennoch erkennt man wo die Fehler stattgefunden haben. +Dies definieren wir als Locator. \end{enumerate} +Nun haben wir mit Hilfe der Fourietransformation die 3 Fehlerstellen durch das Syndrom lokalisiert, +jetzt gilt es nur noch diese zu korrigieren und wir haben unser originales Signal wieder. \begin{figure} \centering diff --git a/buch/papers/reedsolomon/figures/polynom2.pdf b/buch/papers/reedsolomon/figures/polynom2.pdf index dd6cdd3..55a50ac 100644 Binary files a/buch/papers/reedsolomon/figures/polynom2.pdf and b/buch/papers/reedsolomon/figures/polynom2.pdf differ diff --git a/buch/papers/reedsolomon/idee.tex b/buch/papers/reedsolomon/idee.tex index 519e642..8ad3d27 100644 --- a/buch/papers/reedsolomon/idee.tex +++ b/buch/papers/reedsolomon/idee.tex @@ -1,5 +1,5 @@ % -% idee.tex -- Beispiel-File für das Paper +% idee.tex -- Polynom Idee % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % @@ -14,15 +14,19 @@ Das Problem liegt darin Informationen, Zahlen, zu Übertragen und Fehler zu erkennen. Beim Reed-Solomon-Code kann man nicht nur Fehler erkennen, man kann sogar einige Fehler korrigieren. -Der unterschied des Fehler erkennen und korrigiren, ist das beim Erkennen nur die Frage kommt hat es Fehler oder keine, -beim korrigieren muss man den Fehler erkennun und dann zusätzlich noch den original Wert rekonstruieren. -Auch eine variante wäre es die Daten nach einem Fehler einfach nochmals zu senden, was bei Reed-Solomon-Code-Anwendungen nicht immer sinnvolll ist. \ref(reedsolomon:section:anwendung) +Der unterschied des Fehler erkennen und korrigiren, ist das beim Erkennen nur die Frage beantwortet wird mit: Ist die Übertragung fehlerhaft oder nicht? +Beim Korrigieren werden Fehler erkennt und dann zusätzlich noch den original Wert rekonstruieren. +Auch eine Variante wäre es die Daten nach einem Fehler nachdem Fehlerhaften senden, nochmals versenden(auch hier wieder doppelt und dreifach Sendung), +was bei Reed-Solomon-Code-Anwendungen nicht immer sinnvoll ist. +\externaldocument{papers/reedsolomon/anwendungen} +\ref{reedsolomon:section:anwendung} +\subsection{Polynom-Ansatz +\label{reedsolomon:section:polynomansatz}} \rhead{Polynom-Ansatz} -Eine Idee ist aus den Daten -ein Polynom zu bilden. +Eine Idee ist aus den Daten ein Polynom zu bilden. Diese Polynomfunktion bei bestimmten Werten, ausrechnet und diese Punkte dann überträgt. -Nehmen wir als Beispiel die Zahlen \textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5}, +\begin{beispiel} Nehmen wir die Zahlen \textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5}, welche uns dann das Polynom \begin{equation} p(x) @@ -31,7 +35,8 @@ p(x) \label{reedsolomon:equation1} \end{equation} ergeben. -Übertragen werden nun die Werte dieses Polynomes an den Stellen 1, 2, 3\dots 7 dieses Polynomes. +Übertragen werden nun die \textcolor{darkgreen}{grünen Werte} +dieses \textcolor{blue}{blauen Polynomes} an den Stellen 1, 2, 3\dots 7 dieses Polynomes. Grafisch sieht man dies dann in Abbildung \ref{fig:polynom}, mit den Punkten, $p(1),p(2),...,p(7) = (\textcolor{darkgreen}{8}, \textcolor{darkgreen}{15}, \textcolor{darkgreen}{26}, @@ -39,9 +44,11 @@ mit den Punkten, $p(1),p(2),...,p(7) = (\textcolor{darkgreen}{8}, \textcolor{darkgreen}{83}, \textcolor{darkgreen}{110})$ Wenn ein Fehler sich in die Übertragung eingeschlichen hat, muss der Leser/Empfänger diesen erkennen und das Polynom rekonstruieren. Der Leser/Empfänger weiss, den Grad des Polynoms und dessen Werte übermittelt wurden. +Die Farbe blau brauchen wir für die \textcolor{blue}{Daten} welche wir mit der Farbe grün \textcolor{darkgreen}{Übermitteln}. +\end{beispiel} -\subsection{Beispiel} -Für das Beispiel aus der Gleichung \eqref{reedsolomon:equation1}, +\begin{beispiel} +Aus der Gleichung \eqref{reedsolomon:equation1}, ist ein Polynome zweiten Grades durch drei Punkte eindeutig bestimmbar. Hat es Fehler in der Übertragunge gegeben,(Bei Abbildung \ref{fig:polynom}\textcolor{red}{roten Punkte}) kann man diese erkennen, da alle Punkte, die korrekt sind, auf dem Polynom liegen müssen. @@ -51,29 +58,40 @@ Bei 2 Fehlern kann man noch eindeutig bestimmen, dass das Polynom mit 4 Punkten, gegenüber dem mit 5 Punkten falsch liegt.\ref{fig:polynom} Werden es mehr Fehler kann nur erkennt werden, dass das Polynom nicht stimmt. Das orginale Polynom kann aber nicht mehr gefunden werden. -Dafür sind mehr übertragene Werte nötig. +Da das Konkurenzpolynom, grau gestrichelt in Abbildung \ref{fig:polynom}, das orginal fehlleited. +Um das Konkurenzpolynom auszuschliessen, währen mehr \textcolor{darkgreen}{Übertragungspunkte} nötig. +\end{beispiel} \begin{figure} \centering \includegraphics[width=\textwidth]{papers/reedsolomon/figures/polynom2} - %\input{papers/reedsolomon/images/polynom2.tex} - \caption{Polynom $p(x)$ \eqref{reedsolomon:equation1}} + %\input{papers/reedsolomon/tikz/polynom2.tex} + \caption{Polynom $p(x)$ von der Gleichung\eqref{reedsolomon:equation1}} \label{fig:polynom} \end{figure} -\section{Fehlerbestimmung -\label{reedsolomon:section:Fehlerbestimmmung}} -So wird ein Muster indentifiziert, welches genau vorherbestimmen kann, -wie gross das Polynom sein muss und wie viele Übertragungspunkte gegeben werden müssen. -Um zu bestimmen wie viel Fehler erkennt und korriegiert werden können. -Die Anzahl Zahlen (Daten, ab hier verwenden wir das Wort Nutzlast), -die Entschlüsselt werden sollen, brauchen die gleiche Anzahl an Polynomgraden, beginnend bei Grad 0. ( \( k-1 \) ) -Für die Anzahl an Übertragungspunkte, muss bestimmt werden wieviel Fehler erkennt und korrigiert werden sollen. -Mit Hilfe der Tabelle, sieht man das es bei $t$ Fehlern und $k$ Nutzlast Zahlen, -$k+2t$ Punkte übertragen werden müssen. +\section{Fehlerkorekturstellen bestimmen +\label{reedsolomon:section:Fehlerkorrekturstellen}} +Um zu bestimmen wieviel zusätzliche \textcolor{darkgreen}{Übertragungspunkte} notwendig sind, die dann Fehler korrigieren, +muss man zuerst Wissen wieviel \textcolor{blue}{Daten} gesendet und wieviel \textcolor{red}{Fehler} erkennt werden sollen. +Die Anzahl \textcolor{blue}{Daten} (ab hier verwenden wir das Wort Nutzlast), die als Polynomkoeffizente $k$ übergeben werden, +brauchen die gleiche Anzahl an Polynomgraden, beginnend bei Grad 0 somit ergibt sich der Polynomgrad mit $k-1$. +Für die Anzahl der Fehler $t$, welche korrigiert werden können, gehen wir zum Beispiel. +\begin{beispiel} von den Polynom \ref{reedsolomon:equation1} in, welchem wir 7 \textcolor{darkgreen}{Übertragungspunkte} senden. +Durch 3 Punkte wird das Polyom eindeutig bestimmt, nun haben wir mehrere Konkurenzpolynome, doch mit maximal 2 Fehler liegen auf einem Konkurenzpolynom, +maximal 4 Punkte und auf unserem orginal 5 Punkte. Ansonsten hatt es mehr Fehler oder unser Konkurenzpolynom ist das gleiche wie das Original. +Somit können wir nun bestimmen, dass von den \textcolor{darkgreen}{7 Übertragungspunkten$u$} bis zu 2 Fehler korrigiert werden können und 4 Übertragungspunkte zusätzlich gesendet werden müssen. +\end{beispiel} +Durch das erkennen des Schemas in der Tabelle\ref{tabel:fehlerkorrekturstellen} +\begin{equation} + \frac{\textcolor{darkgreen}{u}-\textcolor{blue}{k}}{\textcolor{red}{t}} + =2 + \label{reedsolomon:equation2} +\end{equation} +zeigt sich das es $k+2t$ Übertragungspunkte braucht. \begin{center} - \begin{tabular}{ c c c } + \begin{tabular}{ c c | c} \hline Nutzlas & Fehler & Übertragen \\ \hline @@ -84,12 +102,11 @@ $k+2t$ Punkte übertragen werden müssen. $k$ & $t$ & $k+2t$ Werte eines Polynoms vom Grad $k-1$ \\ \hline \end{tabular} + Fehlerkorrekturstellen Bestimmung TODO: Tabellenreferenz + \label{tabel:fehlerkorrekturstellen} \end{center} -Ein toller Nebeneffekt ist das dadurch auch $2t$ Fehler erkannt werden. -Um zurück auf unser Beispiel zu kommen, -können von den 7 Übertragungspunkten bis zu $2t = 2\cdot2 = 4 $ Punkten falsch liegen -und es wird kein eindeutiges Polynom zweiten Grades erkannt, und somit die Nutzlast Daten als fehlerhaft deklariert. +Ein Nebeneffekt ist das dadurch auch $2t$ Fehler erkannt werden können, nicht aber korrigiert. Um aus den Übertragenen Zahlen wieder die Nutzlastzahlen zu bekommen könnte man eine Polynominterpolation anwenden, doch die Punkte mit Polynominterpolation zu einem Polynom zu rekonstruieren ist schwierig und Fehleranfällig. diff --git a/buch/papers/reedsolomon/packages.tex b/buch/papers/reedsolomon/packages.tex index b84e228..40c6ea3 100644 --- a/buch/papers/reedsolomon/packages.tex +++ b/buch/papers/reedsolomon/packages.tex @@ -10,3 +10,5 @@ \usepackage{pgfplots} \usepackage{filecontents} +\usepackage{xr} + diff --git a/buch/papers/reedsolomon/standalone/standalone.pdf b/buch/papers/reedsolomon/standalone/standalone.pdf index a984f35..1f2f0b9 100644 Binary files a/buch/papers/reedsolomon/standalone/standalone.pdf and b/buch/papers/reedsolomon/standalone/standalone.pdf differ diff --git a/buch/papers/reedsolomon/tikz/polynom2.tex b/buch/papers/reedsolomon/tikz/polynom2.tex index 456e067..47dc679 100644 --- a/buch/papers/reedsolomon/tikz/polynom2.tex +++ b/buch/papers/reedsolomon/tikz/polynom2.tex @@ -29,9 +29,14 @@ \def\hellpunkt#1{ \fill[color=lightgray] #1 circle[radius=0.08]; - \draw #1 circle[radius=0.07]; + \draw[gray] #1 circle[ radius=0.07]; } + \draw[color=gray,line width=1pt,dashed] + plot[domain=0.5:7, samples=100] + ({\x},{(7.832*\x^2-51.5*\x+121.668)/\teiler}); + + \punkt{(1,8/\teiler)} \hellpunkt{(2,15/\teiler)} \hellpunkt{(3,26/\teiler)} @@ -40,9 +45,7 @@ \punkt{(6,83/\teiler)} \punkt{(7,110/\teiler)} - \draw[color=gray,line width=1pt,dashed] - plot[domain=0.5:7, samples=100] - ({\x},{(7.832*\x^2-51.5*\x+121.668)/\teiler}); + \def\erpunkt#1{ \fill[color=red] #1 circle[radius=0.08]; -- cgit v1.2.1