mirror of https://github.com/2martens/uni.git
RS: RS-Bearbeitungen hinzugefügt
This commit is contained in:
Jim Martens
parent
7d291303dc
commit
9a438bd2ec
|
|
@ -0,0 +1,134 @@
|
|||
\documentclass[10pt,a4paper,oneside,ngerman,numbers=noenddot]{scrartcl}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[ngerman]{babel}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{paralist}
|
||||
\usepackage{qtree}
|
||||
\usepackage[locale=DE,exponent-product=\cdot ,detect-all]{siunitx}
|
||||
\usepackage{tikz}
|
||||
\usepackage[scaled=0.78]{luximono}
|
||||
\usepackage{listings}
|
||||
\usepackage{subfigure}
|
||||
\usetikzlibrary{automata,matrix,fadings,calc,positioning,decorations.pathreplacing,decorations.text,arrows}
|
||||
\pagenumbering{arabic}
|
||||
\def\thesection{10.\arabic{section})}
|
||||
\def\thesubsection{(\alph{subsection})}
|
||||
\def\thesubsubsection{(\arabic{subsubsection})}
|
||||
\renewcommand{\labelenumi}{(\roman{enumi})}
|
||||
\hyphenation{Nach-komma-stel-len}
|
||||
\lstnewenvironment{java}[1][]{%
|
||||
\lstset{basicstyle=\ttfamily ,backgroundcolor=\color[gray]{.95},columns=flexible,fontadjust=true,language=Java,tabsize=4,numbers=none,#1}%
|
||||
}{%
|
||||
}
|
||||
\graphicspath{{D:/Users/Jim-Studium/Pictures/Studium/RS/}}
|
||||
|
||||
\tikzstyle{huffmanNodes}=[matrix of nodes,
|
||||
nodes={circle,thin,draw=black!20,minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\tikzstyle{huffmanBase}=[matrix of nodes,
|
||||
nodes={minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\begin{document}
|
||||
\author{Jim Martens (Matrikelnummer 6420323) \and Marlo Kornblum (Matrikelnummer 6427301)}
|
||||
\title{Rechnerstrukturen Aufgabenblatt 10}
|
||||
\maketitle
|
||||
|
||||
\section{} %10.1
|
||||
\subsection{} %a
|
||||
Der Ampelautomat gibt den Befehl zum Zurücksetzen des Zählers an den Zähler. Der Zähler wiederum gibt den Zählerstand an den Ampelautomat.
|
||||
\subsection{} %b
|
||||
Wenn sich der Hauptautomat im Zustand rot-gelb befindet, dann erfährt er bei der nächsten Taktflanke vom Zähler den aktuellen Stand. Auf diesen kann er aber erst bei der nächsten Vorderflanke reagieren und einen Reset schicken, falls der eingestellte Maximalwert erreicht wurde.
|
||||
Gleichzeitig wird er also bei der zweiten Taktvorderflanke, falls dem so war, in den Zustand grün wechseln und dem Zähler einen neuen Zählauftrag geben. Dieser erfährt davon erst im darauffolgenden Takt und startet dann den Zähler.
|
||||
In jedem Takt gibt der Zähler den Zählerstand an den Hauptautomaten. Dieser kann darauf erst im nachfolgenden Takt reagieren.
|
||||
|
||||
Sind die $30$ Sekunden vorbei, dann wird der Zähler erst im nachfolgenden Takt diese Info weitergeben können und der Hauptautomat erst im darauffolgenden Takt darauf reagieren können und somit grün verlassen.
|
||||
\subsection{} %c
|
||||
Der Zähler arbeitet mit der Vorderflanke und der Hauptautomat mit der Rückflanke.
|
||||
|
||||
Der Hauptautomat befindet sich in der Phase rot-gelb und wartet auf das Erreichen des Maximalwertes vom Zähler. In der Vorderflanke meldet der Zähler den Zählerstand. In der Rückflanke prüft der Hauptautomat diesen auf Erreichen des Maximalwertes. Falls dem so ist, schaltet er auf grün und setzt den Zähler mit einem neuen Maximalwert zurück. Diesen bekommt der Zähler bei der nächsten Vorderflanke.
|
||||
|
||||
Somit liegt der Zähler auch hier einen Takt hinten, da er die Lücke zwischen zwei Takten noch gar nicht mitzählen konnte. Der Hauptautomat erfährt nun, dass der Zähler bei $30$ angelangt ist und schaltet auf gelb. Gleichzeitig wird der Zähler erneut zurückgesetzt.
|
||||
\subsection{} %d
|
||||
\begin{tikzpicture}[shorten >=1pt,node distance=2.0cm,on grid,auto,/tikz/initial text=,>=stealth']
|
||||
\tikzset{every state/.style={minimum size=5.0em}}
|
||||
\node[state,initial] (Z0) {$Z_{0}$};
|
||||
\node[state] (Z1) [below right=2.6 and 3 of Z0] {$Z_{1}$};
|
||||
\node[state] (Z2) [below left=2.6 and 3 of Z1] {$Z_{2}$};
|
||||
\node[state] (Z3) [above left=2.6 and 3 of Z2] {$Z_{3}$};
|
||||
|
||||
\node (z0) [below=0.2 of Z0] {\tiny{$00$}};
|
||||
\node (z1) [below=0.2 of Z1] {\tiny{$01$}};
|
||||
\node (z2) [below=0.2 of Z2] {\tiny{$10$}};
|
||||
\node (z3) [below=0.2 of Z3] {\tiny{$11$}};
|
||||
|
||||
\path[every node/.style={font=\scriptsize}]
|
||||
(Z0) edge[loop above,->] node [near start] {$i<30000$} (Z0)
|
||||
(Z0) edge[->] node [near start] {$i=30000;$ reset i} (Z1)
|
||||
(Z1) edge[loop right,->] node [near start] {$i<3000$} (Z1)
|
||||
(Z1) edge[->] node [near start] {$i=3000;$ reset i} (Z2)
|
||||
(Z2) edge[loop below,->] node [near start] {$i<35000$} (Z2)
|
||||
(Z2) edge[->] node [near start] {$i=35000;$ reset i} (Z3)
|
||||
(Z3) edge[loop left,->] node [near start] {$i<5000$} (Z3)
|
||||
(Z3) edge[->] node [near start] {$i=5000;$ reset i} (Z0);
|
||||
|
||||
\draw (-0.8,-0.3) -- +(1.67,0);
|
||||
\draw (2.2,-2.9) -- +(1.67,0);
|
||||
\draw (-0.80,-5.5) -- +(1.67,0);
|
||||
\draw (-3.83,-2.9) -- +(1.67,0);
|
||||
|
||||
\node (z00) [below=0.45 of Z0] {\small{$r;i++$}};
|
||||
\node (z01) [below=0.45 of Z1] {\small{$r,ge;i++$}};
|
||||
\node (z02) [below=0.45 of Z2] {\small{$gr;i++$}};
|
||||
\node (z03) [below=0.45 of Z3] {\small{$ge;i++$}};
|
||||
\end{tikzpicture}
|
||||
|
||||
\section{} %10.2
|
||||
\subsection{} %a
|
||||
\begin{tabular}{cccc||cc}
|
||||
$z_{1}$ & $z_{0}$ & $x_{1}$ & $x_{0}$ & $z_{1}^{+}$ & $z_{0}^{+}$ \\
|
||||
\hline
|
||||
0 & 0 & * & 1 & 0 & 0 \\
|
||||
0 & 0 & * & 0 & 0 & 1 \\
|
||||
0 & 1 & * & 0 & 0 & 1 \\
|
||||
0 & 1 & 1 & 1 & 1 & 0 \\
|
||||
0 & 1 & 0 & 1 & 1 & 1 \\
|
||||
1 & 0 & 1 & * & 1 & 0 \\
|
||||
1 & 0 & 0 & 0 & 0 & 1 \\
|
||||
1 & 0 & 0 & 1 & 0 & 0 \\
|
||||
1 & 1 & * & 1 & 1 & 1 \\
|
||||
1 & 1 & * & 0 & 1 & 0 \\
|
||||
\end{tabular}
|
||||
|
||||
\begin{alignat*}{2}
|
||||
\delta(Z_{0}, x_{0}) &=& Z_{0}\\
|
||||
\delta(Z_{0}, \overline{x_{0}}) &=& Z_{1}\\
|
||||
\delta(Z_{1}, \overline{x_{1}}, x_{0}) &=& Z_{3}\\
|
||||
\delta(Z_{1}, x_{1}, x_{0}) &=& Z_{2}\\
|
||||
\delta(Z_{1}, \overline{x_{0}}) &=& Z_{1}\\
|
||||
\delta(Z_{2}, \overline{x_{1}}, x_{0}) &=& Z_{0}\\
|
||||
\delta(Z_{2}, \overline{x_{1}}, \overline{x_{0}}) &=& Z_{1}\\
|
||||
\delta(Z_{2}, x_{1}) &=& Z_{2}\\
|
||||
\delta(Z_{3}, x_{0}) &=& Z_{3}\\
|
||||
\delta(Z_{3}, \overline{x_{0}}) &=& Z_{2}
|
||||
\end{alignat*}
|
||||
\subsection{} %b
|
||||
\begin{tabular}{cc||cc}
|
||||
$z_{1}$ & $z_{0}$ & $y_{1}$ & $y_{0}$ \\
|
||||
\hline
|
||||
0 & 0 & 1 & 0 \\
|
||||
0 & 1 & 0 & 1 \\
|
||||
1 & 0 & 0 & 1 \\
|
||||
1 & 1 & 1 & 1 \\
|
||||
\end{tabular}
|
||||
|
||||
\begin{alignat*}{2}
|
||||
\lambda(Z) &=& (y_{1}, y_{0}) \\
|
||||
\lambda(Z_{0}) &=& (1, 0) \\
|
||||
\lambda(Z_{1}) &=& (0, 1) \\
|
||||
\lambda(Z_{2}) &=& (0, 1) \\
|
||||
\lambda(Z_{3}) &=& (1, 1) \\
|
||||
\end{alignat*}
|
||||
\subsection{} %c
|
||||
In allen Zuständen gibt es mindestens einen Übergang für eine Eingangsbelegung und ebenso maximal eine. Demzufolge gibt es genau einen Übergang pro Eingangsbelegung. Unabhängig davon ist es jedoch möglich, dass mehrere Belegungen den gleichen Ausgang benutzen.
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,202 @@
|
|||
\documentclass[10pt,a4paper,oneside,ngerman,numbers=noenddot]{scrartcl}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[ngerman]{babel}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{paralist}
|
||||
\usepackage{qtree}
|
||||
\usepackage[locale=DE,exponent-product=\cdot ,detect-all]{siunitx}
|
||||
\usepackage{tikz}
|
||||
\usepackage[scaled=0.78]{luximono}
|
||||
\usepackage{listings}
|
||||
\usepackage{subfigure}
|
||||
\usetikzlibrary{automata,matrix,fadings,calc,positioning,decorations.pathreplacing,decorations.text,arrows}
|
||||
\pagenumbering{arabic}
|
||||
\def\thesection{11.\arabic{section})}
|
||||
\def\thesubsection{(\alph{subsection})}
|
||||
\def\thesubsubsection{(\arabic{subsubsection})}
|
||||
\renewcommand{\labelenumi}{(\roman{enumi})}
|
||||
\hyphenation{Nach-komma-stel-len}
|
||||
\lstnewenvironment{java}[1][]{%
|
||||
\lstset{basicstyle=\ttfamily ,backgroundcolor=\color[gray]{.95},columns=flexible,fontadjust=true,language=Java,tabsize=4,numbers=none,#1}%
|
||||
}{%
|
||||
}
|
||||
\graphicspath{{D:/Users/Jim-Studium/Pictures/Studium/RS/}}
|
||||
|
||||
\tikzstyle{huffmanNodes}=[matrix of nodes,
|
||||
nodes={circle,thin,draw=black!20,minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\tikzstyle{huffmanBase}=[matrix of nodes,
|
||||
nodes={minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\begin{document}
|
||||
\author{Jim Martens (Matrikelnummer 6420323) \and Marlo Kornblum (Matrikelnummer 6427301)}
|
||||
\title{Rechnerstrukturen Aufgabenblatt 11}
|
||||
\maketitle
|
||||
|
||||
\section{} %11.1
|
||||
\subsection{} %a
|
||||
30
|
||||
\subsection{} %b
|
||||
40
|
||||
\subsection{} %c
|
||||
50
|
||||
\subsection{} %d
|
||||
30
|
||||
\subsection{} %e
|
||||
40
|
||||
\section{} %11.2
|
||||
\begin{figure}[h]
|
||||
7 Befehle mit einer 5-bit Registernummer und einer 24-bit Adresse:\\
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(12,0); %untere Kante
|
||||
\draw (0,0) -- +(0,0.5); %linke Kante
|
||||
\draw (0,0.5) -- +(12,0); %obere Kante
|
||||
\draw (12,0) -- +(0,0.5); %rechte Kante
|
||||
\draw (1.5,0) -- +(0,0.5); %rechte Kante von Opcode
|
||||
\draw (4.0,0) -- +(0,0.5); %rechte Kante der Reg-nr.
|
||||
\node at ++(0.7, 0.2) (opcode) {Opcode}; %Opcode
|
||||
\node at ++(2.2, 0.2) (regNr) {Reg-Nr.}; %Reg-Nr.
|
||||
\node at ++(6.2, 0.2) (address) {Adresse}; %Adresse
|
||||
\node at ++(0.2,-0.4) (31) {$31$};
|
||||
\node at ++(1.3,-0.4) (29) {$29$};
|
||||
\node at ++(1.7,-0.4) (28) {$28$};
|
||||
\node at ++(4.8,-0.4) (24) {$24$};
|
||||
\node at ++(5.2,-0.4) (23) {$23$};
|
||||
\node at ++(11.8,-0.4) (0) {$0$};
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[h]
|
||||
100 Befehle mit zwei 5-bit Registernummern und einem Adressoffset:\\
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(12,0); %untere Kante
|
||||
\draw (0,0) -- +(0,0.5); %linke Kante
|
||||
\draw (0,0.5) -- +(12,0); %obere Kante
|
||||
\draw (12,0) -- +(0,0.5); %rechte Kante
|
||||
\draw (1.5,0) -- +(0,0.5); %rechte Kante von Opcode1
|
||||
\draw (5.0,0) -- +(0,0.5); %rechte Kante von Opcode2
|
||||
\draw (7.5,0) -- +(0,0.5); %rechte Kante von Reg-Nr.
|
||||
\draw (10.0,0) -- +(0,0.5); %rechte Kante von Reg-Nr. 2
|
||||
\node at ++(0.7,0.2) (opcode1) {$111$}; %Opcode1
|
||||
\node at ++(2.2,0.2) (opcode2) {Opcode}; %Opcode2
|
||||
\node at ++(5.7,0.2) (regNr) {Reg-Nr.}; %Reg-Nr.
|
||||
\node at ++(8.4,0.2) (regNr2) {Reg-Nr. 2}; %Reg-Nr. 2
|
||||
\node at ++(10.7,0.2) (offset) {Offset}; %Adressoffset
|
||||
\node at ++(0.2,-0.4) (31) {$31$};
|
||||
\node at ++(1.3,-0.4) (29) {$29$};
|
||||
\node at ++(1.7,-0.4) (28) {$28$};
|
||||
\node at ++(4.8,-0.4) (22) {$22$};
|
||||
\node at ++(5.2,-0.4) (21) {$21$};
|
||||
\node at ++(7.3,-0.4) (17) {$17$};
|
||||
\node at ++(7.7,-0.4) (16) {$16$};
|
||||
\node at ++(9.8,-0.4) (12) {$12$};
|
||||
\node at ++(10.2,-0.4) (11) {$11$};
|
||||
\node at ++(11.8,-0.4) (0) {$0$};
|
||||
\end{tikzpicture}\\
|
||||
Es stehen maximal $12$ Bit für den Adressoffset zur Verfügung.
|
||||
\end{figure}
|
||||
\begin{figure}[h]
|
||||
30 Befehle ohne Adressen oder Registerangaben:\\
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(12,0); %untere Kante
|
||||
\draw (0,0) -- +(0,0.5); %linke Kante
|
||||
\draw (0,0.5) -- +(12,0); %obere Kante
|
||||
\draw (12,0) -- +(0,0.5); %rechte Kante
|
||||
\draw (1.5,0) -- +(0,0.5); %rechte Kante von Opcode1
|
||||
\draw (5.0,0) -- +(0,0.5); %rechte Kante von Opcode2
|
||||
\draw (7.5,0) -- +(0,0.5); %rechte Kante von Opcode3
|
||||
\node at ++(0.7,0.2) (opcode1) {$111$}; %Opcode1
|
||||
\node at ++(2.2,0.2) (opcode2) {$1111111$}; %Opcode2
|
||||
\node at ++(5.7,0.2) (opcode3) {Opcode}; %Opcode3
|
||||
\node at ++(8.5,0.2) (undefined) {unbenutzt}; %unbenutzt
|
||||
\node at ++(0.2,-0.4) (31) {$31$};
|
||||
\node at ++(1.3,-0.4) (29) {$29$};
|
||||
\node at ++(1.7,-0.4) (28) {$28$};
|
||||
\node at ++(4.8,-0.4) (22) {$22$};
|
||||
\node at ++(5.2,-0.4) (21) {$21$};
|
||||
\node at ++(7.3,-0.4) (17) {$17$};
|
||||
\node at ++(7.7,-0.4) (16) {$16$};
|
||||
\node at ++(11.8,-0.4) (0) {$0$};
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
|
||||
Für die ersten sieben Befehle wird ein dreistelliger Opcode benötigt, um sie voneinander unterscheiden zu können. Da 8 Codes maximal möglich sind, aber nur 7 benutzt werden, kann die achte Kombination (111) als Differenzierung zur zweiten Befehlsanzahl fungieren.
|
||||
|
||||
Bei dieser wird ein siebenstelliger Opcode benötigt, um die 100 Befehle voneinander unterscheiden zu können. Ein sechsstelliger Opcode würde nur $64$ Befehle unterstützen.
|
||||
Die 5-bit Registernummern sind in den beiden ersten Fällen vorhanden und verbrauchen beide Male 5-bit.
|
||||
|
||||
Der Adressoffset benutzt schließlich die übrigen unbenutzten Stellen.
|
||||
|
||||
Die 30 restlichen Befehle werden von den vorigen beiden Befehlstypen durch die drei Einsen der ersten Befehle und weiteren 7 Einsen der zweiten Befehle (beide Male sind diese Werte ohne Benutzung) unterschieden. Innerhalb dieser 30 Befehle wird mithilfe eines 5-bit Opcodes zwischen den einzelnen Befehlen unterschieden.
|
||||
\section{} %11.3
|
||||
Wir nehmen an, dass anstatt 12-bit tatsächlich die 8-bit Kodierung gemeint war. Schließlich kann der Immediate-Wert nur 8-bit darstellen. Daher werden wir im Folgenden die Werte in 8-bit Kodierung darstellen.
|
||||
\subsection{} %a
|
||||
$185_{10} = 10111001_{2}, rot = 16$
|
||||
\subsection{} %b
|
||||
$355_{10} = 101100011_{2}$
|
||||
Kann nicht dargestellt werden, da der Immediate-Wert nur 8-bit nutzen kann. Hier werden jedoch 9-bit benutzt.
|
||||
\subsection{} %c
|
||||
$1576_{10} = 11000101_{2}, rot = n.e.$
|
||||
Kann nicht dargestellt werden, da durch Rotierung des Immediate-Wertes nicht die nötige Position erreicht werden kann. Es würde eine ungerade Anzahl an Schiebeoperationen erfordern, was hier nicht möglich ist.
|
||||
\subsection{} %d
|
||||
$1584_{10} = 01100011_{2}, rot = 14$
|
||||
\subsection{} %e
|
||||
$2415919104_{10} = 00001001_{2}, rot = 2$
|
||||
\section{} %11.4
|
||||
\subsection{} %a
|
||||
0-Adress-Maschine:\\
|
||||
PUSH F\\
|
||||
PUSH E\\
|
||||
PUSH D\\
|
||||
MUL\\
|
||||
ADD\\
|
||||
PUSH C\\
|
||||
PUSH B\\
|
||||
MUL\\
|
||||
PUSH A\\
|
||||
SUB\\
|
||||
DIV\\
|
||||
POP R\\
|
||||
\\
|
||||
1-Adress-Maschine:\\
|
||||
LOAD D\\
|
||||
MUL E\\
|
||||
ADD F\\
|
||||
STORE G\\
|
||||
LOAD B\\
|
||||
MUL C\\
|
||||
STORE H\\
|
||||
LOAD A\\
|
||||
SUB H\\
|
||||
DIV G\\
|
||||
STORE R\\
|
||||
\\
|
||||
2-Adress-Maschine:\\
|
||||
MOV G,D\\
|
||||
MUL G,E\\
|
||||
ADD G,F\\
|
||||
MOV H,B\\
|
||||
MUL H,C\\
|
||||
SUB A,H\\
|
||||
DIV A,G\\
|
||||
MOV R,A\\
|
||||
\\
|
||||
3-Adress-Maschine:\\
|
||||
LOAD X,D\\
|
||||
LOAD Y,E\\
|
||||
MUL X,X,Y\\
|
||||
LOAD Y,F\\
|
||||
ADD X,X,Y\\
|
||||
LOAD Y,B\\
|
||||
LOAD Z,C\\
|
||||
MUL Y,Y,Z\\
|
||||
LOAD Z,A\\
|
||||
SUB Z,Z,Y\\
|
||||
DIV Z,Z,X\\
|
||||
STORE R,Z\\
|
||||
\subsection{} %b
|
||||
Die 0-Adress-Maschine verbraucht $212$ Bit. Die 1-Adress-Maschine verbraucht $236$ Bit. Die 2-Adress-Maschine verbraucht $208$ Bit. Die 3-Adress-Maschine verbraucht $220$ Bit.
|
||||
|
||||
Die 2-Adress-Maschine verbraucht am wenigsten Bit.
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,153 @@
|
|||
\documentclass[10pt,a4paper,oneside,ngerman,numbers=noenddot]{scrartcl}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[ngerman]{babel}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{paralist}
|
||||
\usepackage{qtree}
|
||||
\usepackage[locale=DE,exponent-product=\cdot ,detect-all]{siunitx}
|
||||
\usepackage{tikz}
|
||||
\usepackage[scaled=0.78]{luximono}
|
||||
\usepackage{listings}
|
||||
\usepackage{subfigure}
|
||||
\usetikzlibrary{automata,matrix,fadings,calc,positioning,decorations.pathreplacing,decorations.text,arrows}
|
||||
\pagenumbering{arabic}
|
||||
\def\thesection{12.\arabic{section})}
|
||||
\def\thesubsection{(\alph{subsection})}
|
||||
\def\thesubsubsection{(\arabic{subsubsection})}
|
||||
\renewcommand{\labelenumi}{(\roman{enumi})}
|
||||
\hyphenation{Nach-komma-stel-len}
|
||||
\lstnewenvironment{java}[1][]{%
|
||||
\lstset{basicstyle=\ttfamily ,backgroundcolor=\color[gray]{.95},columns=flexible,fontadjust=true,language=Java,tabsize=4,numbers=none,#1}%
|
||||
}{%
|
||||
}
|
||||
\graphicspath{{D:/Users/Jim-Studium/Pictures/Studium/RS/}}
|
||||
|
||||
\tikzstyle{huffmanNodes}=[matrix of nodes,
|
||||
nodes={circle,thin,draw=black!20,minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\tikzstyle{huffmanBase}=[matrix of nodes,
|
||||
nodes={minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\begin{document}
|
||||
\author{Jim Martens (Matrikelnummer 6420323) \and Marlo Kornblum (Matrikelnummer 6427301)}
|
||||
\title{Rechnerstrukturen Aufgabenblatt 12}
|
||||
\maketitle
|
||||
|
||||
\section{} %12.1
|
||||
\subsection{} %a
|
||||
Ziel: $0x100$ \\
|
||||
Resultat in Ziel: $0x0000CB03$
|
||||
\subsection{} %b
|
||||
Ziel: $0x104$ \\
|
||||
Resultat in Ziel: $0x000000BF$
|
||||
\subsection{} %c
|
||||
Ziel: $0x108$ \\
|
||||
Resultat in Ziel: $0x00246000$
|
||||
\subsection{} %d
|
||||
Ziel: $0x10C$ \\
|
||||
Resultat in Ziel: $0x00000043$
|
||||
\subsection{} %e
|
||||
Ziel: $\%ecx$ \\
|
||||
Resultat in Ziel: $0x0000000B$
|
||||
\subsection{} %f
|
||||
Ziel: $\%eax$ \\
|
||||
Resultat in Ziel: $0x000000FC$
|
||||
\section{} %12.2
|
||||
xorl $<Rb>, <Rb>$
|
||||
\section{} %12.3
|
||||
movl: $\%eip, \%eax$
|
||||
\section{} %12.4
|
||||
pushl $\%ebp$\\
|
||||
movl $\%esp, \%ebp$ \\
|
||||
movl $8(\%ebp), \%eax$ \\
|
||||
subl $\$32, \%eax$ \\
|
||||
imull $\$142, \%eax$ \\
|
||||
movl $\%ebp, \%esp$ \\
|
||||
popl $\%ebp$ \\
|
||||
ret
|
||||
\section{} %12.5
|
||||
\subsection{} %a
|
||||
Der Reihenfolge nach speichert das Assemblerprogramm zunächst das Register $\%ebx$ auf den Stack. Anschließend wird vom Register $\%esp$ die dezimale Konstante $24$ abgezogen und der neue Wert in $\%esp$ gespeichert. Nun wird der Inhalt, der ausgehend vom Wert von $\%esp$ plus einem Offset von $32$ im Stack steht, in das Register $\%ebx$ geschrieben. Im nächsten Schritt wird der Inhalt, der ausgehend vom Wert von $\%esp$ plus einem Offset von $36$ im Stack liegt, in das Register $\%edx$ kopiert.
|
||||
|
||||
In das Register $\%eax$ wird schließlich die Konstante $1$ gespeichert. Nun wird das Register $\%edx$ mit sich selbst verglichen. Dabei werden nur die Flags gesetzt, das Ergebnis selber wird jedoch verworfen. Das hier interessante (wegen dem nachfolgenden Jumpbefehl) Zero Flag wird gesetzt, wenn $\%edx$ gleich $0$ ist, ansonsten wird es nicht gesetzt.
|
||||
|
||||
In der darauffolgenden Zeile folgt ein Jumpbefehl, der nur ausgeführt wird, wenn das Zero Flag gesetzt ist. Falls dem so ist, wird zu L2 gesprungen.
|
||||
|
||||
Dort wird zum Inhalt des Registers $\%esp$ die dezimale Konstante $24$ hinzugezählt. Anschließend wird das oberste Element des Stacks in das Register $\%ebx$ gespeichert und der aktuelle Aufruf beendet.
|
||||
|
||||
Wenn nicht gesprungen wird, dann wird dieses hier ausgeführt:
|
||||
|
||||
Nun wird $1$ vom Inhalt von $\%edx$ subtrahiert. Anschließend wird der Inhalt von $\%edx$ an die Stelle geschrieben, die in $\%esp$ plus dem Offset von $4$ referenziert ist. Dann wird der Inhalt von $\%ebx$ an die Stelle geschrieben, die in $\%esp$ referenziert ist.
|
||||
|
||||
Jetzt ruft sich myst rekursiv erneut auf. Sobald bei einem Aufruf von myst das Register $\%edx$ gleich $0$ ist, wird der Aufruf beendet und abschließend in $\%eax$ das Produkt der Werte von $\%eax$ und $\%ebx$ geschrieben.
|
||||
|
||||
Damit wird das Unterprogramm beendet.
|
||||
\subsection{} %b
|
||||
Das Unterprogramm berechnet $5^{2}=25$. Allgemeiner wird $a^{b}$ berechnet.
|
||||
\subsection{} %c
|
||||
\begin{tikzpicture}
|
||||
\draw (0, 0) -- +(2, 0);
|
||||
\draw (0, 0.5) -- +(2, 0);
|
||||
\draw (0, 1) -- +(2, 0);
|
||||
\draw (0, 1.5) -- +(2, 0);
|
||||
\draw (0, 2) -- +(2, 0);
|
||||
\draw (0, 2.5) -- +(2, 0);
|
||||
\draw (0, 3) -- +(2, 0);
|
||||
\draw (0, 3.5) -- +(2, 0);
|
||||
\draw (0, 4) -- +(2, 0);
|
||||
\draw (0, 4.5) -- +(2, 0);
|
||||
\draw (0, 5) -- +(2, 0);
|
||||
\draw (0, 5.5) -- +(2, 0);
|
||||
\draw (0, 6) -- +(2, 0);
|
||||
\draw (0, 6.5) -- +(2, 0);
|
||||
\draw (0, 7) -- +(2, 0);
|
||||
\draw (0, 7.5) -- +(2, 0);
|
||||
\draw (0, 8) -- +(2, 0);
|
||||
\draw (0, 8.5) -- +(2, 0);
|
||||
\draw (0, 9) -- +(2, 0);
|
||||
\draw (0, 9.5) -- +(2, 0);
|
||||
\draw (0, 10) -- +(2, 0);
|
||||
\draw (0, 10.5) -- +(2, 0);
|
||||
\draw (0, 11) -- +(2, 0);
|
||||
\draw (0, 11.5) -- +(2, 0);
|
||||
\draw (0, 12) -- +(2, 0);
|
||||
\draw (0, 12.5) -- +(2, 0);
|
||||
\draw (0, 13) -- +(2, 0);
|
||||
%\draw (0, 13.5) -- +(2, 0);
|
||||
\draw (0,0) -- +(0, 13.5);
|
||||
\draw (2,0) -- +(0, 13.5);
|
||||
|
||||
\node (other) at (1, 13.25) {...};
|
||||
\node (b1) at (1, 12.75) {b = 2};
|
||||
\node (a1) at (1, 12.25) {a = 5};
|
||||
\node (main) at (1, 11.75) {$\%eip$ main()};
|
||||
\node (ebx1) at (1, 11.25) {$\%ebx \rightarrow ?$};
|
||||
\node (other2) at (1, 10.75) {...};
|
||||
\node (other3) at (1, 10.25) {...};
|
||||
\node (other4) at (1, 9.75) {...};
|
||||
\node (other5) at (1, 9.25) {...};
|
||||
\node (b2) at (1, 8.75) {b = 1};
|
||||
\node (a2) at (1, 8.25) {a = 5};
|
||||
\node (myst1) at (1, 7.75) {$\%eip$ myst()};
|
||||
\node (ebx2) at (1, 7.25) {$\%ebx \rightarrow 5$};
|
||||
\node (other6) at (1, 6.75) {...};
|
||||
\node (other7) at (1, 6.25) {...};
|
||||
\node (other8) at (1, 5.75) {...};
|
||||
\node (other9) at (1, 5.25) {...};
|
||||
\node (b3) at (1, 4.75) {b = 0};
|
||||
\node (a3) at (1, 4.25) {a = 5};
|
||||
\node (myst2) at (1, 3.75) {$\%eip$ myst()};
|
||||
\node (ebx3) at (1, 3.25) {$\%ebx \rightarrow 5$};
|
||||
\node (other10) at (1, 2.75) {...};
|
||||
\node (other11) at (1, 2.25) {...};
|
||||
\node (other12) at (1, 1.75) {...};
|
||||
\node (other13) at (1, 1.25) {...};
|
||||
\node (other14) at (1, 0.75) {...};
|
||||
\node (other15) at (1, 0.25) {...};
|
||||
\node (esp) at (-0.75, 0.25) {$\%esp \rightarrow$};
|
||||
\end{tikzpicture}
|
||||
|
||||
\subsection{} %d
|
||||
Es würde eine Endlosschleife entstehen, da der zweite Parameter nie $0$ wird und damit nie die Sprungmarke L2 erreicht wird, wodurch nie ein Aufruf beendet würde und unendlich viele Rekursionsschritte durchgeführt würden.
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,144 @@
|
|||
\documentclass[10pt,a4paper,oneside,ngerman,numbers=noenddot]{scrartcl}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[ngerman]{babel}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{paralist}
|
||||
\usepackage[locale=DE,exponent-product=\cdot,detect-all]{siunitx}
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{matrix,fadings,calc,positioning,decorations.pathreplacing,arrows}
|
||||
\pagenumbering{arabic}
|
||||
\def\thesection{3.\arabic{section})}
|
||||
\def\thesubsection{(\alph{subsection})}
|
||||
\def\thesubsubsection{(\arabic{subsubsection})}
|
||||
\renewcommand{\labelenumi}{(\roman{enumi})}
|
||||
\hyphenation{Nach-komma-stel-len}
|
||||
|
||||
\begin{document}
|
||||
\author{Jim Martens (Matrikelnummer 6420323) \and Marlo Kornblum (Matrikelnummer 6427301)}
|
||||
\title{Rechnerstrukturen Aufgabenblatt 3}
|
||||
\maketitle
|
||||
|
||||
\section{}%3.1
|
||||
\subsection{}
|
||||
\begin{alignat}{3}
|
||||
&\; 1385_{10} &-&\: 532_{10} &=&\: x\\
|
||||
\intertext{Bildung des 10-Komplements des Subtrahenden:}
|
||||
\Leftrightarrow &\; 1385_{10} &+&\: K_{10}(532)_{10} &=&\: x\\
|
||||
\Leftrightarrow &\; 1385_{10} &+&\: 9467_{10} &=&\: 10852_{10}\\
|
||||
\intertext{Streichen der führenden Eins}
|
||||
\Leftrightarrow &\; 1385_{10} &-&\: 532_{10} &=&\: \underline{\underline{0852_{10}}}
|
||||
\end{alignat}
|
||||
\subsection{}
|
||||
\begin{alignat}{3}
|
||||
&\; 372_{10} &-&\: 687_{10} &=&\: x\\
|
||||
\intertext{Bildung des 10-Komplements des Subtrahenden:}
|
||||
\Leftrightarrow &\; 372_{10} &+&\: K_{10}(0687)_{10} &=&\: x\\
|
||||
\Leftrightarrow &\; 372_{10} &+&\: 9313_{10} &=&\: 9685_{10}\\
|
||||
\intertext{Bildung des 10-Komplements des Ergebnisses}
|
||||
\Leftrightarrow &\; K_{10}(9685)_{10} &=&\: \underline{\underline{-0315_{10}}}
|
||||
\end{alignat}
|
||||
\subsection{}
|
||||
\begin{alignat}{3}
|
||||
&\; 1385_{10} &-&\: 532_{10} &=&\: x\\
|
||||
\intertext{Umwandlung in Dualzahlen:}
|
||||
&\; 10101101001_{2} &-&\: 1000010100_{2} &=&\: x\\
|
||||
\intertext{Bildung des 2-Komplements des Subtrahenden:}
|
||||
\Leftrightarrow &\; 10101101001_{2} &+&\: K_{2}(001000010100)_{2} &=&\: x\\
|
||||
\Leftrightarrow &\; 10101101001_{2} &+&\: 110111101100_{2} &=&\: 101101010101_{2}\\
|
||||
\intertext{Streichen der führenden Eins}
|
||||
\Leftrightarrow &\; 10101101001_{2} &-&\: 001000010100_{2} &=&\: \underline{\underline{001101010101_{2}}}
|
||||
\end{alignat}
|
||||
\subsection{}
|
||||
\begin{alignat}{3}
|
||||
&\; 372_{10} &-&\: 687_{10} &=&\: x\\
|
||||
\intertext{Umwandlung in Dualzahlen:}
|
||||
&\; 101110100_{2} &-&\: 1010101111_{2} &=&\: x\\
|
||||
\intertext{Bildung des 2-Komplements:}
|
||||
\Leftrightarrow &\; 101110100_{2} &+&\: K_{2}(001010101111)_{2} &=&\: x\\
|
||||
\Leftrightarrow &\; 000101110100_{2} &+&\: 110101010001_{2} &=&\: 111011000101_{2}\\
|
||||
\intertext{Bildung des 2-Komplements des Ergebnisses}
|
||||
\Leftrightarrow &\; K_{2}(111011000101)_{2} &=&\: \underline{\underline{000100111011_{2}}}
|
||||
\end{alignat}
|
||||
\section{}%3.2 _todo
|
||||
\subsection{}
|
||||
$(6,9242 \mid 4)_{10}$
|
||||
\subsection{}
|
||||
$(-1,100101 \mid -10)_{2}$
|
||||
\subsection{}
|
||||
$(-2,D4A \mid B)_{16}$
|
||||
\section{}%3.3
|
||||
\subsection{}
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(8,0); %untere Kante
|
||||
\draw (0,0) -- +(0,0.5); %linke Kante
|
||||
\draw (0,0.5) -- +(8,0); %obere Kante
|
||||
\draw (8,0) -- +(0,0.5); %rechte Kante
|
||||
\draw (0.25,0) -- +(0,0.5); %rechte Kante von Vorzeichen
|
||||
\draw (2.25,0) -- +(0,0.5); %rechte Kante des Exponenten
|
||||
\node at ++(0.15,0.2) (sign) {$0$}; %Vorzeichen
|
||||
\node at ++(1.25,0.2) (exp) {$0000\, 0110$}; %Exponent
|
||||
\node at ++(5.125,0.2) (mantisse) {$011\, 0110\, 0000\, 0000\, 0000\, 0000$}; %Mantisse
|
||||
\end{tikzpicture}
|
||||
\subsection{}
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(8,0); %untere Kante
|
||||
\draw (0,0) -- +(0,0.5); %linke Kante
|
||||
\draw (0,0.5) -- +(8,0); %obere Kante
|
||||
\draw (8,0) -- +(0,0.5); %rechte Kante
|
||||
\draw (0.25,0) -- +(0,0.5); %rechte Kante von Vorzeichen
|
||||
\draw (2.25,0) -- +(0,0.5); %rechte Kante des Exponenten
|
||||
\node at ++(0.15,0.2) (sign) {$1$}; %Vorzeichen
|
||||
\node at ++(1.25,0.2) (exp) {$0000\, 0111$}; %Exponent
|
||||
\node at ++(5.125,0.2) (mantisse) {$010\, 1000\, 1010\, 0000\, 0000\, 0000$}; %Mantisse
|
||||
\end{tikzpicture}
|
||||
\section{}%3.4
|
||||
$8,626 \cdot 10^{5} + 9,9442 \cdot 10^{7}$\\
|
||||
|
||||
\subsection{}
|
||||
Skalierung des kleineren Summanden, bis beide Exponenten gleich sind:
|
||||
$0,08626 \cdot 10^{7} + 9,9442 \cdot 10^{7}$.\\
|
||||
Daraus folgt:
|
||||
\begin{alignat}{2}
|
||||
&\; 0,08626 \cdot 10^{7} + 9,9442 \cdot 10^{7} &=&\: x\\
|
||||
\Leftrightarrow &\; 0,08626 \cdot 10^{7} + 9,94420 \cdot 10^{7} &=&\: 10,03046 \cdot 10^{7}
|
||||
\end{alignat}
|
||||
Daraus ergibt sich dieses normalisierte Ergebnis:\\
|
||||
$1,003046 \cdot 10^{8}$\\
|
||||
Gerundet ergibt sich:\\
|
||||
$1,0030 \cdot 10^{8}$\\
|
||||
\subsection{}
|
||||
Skalierung des kleineren Summanden, bis beide Exponenten gleich sind:
|
||||
$0,0863 \cdot 10^{7} + 9,9442 \cdot 10^{7}$.\\
|
||||
Daraus folgt:
|
||||
\begin{alignat}{2}
|
||||
&\; 0,0863 \cdot 10^{7} + 9,9442 \cdot 10^{7} &=&\: x\\
|
||||
\Leftrightarrow &\; 0,0863 \cdot 10^{7} + 9,9442 \cdot 10^{7} &=&\: 10,0305 \cdot 10^{7}
|
||||
\end{alignat}
|
||||
Daraus ergibt sich dieses normalisierte Ergebnis:\\
|
||||
$1,00305 \cdot 10^{8}$\\
|
||||
Gerundet ergibt sich:\\
|
||||
$1,0031 \cdot 10^{8}$\\
|
||||
\subsection{}
|
||||
Bei Zahlen mit vielen Nachkommastellen ist eine Rundung nach jedem Schritt vorteilhafter, weil dadurch kompletter Präzisionsverlust vermieden werden kann (mehr Nachkommastellen als sinnvoll darstellbar).
|
||||
|
||||
Bei Zahlen mit wenigen Nachkommastellen eignet sich das Verfahren mit einmaliger Rundung am Ende besser, da hierbei ohne Genauigkeitsverlust (zu viele Nachkommastellen) ein relativ gutes Ergebnis ermittelt werde kann.
|
||||
|
||||
Anhand des Beispiels wird sichtbar, dass die Entscheidung Auswirkungen auf das Ergebnis haben können (in diesem Fall im Wert von $10000$), welche je nach Kontext trivial sind oder bereits katastrophale Ausmaße annehmen können.
|
||||
|
||||
Von daher hängt die Wahl des "`perfekten"' Verfahrens davon ab, was man machen muss.
|
||||
\section{}%3.5
|
||||
$5,6538 \cdot 10^{7} * 3,1415 \cdot 10^{4}$\\
|
||||
Daraus ergibt sich:\\
|
||||
\begin{alignat}{2}
|
||||
&\; 5,6538 \cdot 10^{7} * 3,1415 \cdot 10^{4} &=&\: x\\
|
||||
\Leftrightarrow &\; 5,6538 \cdot 10^{7} * 3,1415 \cdot 10^{4} &=& (5,6538 \cdot 3,1415) \cdot 10^{7+4}\\
|
||||
\Leftrightarrow &\; 5,6538 \cdot 10^{7} * 3,1415 \cdot 10^{4} &=& 17,7614127 \cdot 10^{11}
|
||||
\end{alignat}
|
||||
Daraus ergibt sich dieses normalisierte Ergebnis:\\
|
||||
$1,77614127 \cdot 10^{12}$\\
|
||||
Gerundet ergibt sich:\\
|
||||
$1,7761 \cdot 10^{12}$
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,116 @@
|
|||
\documentclass[10pt,a4paper,oneside,ngerman,numbers=noenddot]{scrartcl}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[ngerman]{babel}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{paralist}
|
||||
\usepackage[locale=DE,exponent-product=\cdot,detect-all]{siunitx}
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{matrix,fadings,calc,positioning,decorations.pathreplacing,arrows}
|
||||
\pagenumbering{arabic}
|
||||
\def\thesection{4.\arabic{section})}
|
||||
\def\thesubsection{(\alph{subsection})}
|
||||
\def\thesubsubsection{(\arabic{subsubsection})}
|
||||
\renewcommand{\labelenumi}{(\roman{enumi})}
|
||||
\hyphenation{Nach-komma-stel-len}
|
||||
|
||||
\begin{document}
|
||||
\author{Jim Martens (Matrikelnummer 6420323) \and Marlo Kornblum (Matrikelnummer 6427301)}
|
||||
\title{Rechnerstrukturen Aufgabenblatt 4}
|
||||
\maketitle
|
||||
|
||||
\section{}%4.1
|
||||
\subsection{} %a
|
||||
Der Nachteil dieser Methode liegt offensichtlich darin, dass in Zahlenbereichen mit vielen darstellbaren Zahlen zwei ganz unterschiedliche Zahlen als gleich angesehen werden.
|
||||
|
||||
Außerdem hängt die Feststellung der Gleichheit vom jeweiligen Nutzer ab, wodurch auf zwei unterschiedlichen Rechnern mit unterschiedlichen festgelegten Toleranzgrenzen die gleichen Zahlen unterschiedliche Vergleichsergebnisse liefern.
|
||||
|
||||
Ein Vergleich ist demzufolge also nicht eindeutig, sondern hängt von anderen Faktoren ab.
|
||||
\subsection{} %b
|
||||
Die relative Abweichung lässt sich wie folgt berechnen:
|
||||
\begin{alignat*}{2}
|
||||
y &=& |\frac{x - rd(x)}{x}|
|
||||
\end{alignat*}
|
||||
Bei diesem Beispiel sei $x$ die echte Gleitkommazahl und $rd(x)$ die gerundete Gleitkommazahl.
|
||||
|
||||
Als gleich können beide Werte angesehen werden, wenn folgendes gilt:
|
||||
\begin{alignat*}{2}
|
||||
y &\leq & \varepsilon = 0.5 \cdot b^{-t-1}
|
||||
\end{alignat*}
|
||||
$t$ sei die Anzahl der Bits für die Mantisse (bei Float $23$ und bei Double $52$) und $b$ die Basis der Gleitkommadarstellung.
|
||||
|
||||
\subsection{} %c
|
||||
Dieses Verfahren hat den großen Nachteil, dass die Anzahl an Nachkommastellen bei der Berechnung der relativen Abweichung sehr groß wird.
|
||||
|
||||
%todo, beheben
|
||||
\section{}%4.2
|
||||
\subsection{} %a
|
||||
Die CRLF SP Lösung CRLF SP SP der CRLF SP SP SP Übungsaufgabe CRLF SP SP SP SP liegt CRLF SP SP SP SP SP vor CRLF SP SP SP SP SP SP Ihnen!
|
||||
\subsection{} %b
|
||||
Der Rechner wurde auf Windows erstellt, da Unix-basierte Rechner nur LF nutzen, wohingegen Windows CRLF nutzt. Ab Mac OS X wird dort auch LF genutzt, vorher (bis Mac OS 9) wurde CR genutzt.
|
||||
\section{}%4.3
|
||||
\subsection{} %a
|
||||
Bei Kodierung nach ISO-8559-1 belegt dieser Text $1 \cdot 800000 = 800000$ Byte, da jedes Zeichen mit einem Byte kodiert wird. In direkter Unicode-Kodierung belegt der Text $2 \cdot 800000 = 1600000$ Byte, da Unicode jedes Zeichen mit zwei Byte kodiert.
|
||||
|
||||
In UTF-8 werden Zeichen mit Vielfachen von $8$ Bit kodiert. Dabei werden alle Zeichen, die auch im ISO-8559-1 vorhanden sind mit einem Byte kodiert ($8$ Bits) und alle deutschen Sonderzeichen (Umlaute, ß) mit zwei Byte.
|
||||
|
||||
Demnach ergeben sich $814240$ Byte.
|
||||
\subsection{} %b
|
||||
Der Bereich für CJK-Symbole umfasst $3400_{16}$ bis $4DBF_{16}$ und $4E00_{16}$ bis $9FCF_{16}$. Das sind $27536$ Symbole.
|
||||
\subsection{} %c
|
||||
Bei direkter Unicode-Kodierung belegt der chinesische Text $2 \cdot 800000 = 1600000$ Bytes, da hier jedes Zeichen mit zwei Byte kodiert ist.
|
||||
|
||||
Bei UTF-8 belegt ein chinesisches Zeichen 3 Byte. Demzufolge belegt der Text in UTF-8-Kodierung $3 \cdot 800000 = 2400000$ Byte.
|
||||
\section{}%4.4
|
||||
\subsection{} %a
|
||||
$y = (x \ll 3) + (x \ll 1)$
|
||||
\subsection{} %b
|
||||
$y = (x \ll 4) + (x \ll 3) + (x \ll 2) + (x \ll 1)$
|
||||
\subsection{} %c
|
||||
$y = (x \ll 4) - (x \ll 6)$
|
||||
\subsection{} %d
|
||||
$a = (x+6)$\\
|
||||
$y = 60 \cdot a$\\
|
||||
$y = (a \ll 5) + (a \ll 4) + (a \ll 3) + (a \ll 2)$\\
|
||||
%$y = ((x+6) \ll 5) + ((x+6) \ll 4) + ((x+6) \ll 3) + ((x+6) \ll 2)$
|
||||
\section{}%4.5
|
||||
\subsection{} %a
|
||||
$\text{bitNor}(x,y) := \sim x \, \& \sim y$
|
||||
\subsection{} %b
|
||||
$\text{bitXor}(x,y) := \sim(\sim(\sim x \,\&\, y) \,\&\, \sim(\sim y \,\&\, x))$
|
||||
\subsection{} %c
|
||||
$\text{rotateRight}(x,n) := (x\ggg n)_{2} \,|\, (x\ll (0111111_{2} \& (100000_{2} + (\sim n + 1)_{2})))$
|
||||
\subsection{} %d
|
||||
$\text{abs}(x) := (x \: \hat{} \: (x \gg 31)) + \sim (x \gg 31) + 1$ %todo
|
||||
|
||||
Die Funktion liefert für den Eingabewert $-2^{31}$ wieder denselben Wert. Das liegt daran, dass die größte positive darstellbare Zahl bei 32-bit und 2-Komplement $2^{31}-1$ ist. Bei der angegebenen Funktion wird zunächst der Wert mithilfe des Arithmetic Shift Right um 31 bit nach rechts verschoben. Ist die Zahl positiv ist die entstehende Bitmaske 0...0 und andernfalls 1...1. Diese Bitmaske wird nun xor mit dem Ausgangswert genommen. Dadurch wird der Ausgangswert wenn er negativ ist negiert und andernfalls so belassen, wie er ist.
|
||||
|
||||
Anschließend wird das 2-Komplement der vorher erzeugten Bitmaske addiert. Ist der Ausgangswert positiv, dann wird $0$ addiert, wodurch der Ausgangswert bestehen bleibt. Ist der Ausgangswert negativ, dann bekommt man dadurch den positiven Wert. Dies funktioniert für alle negativen Zahlen größer oder gleich $-2^{31}+1$. Noch einmal an den Beispielen $-2^{31}$(1) und $-1$(2) veranschaulicht:
|
||||
|
||||
Beispiel (1)
|
||||
\begin{alignat*}{2}
|
||||
&\; x &=& 10...0 \\
|
||||
&\; a &=& x \gg 31 \\
|
||||
\Leftrightarrow &\; a &=& 11...1 \\
|
||||
&\; b &=& a \, \hat{}\, x \\
|
||||
\Leftrightarrow &\; b &=& 01...1 \\
|
||||
&\; c &=& b + K_{2}(a)_{2} \\
|
||||
\Leftrightarrow &\; c &=& 01...1 + 00...01 \\
|
||||
\Leftrightarrow &\; c &=& 10...0
|
||||
\end{alignat*}
|
||||
|
||||
Beispiel (2)
|
||||
\begin{alignat*}{2}
|
||||
&\; x &=& 11...1 \\
|
||||
&\; a &=& x \gg 31 \\
|
||||
\Leftrightarrow &\; a &=& 11...1 \\
|
||||
&\; b &=& a \, \hat{}\, x \\
|
||||
\Leftrightarrow &\; b &=& 00...0 \\
|
||||
&\; c &=& b + K_{2}(a)_{2} \\
|
||||
\Leftrightarrow &\; c &=& 00...0 + 00...01 \\
|
||||
\Leftrightarrow &\; c &=& 00...01
|
||||
\end{alignat*}
|
||||
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,823 @@
|
|||
\documentclass[10pt,a4paper,oneside,ngerman,numbers=noenddot]{scrartcl}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[ngerman]{babel}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{paralist}
|
||||
\usepackage[locale=DE,exponent-product=\cdot ,detect-all]{siunitx}
|
||||
\usepackage{tikz}
|
||||
\usepackage[scaled=0.78]{luximono}
|
||||
\usepackage{listings}
|
||||
\usetikzlibrary{automata,matrix,fadings,calc,positioning,decorations.pathreplacing,decorations.text,arrows}
|
||||
\pagenumbering{arabic}
|
||||
\def\thesection{5.\arabic{section})}
|
||||
\def\thesubsection{(\alph{subsection})}
|
||||
\def\thesubsubsection{(\arabic{subsubsection})}
|
||||
\renewcommand{\labelenumi}{(\roman{enumi})}
|
||||
\hyphenation{Nach-komma-stel-len}
|
||||
\lstnewenvironment{java}[1][]{%
|
||||
\lstset{basicstyle=\ttfamily ,backgroundcolor=\color[gray]{.95},columns=flexible,fontadjust=true,language=Java,tabsize=4,numbers=none,#1}%
|
||||
}{%
|
||||
}
|
||||
|
||||
\tikzstyle{huffmanNodes}=[matrix of nodes,
|
||||
nodes={circle,thin,draw=black!20,minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\tikzstyle{huffmanBase}=[matrix of nodes,
|
||||
nodes={minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\begin{document}
|
||||
\author{Jim Martens (Matrikelnummer 6420323) \and Marlo Kornblum (Matrikelnummer 6427301)}
|
||||
\title{Rechnerstrukturen Aufgabenblatt 5}
|
||||
\maketitle
|
||||
|
||||
\section{}%5.1
|
||||
\begin{java}
|
||||
int a1, a2, a3, a4;
|
||||
|
||||
int b1 = (a1 << 2) | (a2 >>> 4);
|
||||
int b2 = (a2 << 4) | (a3 >>> 2);
|
||||
int b3 = (a3 << 6) | a4;
|
||||
|
||||
...
|
||||
\end{java}
|
||||
\section{}%5.2
|
||||
\subsection{} %a
|
||||
Der Code hat $\frac{360^{\circ}}{15^{\circ}}=24$ Codewörter.
|
||||
\subsection{} %b
|
||||
Wir starten mit den beiden Symbolen $0$ und $1$. Daraus ergibt sich in weiteren Schritten folgendes:
|
||||
\begin{alignat*}{2}
|
||||
0, & 1 \\
|
||||
00, 01, & 11, 10 \\
|
||||
000, 001, 011, 010, & 110, 111, 101, 100 \\
|
||||
0000, 0001, 0011, 0010, 0110, 0111, 0101, 0100, & 1100, 1101, 1111, 1110, 1010, 1011, 1001, 1000
|
||||
\end{alignat*}
|
||||
\begin{alignat*}{2}
|
||||
\intertext{Nachfolgend der erste Teil der fünften Zeile}
|
||||
00000, 000001, 00011, 00010, 00110, 00111, 00101, 00100,\\
|
||||
\intertext{Der zweite Teil}
|
||||
01100, 01101, 01111, 01110, 01010, 01011, 01001, 01000,\\
|
||||
\intertext{Und nun der dritte Teil} 11000, 11001, 11011, 11010, 11110, 11111, 11101, 11100,\\
|
||||
\intertext{Abschließend der vierte Teil}
|
||||
10100, 10101, 10111, 10110, 10010, 10011, 10001, 10000
|
||||
\end{alignat*}
|
||||
Die zuletzt dargestellte Zeile enthält $32$ Codewörter. Da wir nur $24$ benötigen, werden beidseitig $4$ weggestrichen. Damit bleibt die Fano-Bedingung erfüllt. Das Ergebnis ist dieser Code (auf zwei Zeileneinträge aufgeteilt):
|
||||
|
||||
\begin{alignat*}{2}
|
||||
00110, 00111, 00101, 00100, 01100, 01101, 01111, 01110, 01010, 01011, 01001, 01000, \\
|
||||
11000, 11001, 11011, 11010, 11110, 11111, 11101, 11100, 10100, 10101, 10111, 10110
|
||||
\end{alignat*}
|
||||
\section{}%5.3
|
||||
\subsection{} %a
|
||||
|
||||
Es ergeben sich die folgenden Codes:
|
||||
|
||||
\begin{tikzpicture}
|
||||
\matrix[matrix of nodes]
|
||||
{a & b & c & d & e & f & g & h & i & j & k & l \\
|
||||
$100$ & $01011$ & $01000$ & $00$ & $010010$ & $1011$ & $111$ & $010011$ & $01010$ & $110$ & $011$ & $1010$ \\};
|
||||
\end{tikzpicture}
|
||||
|
||||
\begin{figure}[hbp]
|
||||
Erster Schritt:
|
||||
|
||||
\begin{tikzpicture}[shorten >=1pt,node distance=1.1cm,on grid,auto,/tikz/initial text=]
|
||||
%level 1
|
||||
\node[state] (e) {e};
|
||||
\node[state] (h) [right=of e] {h};
|
||||
\node[state] (b) [right=of h] {b};
|
||||
\node[state] (i) [right=of b] {i};
|
||||
\node[state] (c) [right=of i] {c};
|
||||
\node[state] (f) [right=of c] {f};
|
||||
\node[state] (l) [right=of f] {l};
|
||||
\node[state] (g) [right=of l] {g};
|
||||
\node[state] (j) [right=of g] {j};
|
||||
\node[state] (a) [right=of j] {a};
|
||||
\node[state] (k) [right=of a] {k};
|
||||
\node[state] (d) [right=of k] {d};
|
||||
|
||||
%level 2
|
||||
\node[state] (eh) [above right=1.3 and 0.6 of e] {$0.04$};
|
||||
|
||||
\path[every node/.style={font=\scriptsize}]
|
||||
(e) edge node [near start] {$1$} (eh)
|
||||
(h) edge node [near end] {$0$} (eh);
|
||||
%\draw[decorate,decoration={text along path,text=$0.02$] (e) -- (eh);
|
||||
|
||||
\node (pE) [below=0.7 of e] {0.02};
|
||||
\node (pH) [below=0.7 of h] {0.02};
|
||||
\node (pB) [below=0.7 of b] {0.03};
|
||||
\node (pI) [below=0.7 of i] {0.03};
|
||||
\node (pC) [below=0.7 of c] {0.05};
|
||||
\node (pF) [below=0.7 of f] {0.05};
|
||||
\node (pL) [below=0.7 of l] {0.06};
|
||||
\node (pG) [below=0.7 of g] {0.1};
|
||||
\node (pJ) [below=0.7 of j] {0.1};
|
||||
\node (pA) [below=0.7 of a] {0.12};
|
||||
\node (pK) [below=0.7 of k] {0.12};
|
||||
\node (pD) [below=0.7 of d] {0.3};
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[hbp]
|
||||
Zweiter Schritt:
|
||||
|
||||
\begin{tikzpicture}[shorten >=1pt,node distance=1.1cm,on grid,auto,/tikz/initial text=, nodes={decoration={}} ]
|
||||
%level 1
|
||||
\node[state] (e2) {e};
|
||||
\node[state] (h2) [right=of e2] {h};
|
||||
|
||||
%level 2
|
||||
\node[state] (eh2) [above right=1.3 and 0.6 of e2] {$0.04$};
|
||||
\node[state] (i2) [left=of eh2] {i};
|
||||
\node[state] (b2) [left=of i2] {b};
|
||||
\node[state] (c2) [right=of eh] {c};
|
||||
\node[state] (f2) [right=of c2] {f};
|
||||
\node[state] (l2) [right=of f2] {l};
|
||||
\node[state] (g2) [right=of l2] {g};
|
||||
\node[state] (j2) [right=of g2] {j};
|
||||
\node[state] (a2) [right=of j2] {a};
|
||||
\node[state] (k2) [right=of a2] {k};
|
||||
\node[state] (d2) [right=of k2] {d};
|
||||
|
||||
%level 3
|
||||
\node[state] (bi2) [above right=1.3 and 0.6 of b2] {$0.06$};
|
||||
|
||||
\path[every node/.style={font=\scriptsize}]
|
||||
(e2) edge node [near start] {$1$} (eh2)
|
||||
(h2) edge node [near end] {$0$} (eh2)
|
||||
(b2) edge node [near start] {$1$} (bi2)
|
||||
(i2) edge node [near end] {$0$} (bi2);
|
||||
|
||||
\node (pE2) [below=0.7 of e2] {0.02};
|
||||
\node (pH2) [below=0.7 of h2] {0.02};
|
||||
\node (pB2) [below=0.7 of b2] {0.03};
|
||||
\node (pI2) [below=0.7 of i2] {0.03};
|
||||
\node (pC2) [below=0.7 of c2] {0.05};
|
||||
\node (pF2) [below=0.7 of f2] {0.05};
|
||||
\node (pL2) [below=0.7 of l2] {0.06};
|
||||
\node (pG2) [below=0.7 of g2] {0.1};
|
||||
\node (pJ2) [below=0.7 of j2] {0.1};
|
||||
\node (pA2) [below=0.7 of a2] {0.12};
|
||||
\node (pK2) [below=0.7 of k2] {0.12};
|
||||
\node (pD2) [below=0.7 of d2] {0.3};
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[hbp]
|
||||
Dritter Schritt:
|
||||
|
||||
\begin{tikzpicture}[shorten >=1pt,node distance=1.1cm,on grid,auto,/tikz/initial text=]
|
||||
%level 2
|
||||
\node[state] (e3) {e};
|
||||
\node[state] (h3) [right=of e3] {h};
|
||||
\node[state] (b3) [right=2.2 of h3] {b};
|
||||
\node[state] (i3) [right=of b3] {i};
|
||||
|
||||
%level 3
|
||||
\node[state] (eh3) [above right=1.3 and 0.6 of e3] {$0.04$};
|
||||
\node[state] (c3) [right=of eh3] {c};
|
||||
\node[state] (f3) [right=of c3] {f};
|
||||
\node[state] (bi3) [right=of f3] {$0.06$};
|
||||
\node[state] (l3) [right=of bi3] {l};
|
||||
\node[state] (g3) [right=of l3] {g};
|
||||
\node[state] (j3) [right=of g3] {j};
|
||||
\node[state] (a3) [right=of j3] {a};
|
||||
\node[state] (k3) [right=of a3] {k};
|
||||
\node[state] (d3) [right=of k3] {d};
|
||||
|
||||
%level 4
|
||||
\node[state] (ehc3) [above right=1.3 and 0.6 of eh3] {$0.09$};
|
||||
|
||||
\path[every node/.style={font=\scriptsize}]
|
||||
(e3) edge node [near start] {$1$} (eh3)
|
||||
(h3) edge node [near end] {$0$} (eh3)
|
||||
(b3) edge node [near start] {$1$} (bi3)
|
||||
(i3) edge node [near end] {$0$} (bi3)
|
||||
(eh3) edge node [near start] {$1$} (ehc3)
|
||||
(c3) edge node [near end] {$0$} (ehc3);
|
||||
|
||||
\node (pE3) [below=0.7 of e3] {0.02};
|
||||
\node (pH3) [below=0.7 of h3] {0.02};
|
||||
\node (pB3) [below=0.7 of b3] {0.03};
|
||||
\node (pI3) [below=0.7 of i3] {0.03};
|
||||
\node (pC3) [below=0.7 of c3] {0.05};
|
||||
\node (pF3) [below=0.7 of f3] {0.05};
|
||||
\node (pL3) [below=0.7 of l3] {0.06};
|
||||
\node (pG3) [below=0.7 of g3] {0.1};
|
||||
\node (pJ3) [below=0.7 of j3] {0.1};
|
||||
\node (pA3) [below=0.7 of a3] {0.12};
|
||||
\node (pK3) [below=0.7 of k3] {0.12};
|
||||
\node (pD3) [below=0.7 of d3] {0.3};
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[hbp]
|
||||
Vierter Schritt:
|
||||
|
||||
\begin{tikzpicture}[shorten >=1pt,node distance=1.1cm,on grid,auto,/tikz/initial text=]
|
||||
%level 2
|
||||
\node[state] (e4) {e};
|
||||
\node[state] (h4) [right=of e4] {h};
|
||||
|
||||
%level 3
|
||||
\node[state] (eh4) [above right=1.3 and 0.6 of e4] {$0.04$};
|
||||
\node[state] (i4) [left=of eh4] {i};
|
||||
\node[state] (b4) [left=of i4] {b};
|
||||
\node[state] (c4) [right=of eh4] {c};
|
||||
|
||||
%level 4
|
||||
\node[state] (bi4) [above right=1.3 and 0.6 of b4] {$0.06$};
|
||||
\node[state] (l4) [left=of bi4] {l};
|
||||
\node[state] (f4) [left=of l4] {f};
|
||||
\node[state] (ehc4) [right=2.2 of bi4] {$0.09$};
|
||||
\node[state] (g4) [right=of ehc4] {g};
|
||||
\node[state] (j4) [right=of g4] {j};
|
||||
\node[state] (a4) [right=of j4] {a};
|
||||
\node[state] (k4) [right=of a4] {k};
|
||||
\node[state] (d4) [right=of k4] {d};
|
||||
|
||||
%level 5
|
||||
\node[state] (fl4) [above right=1.3 and 0.6 of f4] {$0.11$};
|
||||
|
||||
\path[every node/.style={font=\scriptsize}]
|
||||
(e4) edge node [near start] {$1$} (eh4)
|
||||
(h4) edge node [near end] {$0$} (eh4)
|
||||
(b4) edge node [near start] {$1$} (bi4)
|
||||
(i4) edge node [near end] {$0$} (bi4)
|
||||
(eh4) edge node [near start] {$1$} (ehc4)
|
||||
(c4) edge node [near end] {$0$} (ehc4)
|
||||
(f4) edge node [near start] {$1$} (fl4)
|
||||
(l4) edge node [near end] {$0$} (fl4);
|
||||
|
||||
\node (pE4) [below=0.7 of e4] {0.02};
|
||||
\node (pH4) [below=0.7 of h4] {0.02};
|
||||
\node (pB4) [below=0.7 of b4] {0.03};
|
||||
\node (pI4) [below=0.7 of i4] {0.03};
|
||||
\node (pC4) [below=0.7 of c4] {0.05};
|
||||
\node (pF4) [below=0.7 of f4] {0.05};
|
||||
\node (pL4) [below=0.7 of l4] {0.06};
|
||||
\node (pG4) [below=0.7 of g4] {0.1};
|
||||
\node (pJ4) [below=0.7 of j4] {0.1};
|
||||
\node (pA4) [below=0.7 of a4] {0.12};
|
||||
\node (pK4) [below=0.7 of k4] {0.12};
|
||||
\node (pD4) [below=0.7 of d4] {0.3};
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[hbp]
|
||||
Fünfter Schritt:
|
||||
|
||||
\begin{tikzpicture}[shorten >=1pt,node distance=1.1cm,on grid,auto,/tikz/initial text=]
|
||||
%level 3
|
||||
\node[state] (e6) {e};
|
||||
\node[state] (h6) [right=of e6] {h};
|
||||
|
||||
%level 4
|
||||
\node[state] (eh6) [above right=1.3 and 0.6 of e6] {$0.04$};
|
||||
\node[state] (i6) [left=of eh6] {i};
|
||||
\node[state] (b6) [left=of i6] {b};
|
||||
\node[state] (c6) [right=of eh6] {c};
|
||||
\node[state] (f6) [right=2.2 of c6] {f};
|
||||
\node[state] (l6) [right=of f6] {l};
|
||||
|
||||
%level 5
|
||||
\node[state] (bi6) [above right=1.3 and 0.6 of b6] {$0.06$};
|
||||
\node[state] (ehc6) [right=2.2 of bi6] {$0.09$};
|
||||
\node[state] (g6) [right=of ehc6] {j};
|
||||
\node[state] (j6) [right=of g6] {g};
|
||||
\node[state] (fl6) [right=of j6] {$0.11$};
|
||||
\node[state] (a6) [right=of fl6] {a};
|
||||
\node[state] (k6) [right=of a6] {k};
|
||||
\node[state] (d6) [right=of k6] {d};
|
||||
|
||||
%level 6
|
||||
\node[state] (ehcbi6) [above right=1.3 and 1.1 of bi6] {$0.15$};
|
||||
|
||||
\path[every node/.style={font=\scriptsize}]
|
||||
(e6) edge node [near start] {$1$} (eh6)
|
||||
(h6) edge node [near end] {$0$} (eh6)
|
||||
(b6) edge node [near start] {$1$} (bi6)
|
||||
(i6) edge node [near end] {$0$} (bi6)
|
||||
(eh6) edge node [near start] {$1$} (ehc6)
|
||||
(c6) edge node [near end] {$0$} (ehc6)
|
||||
(f6) edge node [near start] {$1$} (fl6)
|
||||
(l6) edge node [near end] {$0$} (fl6)
|
||||
(bi6) edge node [near start] {$1$} (ehcbi6)
|
||||
(ehc6) edge node [near end] {$0$} (ehcbi6);
|
||||
|
||||
\node (pE6) [below=0.7 of e6] {0.02};
|
||||
\node (pH6) [below=0.7 of h6] {0.02};
|
||||
\node (pB6) [below=0.7 of b6] {0.03};
|
||||
\node (pI6) [below=0.7 of i6] {0.03};
|
||||
\node (pC6) [below=0.7 of c6] {0.05};
|
||||
\node (pF6) [below=0.7 of f6] {0.05};
|
||||
\node (pL6) [below=0.7 of l6] {0.06};
|
||||
\node (pG6) [below=0.7 of g6] {0.1};
|
||||
\node (pJ6) [below=0.7 of j6] {0.1};
|
||||
\node (pA6) [below=0.7 of a6] {0.12};
|
||||
\node (pK6) [below=0.7 of k6] {0.12};
|
||||
\node (pD6) [below=0.7 of d6] {0.3};
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[hbp]
|
||||
Sechster Schritt:
|
||||
|
||||
\begin{tikzpicture}[shorten >=1pt,node distance=1.1cm,on grid,auto,/tikz/initial text=]
|
||||
%level 3
|
||||
\node[state] (e7) {e};
|
||||
\node[state] (h7) [right=of e7] {h};
|
||||
|
||||
%level 4
|
||||
\node[state] (eh7) [above right=1.3 and 0.6 of e7] {$0.04$};
|
||||
\node[state] (i7) [left=of eh7] {i};
|
||||
\node[state] (b7) [left=of i7] {b};
|
||||
\node[state] (c7) [right=of eh7] {c};
|
||||
|
||||
%level 5
|
||||
\node[state] (bi7) [above right=1.3 and 0.6 of b7] {$0.06$};
|
||||
\node[state] (l7) [left=1.65 of bi7] {l};
|
||||
\node[state] (f7) [left=of l7] {f};
|
||||
\node[state] (ehc7) [right=2.2 of bi7] {$0.09$};
|
||||
|
||||
%level 6
|
||||
\node[state] (ehcbi7) [above right=1.3 and 1.1 of bi7] {$0.15$};
|
||||
\node[state] (k7) [left=of ehcbi7] {k};
|
||||
\node[state] (a7) [left=of k7] {a};
|
||||
\node[state] (fl7) [left=of a7] {$0.11$};
|
||||
\node[state] (j7) [left=of fl7] {j};
|
||||
\node[state] (g7) [left=of j7] {g};
|
||||
\node[state] (d7) [right=of ehcbi7] {d};
|
||||
|
||||
%level 7
|
||||
\node[state] (gj7) [above right=1.3 and 0.6 of g7] {$0.2$};
|
||||
|
||||
\path[every node/.style={font=\scriptsize}]
|
||||
(e7) edge node [near start] {$1$} (eh7)
|
||||
(h7) edge node [near end] {$0$} (eh7)
|
||||
(b7) edge node [near start] {$1$} (bi7)
|
||||
(i7) edge node [near end] {$0$} (bi7)
|
||||
(eh7) edge node [near start] {$1$} (ehc7)
|
||||
(c7) edge node [near end] {$0$} (ehc7)
|
||||
(f7) edge node [near start] {$1$} (fl7)
|
||||
(l7) edge node [near end] {$0$} (fl7)
|
||||
(bi7) edge node [near start] {$1$} (ehcbi7)
|
||||
(ehc7) edge node [near end] {$0$} (ehcbi7)
|
||||
(g7) edge node [near start] {$1$} (gj7)
|
||||
(j7) edge node [near end] {$0$} (gj7);
|
||||
|
||||
\node (pE7) [below=0.7 of e7] {0.02};
|
||||
\node (pH7) [below=0.7 of h7] {0.02};
|
||||
\node (pB7) [below=0.7 of b7] {0.03};
|
||||
\node (pI7) [below=0.7 of i7] {0.03};
|
||||
\node (pC7) [below=0.7 of c7] {0.05};
|
||||
\node (pF7) [below=0.7 of f7] {0.05};
|
||||
\node (pL7) [below=0.7 of l7] {0.06};
|
||||
\node (pG7) [below=0.7 of g7] {0.1};
|
||||
\node (pJ7) [below=0.7 of j7] {0.1};
|
||||
\node (pA7) [below=0.7 of a7] {0.12};
|
||||
\node (pK7) [below=0.7 of k7] {0.12};
|
||||
\node (pD7) [below=0.7 of d7] {0.3};
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[hbp]
|
||||
Siebter Schritt:
|
||||
|
||||
\begin{tikzpicture}[shorten >=1pt,node distance=1.1cm,on grid,auto,/tikz/initial text=]
|
||||
%level 4
|
||||
\node[state] (e8) {e};
|
||||
\node[state] (h8) [right=of e8] {h};
|
||||
|
||||
%level 5
|
||||
\node[state] (eh8) [above right=1.3 and 0.6 of e8] {$0.04$};
|
||||
\node[state] (i8) [left=of eh8] {i};
|
||||
\node[state] (b8) [left=of i8] {b};
|
||||
\node[state] (c8) [right=of eh8] {c};
|
||||
|
||||
%level 6
|
||||
\node[state] (bi8) [above right=1.3 and 0.6 of b8] {$0.06$};
|
||||
\node[state] (l8) [left=1.65 of bi8] {l};
|
||||
\node[state] (f8) [left=of l8] {f};
|
||||
\node[state] (ehc8) [right=2.2 of bi8] {$0.09$};
|
||||
\node[state] (g8) [right=1.1 of ehc8] {g};
|
||||
\node[state] (j8) [right=of g8] {j};
|
||||
|
||||
%level 7
|
||||
\node[state] (gj8) [above right=1.3 and 0.6 of g8] {$0.2$};
|
||||
\node[state] (ehcbi8) [left=2.75 of gj8] {$0.15$};
|
||||
\node[state] (k8) [left=of ehcbi8] {k};
|
||||
\node[state] (a8) [left=of k8] {a};
|
||||
\node[state] (fl8) [left=1.1 of a8] {$0.11$};
|
||||
\node[state] (d8) [right=of gj8] {d};
|
||||
|
||||
%level 8
|
||||
\node[state] (fla8) [above right=1.3 and 0.6 of fl8] {$0.23$};
|
||||
|
||||
\path[every node/.style={font=\scriptsize}]
|
||||
(e8) edge node [near start] {$1$} (eh8)
|
||||
(h8) edge node [near end] {$0$} (eh8)
|
||||
(b8) edge node [near start] {$1$} (bi8)
|
||||
(i8) edge node [near end] {$0$} (bi8)
|
||||
(eh8) edge node [near start] {$1$} (ehc8)
|
||||
(c8) edge node [near end] {$0$} (ehc8)
|
||||
(f8) edge node [near start] {$1$} (fl8)
|
||||
(l8) edge node [near end] {$0$} (fl8)
|
||||
(bi8) edge node [near start] {$1$} (ehcbi8)
|
||||
(ehc8) edge node [near end] {$0$} (ehcbi8)
|
||||
(g8) edge node [near start] {$1$} (gj8)
|
||||
(j8) edge node [near end] {$0$} (gj8)
|
||||
(fl8) edge node [near start] {$1$} (fla8)
|
||||
(a8) edge node [near end] {$0$} (fla8);
|
||||
|
||||
\node (pE8) [below=0.7 of e8] {0.02};
|
||||
\node (pH8) [below=0.7 of h8] {0.02};
|
||||
\node (pB8) [below=0.7 of b8] {0.03};
|
||||
\node (pI8) [below=0.7 of i8] {0.03};
|
||||
\node (pC8) [below=0.7 of c8] {0.05};
|
||||
\node (pF8) [below=0.7 of f8] {0.05};
|
||||
\node (pL8) [below=0.7 of l8] {0.06};
|
||||
\node (pG8) [below=0.7 of g8] {0.1};
|
||||
\node (pJ8) [below=0.7 of j8] {0.1};
|
||||
\node (pA8) [below=0.7 of a8] {0.12};
|
||||
\node (pK8) [below=0.7 of k8] {0.12};
|
||||
\node (pD8) [below=0.7 of d8] {0.3};
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[hbp]
|
||||
Achter Schritt:
|
||||
|
||||
\begin{tikzpicture}[shorten >=1pt,node distance=1.1cm,on grid,auto,/tikz/initial text=]
|
||||
%level 4
|
||||
\node[state] (e9) {e};
|
||||
\node[state] (h9) [right=of e9] {h};
|
||||
|
||||
%level 5
|
||||
\node[state] (eh9) [above right=1.3 and 0.6 of e9] {$0.04$};
|
||||
\node[state] (i9) [left=of eh9] {i};
|
||||
\node[state] (b9) [left=of i9] {b};
|
||||
\node[state] (c9) [right=of eh9] {c};
|
||||
\node[state] (f9) [right=2.2 of c9] {f};
|
||||
\node[state] (l9) [right=of f9] {l};
|
||||
|
||||
%level 6
|
||||
\node[state] (bi9) [above right=1.3 and 0.6 of b9] {$0.06$};
|
||||
\node[state] (ehc9) [right=2.2 of bi9] {$0.09$};
|
||||
\node[state] (g9) [right=1.1 of ehc9] {g};
|
||||
\node[state] (j9) [right=of g9] {j};
|
||||
\node[state] (fl9) [right=of j9] {$0.11$};
|
||||
\node[state] (a9) [right=of fl9] {a};
|
||||
|
||||
%level 7
|
||||
\node[state] (gj9) [above right=1.3 and 0.6 of g9] {$0.2$};
|
||||
\node[state] (ehcbi9) [left=2.75 of gj9] {$0.15$};
|
||||
\node[state] (k9) [left=of ehcbi9] {k};
|
||||
\node[state] (fla9) [right=2.2 of gj9] {$0.23$};
|
||||
\node[state] (d9) [right=of fla9] {d};
|
||||
|
||||
%level 8
|
||||
\node[state] (kehcbi9) [above right=1.3 and 0.6 of k9] {$0.27$};
|
||||
|
||||
\path[every node/.style={font=\scriptsize}]
|
||||
(e9) edge node [near start] {$1$} (eh9)
|
||||
(h9) edge node [near end] {$0$} (eh9)
|
||||
(b9) edge node [near start] {$1$} (bi9)
|
||||
(i9) edge node [near end] {$0$} (bi9)
|
||||
(eh9) edge node [near start] {$1$} (ehc9)
|
||||
(c9) edge node [near end] {$0$} (ehc9)
|
||||
(f9) edge node [near start] {$1$} (fl9)
|
||||
(l9) edge node [near end] {$0$} (fl9)
|
||||
(bi9) edge node [near start] {$1$} (ehcbi9)
|
||||
(ehc9) edge node [near end] {$0$} (ehcbi9)
|
||||
(g9) edge node [near start] {$1$} (gj9)
|
||||
(j9) edge node [near end] {$0$} (gj9)
|
||||
(fl9) edge node [near start] {$1$} (fla9)
|
||||
(a9) edge node [near end] {$0$} (fla9)
|
||||
(k9) edge node [near start] {$1$} (kehcbi9)
|
||||
(ehcbi9) edge node [near end] {$0$} (kehcbi9);
|
||||
|
||||
\node (pE9) [below=0.7 of e9] {0.02};
|
||||
\node (pH9) [below=0.7 of h9] {0.02};
|
||||
\node (pB9) [below=0.7 of b9] {0.03};
|
||||
\node (pI9) [below=0.7 of i9] {0.03};
|
||||
\node (pC9) [below=0.7 of c9] {0.05};
|
||||
\node (pF9) [below=0.7 of f9] {0.05};
|
||||
\node (pL9) [below=0.7 of l9] {0.06};
|
||||
\node (pG9) [below=0.7 of g9] {0.1};
|
||||
\node (pJ9) [below=0.7 of j9] {0.1};
|
||||
\node (pA9) [below=0.7 of a9] {0.12};
|
||||
\node (pK9) [below=0.7 of k9] {0.12};
|
||||
\node (pD9) [below=0.7 of d9] {0.3};
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[hbp]
|
||||
Neunter Schritt:
|
||||
|
||||
\begin{tikzpicture}[shorten >=1pt,node distance=1.1cm,on grid,auto,/tikz/initial text=]
|
||||
|
||||
|
||||
%level 8
|
||||
\node[state] [above right=5.2 and 3.6] (kehcbi10) {$0.27$};
|
||||
\node[state] (fla10) [left=2.2 of kehcbi10] {$0.23$};
|
||||
\node[state] (gj10) [left=2.2 of fla10] {$0.2$};
|
||||
\node[state] (d10) [right=of kehcbi10] {d};
|
||||
|
||||
%level 7
|
||||
\node[state] (ehcbi10) [below right=1.3 and 0.6 of kehcbi10] {$0.15$};
|
||||
\node[state] (k10) [left=of ehcbi10] {k};
|
||||
\node[state] (a10) [left=of k10] {a};
|
||||
\node[state] (fl10) [left=of a10] {$0.11$};
|
||||
\node[state] (j10) [left=of fl10] {j};
|
||||
\node[state] (g10) [left=of j10] {g};
|
||||
|
||||
\node[state] (bi10) [below left=1.3 and 1.2 of ehcbi10] {$0.06$};
|
||||
\node[state] (l10) [left=1.65 of bi10] {l};
|
||||
\node[state] (f10) [left=of l10] {f};
|
||||
\node[state] (ehc10) [right=2.2of bi10] {$0.09$};
|
||||
|
||||
\node[state] (eh10) [below left=1.3 and 0.6 of ehc10] {$0.04$};
|
||||
\node[state] (i10) [left=of eh10] {i};
|
||||
\node[state] (b10) [left=of i10] {b};
|
||||
\node[state] (c10) [right=of eh10] {c};
|
||||
|
||||
\node[state] (e10) [below left=1.3 and 0.6 of eh10] {e};
|
||||
\node[state] (h10) [right=of e10] {h};
|
||||
|
||||
%level 9
|
||||
\node[state] (gjfla10) [above right=1.3 and 1.1 of gj10] {$0.43$};
|
||||
|
||||
\path[every node/.style={font=\scriptsize}]
|
||||
(e10) edge node [near start] {$1$} (eh10)
|
||||
(h10) edge node [near end] {$0$} (eh10)
|
||||
(b10) edge node [near start] {$1$} (bi10)
|
||||
(i10) edge node [near end] {$0$} (bi10)
|
||||
(eh10) edge node [near start] {$1$} (ehc10)
|
||||
(c10) edge node [near end] {$0$} (ehc10)
|
||||
(f10) edge node [near start] {$1$} (fl10)
|
||||
(l10) edge node [near end] {$0$} (fl10)
|
||||
(bi10) edge node [near start] {$1$} (ehcbi10)
|
||||
(ehc10) edge node [near end] {$0$} (ehcbi10)
|
||||
(g10) edge node [near start] {$1$} (gj10)
|
||||
(j10) edge node [near end] {$0$} (gj10)
|
||||
(fl10) edge node [near start] {$1$} (fla10)
|
||||
(a10) edge node [near end] {$0$} (fla10)
|
||||
(k10) edge node [near start] {$1$} (kehcbi10)
|
||||
(ehcbi10) edge node [near end] {$0$} (kehcbi10)
|
||||
(gj10) edge node [near start] {$1$} (gjfla10)
|
||||
(fla10) edge node [near end] {$0$} (gjfla10);
|
||||
|
||||
\node (pE10) [below=0.7 of e10] {0.02};
|
||||
\node (pH10) [below=0.7 of h10] {0.02};
|
||||
\node (pB10) [below=0.7 of b10] {0.03};
|
||||
\node (pI10) [below=0.7 of i10] {0.03};
|
||||
\node (pC10) [below=0.7 of c10] {0.05};
|
||||
\node (pF10) [below=0.7 of f10] {0.05};
|
||||
\node (pL10) [below=0.7 of l10] {0.06};
|
||||
\node (pG10) [below=0.7 of g10] {0.1};
|
||||
\node (pJ10) [below=0.7 of j10] {0.1};
|
||||
\node (pA10) [below=0.7 of a10] {0.12};
|
||||
\node (pK10) [below=0.7 of k10] {0.12};
|
||||
\node (pD10) [below=0.7 of d10] {0.3};
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[hbp]
|
||||
Zehnter Schritt:
|
||||
|
||||
\begin{tikzpicture}[shorten >=1pt,node distance=1.1cm,on grid,auto,/tikz/initial text=]
|
||||
%level 4
|
||||
\node[state] (e11) {e};
|
||||
\node[state] (h11) [right=of e11] {h};
|
||||
|
||||
%level 5
|
||||
\node[state] (eh11) [above right=1.3 and 0.6 of e11] {$0.04$};
|
||||
\node[state] (i11) [left=of eh11] {i};
|
||||
\node[state] (b11) [left=of i11] {b};
|
||||
\node[state] (c11) [right=of eh11] {c};
|
||||
\node[state] (f11) [right=2.2 of c11] {f};
|
||||
\node[state] (l11) [right=of f11] {l};
|
||||
|
||||
%level 6
|
||||
\node[state] (bi11) [above right=1.3 and 0.6 of b11] {$0.06$};
|
||||
|
||||
\node[state] (ehc11) [right=2.2 of bi11] {$0.09$};
|
||||
\node[state] (g11) [right=of ehc11] {g};
|
||||
\node[state] (j11) [right=of g11] {j};
|
||||
\node[state] (fl11) [right=of j11] {$0.11$};
|
||||
\node[state] (a11) [right=of fl11] {a};
|
||||
|
||||
%level 7
|
||||
\node[state] (ehcbi11) [above right=1.3 and 1.2 of bi11] {$0.15$};
|
||||
\node[state] (k11) [left=of ehcbi11] {k};
|
||||
\node[state] (gj11) [right=2.75 of ehcbi11] {$0.2$};
|
||||
\node[state] (fla11) [right=2.2 of gj11] {$0.23$};
|
||||
|
||||
%level 8
|
||||
\node[state] (kehcbi11) [above right=1.3 and 0.6 of k11] {$0.27$};
|
||||
|
||||
\node[state] (d11) [right=of kehcbi11] {d};
|
||||
\node[state] (gjfla11) [right=3.3 of d11] {$0.43$};
|
||||
|
||||
%level 9
|
||||
\node[state] (kehcbid11) [above right=1.3 and 0.6 of kehcbi11] {$0.57$};
|
||||
|
||||
\path[every node/.style={font=\scriptsize}]
|
||||
(e11) edge node [near start] {$1$} (eh11)
|
||||
(h11) edge node [near end] {$0$} (eh11)
|
||||
(b11) edge node [near start] {$1$} (bi11)
|
||||
(i11) edge node [near end] {$0$} (bi11)
|
||||
(eh11) edge node [near start] {$1$} (ehc11)
|
||||
(c11) edge node [near end] {$0$} (ehc11)
|
||||
(f11) edge node [near start] {$1$} (fl11)
|
||||
(l11) edge node [near end] {$0$} (fl11)
|
||||
(bi11) edge node [near start] {$1$} (ehcbi11)
|
||||
(ehc11) edge node [near end] {$0$} (ehcbi11)
|
||||
(g11) edge node [near start] {$1$} (gj11)
|
||||
(j11) edge node [near end] {$0$} (gj11)
|
||||
(fl11) edge node [near start] {$1$} (fla11)
|
||||
(a11) edge node [near end] {$0$} (fla11)
|
||||
(k11) edge node [near start] {$1$} (kehcbi11)
|
||||
(ehcbi11) edge node [near end] {$0$} (kehcbi11)
|
||||
(gj11) edge node [near start] {$1$} (gjfla11)
|
||||
(fla11) edge node [near end] {$0$} (gjfla11)
|
||||
(kehcbi11) edge node [near start] {$1$} (kehcbid11)
|
||||
(d11) edge node [near end] {$0$} (kehcbid11);
|
||||
|
||||
\node (pE11) [below=0.7 of e11] {0.02};
|
||||
\node (pH11) [below=0.7 of h11] {0.02};
|
||||
\node (pB11) [below=0.7 of b11] {0.03};
|
||||
\node (pI11) [below=0.7 of i11] {0.03};
|
||||
\node (pC11) [below=0.7 of c11] {0.05};
|
||||
\node (pF11) [below=0.7 of f11] {0.05};
|
||||
\node (pL11) [below=0.7 of l11] {0.06};
|
||||
\node (pG11) [below=0.7 of g11] {0.1};
|
||||
\node (pJ11) [below=0.7 of j11] {0.1};
|
||||
\node (pA11) [below=0.7 of a11] {0.12};
|
||||
\node (pK11) [below=0.7 of k11] {0.12};
|
||||
\node (pD11) [below=0.7 of d11] {0.3};
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[h!]
|
||||
Elfter Schritt:
|
||||
|
||||
\begin{tikzpicture}[shorten >=1pt,node distance=1.1cm,on grid,auto,/tikz/initial text=]
|
||||
%level 4
|
||||
\node[state] (e12) {e};
|
||||
\node[state] (h12) [right=of e12] {h};
|
||||
|
||||
%level 5
|
||||
\node[state] (eh12) [above right=1.3 and 0.6 of e12] {$0.04$};
|
||||
\node[state] (i12) [left=of eh12] {i};
|
||||
\node[state] (b12) [left=of i12] {b};
|
||||
\node[state] (c12) [right=of eh12] {c};
|
||||
|
||||
%level 6
|
||||
\node[state] (bi12) [above right=1.3 and 0.6 of b12] {$0.06$};
|
||||
\node[state] (l12) [left=1.65 of bi12] {l};
|
||||
\node[state] (f12) [left=of l12] {f};
|
||||
\node[state] (ehc12) [right=2.2 of bi12] {$0.09$};
|
||||
|
||||
%level 7
|
||||
\node[state] (ehcbi12) [above right=1.3 and 1.2 of bi12] {$0.15$};
|
||||
\node[state] (k12) [left=of ehcbi12] {k};
|
||||
\node[state] (a12) [left=of k12] {a};
|
||||
\node[state] (fl12) [left=of a12] {$0.11$};
|
||||
\node[state] (j12) [left=of fl12] {j};
|
||||
\node[state] (g12) [left=of j12] {g};
|
||||
|
||||
%level 8
|
||||
\node[state] (kehcbi12) [above right=1.3 and 0.6 of k12] {$0.27$};
|
||||
\node[state] (fla12) [left=2.2 of kehcbi12] {$0.23$};
|
||||
\node[state] (gj12) [left=2.2 of fla12] {$0.2$};
|
||||
|
||||
\node[state] (d12) [right=of kehcbi12] {d};
|
||||
|
||||
%level 9
|
||||
\node[state] (kehcbid12) [above right=1.3 and 0.6 of kehcbi12] {$0.57$};
|
||||
\node[state] (gjfla12) [left=3.95 of kehcbid12] {$0.43$};
|
||||
|
||||
%level 10
|
||||
\node[state] (gjflakehcbid12) [above right=1.3 and 2.0 of gjfla12] {$1$};
|
||||
|
||||
\path[every node/.style={font=\scriptsize}]
|
||||
(e12) edge node [near start] {$1$} (eh12)
|
||||
(h12) edge node [near end] {$0$} (eh12)
|
||||
(b12) edge node [near start] {$1$} (bi12)
|
||||
(i12) edge node [near end] {$0$} (bi12)
|
||||
(eh12) edge node [near start] {$1$} (ehc12)
|
||||
(c12) edge node [near end] {$0$} (ehc12)
|
||||
(f12) edge node [near start] {$1$} (fl12)
|
||||
(l12) edge node [near end] {$0$} (fl12)
|
||||
(bi12) edge node [near start] {$1$} (ehcbi12)
|
||||
(ehc12) edge node [near end] {$0$} (ehcbi12)
|
||||
(g12) edge node [near start] {$1$} (gj12)
|
||||
(j12) edge node [near end] {$0$} (gj12)
|
||||
(fl12) edge node [near start] {$1$} (fla12)
|
||||
(a12) edge node [near end] {$0$} (fla12)
|
||||
(k12) edge node [near start] {$1$} (kehcbi12)
|
||||
(ehcbi12) edge node [near end] {$0$} (kehcbi12)
|
||||
(gj12) edge node [near start] {$1$} (gjfla12)
|
||||
(fla12) edge node [near end] {$0$} (gjfla12)
|
||||
(kehcbi12) edge node [near start] {$1$} (kehcbid12)
|
||||
(d12) edge node [near end] {$0$} (kehcbid12)
|
||||
(gjfla12) edge node [near start] {$1$} (gjflakehcbid12)
|
||||
(kehcbid12) edge node [near end] {$0$} (gjflakehcbid12);
|
||||
|
||||
\node (pE12) [below=0.7 of e12] {0.02};
|
||||
\node (pH12) [below=0.7 of h12] {0.02};
|
||||
\node (pB12) [below=0.7 of b12] {0.03};
|
||||
\node (pI12) [below=0.7 of i12] {0.03};
|
||||
\node (pC12) [below=0.7 of c12] {0.05};
|
||||
\node (pF12) [below=0.7 of f12] {0.05};
|
||||
\node (pL12) [below=0.7 of l12] {0.06};
|
||||
\node (pG12) [below=0.7 of g12] {0.1};
|
||||
\node (pJ12) [below=0.7 of j12] {0.1};
|
||||
\node (pA12) [below=0.7 of a12] {0.12};
|
||||
\node (pK12) [below=0.7 of k12] {0.12};
|
||||
\node (pD12) [below=0.7 of d12] {0.3};
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
|
||||
\subsection{} %b
|
||||
Mittlerer Informationsgehalt(Entropie):
|
||||
\begin{alignat*}{2}
|
||||
H &=& \sum\limits_{i} p_{i} \cdot \log_{2}{(\frac{1}{p_{i}})} \\
|
||||
H &=& 2 \cdot (0.02 \cdot \log_{2}{(\frac{1}{0.02})}) + 2 \cdot (0.03 \cdot \log_{2}{(\frac{1}{0.03})}) \\
|
||||
& & + 2 \cdot (0.05 \cdot \log_{2}{(\frac{1}{0.05})}) + (0.06 \cdot \log_{2}{(\frac{1}{0.06})})\\
|
||||
& & + 2 \cdot ( 0.1 \cdot \log_{2}{(\frac{1}{0.1})}) + 2 \cdot (0.12 \cdot \log_{2}{(\frac{1}{0.12})})\\
|
||||
& & + (0.3 \cdot \log_{2}{(\frac{1}{0.3})}) \\
|
||||
H &\approx & 3.125 Bit
|
||||
\end{alignat*}
|
||||
\section{}%5.4
|
||||
\subsection{} %a
|
||||
Codieren von Dezimalziffern auf 4-Bit Binärwörter:
|
||||
|
||||
\begin{tikzpicture}
|
||||
\matrix[matrix of math nodes,nodes={rectangle,thin,draw=black!20,minimum size=10mm}]
|
||||
{0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
|
||||
0000 & 0001 & 0010 & 0011 & 0100 & 0101 & 0110 & 0111 & 1000 & 1001 \\};
|
||||
\end{tikzpicture}
|
||||
|
||||
4 Binärstellen für 10 Codewörter, da maximal mögliche Anzahl von Codewörtern gleich $2^{4} = 16$.
|
||||
\begin{alignat*}{2}
|
||||
H_{0} &=& \log_{2}{(2^{4})} \\
|
||||
H_{0} &=& 4 \\
|
||||
\intertext{Also 4 Bit möglicher Informationsgehalt}
|
||||
H &=& 10 \cdot \frac{1}{10} \cdot \log_{2}{(\frac{1}{\frac{1}{10}})} \\
|
||||
H &\approx & 3.32 \\
|
||||
\intertext{Also rund 3.32 Bit mittlerer Informationsgehalt}
|
||||
R &=& H_{0} - H \\
|
||||
R &=& 0.68
|
||||
\intertext{Also rund 0.68 Bit Redundanz}
|
||||
\end{alignat*}
|
||||
\subsection{} %b
|
||||
Es werden $7$ bits verbraucht, da $2^{7}=128$ und $2^{6}=64$.
|
||||
|
||||
Code für die Dezimalziffern 0-9:
|
||||
|
||||
\begin{tikzpicture}
|
||||
\matrix[matrix of math nodes,nodes={rectangle,thin,draw=black!20,minimum size=10mm}]
|
||||
{0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
|
||||
00 & 01 & 02 & 03 & 04 & 05 & 06 & 07 & 08 & 09 \\};
|
||||
\end{tikzpicture}
|
||||
|
||||
Aufgrund der Formel $H_{0}=\log_{2}{2^{7}}=7$ gilt, dass der mögliche Informationsgehalt $7$ beträgt.
|
||||
|
||||
Aufgrund der Formel $H = 100 \cdot \frac{1}{100} \cdot \log_{2}{(\frac{1}{\frac{1}{100}})} \approx 6.43$ beträgt der mittlere Informationsgehalt rund $6.43$ Bit.
|
||||
|
||||
Daraus ergibt sich sich die Redundanz $R = H_{0} - H \approx 0.36$ Bit.
|
||||
|
||||
Die Redundanz bezogen auf eine einzelne Dezimalziffer beträgt $\frac{R}{2} \approx 0.18$ Bit.
|
||||
\subsection{} %c
|
||||
Bei Gruppen von drei Ziffern werden $10$ Bits belegt. Dies ergibt sich aus $2^{10} = 1024$.
|
||||
Demzufolge ist der mögliche Informationsgehalt $H_{0}$ für alle Codewörter gleich $10$ Bit. Bezogen auf eine einzelne Ziffer sind es $\frac{10}{3}$ Bit.
|
||||
|
||||
Bei Gruppen von vier Ziffern werden $14$ Bits belegt. Dies ergibt sich aus $2^{14}=16384$.
|
||||
Demzufolge ist der mögliche Informationsgehalt gleich $14$ Bit. Bezogen auf eine einzelne Ziffer sind es $\frac{14}{4} = \frac{7}{2} = 3.5$ Bit.
|
||||
|
||||
Die Unterschiede ergeben sich durch unterschiedlich viele Kombinationsmöglichkeiten. Bei 3-er Gruppen gibt es $3^{10}$ und bei 4-er Gruppen $4^{10}$ mögliche Kombinationen. Demzufolge ist der mögliche Informationsgehalt pro Ziffer bei letzterem minimal höher.
|
||||
\subsection{} %d
|
||||
|
||||
Code mit variabler Länge (Fano-Code):
|
||||
|
||||
\begin{tikzpicture}
|
||||
\matrix[matrix of nodes]
|
||||
{0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
|
||||
$000$ & $001$ & $010$ & $0110$ & $0111$ & $100$ & $101$ & $110$ & $1110$ & $1111$ \\};
|
||||
\end{tikzpicture}
|
||||
|
||||
\begin{alignat*}{2}
|
||||
P(a_{i}) &=& \frac{1}{10} \\
|
||||
H_{0} &=& (\frac{1}{10} \cdot 4 \cdot \log_{2}{2^{4}}) + (\frac{1}{10} \cdot 6 \cdot \log_{2}{2^{3}}) \\
|
||||
H_{0} &\approx & 1.6 + 1.8 = 3.4\, \text{Bit} \\
|
||||
R &=& H_{0} - H \\
|
||||
R &=& 0.08\,\text{Bit}
|
||||
\end{alignat*}
|
||||
|
||||
Der Code wird noch effizienter, wenn x-beliebige Codewörter zu einem Codewort zusammengefasst werden.
|
||||
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,112 @@
|
|||
\documentclass[10pt,a4paper,oneside,ngerman,numbers=noenddot]{scrartcl}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[ngerman]{babel}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{paralist}
|
||||
\usepackage[locale=DE,exponent-product=\cdot ,detect-all]{siunitx}
|
||||
\usepackage{tikz}
|
||||
\usepackage[scaled=0.78]{luximono}
|
||||
\usepackage{listings}
|
||||
\usetikzlibrary{automata,matrix,fadings,calc,positioning,decorations.pathreplacing,decorations.text,arrows}
|
||||
\pagenumbering{arabic}
|
||||
\def\thesection{6.\arabic{section})}
|
||||
\def\thesubsection{(\alph{subsection})}
|
||||
\def\thesubsubsection{(\arabic{subsubsection})}
|
||||
\renewcommand{\labelenumi}{(\roman{enumi})}
|
||||
\hyphenation{Nach-komma-stel-len}
|
||||
\lstnewenvironment{java}[1][]{%
|
||||
\lstset{basicstyle=\ttfamily ,backgroundcolor=\color[gray]{.95},columns=flexible,fontadjust=true,language=Java,tabsize=4,numbers=none,#1}%
|
||||
}{%
|
||||
}
|
||||
|
||||
\tikzstyle{huffmanNodes}=[matrix of nodes,
|
||||
nodes={circle,thin,draw=black!20,minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\tikzstyle{huffmanBase}=[matrix of nodes,
|
||||
nodes={minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\begin{document}
|
||||
\author{Jim Martens (Matrikelnummer 6420323) \and Marlo Kornblum (Matrikelnummer 6427301)}
|
||||
\title{Rechnerstrukturen Aufgabenblatt 6}
|
||||
\maketitle
|
||||
|
||||
\section{}%6.1
|
||||
Ein zyklisch-einschrittiger Code hat die Voraussetzung, dass pro Schritt nur ein Bit verändert wird. Zusätzlich dürfen das erste und letzte Codewort ebenfalls nur ein Bit auseinanderliegen.
|
||||
|
||||
Bei einem Binärcode ergibt sich die Bedingung, dass die Parität pro Schritt wechselt. Startet man mit 0 und damit gerader Parität, muss als nächstes eine ungerade Parität folgen. Darauf wieder eine gerade und so weiter.
|
||||
|
||||
Die Eigenschaft des zyklischen Codes besagt, dass man an jeder beliebigen Stelle anfangen kann und immer die Bedingung erfüllt ist. Hätte man einen Code mit einer ungeraden Anzahl von Codewörtern, dann würde entweder eine gerade Parität auf eine gerade Parität folgen oder eine ungerade Parität auf eine ungerade Parität, womit der Abstand an mindestens einer Stelle von eins verschieden sein.
|
||||
|
||||
\section{}%6.2
|
||||
\subsection{} %a
|
||||
Die Minimaldistanz $d$ dieses Codes ist $4$, da $4$ Bits geändert werden müssen, um ein neues gültiges Codewort zu erzeugen. Man stelle sich einen Code vor, bei dem überall eine $0$ steht. Setzt man nun ein Datenbit auf $1$, so muss das zugehörige Zeilen- und Spaltenparitätsbit auf $1$ gesetzt werden und das Paritätsbit über die Spaltenparitätsbits muss auch auf $1$ gesetzt werden. Es werden also $4$ Bits verändert.
|
||||
\subsection{} %b
|
||||
Es können nur 1-bit Fehler erkannt und korrigiert werden. Dies liegt daran, dass nur solche Fehler eindeutig lokalisierbar sind. Zweibit- und 3-bit Fehler können noch erkannt werden, aber nicht mehr korrigiert. Dies liegt daran, dass schon bei 2-bit Fehlern mehrere Fehlerkonstellationen zum selben Ergebnis führen.
|
||||
|
||||
4-bit Fehler können nur noch teilweise erkannt werden. Rechteckige 4-bit Fehler können gar nicht mehr erkannt werden.
|
||||
\subsection{} %c
|
||||
Ein Beispiel eines nicht erkannten 4-bit Fehlers ist ein Fehler in den Bits $d_{0,0},d_{0,1},d_{1,0},d_{1,1}$.
|
||||
\subsection{} %d
|
||||
Es gibt $81 \cdot 16 \cdot 16 \cdot 1 = 20736$ Möglichkeiten 4-bit Fehler zu erzeugen, die nicht erkannt werden.
|
||||
|
||||
Das liegt daran, dass das erste Fehlerbit komplett frei in $81$ Feldern gesetzt werden kann. Das zweite Bit hat nun von jeweils $9$ Zeilen und $9$ Spalten noch $8$ Zeilen bzw. $8$ Spalten zur Verfügung. Demzufolge bleiben also $16$ Möglichkeiten. Das dritte Bit hat nun noch je nach Wahl des zweiten Bits $8$ Zeilen und $2$ Spalten zur Verfügung oder $8$ Spalten und $2$ Zeilen. Dies sind also $16$ Möglichkeiten. Das letzte Bit hat nur noch eine einzige mögliche Position.
|
||||
|
||||
Insgesamt gibt es $\binom{81}{4} = 1663740$ Möglichkeiten einen 4-bit Fehler zu erzeugen. In Relation zur Gesamtzahl der möglichen 4-bit Fehler machen die nicht erkennbaren Fehler $\frac{20736}{1663740} \cdot 100 \approx 1.2$ Prozent aus.
|
||||
|
||||
\section{}%6.3
|
||||
\subsection{} %a
|
||||
Anhand eines Beispiels werden wir zeigen, wie ein auftretender 1-bit Fehler lokalisiert und damit korrigiert werden kann. Dazu werden die Codewortstelle $c_{7}$ des 9. Codewortes verfälscht und anschließend die Prüfbits gebildet.\\\\
|
||||
Legende:\\
|
||||
R bedeutet richtig. FoP bedeutet Fehler ohne aktualisierte Prüfbits und FmP bedeutet Fehler mit aktualisierten Prüfbits.
|
||||
|
||||
\begin{tabular}{cccccccc}
|
||||
& $c_{1}$ & $c_{2}$ & $c_{3}$ & $c_{4}$ & $c_{5}$ & $c_{6}$ & $c_{7}$ \\
|
||||
Nr. & $p_{1}$ & $p_{2}$ & $d_{1}$ & $p_{3}$ & $d_{2}$ & $d_{3}$ & $d_{4}$ \\
|
||||
\hline
|
||||
8R & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
|
||||
8FoP & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\
|
||||
8FmP & 0 & 0 & 1 & 1 & 0 & 0 & 1
|
||||
\end{tabular}
|
||||
|
||||
Das Prüfwort gebildet von 8FoP: $x_{a} = 1, x_{b} = 1, x_{c} = 1$.
|
||||
|
||||
Anhand des Prüfwortes lässt sich die Stelle des Fehlers bestimmen. Nimmt man $x_{a}$ als $2^{0}$-Stelle, $x_{b}$ als $2^{1}$ und $x_{c}$ als $2^{2}$ Stelle an, so ergibt sich immer die binäre Ganzzahl der betroffenen Stelle.
|
||||
|
||||
Wichtig hierbei ist, dass das Prüfwort von dem fehlerhaften Code gebildet wird, bei dem die Prüfbits nicht angepasst wurden.
|
||||
\subsection{} %b
|
||||
Mithilfe des angegebenen Schemas für Hamming-Codes kann sowohl die Generatormatrix als auch die Prüfmatrix abgelesen werden.
|
||||
|
||||
Dazu muss man wissen, dass die Prüfbits so gebildet werden:
|
||||
|
||||
Das erste Prüfbit $p_{1}$ enthält auf diesem Schema alle Datenbits, die mit einem roten Punkt markiert sind.
|
||||
|
||||
Das zweite Prüfbit enthält auf diesem Schema alle Datenbits, die mit einem grünen Punkt markiert sind.
|
||||
|
||||
Das dritte Prüfbit enthält auf diesem Schema alle Datenbits, die mit einem dunkelblauen Punkt markiert sind.
|
||||
|
||||
Die Generatormatrix bildet sich zeilenweise über die angegebenen Prüf- und Datenbits. Die erste Zeile entspricht dem ersten Prüfbit, die zweite Zeile dem zweiten Prüfbit, die dritte Zeile dem ersten Datenbit, und so weiter.
|
||||
|
||||
Die Einsen in jeder Zeile markieren welche Datenbits enthalten sind. In den Datenbitzeilen gibt es logischerweise nur eine Eins und zwar bei der Zeile des jeweiligen Datenbits, welches sich selbst enthält. In den Prüfbitzeilen gibt es dort Einsen, wo Datenbits in die Berechnung einfließen.
|
||||
|
||||
Als Generatormatrix ergibt sich bei einem ($7,4$)-Hamming-Code demnach also folgende Matrix:
|
||||
|
||||
\begin{equation*}
|
||||
G := \begin{pmatrix} 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
|
||||
\end{equation*}
|
||||
|
||||
Die Kontrollmatrix lässt sich noch einfacher ablesen. In der ersten Zeile steht überall dort eine Eins, wo auf dem Schema ein roter Punkt ist. Der Rest der Zeile wir mit 0 aufgefüllt.
|
||||
|
||||
In der zweiten Zeile steht überall dort eine Eins, wo auf dem Schema ein grüner Punkt ist. Der Rest der Zeile wir mit 0 aufgefüllt.
|
||||
|
||||
In der dritten Zeile steht überall dort eine Eins, wo auf dem Schema ein dunkelblauer Punkt ist. Der Rest der Zeile wir mit 0 aufgefüllt.
|
||||
|
||||
Daraus ergibt sich für ein ($7,4$)-Hamming-Code Folgendes:
|
||||
|
||||
\begin{equation*}
|
||||
H := \begin{pmatrix} 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 & 1 & 1 \end{pmatrix}
|
||||
\end{equation*}
|
||||
|
||||
Die Kontrollmatrix hat dabei so viele Spalten, wie die Generatormatrix Zeilen hat.
|
||||
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,402 @@
|
|||
\documentclass[10pt,a4paper,oneside,ngerman,numbers=noenddot]{scrartcl}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[ngerman]{babel}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{paralist}
|
||||
\usepackage{qtree}
|
||||
\usepackage[locale=DE,exponent-product=\cdot ,detect-all]{siunitx}
|
||||
\usepackage{tikz}
|
||||
\usepackage[scaled=0.78]{luximono}
|
||||
\usepackage{listings}
|
||||
\usetikzlibrary{automata,matrix,fadings,calc,positioning,decorations.pathreplacing,decorations.text,arrows}
|
||||
\pagenumbering{arabic}
|
||||
\def\thesection{7.\arabic{section})}
|
||||
\def\thesubsection{(\alph{subsection})}
|
||||
\def\thesubsubsection{(\arabic{subsubsection})}
|
||||
\renewcommand{\labelenumi}{(\roman{enumi})}
|
||||
\hyphenation{Nach-komma-stel-len}
|
||||
\lstnewenvironment{java}[1][]{%
|
||||
\lstset{basicstyle=\ttfamily ,backgroundcolor=\color[gray]{.95},columns=flexible,fontadjust=true,language=Java,tabsize=4,numbers=none,#1}%
|
||||
}{%
|
||||
}
|
||||
\graphicspath{{D:/Users/Jim-Studium/Pictures/Studium/RS/}}
|
||||
|
||||
\tikzstyle{huffmanNodes}=[matrix of nodes,
|
||||
nodes={circle,thin,draw=black!20,minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\tikzstyle{huffmanBase}=[matrix of nodes,
|
||||
nodes={minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\begin{document}
|
||||
\author{Jim Martens (Matrikelnummer 6420323) \and Marlo Kornblum (Matrikelnummer 6427301)}
|
||||
\title{Rechnerstrukturen Aufgabenblatt 6}
|
||||
\maketitle
|
||||
|
||||
\section{} %7.1
|
||||
\subsection{} %a
|
||||
DNF:\\
|
||||
\begin{alignat*}{2}
|
||||
f(x) &=& ((\overline{x_{3}} \vee \overline{x_{2}}) \wedge (\overline{x_{2}} \vee \overline{x_{1}}))\vee ((x_{3} \vee \overline{x_{2}}) \wedge (\overline{x_{2}} \vee \overline{x_{1}})) \\
|
||||
&& \vee ((x_{3} \vee x_{2}) \wedge (x_{2} \vee \overline{x_{1}})) \vee ((x_{3} \vee x_{2}) \wedge (x_{2} \vee x_{1}))
|
||||
\end{alignat*}\\
|
||||
KNF:\\
|
||||
\begin{alignat*}{2}
|
||||
f(x) &=& ((x_{3} \vee x_{2}) \wedge (x_{2} \vee x_{1})) \wedge ((x_{3} \vee x_{2}) \wedge (x_{2} \vee \overline{x_{1}})) \wedge ((x_{3} \vee \overline{x_{2}}) \\
|
||||
&& \wedge (x_{2} \vee x_{1})) \wedge ((x_{3} \vee \overline{x_{2}}) \wedge (x_{2} \vee \overline{x_{1}})) \wedge ((\overline{x_{3}} \vee x_{2}) \wedge (x_{2} \vee \overline{x_{1}}))
|
||||
\end{alignat*}\\
|
||||
Reed-Muller Form:\\
|
||||
\begin{alignat*}{2}
|
||||
f(x) &=& (x_{3} \vee \overline{x_{2}}) \wedge (x_{2} \vee \overline{x_{1}}) \\
|
||||
&=& (x_{3} \oplus \overline{x_{2}} \oplus x_{3}\overline{x_{2}}) \wedge (x_{2} \oplus \overline{x_{1}} \oplus x_{2}\overline{x_{1}}) \\
|
||||
&=& (x_{3} \oplus (x_{2} \oplus 1) \oplus x_{3}(x_{2} \oplus 1)) \wedge (x_{2} \oplus (x_{1} \oplus 1) \oplus x_{2}(x_{1} \oplus 1)) \\
|
||||
&=& (x_{3} \oplus x_{2} \oplus 1 \oplus x_{3}x_{2} \oplus x_{3}) \wedge (x_{2} \oplus x_{1} \oplus 1 \oplus x_{2}x_{1} \oplus x_{2}) \\
|
||||
&=& (x_{2} \oplus 1 \oplus x_{3}x_{2}) \wedge (x_{1} \oplus 1 \oplus x_{2}x_{1})
|
||||
\end{alignat*}
|
||||
\subsection{} %b
|
||||
DNF:\\
|
||||
\begin{alignat*}{2}
|
||||
g(x) &=& (\overline{x_{3}} \oplus x_{1}) \vee (x_{3} \oplus \overline{x_{1}})
|
||||
\end{alignat*}\\
|
||||
KNF:\\
|
||||
\begin{alignat*}{2}
|
||||
g(x) &=& (x_{3} \oplus x_{1}) \wedge (\overline{x_{3}} \oplus \overline{x_{1}})
|
||||
\end{alignat*}\\
|
||||
Reed-Muller Form:\\
|
||||
\begin{alignat*}{2}
|
||||
g(x) &=& \overline{x_{3}} \oplus \overline{x_{1}} \\
|
||||
&=& (x_{3} \oplus 1) \oplus (x_{1} \oplus 1) \\
|
||||
&=& x_{3} \oplus 1 \oplus x_{1} \oplus 1 \\
|
||||
&=& x_{3} \oplus x_{1}
|
||||
\end{alignat*}
|
||||
\section{} %7.2
|
||||
\subsection{} %a
|
||||
AND: $a \wedge b = \overline{(\overline{a \wedge b}) \wedge (\overline{a \wedge b})}$\\
|
||||
ODER: $a \vee b = \overline{\overline{a} \wedge \overline{b}} = \overline{(\overline{a \wedge a}) \wedge (\overline{b \wedge b})}$\\
|
||||
Negation: $\overline{a} = \overline{a \wedge a}$\\
|
||||
\subsection{} %b
|
||||
\begin{alignat*}{2}
|
||||
f(x_{3}, x_{2}, x_{1}) &=&& (\overline{x_{3}}(\overline{x_{2}} \vee x_{1})) \vee (x_{1}(\overline{x_{2}} \vee x_{1})) \\
|
||||
&=&& ((\overline{x_{3}} \wedge \overline{x_{2}}) \vee \overline{x_{3}}x_{1}) \vee (x_{1}\overline{x_{2}} \vee x_{1}x_{1})
|
||||
%&=&& ( \overline{( \overline{( \overline{x_{3}} \wedge \overline{x_{2}} ) \wedge ( \overline{x_{3}} \wedge \overline{x_{2}} ) } ) \wedge ( \overline{( \overline{x_{3}}x_{1} ) \wedge ( \overline{x_{3}}x_{1} )} )} ) \vee \\
|
||||
%&&&( \overline{ (\overline{ (x_{1} \wedge \overline{x_{2}}) \wedge (x_{1} \wedge \overline{x_{2}})}) \wedge (\overline{ (x_{1} \wedge x_{1}) \wedge (x_{1} \wedge x_{1})}) } ) \\
|
||||
%
|
||||
%&=&& (\overline{(\overline{( (\overline{x_{3} \wedge x_{3}}) \wedge (\overline{x_{2} \wedge x_{2}}))} \wedge ((\overline{x_{3} \wedge x_{3}}) \wedge (\overline{x_{2} \wedge x_{2}) )}) \wedge (\overline{( (\overline{x_{3} \wedge x_{3}}) \wedge x_{1}) \wedge ( (\overline{x_{3} \wedge x_{3}}) \wedge x_{1})})}) \vee \\
|
||||
%&&&(\overline{(\overline{(x_{1} \wedge (\overline{x_{2} \wedge x_{2}})) \wedge (x_{1} \wedge (\overline{x_{2} \wedge x_{2}}) )}) \wedge (\overline{ (x_{1} \wedge x_{1}) \wedge (x_{1} \wedge x_{1})})}) \\
|
||||
%
|
||||
%&=&& \overline{(\overline{(\overline{( (\overline{x_{3} \wedge x_{3}}) \wedge (\overline{x_{2} \wedge x_{2}}))} \wedge ((\overline{x_{3} \wedge x_{3}}) \wedge (\overline{x_{2} \wedge x_{2}) )}) \wedge (\overline{( (\overline{x_{3} \wedge x_{3}}) \wedge x_{1}) \wedge ( (\overline{x_{3} \wedge x_{3}}) \wedge x_{1})})}) \wedge (\overline{(\overline{( (\overline{x_{3} \wedge x_{3}}) \wedge (\overline{x_{2} \wedge x_{2}}))} \wedge ((\overline{x_{3} \wedge x_{3}}) \wedge (\overline{x_{2} \wedge x_{2}) )}) \wedge (\overline{( (\overline{x_{3} \wedge x_{3}}) \wedge x_{1}) \wedge ( (\overline{x_{3} \wedge x_{3}}) \wedge x_{1})})})} \wedge \\
|
||||
%&&&\overline{(\overline{(\overline{(x_{1} \wedge (\overline{x_{2} \wedge x_{2}})) \wedge (x_{1} \wedge (\overline{x_{2} \wedge x_{2}}) )}) \wedge (\overline{ (x_{1} \wedge x_{1}) \wedge (x_{1} \wedge x_{1})})}) \wedge (\overline{(\overline{(x_{1} \wedge (\overline{x_{2} \wedge x_{2}})) \wedge (x_{1} \wedge (\overline{x_{2} \wedge x_{2}}) )}) \wedge (\overline{ (x_{1} \wedge x_{1}) \wedge (x_{1} \wedge x_{1})})})}
|
||||
%wie soll das bitte gehen? (Trick17?)
|
||||
\end{alignat*}
|
||||
|
||||
\section{} %7.3
|
||||
\subsection{} %a
|
||||
\begin{tabular}{r|ccccc}
|
||||
dez.& $x_{3}$ & $x_{2}$ & $x_{1}$ & $x_{0}$ & A \\
|
||||
\hline
|
||||
0 & 0 & 0 & 0 & 0 & 1 \\
|
||||
1 & 0 & 0 & 0 & 1 & 0 \\
|
||||
2 & 0 & 0 & 1 & 0 & 1 \\
|
||||
3 & 0 & 0 & 1 & 1 & 1 \\
|
||||
4 & 0 & 1 & 0 & 0 & 0 \\
|
||||
5 & 0 & 1 & 0 & 1 & 1 \\
|
||||
6 & 0 & 1 & 1 & 0 & 1 \\
|
||||
7 & 0 & 1 & 1 & 1 & 1 \\
|
||||
8 & 1 & 0 & 0 & 0 & 1 \\
|
||||
9 & 1 & 0 & 0 & 1 & 1 \\
|
||||
10 & 1 & 0 & 1 & 0 & * \\
|
||||
11 & 1 & 0 & 1 & 1 & * \\
|
||||
12 & 1 & 1 & 0 & 0 & * \\
|
||||
13 & 1 & 1 & 0 & 1 & * \\
|
||||
14 & 1 & 1 & 1 & 0 & * \\
|
||||
15 & 1 & 1 & 1 & 1 & *
|
||||
\end{tabular}
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(2,0); %obere Kante
|
||||
\draw (0,0) -- +(0,-2); %linke Kante
|
||||
\draw (1,0) -- +(0,-2); %mittlere vertikale Kante
|
||||
\draw (0,-1) -- +(2,0); %mittlere horizontale Kante
|
||||
\draw (0,-2) -- +(2,0); %untere Kante
|
||||
\draw (2,0) -- +(0,-2); %rechte Kante
|
||||
\draw (0.5,0) -- +(0,-2); %linke mittlere Kante
|
||||
\draw (1.5,0) -- +(0,-2); %rechte mittlere Kante
|
||||
\draw (0,-0.5) -- +(2,0); %obere mittlere Kante
|
||||
\draw (0,-1.5) -- +(2,0); %untere mittlere Kante
|
||||
\draw (0,0) -- +(-0.5,0.5); %schräger Strich
|
||||
|
||||
\node at ++(0.2,0.25) (start11) {$00$}; %Variablenzeile 0
|
||||
\node at ++(0.7,0.25) (start12) {$01$};
|
||||
\node at ++(1.2,0.25) (start13) {$11$};
|
||||
\node at ++(1.7,0.25) (start14) {$10$};
|
||||
|
||||
\node at ++(-0.25,-0.2) (start10) {$00$}; %Variablenspalte 0
|
||||
\node at ++(-0.25,-0.7) (start20) {$01$};
|
||||
\node at ++(-0.25,-1.2) (start30) {$11$};
|
||||
\node at ++(-0.25,-1.7) (start40) {$10$};
|
||||
|
||||
\node at ++(-0.8,0.2) (var1) {$x3\,x2$};
|
||||
\node at ++(0.0,0.6) (var1) {$x1\,x0$};
|
||||
|
||||
\node at ++(0.25, -0.25) (row11) {$1$}; %erste Zeile
|
||||
\node at ++(0.75, -0.25) (row12) {$0$};
|
||||
\node at ++(1.25, -0.25) (row13) {$1$};
|
||||
\node at ++(1.75, -0.25) (row14) {$1$};
|
||||
|
||||
\node at ++(0.25, -0.75) (row21) {$0$}; %zweite Zeile
|
||||
\node at ++(0.75, -0.75) (row22) {$1$};
|
||||
\node at ++(1.25, -0.75) (row23) {$1$};
|
||||
\node at ++(1.75, -0.75) (row24) {$1$};
|
||||
|
||||
\node at ++(0.25, -1.25) (row31) {$*$}; %dritte Zeile
|
||||
\node at ++(0.75, -1.25) (row32) {$*$};
|
||||
\node at ++(1.25, -1.25) (row33) {$*$};
|
||||
\node at ++(1.75, -1.25) (row34) {$*$};
|
||||
|
||||
\node at ++(0.25, -1.75) (row41) {$1$}; %vierte Zeile
|
||||
\node at ++(0.75, -1.75) (row42) {$1$};
|
||||
\node at ++(1.25, -1.75) (row43) {$*$};
|
||||
\node at ++(1.75, -1.75) (row44) {$*$};
|
||||
\end{tikzpicture}
|
||||
|
||||
\begin{figure}[h]
|
||||
\begin{tabular}{r|ccccc}
|
||||
dez.& $x_{3}$ & $x_{2}$ & $x_{1}$ & $x_{0}$ & B \\
|
||||
\hline
|
||||
0 & 0 & 0 & 0 & 0 & 1 \\
|
||||
1 & 0 & 0 & 0 & 1 & 1 \\
|
||||
2 & 0 & 0 & 1 & 0 & 1 \\
|
||||
3 & 0 & 0 & 1 & 1 & 1 \\
|
||||
4 & 0 & 1 & 0 & 0 & 1 \\
|
||||
5 & 0 & 1 & 0 & 1 & 0 \\
|
||||
6 & 0 & 1 & 1 & 0 & 0 \\
|
||||
7 & 0 & 1 & 1 & 1 & 1 \\
|
||||
8 & 1 & 0 & 0 & 0 & 1 \\
|
||||
9 & 1 & 0 & 0 & 1 & 1 \\
|
||||
10 & 1 & 0 & 1 & 0 & * \\
|
||||
11 & 1 & 0 & 1 & 1 & * \\
|
||||
12 & 1 & 1 & 0 & 0 & * \\
|
||||
13 & 1 & 1 & 0 & 1 & * \\
|
||||
14 & 1 & 1 & 1 & 0 & * \\
|
||||
15 & 1 & 1 & 1 & 1 & *
|
||||
\end{tabular}
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(2,0); %obere Kante
|
||||
\draw (0,0) -- +(0,-2); %linke Kante
|
||||
\draw (1,0) -- +(0,-2); %mittlere vertikale Kante
|
||||
\draw (0,-1) -- +(2,0); %mittlere horizontale Kante
|
||||
\draw (0,-2) -- +(2,0); %untere Kante
|
||||
\draw (2,0) -- +(0,-2); %rechte Kante
|
||||
\draw (0.5,0) -- +(0,-2); %linke mittlere Kante
|
||||
\draw (1.5,0) -- +(0,-2); %rechte mittlere Kante
|
||||
\draw (0,-0.5) -- +(2,0); %obere mittlere Kante
|
||||
\draw (0,-1.5) -- +(2,0); %untere mittlere Kante
|
||||
\draw (0,0) -- +(-0.5,0.5); %schräger Strich
|
||||
|
||||
\node at ++(0.2,0.25) (start11) {$00$}; %Variablenzeile 0
|
||||
\node at ++(0.7,0.25) (start12) {$01$};
|
||||
\node at ++(1.2,0.25) (start13) {$11$};
|
||||
\node at ++(1.7,0.25) (start14) {$10$};
|
||||
|
||||
\node at ++(-0.25,-0.2) (start10) {$00$}; %Variablenspalte 0
|
||||
\node at ++(-0.25,-0.7) (start20) {$01$};
|
||||
\node at ++(-0.25,-1.2) (start30) {$11$};
|
||||
\node at ++(-0.25,-1.7) (start40) {$10$};
|
||||
|
||||
\node at ++(-0.8,0.2) (var1) {$x3\,x2$};
|
||||
\node at ++(0.0,0.6) (var1) {$x1\,x0$};
|
||||
|
||||
\node at ++(0.25, -0.25) (row11) {$1$}; %erste Zeile
|
||||
\node at ++(0.75, -0.25) (row12) {$1$};
|
||||
\node at ++(1.25, -0.25) (row13) {$1$};
|
||||
\node at ++(1.75, -0.25) (row14) {$1$};
|
||||
|
||||
\node at ++(0.25, -0.75) (row21) {$1$}; %zweite Zeile
|
||||
\node at ++(0.75, -0.75) (row22) {$0$};
|
||||
\node at ++(1.25, -0.75) (row23) {$1$};
|
||||
\node at ++(1.75, -0.75) (row24) {$0$};
|
||||
|
||||
\node at ++(0.25, -1.25) (row31) {$*$}; %dritte Zeile
|
||||
\node at ++(0.75, -1.25) (row32) {$*$};
|
||||
\node at ++(1.25, -1.25) (row33) {$*$};
|
||||
\node at ++(1.75, -1.25) (row34) {$*$};
|
||||
|
||||
\node at ++(0.25, -1.75) (row41) {$1$}; %vierte Zeile
|
||||
\node at ++(0.75, -1.75) (row42) {$1$};
|
||||
\node at ++(1.25, -1.75) (row43) {$*$};
|
||||
\node at ++(1.75, -1.75) (row44) {$*$};
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
\subsection{} %b
|
||||
A:\\
|
||||
Es lassen sich vier Schleifen bilden. Man kann die dritte und vierte Spalte und dritte und vierte Zeile jeweils komplett in eine Schleife packen. Zudem kann die 1 ganz oben links mit der 1 ganz unten links verbunden werden. Außerdem lassen sich die 1 in der zweiten Zeile und Spalte mit dem Don't-Care-Term in der zweiten Spalte und dritten Zeile verbinden.
|
||||
|
||||
Daraus ergibt sich dieser Term für A:
|
||||
\begin{equation*}
|
||||
A(x_{3},x_{2},x_{1},x_{0}) = (x_{3}) \vee (x_{1}) \vee (x_{2}\overline{x_{1}}x_{0}) \vee (\overline{x_{2}} \wedge \overline{x_{1}} \wedge \overline{x_{0}})
|
||||
\end{equation*}
|
||||
\\
|
||||
B:\\
|
||||
Es lassen sich auch hier wieder vier Schleifen bilden. Man kann die dritte und vierte Zeile komplett in eine Schleife üacken. Zudem kann man die erste Zeile, erste Spalte und dritte Spalte jeweils in eine Schleife packen.
|
||||
|
||||
Daraus ergibt sich dieser Term für B:
|
||||
\begin{equation*}
|
||||
B(x_{3},x_{2},x_{1},x_{0}) = (x_{3}) \vee (x_{1}x_{0}) \vee (\overline{x_{3}} \wedge \overline{x_{2}}) \vee (\overline{x_{1}} \wedge \overline{x_{0}})
|
||||
\end{equation*}
|
||||
\section{} %7.4
|
||||
\subsection{} %a
|
||||
\begin{tabular}{r|ccccc}
|
||||
dez.& $x_{3}$ & $x_{2}$ & $x_{1}$ & $x_{0}$ & y \\
|
||||
\hline
|
||||
0 & 0 & 0 & 0 & 0 & 0 \\
|
||||
1 & 0 & 0 & 0 & 1 & 0 \\
|
||||
2 & 0 & 0 & 1 & 0 & 0 \\
|
||||
3 & 0 & 0 & 1 & 1 & 0 \\
|
||||
4 & 0 & 1 & 0 & 0 & 0 \\
|
||||
5 & 0 & 1 & 0 & 1 & 1 \\
|
||||
6 & 0 & 1 & 1 & 0 & 0 \\
|
||||
7 & 0 & 1 & 1 & 1 & 1 \\
|
||||
8 & 1 & 0 & 0 & 0 & 0 \\
|
||||
9 & 1 & 0 & 0 & 1 & 0 \\
|
||||
10 & 1 & 0 & 1 & 0 & 0 \\
|
||||
11 & 1 & 0 & 1 & 1 & 0 \\
|
||||
12 & 1 & 1 & 0 & 0 & 1 \\
|
||||
13 & 1 & 1 & 0 & 1 & 1 \\
|
||||
14 & 1 & 1 & 1 & 0 & 1 \\
|
||||
15 & 1 & 1 & 1 & 1 & 1
|
||||
\end{tabular}
|
||||
\subsection{} %b
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(2,0); %obere Kante
|
||||
\draw (0,0) -- +(0,-2); %linke Kante
|
||||
\draw (1,0) -- +(0,-2); %mittlere vertikale Kante
|
||||
\draw (0,-1) -- +(2,0); %mittlere horizontale Kante
|
||||
\draw (0,-2) -- +(2,0); %untere Kante
|
||||
\draw (2,0) -- +(0,-2); %rechte Kante
|
||||
\draw (0.5,0) -- +(0,-2); %linke mittlere Kante
|
||||
\draw (1.5,0) -- +(0,-2); %rechte mittlere Kante
|
||||
\draw (0,-0.5) -- +(2,0); %obere mittlere Kante
|
||||
\draw (0,-1.5) -- +(2,0); %untere mittlere Kante
|
||||
\draw (0,0) -- +(-0.5,0.5); %schräger Strich
|
||||
|
||||
\node at ++(0.2,0.25) (start11) {$00$}; %Variablenzeile 0
|
||||
\node at ++(0.7,0.25) (start12) {$01$};
|
||||
\node at ++(1.2,0.25) (start13) {$11$};
|
||||
\node at ++(1.7,0.25) (start14) {$10$};
|
||||
|
||||
\node at ++(-0.25,-0.2) (start10) {$00$}; %Variablenspalte 0
|
||||
\node at ++(-0.25,-0.7) (start20) {$01$};
|
||||
\node at ++(-0.25,-1.2) (start30) {$11$};
|
||||
\node at ++(-0.25,-1.7) (start40) {$10$};
|
||||
|
||||
\node at ++(-0.8,0.2) (var1) {$x3\,x2$};
|
||||
\node at ++(0.0,0.6) (var1) {$x1\,x0$};
|
||||
|
||||
\node at ++(0.25, -0.25) (row11) {$0$}; %erste Zeile
|
||||
\node at ++(0.75, -0.25) (row12) {$0$};
|
||||
\node at ++(1.25, -0.25) (row13) {$0$};
|
||||
\node at ++(1.75, -0.25) (row14) {$0$};
|
||||
|
||||
\node at ++(0.25, -0.75) (row21) {$0$}; %zweite Zeile
|
||||
\node at ++(0.75, -0.75) (row22) {$1$};
|
||||
\node at ++(1.25, -0.75) (row23) {$1$};
|
||||
\node at ++(1.75, -0.75) (row24) {$0$};
|
||||
|
||||
\node at ++(0.25, -1.25) (row31) {$1$}; %dritte Zeile
|
||||
\node at ++(0.75, -1.25) (row32) {$1$};
|
||||
\node at ++(1.25, -1.25) (row33) {$1$};
|
||||
\node at ++(1.75, -1.25) (row34) {$1$};
|
||||
|
||||
\node at ++(0.25, -1.75) (row41) {$0$}; %vierte Zeile
|
||||
\node at ++(0.75, -1.75) (row42) {$0$};
|
||||
\node at ++(1.25, -1.75) (row43) {$0$};
|
||||
\node at ++(1.75, -1.75) (row44) {$0$};
|
||||
\end{tikzpicture}
|
||||
\subsection{} %c
|
||||
Es können zwei Schleifen gebildet werden. Zunächst kann die dritte Zeile komplett als Schleife genommen werden. Außerdem können die beiden Einsen in der zweiten Zeile und die mittleren Einsen in der dritten Zeile in eine Schleife gepackt werden.
|
||||
|
||||
Daraus ergibt sich folgender Term für y:
|
||||
\begin{equation*}
|
||||
y(x_{3},x_{2},x_{1},x_{0}) = (x_{3}x_{2}) \vee (x_{2}x_{0})
|
||||
\end{equation*}
|
||||
|
||||
\subsection{} %d
|
||||
\includegraphics[scale=.5]{Uebung7-Schaltplan.png}
|
||||
\\\\\\\\\\\\\\\\\\\\\\\\
|
||||
\subsection{} %e
|
||||
\begin{figure}[h!bp]
|
||||
\begin{tikzpicture}[shorten >=1pt,node distance=1.1cm,on grid,auto,/tikz/initial text=]
|
||||
%level 4
|
||||
\node[state] (f1) {$1$};
|
||||
%\node[state] (f2) [right=1 of f1] {$1$};
|
||||
\node[state] (f3) [right=2 of f1] {$0$};
|
||||
%\node[state] (f4) [right=3 of f1] {$0$};
|
||||
%\node[state] (f5) [right=4 of f1] {$1$};
|
||||
%\node[state] (f6) [right=5 of f1] {$1$};
|
||||
%\node[state] (f7) [right=6 of f1] {$0$};
|
||||
%\node[state] (f8) [right=7 of f1] {$0$};
|
||||
\node[state] (f9) [right=4 of f1] {$1$};
|
||||
\node[state] (f10) [right=5 of f1] {$0$};
|
||||
\node[state] (f11) [right=6 of f1] {$0$};
|
||||
%\node[state] (f12) [right=11 of f1] {$0$};
|
||||
\node[state] (f13) [right=8 of f1] {$1$};
|
||||
\node[state] (f14) [right=9 of f1] {$0$};
|
||||
%\node[state] (f15) [right=14 of f1] {$0$};
|
||||
\node[state] (f16) [right=11 of f1] {$0$};
|
||||
|
||||
%level 3
|
||||
%\node[state] (x31) [above right=1.3 and 0.5 of f1] {$x_{3}$};
|
||||
%\node[state] (x32) [right=2 of x31] {$x_{3}$};
|
||||
%\node[state] (x33) [above right=1.3 and 0.5 of f5] {$x_{3}$};
|
||||
%\node[state] (x34) [right=2 of x33] {$x_{3}$};
|
||||
\node[state] (x35) [above right=1.3 and 0.5 of f9] {$x_{3}$};
|
||||
%\node[state] (x36) [right=2 of x35] {$x_{3}$};
|
||||
\node[state] (x37) [right=4 of x35] {$x_{3}$};
|
||||
%\node[state] (x38) [right=6 of x35] {$x_{3}$};
|
||||
|
||||
%level 2
|
||||
\node[state] (x23) [above right=1.3 and 1 of x35] {$x_{2}$};
|
||||
%\node[state] (x22) [left=4 of x23] {$x_{2}$};
|
||||
\node[state] (x21) [left=4 of x23] {$x_{2}$};
|
||||
\node[state] (x24) [right=4 of x23] {$x_{2}$};
|
||||
|
||||
%level 1
|
||||
%\node[state] (x11) [above right=1.3 and 2 of x21] {$x_{1}$};
|
||||
\node[state] (x12) [above right=1.3 and 2 of x23] {$x_{1}$};
|
||||
|
||||
%level 0
|
||||
\node[state] (x0) [above left=1.3 and 4 of x12] {$x_{0}$};
|
||||
|
||||
\path[every node/.style={font=\scriptsize}]
|
||||
(f1) edge node [near start] {$1$} (x21)
|
||||
%(f2) edge node [near end] {$0$} (x31)
|
||||
(f3) edge node [near end] {$0$} (x21)
|
||||
%(f4) edge node [near end] {$0$} (x32)
|
||||
%(f5) edge node [near start] {$1$} (x22)
|
||||
%(f6) edge node [near end] {$0$} (x33)
|
||||
%(f7) edge node [near end] {$0$} (x22)
|
||||
%(f8) edge node [near end] {$0$} (x34)
|
||||
(f9) edge node [near start] {$1$} (x35)
|
||||
(f10) edge node [near end] {$0$} (x35)
|
||||
(f11) edge node [near end] {$0$} (x23)
|
||||
%(f12) edge node [near end] {$0$} (x36)
|
||||
(f13) edge node [near start] {$1$} (x37)
|
||||
(f14) edge node [near end] {$0$} (x37)
|
||||
%(f15) edge node [near start] {$1$} (x38)
|
||||
(f16) edge node [near end] {$0$} (x24)
|
||||
%(x31) edge node [near start] {$1$} (x21)
|
||||
%(x32) edge node [near end] {$0$} (x21)
|
||||
%(x33) edge node [near start] {$1$} (x22)
|
||||
%(x34) edge node [near end] {$0$} (x22)
|
||||
(x35) edge node [near start] {$1$} (x23)
|
||||
%(x36) edge node [near end] {$0$} (x23)
|
||||
(x37) edge node [near start] {$1$} (x24)
|
||||
%(x38) edge node [near end] {$0$} (x24)
|
||||
(x21) edge node [near start] {$1$} (x0)
|
||||
%(x22) edge node [near end] {$0$} (x11)
|
||||
(x23) edge node [near start] {$1$} (x12)
|
||||
(x24) edge node [near end] {$0$} (x12)
|
||||
%(x11) edge node [near start] {$1$} (x0)
|
||||
(x12) edge node [near end] {$0$} (x0);
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,154 @@
|
|||
\documentclass[10pt,a4paper,oneside,ngerman,numbers=noenddot]{scrartcl}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[ngerman]{babel}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{paralist}
|
||||
\usepackage{qtree}
|
||||
\usepackage[locale=DE,exponent-product=\cdot ,detect-all]{siunitx}
|
||||
\usepackage{tikz}
|
||||
\usepackage[scaled=0.78]{luximono}
|
||||
\usepackage{listings}
|
||||
\usetikzlibrary{automata,matrix,fadings,calc,positioning,decorations.pathreplacing,decorations.text,arrows}
|
||||
\pagenumbering{arabic}
|
||||
\def\thesection{8.\arabic{section})}
|
||||
\def\thesubsection{(\alph{subsection})}
|
||||
\def\thesubsubsection{(\arabic{subsubsection})}
|
||||
\renewcommand{\labelenumi}{(\roman{enumi})}
|
||||
\hyphenation{Nach-komma-stel-len}
|
||||
\lstnewenvironment{java}[1][]{%
|
||||
\lstset{basicstyle=\ttfamily ,backgroundcolor=\color[gray]{.95},columns=flexible,fontadjust=true,language=Java,tabsize=4,numbers=none,#1}%
|
||||
}{%
|
||||
}
|
||||
\graphicspath{{D:/Users/Jim-Studium/Pictures/Studium/RS/}}
|
||||
|
||||
\tikzstyle{huffmanNodes}=[matrix of nodes,
|
||||
nodes={circle,thin,draw=black!20,minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\tikzstyle{huffmanBase}=[matrix of nodes,
|
||||
nodes={minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\begin{document}
|
||||
\author{Jim Martens (Matrikelnummer 6420323) \and Marlo Kornblum (Matrikelnummer 6427301)}
|
||||
\title{Rechnerstrukturen Aufgabenblatt 8}
|
||||
\maketitle
|
||||
|
||||
\section{} %8.1
|
||||
1-bit Volladdierer nur mit 2:1 Multiplexern:\\
|
||||
\begin{alignat*}{2}
|
||||
s &=& a \oplus b \oplus c_{i} \\
|
||||
&=& ((a \vee b) \wedge (\overline{a \wedge b})) \oplus c_{i} \\
|
||||
&=& (((a \vee b) \wedge (\overline{a \wedge b})) \vee c_{i}) \wedge (\overline{((a \vee b) \wedge (\overline{a \wedge b})) \wedge c_{i}}) \\
|
||||
c_{0} &=& (ab) \vee (a \vee b)c_{i}
|
||||
\end{alignat*}
|
||||
|
||||
\includegraphics[scale=.5]{Uebung8-Schaltplan.png}
|
||||
\section{} %8.2
|
||||
\subsection{} %a
|
||||
\begin{tabular}{r|ccccc}
|
||||
dez.& $x_{3}$ & $x_{2}$ & $x_{1}$ & $x_{0}$ & A \\
|
||||
\hline
|
||||
0 & 0 & 0 & 0 & 0 & 1 \\
|
||||
1 & 0 & 0 & 0 & 1 & 1 \\
|
||||
2 & 0 & 0 & 1 & 0 & 1 \\
|
||||
3 & 0 & 0 & 1 & 1 & 1 \\
|
||||
4 & 0 & 1 & 0 & 0 & 1 \\
|
||||
5 & 0 & 1 & 0 & 1 & 0 \\
|
||||
6 & 0 & 1 & 1 & 0 & 0 \\
|
||||
7 & 0 & 1 & 1 & 1 & 0 \\
|
||||
8 & 1 & 0 & 0 & 0 & 1 \\
|
||||
9 & 1 & 0 & 0 & 1 & 1 \\
|
||||
10 & 1 & 0 & 1 & 0 & 1 \\
|
||||
11 & 1 & 0 & 1 & 1 & 1 \\
|
||||
12 & 1 & 1 & 0 & 0 & 1 \\
|
||||
13 & 1 & 1 & 0 & 1 & 1 \\
|
||||
14 & 1 & 1 & 1 & 0 & 1 \\
|
||||
15 & 1 & 1 & 1 & 1 & 1
|
||||
\end{tabular}
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(2,0); %obere Kante
|
||||
\draw (0,0) -- +(0,-2); %linke Kante
|
||||
\draw (1,0) -- +(0,-2); %mittlere vertikale Kante
|
||||
\draw (0,-1) -- +(2,0); %mittlere horizontale Kante
|
||||
\draw (0,-2) -- +(2,0); %untere Kante
|
||||
\draw (2,0) -- +(0,-2); %rechte Kante
|
||||
\draw (0.5,0) -- +(0,-2); %linke mittlere Kante
|
||||
\draw (1.5,0) -- +(0,-2); %rechte mittlere Kante
|
||||
\draw (0,-0.5) -- +(2,0); %obere mittlere Kante
|
||||
\draw (0,-1.5) -- +(2,0); %untere mittlere Kante
|
||||
\draw (0,0) -- +(-0.5,0.5); %schräger Strich
|
||||
|
||||
\node at ++(0.2,0.25) (start11) {$00$}; %Variablenzeile 0
|
||||
\node at ++(0.7,0.25) (start12) {$01$};
|
||||
\node at ++(1.2,0.25) (start13) {$11$};
|
||||
\node at ++(1.7,0.25) (start14) {$10$};
|
||||
|
||||
\node at ++(-0.25,-0.2) (start10) {$00$}; %Variablenspalte 0
|
||||
\node at ++(-0.25,-0.7) (start20) {$01$};
|
||||
\node at ++(-0.25,-1.2) (start30) {$11$};
|
||||
\node at ++(-0.25,-1.7) (start40) {$10$};
|
||||
|
||||
\node at ++(-0.8,0.2) (var1) {$x3\,x2$};
|
||||
\node at ++(0.0,0.6) (var1) {$x1\,x0$};
|
||||
|
||||
\node at ++(0.25, -0.25) (row11) {$1$}; %erste Zeile
|
||||
\node at ++(0.75, -0.25) (row12) {$1$};
|
||||
\node at ++(1.25, -0.25) (row13) {$1$};
|
||||
\node at ++(1.75, -0.25) (row14) {$1$};
|
||||
|
||||
\node at ++(0.25, -0.75) (row21) {$1$}; %zweite Zeile
|
||||
\node at ++(0.75, -0.75) (row22) {$0$};
|
||||
\node at ++(1.25, -0.75) (row23) {$0$};
|
||||
\node at ++(1.75, -0.75) (row24) {$0$};
|
||||
|
||||
\node at ++(0.25, -1.25) (row31) {$1$}; %dritte Zeile
|
||||
\node at ++(0.75, -1.25) (row32) {$1$};
|
||||
\node at ++(1.25, -1.25) (row33) {$1$};
|
||||
\node at ++(1.75, -1.25) (row34) {$1$};
|
||||
|
||||
\node at ++(0.25, -1.75) (row41) {$1$}; %vierte Zeile
|
||||
\node at ++(0.75, -1.75) (row42) {$1$};
|
||||
\node at ++(1.25, -1.75) (row43) {$1$};
|
||||
\node at ++(1.75, -1.75) (row44) {$1$};
|
||||
\end{tikzpicture}
|
||||
\subsection{} %b
|
||||
disjunktiv:\\
|
||||
Es lassen sich die erste und vierte Zeile als eine Schleife einzeichnen. Außerdem kann die erste Spalte als eine Schleife genommen werden. Zusätzlich können die die dritte und vierte Zeile komplett als eine Schleife realisiert werden. Es ergibt sich:
|
||||
\begin{equation*}
|
||||
f(x_{3},x_{2},x_{1},x_{0}) = \overline{x_{2}} \vee (\overline{x_{1}}\overline{x_{0}}) \vee x_{3}
|
||||
\end{equation*}
|
||||
|
||||
konjunktiv:\\
|
||||
Die zwei rechten Nullen und die zwei linken Nullen können jeweils in eine Schleife gepackt werden. Es ergibt sich:
|
||||
\begin{equation*}
|
||||
f(x_{3},x_{2},x_{1},x_{0}) = (x_{3} \vee \overline{x_{2}} \vee \overline{x_{0}}) \wedge (x_{3} \vee \overline{x_{2}} \vee \overline{x_{1}})
|
||||
\end{equation*}
|
||||
\subsection{} %c
|
||||
disjunktiv:\\
|
||||
\includegraphics[scale=.5]{Uebung8_2_c1-Schaltplan.png}
|
||||
|
||||
konjunktiv:\\
|
||||
\includegraphics[scale=.5]{Uebung8_2_c2-Schaltplan.png}
|
||||
\section{} %8.3
|
||||
\subsection{} %a
|
||||
\includegraphics[scale=1.0]{Uebung8-Impulsdiagramm.png}
|
||||
|
||||
Es tritt ein statischer 1-Hazard beim Ausgang x auf. Beim Ausgang y tritt kein Hazard auf.
|
||||
\subsection{} %b
|
||||
\includegraphics[scale=1.0]{Uebung8-Impulsdiagramm2.png}
|
||||
|
||||
Es treten keine Hazards auf.
|
||||
\section{} %8.4
|
||||
\subsection{} %a
|
||||
Für $n$ Bits ergibt sich folgende Formel für die Zeitschritte T: $T(n) = 2 \cdot \log_{2} (n) + 4$.
|
||||
\subsection{} %b
|
||||
Dieser Addierer benötigt $\frac{n}{m} + 1$ Zeitschritte für die Blöcke und daher insgesamt $m \cdot (2 \cdot \frac{n}{m} + 1)$ Zeitschritte.
|
||||
|
||||
Um die Verzögerung zu minimieren muss $m = \lfloor \sqrt{n} \rfloor$ gelten.
|
||||
\subsection{} %c
|
||||
Für $n=64$ ergeben sich für den Ripple-Carry-Addierer $64$ Zeiteinheiten und damit $64 \cdot 35ps=2240ps$. Der Carry-Lookahead-Addierer benötigt $2 \cdot \log_{2} (64) + 4 = 2 \cdot 6 + 4 = 12 + 4 = 16$ Zeiteinheiten und damit $16 \cdot 35ps=560ps$.
|
||||
Mit dem Carry-Select-Addierer erreicht man $8 \cdot (2\cdot 8 + 1) = 8 \cdot (16 + 1) = 8 \cdot 17 = 136$ Zeiteinheiten. Somit ergeben sich $136 \cdot 35ps=4760ps$.
|
||||
|
||||
Für $n=256$ ergeben sich für den Ripple-Carry-Addierer $256$ Zeiteinheiten und damit $256 \cdot 35ps=8960ps$. Der Carry-Lookahead-Addierer benötigt $2 \cdot \log_{2} (256) + 4 = 2 \cdot 8 + 4 = 16 + 4 = 20$ Zeiteinheiten und damit $20 \cdot 35ps=700ps$.
|
||||
Mit dem Carry-Select-Addierer erreicht man $16 \cdot (2 \cdot 16 + 1) = 16 \cdot (32 + 1) = 16 \cdot 33 = 528$ Zeiteinheiten. Somit ergeben sich $528 \cdot 35ps=18480ps$.
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,607 @@
|
|||
\documentclass[10pt,a4paper,oneside,ngerman,numbers=noenddot]{scrartcl}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[ngerman]{babel}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{paralist}
|
||||
\usepackage{qtree}
|
||||
\usepackage[locale=DE,exponent-product=\cdot ,detect-all]{siunitx}
|
||||
\usepackage{tikz}
|
||||
\usepackage[scaled=0.78]{luximono}
|
||||
\usepackage{listings}
|
||||
\usepackage{subfigure}
|
||||
\usetikzlibrary{automata,matrix,fadings,calc,positioning,decorations.pathreplacing,decorations.text,arrows}
|
||||
\pagenumbering{arabic}
|
||||
\def\thesection{9.\arabic{section})}
|
||||
\def\thesubsection{(\alph{subsection})}
|
||||
\def\thesubsubsection{(\arabic{subsubsection})}
|
||||
\renewcommand{\labelenumi}{(\roman{enumi})}
|
||||
\hyphenation{Nach-komma-stel-len}
|
||||
\lstnewenvironment{java}[1][]{%
|
||||
\lstset{basicstyle=\ttfamily ,backgroundcolor=\color[gray]{.95},columns=flexible,fontadjust=true,language=Java,tabsize=4,numbers=none,#1}%
|
||||
}{%
|
||||
}
|
||||
\graphicspath{{D:/Users/Jim-Studium/Pictures/Studium/RS/}}
|
||||
|
||||
\tikzstyle{huffmanNodes}=[matrix of nodes,
|
||||
nodes={circle,thin,draw=black!20,minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\tikzstyle{huffmanBase}=[matrix of nodes,
|
||||
nodes={minimum size=10mm,text height=1.5ex,text depth=.25ex,inner sep=-10pt}]
|
||||
\begin{document}
|
||||
\author{Jim Martens (Matrikelnummer 6420323) \and Marlo Kornblum (Matrikelnummer 6427301)}
|
||||
\title{Rechnerstrukturen Aufgabenblatt 9}
|
||||
\maketitle
|
||||
|
||||
\section{} %9.1
|
||||
\includegraphics[scale=1.0]{Uebung9-1.png}
|
||||
\section{} %9.2
|
||||
\subsection{} %a
|
||||
\begin{figure}[h!]
|
||||
Erste Schaltung:
|
||||
\hfill
|
||||
ZweiteSchaltung:
|
||||
|
||||
\subfigure{
|
||||
\begin{tabular}{ccc||c}
|
||||
D & E & CLK & $Q^{+}$ \\
|
||||
\hline
|
||||
* & * & 0 & Q \\
|
||||
* & * & 1 & Q \\
|
||||
* & 0 & $\uparrow$ & Q \\
|
||||
0 & 1 & $\uparrow$ & 1 \\
|
||||
1 & 1 & $\uparrow$ & 1
|
||||
\end{tabular}
|
||||
}\hfill
|
||||
\subfigure{
|
||||
\begin{tabular}{ccc||c}
|
||||
D & E & CLK & $Q^{+}$ \\
|
||||
\hline
|
||||
* & 0 & * & Q \\
|
||||
* & 1 & 0 & Q \\
|
||||
* & 1 & 1 & Q \\
|
||||
0 & $\uparrow$ & 1 & 0 \\
|
||||
1 & $\uparrow$ & 1 & 1 \\
|
||||
0 & 1 & $\uparrow$ & 0 \\
|
||||
1 & 1 & $\uparrow$ & 1 \\
|
||||
0 & $\uparrow$ & $\uparrow$ & 0 \\
|
||||
1 & $\uparrow$ & $\uparrow$ & 1
|
||||
\end{tabular}
|
||||
}
|
||||
\end{figure}
|
||||
\subsection{} %b
|
||||
Bei beiden Schaltungen werden nur bestimmte Dateneingänge gespeichert. Dies wird für Schaltungen eingesetzt, bei denen sich der Zustand nur in festgelegten Intervallen (Takten) ändern soll.
|
||||
\subsection{} %c
|
||||
Der Vorteil der zweiten Lösung ist die Nichtverwendung eines Multiplexers. Das macht die Schaltung weitaus simpler und reduziert womöglich die Gatterlaufzeiten.
|
||||
Auf der anderen Seite ist ein großer Nachteil der zweiten Lösung, dass auch mit E "geclockt" werden kann. Somit ist nicht immer eindeutig, ob immer ein gewollter Takt anliegt.
|
||||
Durch Hazards kann an ungewollten Stellen geschaltet werden.
|
||||
|
||||
Die Vor- und Nachteile der zweiten Lösung, sind die Nach- und Vorteile der ersten Lösung. Negativ ist, dass ein Multiplexer verwendet wird. Positiv ist hingegen, dass der Takt an sich immer eindeutig ist und auch immer eindeutige Daten am Flipflop ankommen. Entweder D (E ist auf 1) oder Q. Ist E auf 0 kann sich der Folgezustand demzufolge gar nicht ändern.
|
||||
\section{} %9.3
|
||||
\subsection{} %a
|
||||
\begin{tikzpicture}[shorten >=1pt,node distance=2.0cm,on grid,auto,/tikz/initial text=,>=stealth']
|
||||
\tikzset{every state/.style={minimum size=3.6em}}
|
||||
\node[state,initial] (Z0) {$Z_{0}$};
|
||||
\node[state] (Z1) [below right=1.3 and 1.5 of Z0] {$Z_{1}$};
|
||||
\node[state] (Z2) [below right=1.3 and 1.5 of Z1] {$Z_{2}$};
|
||||
\node[state] (Z3) [below left=1.3 and 1.5 of Z2] {$Z_{3}$};
|
||||
\node[state] (Z4) [below left=1.3 and 1.5 of Z3] {$Z_{4}$};
|
||||
\node[state] (Z5) [above left=1.3 and 1.5 of Z4] {$Z_{5}$};
|
||||
\node[state] (Z6) [above left=1.3 and 1.5 of Z5] {$Z_{6}$};
|
||||
\node[state] (Z7) [above right=1.3 and 1.5 of Z6] {$Z_{7}$};
|
||||
|
||||
\node (z0) [below=0.2 of Z0] {\tiny{$000$}};
|
||||
\node (z1) [below=0.2 of Z1] {\tiny{$001$}};
|
||||
\node (z2) [below=0.2 of Z2] {\tiny{$010$}};
|
||||
\node (z3) [below=0.2 of Z3] {\tiny{$011$}};
|
||||
\node (z4) [below=0.2 of Z4] {\tiny{$100$}};
|
||||
\node (z5) [below=0.2 of Z5] {\tiny{$101$}};
|
||||
\node (z6) [below=0.2 of Z6] {\tiny{$110$}};
|
||||
\node (z7) [below=0.2 of Z7] {\tiny{$111$}};
|
||||
|
||||
\path[every node/.style={font=\scriptsize}]
|
||||
(Z0) edge[->] (Z1)
|
||||
(Z1) edge[->] (Z2)
|
||||
(Z2) edge[loop right,->] node [near start] {$i=0$} (Z2)
|
||||
(Z2) edge[->] node [near start] {$i=1$} (Z3)
|
||||
(Z3) edge[->] (Z4)
|
||||
(Z4) edge[->] (Z5)
|
||||
(Z5) edge[->] (Z6)
|
||||
(Z6) edge[->] (Z7)
|
||||
(Z7) edge[->] (Z0);
|
||||
|
||||
\draw (-0.55,-0.3) -- +(1.15,0);
|
||||
\draw (0.95,-1.6) -- +(1.15,0);
|
||||
\draw (2.45,-2.9) -- +(1.15,0);
|
||||
\draw (0.95,-4.2) -- +(1.15,0);
|
||||
\draw (-0.55,-5.5) -- +(1.15,0);
|
||||
\draw (-2.05,-4.2) -- +(1.15,0);
|
||||
\draw (-3.55,-2.9) -- +(1.15,0);
|
||||
\draw (-2.05,-1.6) -- +(1.15,0);
|
||||
|
||||
\node (z00) [below=0.45 of Z0] {\small{$r;r$}};
|
||||
\node (z01) [below=0.45 of Z1] {\small{$r,ge;r$}};
|
||||
\node (z02) [below=0.45 of Z2] {\small{$gr;r$}};
|
||||
\node (z03) [below=0.45 of Z3] {\small{$ge;r$}};
|
||||
\node (z04) [below=0.45 of Z4] {\small{$r;r$}};
|
||||
\node (z05) [below=0.45 of Z5] {\small{$r;r,ge$}};
|
||||
\node (z06) [below=0.45 of Z6] {\small{$r;gr$}};
|
||||
\node (z07) [below=0.45 of Z7] {\small{$r;ge$}};
|
||||
\end{tikzpicture}
|
||||
\subsection{} %b
|
||||
\begin{tabular}{c|ccc||ccc|ccc|ccc}
|
||||
i & $z_{2}$ & $z_{1}$ & $z_{0}$ & $z_{2}^{+}$ & $z_{1}^{+}$ & $z_{0}^{+}$ & $rt_{H}$ & $ge_{H}$ & $gr_{H}$ & $rt_{N}$ & $ge_{N}$ & $gr_{N}$\\
|
||||
\hline
|
||||
* & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0\\
|
||||
* & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0\\
|
||||
0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0\\
|
||||
1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0\\
|
||||
* & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0\\
|
||||
* & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0\\
|
||||
* & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0\\
|
||||
* & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1\\
|
||||
* & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0
|
||||
\end{tabular}
|
||||
\subsection{} %c
|
||||
\begin{figure}[hbp]
|
||||
\subfigure{
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(2,0); %obere Kante
|
||||
\draw (0,0) -- +(0,-2); %linke Kante
|
||||
\draw (1,0) -- +(0,-2); %mittlere vertikale Kante
|
||||
\draw (0,-1) -- +(2,0); %mittlere horizontale Kante
|
||||
\draw (0,-2) -- +(2,0); %untere Kante
|
||||
\draw (2,0) -- +(0,-2); %rechte Kante
|
||||
\draw (0.5,0) -- +(0,-2); %linke mittlere Kante
|
||||
\draw (1.5,0) -- +(0,-2); %rechte mittlere Kante
|
||||
\draw (0,-0.5) -- +(2,0); %obere mittlere Kante
|
||||
\draw (0,-1.5) -- +(2,0); %untere mittlere Kante
|
||||
\draw (0,0) -- +(-0.5,0.5); %schräger Strich
|
||||
|
||||
\node at ++(0.2,0.25) (start11) {$00$}; %Variablenzeile 0
|
||||
\node at ++(0.7,0.25) (start12) {$01$};
|
||||
\node at ++(1.2,0.25) (start13) {$11$};
|
||||
\node at ++(1.7,0.25) (start14) {$10$};
|
||||
|
||||
\node at ++(-0.25,-0.2) (start10) {$00$}; %Variablenspalte 0
|
||||
\node at ++(-0.25,-0.7) (start20) {$01$};
|
||||
\node at ++(-0.25,-1.2) (start30) {$11$};
|
||||
\node at ++(-0.25,-1.7) (start40) {$10$};
|
||||
|
||||
\node at ++(-0.8,0.2) (var1) {$i, z_{2}$};
|
||||
\node at ++(0.0,0.6) (var1) {$z_{1}\,z_{0}$};
|
||||
|
||||
\node at ++(0.25, -0.25) (row11) {$0$}; %erste Zeile
|
||||
\node at ++(0.75, -0.25) (row12) {$0$};
|
||||
\node at ++(1.25, -0.25) (row13) {$0$};
|
||||
\node at ++(1.75, -0.25) (row14) {$1$};
|
||||
|
||||
\node at ++(0.25, -0.75) (row21) {$1$}; %zweite Zeile
|
||||
\node at ++(0.75, -0.75) (row22) {$1$};
|
||||
\node at ++(1.25, -0.75) (row23) {$1$};
|
||||
\node at ++(1.75, -0.75) (row24) {$0$};
|
||||
|
||||
\node at ++(0.25, -1.25) (row31) {$1$}; %dritte Zeile
|
||||
\node at ++(0.75, -1.25) (row32) {$1$};
|
||||
\node at ++(1.25, -1.25) (row33) {$1$};
|
||||
\node at ++(1.75, -1.25) (row34) {$0$};
|
||||
|
||||
\node at ++(0.25, -1.75) (row41) {$0$}; %vierte Zeile
|
||||
\node at ++(0.75, -1.75) (row42) {$0$};
|
||||
\node at ++(1.25, -1.75) (row43) {$0$};
|
||||
\node at ++(1.75, -1.75) (row44) {$1$};
|
||||
\end{tikzpicture}
|
||||
}\hfill
|
||||
\subfigure{
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(2,0); %obere Kante
|
||||
\draw (0,0) -- +(0,-2); %linke Kante
|
||||
\draw (1,0) -- +(0,-2); %mittlere vertikale Kante
|
||||
\draw (0,-1) -- +(2,0); %mittlere horizontale Kante
|
||||
\draw (0,-2) -- +(2,0); %untere Kante
|
||||
\draw (2,0) -- +(0,-2); %rechte Kante
|
||||
\draw (0.5,0) -- +(0,-2); %linke mittlere Kante
|
||||
\draw (1.5,0) -- +(0,-2); %rechte mittlere Kante
|
||||
\draw (0,-0.5) -- +(2,0); %obere mittlere Kante
|
||||
\draw (0,-1.5) -- +(2,0); %untere mittlere Kante
|
||||
\draw (0,0) -- +(-0.5,0.5); %schräger Strich
|
||||
|
||||
\node at ++(0.2,0.25) (start11) {$00$}; %Variablenzeile 0
|
||||
\node at ++(0.7,0.25) (start12) {$01$};
|
||||
\node at ++(1.2,0.25) (start13) {$11$};
|
||||
\node at ++(1.7,0.25) (start14) {$10$};
|
||||
|
||||
\node at ++(-0.25,-0.2) (start10) {$00$}; %Variablenspalte 0
|
||||
\node at ++(-0.25,-0.7) (start20) {$01$};
|
||||
\node at ++(-0.25,-1.2) (start30) {$11$};
|
||||
\node at ++(-0.25,-1.7) (start40) {$10$};
|
||||
|
||||
\node at ++(-0.8,0.2) (var1) {$i, z_{2}$};
|
||||
\node at ++(0.0,0.6) (var1) {$z_{1}\,z_{0}$};
|
||||
|
||||
\node at ++(0.25, -0.25) (row11) {$0$}; %erste Zeile
|
||||
\node at ++(0.75, -0.25) (row12) {$1$};
|
||||
\node at ++(1.25, -0.25) (row13) {$0$};
|
||||
\node at ++(1.75, -0.25) (row14) {$1$};
|
||||
|
||||
\node at ++(0.25, -0.75) (row21) {$0$}; %zweite Zeile
|
||||
\node at ++(0.75, -0.75) (row22) {$1$};
|
||||
\node at ++(1.25, -0.75) (row23) {$0$};
|
||||
\node at ++(1.75, -0.75) (row24) {$1$};
|
||||
|
||||
\node at ++(0.25, -1.25) (row31) {$0$}; %dritte Zeile
|
||||
\node at ++(0.75, -1.25) (row32) {$1$};
|
||||
\node at ++(1.25, -1.25) (row33) {$0$};
|
||||
\node at ++(1.75, -1.25) (row34) {$1$};
|
||||
|
||||
\node at ++(0.25, -1.75) (row41) {$0$}; %vierte Zeile
|
||||
\node at ++(0.75, -1.75) (row42) {$1$};
|
||||
\node at ++(1.25, -1.75) (row43) {$0$};
|
||||
\node at ++(1.75, -1.75) (row44) {$1$};
|
||||
\end{tikzpicture}
|
||||
}
|
||||
|
||||
KV-Diagramm für $z_{2}^{+}$
|
||||
\hfill
|
||||
KV-Diagramm für $z_{1}^{+}$
|
||||
\end{figure}
|
||||
\begin{figure}[hbp]
|
||||
\subfigure{
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(2,0); %obere Kante
|
||||
\draw (0,0) -- +(0,-2); %linke Kante
|
||||
\draw (1,0) -- +(0,-2); %mittlere vertikale Kante
|
||||
\draw (0,-1) -- +(2,0); %mittlere horizontale Kante
|
||||
\draw (0,-2) -- +(2,0); %untere Kante
|
||||
\draw (2,0) -- +(0,-2); %rechte Kante
|
||||
\draw (0.5,0) -- +(0,-2); %linke mittlere Kante
|
||||
\draw (1.5,0) -- +(0,-2); %rechte mittlere Kante
|
||||
\draw (0,-0.5) -- +(2,0); %obere mittlere Kante
|
||||
\draw (0,-1.5) -- +(2,0); %untere mittlere Kante
|
||||
\draw (0,0) -- +(-0.5,0.5); %schräger Strich
|
||||
|
||||
\node at ++(0.2,0.25) (start11) {$00$}; %Variablenzeile 0
|
||||
\node at ++(0.7,0.25) (start12) {$01$};
|
||||
\node at ++(1.2,0.25) (start13) {$11$};
|
||||
\node at ++(1.7,0.25) (start14) {$10$};
|
||||
|
||||
\node at ++(-0.25,-0.2) (start10) {$00$}; %Variablenspalte 0
|
||||
\node at ++(-0.25,-0.7) (start20) {$01$};
|
||||
\node at ++(-0.25,-1.2) (start30) {$11$};
|
||||
\node at ++(-0.25,-1.7) (start40) {$10$};
|
||||
|
||||
\node at ++(-0.8,0.2) (var1) {$i, z_{2}$};
|
||||
\node at ++(0.0,0.6) (var1) {$z_{1}\,z_{0}$};
|
||||
|
||||
\node at ++(0.25, -0.25) (row11) {$1$}; %erste Zeile
|
||||
\node at ++(0.75, -0.25) (row12) {$0$};
|
||||
\node at ++(1.25, -0.25) (row13) {$0$};
|
||||
\node at ++(1.75, -0.25) (row14) {$0$};
|
||||
|
||||
\node at ++(0.25, -0.75) (row21) {$1$}; %zweite Zeile
|
||||
\node at ++(0.75, -0.75) (row22) {$0$};
|
||||
\node at ++(1.25, -0.75) (row23) {$0$};
|
||||
\node at ++(1.75, -0.75) (row24) {$1$};
|
||||
|
||||
\node at ++(0.25, -1.25) (row31) {$1$}; %dritte Zeile
|
||||
\node at ++(0.75, -1.25) (row32) {$0$};
|
||||
\node at ++(1.25, -1.25) (row33) {$0$};
|
||||
\node at ++(1.75, -1.25) (row34) {$1$};
|
||||
|
||||
\node at ++(0.25, -1.75) (row41) {$1$}; %vierte Zeile
|
||||
\node at ++(0.75, -1.75) (row42) {$0$};
|
||||
\node at ++(1.25, -1.75) (row43) {$0$};
|
||||
\node at ++(1.75, -1.75) (row44) {$1$};
|
||||
\end{tikzpicture}
|
||||
}\hfill
|
||||
\subfigure{
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(2,0); %obere Kante
|
||||
\draw (0,0) -- +(0,-2); %linke Kante
|
||||
\draw (1,0) -- +(0,-2); %mittlere vertikale Kante
|
||||
\draw (0,-1) -- +(2,0); %mittlere horizontale Kante
|
||||
\draw (0,-2) -- +(2,0); %untere Kante
|
||||
\draw (2,0) -- +(0,-2); %rechte Kante
|
||||
\draw (0.5,0) -- +(0,-2); %linke mittlere Kante
|
||||
\draw (1.5,0) -- +(0,-2); %rechte mittlere Kante
|
||||
\draw (0,-0.5) -- +(2,0); %obere mittlere Kante
|
||||
\draw (0,-1.5) -- +(2,0); %untere mittlere Kante
|
||||
\draw (0,0) -- +(-0.5,0.5); %schräger Strich
|
||||
|
||||
\node at ++(0.2,0.25) (start11) {$00$}; %Variablenzeile 0
|
||||
\node at ++(0.7,0.25) (start12) {$01$};
|
||||
\node at ++(1.2,0.25) (start13) {$11$};
|
||||
\node at ++(1.7,0.25) (start14) {$10$};
|
||||
|
||||
\node at ++(-0.25,-0.2) (start10) {$00$}; %Variablenspalte 0
|
||||
\node at ++(-0.25,-0.7) (start20) {$01$};
|
||||
\node at ++(-0.25,-1.2) (start30) {$11$};
|
||||
\node at ++(-0.25,-1.7) (start40) {$10$};
|
||||
|
||||
\node at ++(-0.8,0.2) (var1) {$i, z_{2}$};
|
||||
\node at ++(0.0,0.6) (var1) {$z_{1}\,z_{0}$};
|
||||
|
||||
\node at ++(0.25, -0.25) (row11) {$1$}; %erste Zeile
|
||||
\node at ++(0.75, -0.25) (row12) {$1$};
|
||||
\node at ++(1.25, -0.25) (row13) {$0$};
|
||||
\node at ++(1.75, -0.25) (row14) {$0$};
|
||||
|
||||
\node at ++(0.25, -0.75) (row21) {$1$}; %zweite Zeile
|
||||
\node at ++(0.75, -0.75) (row22) {$1$};
|
||||
\node at ++(1.25, -0.75) (row23) {$1$};
|
||||
\node at ++(1.75, -0.75) (row24) {$1$};
|
||||
|
||||
\node at ++(0.25, -1.25) (row31) {$1$}; %dritte Zeile
|
||||
\node at ++(0.75, -1.25) (row32) {$1$};
|
||||
\node at ++(1.25, -1.25) (row33) {$1$};
|
||||
\node at ++(1.75, -1.25) (row34) {$1$};
|
||||
|
||||
\node at ++(0.25, -1.75) (row41) {$1$}; %vierte Zeile
|
||||
\node at ++(0.75, -1.75) (row42) {$1$};
|
||||
\node at ++(1.25, -1.75) (row43) {$0$};
|
||||
\node at ++(1.75, -1.75) (row44) {$0$};
|
||||
\end{tikzpicture}
|
||||
}
|
||||
|
||||
KV-Diagramm für $z_{0}^{+}$
|
||||
\hfill
|
||||
KV-Diagramm für $rt_{H}$
|
||||
\end{figure}
|
||||
\begin{figure}[hbp]
|
||||
\subfigure{
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(2,0); %obere Kante
|
||||
\draw (0,0) -- +(0,-2); %linke Kante
|
||||
\draw (1,0) -- +(0,-2); %mittlere vertikale Kante
|
||||
\draw (0,-1) -- +(2,0); %mittlere horizontale Kante
|
||||
\draw (0,-2) -- +(2,0); %untere Kante
|
||||
\draw (2,0) -- +(0,-2); %rechte Kante
|
||||
\draw (0.5,0) -- +(0,-2); %linke mittlere Kante
|
||||
\draw (1.5,0) -- +(0,-2); %rechte mittlere Kante
|
||||
\draw (0,-0.5) -- +(2,0); %obere mittlere Kante
|
||||
\draw (0,-1.5) -- +(2,0); %untere mittlere Kante
|
||||
\draw (0,0) -- +(-0.5,0.5); %schräger Strich
|
||||
|
||||
\node at ++(0.2,0.25) (start11) {$00$}; %Variablenzeile 0
|
||||
\node at ++(0.7,0.25) (start12) {$01$};
|
||||
\node at ++(1.2,0.25) (start13) {$11$};
|
||||
\node at ++(1.7,0.25) (start14) {$10$};
|
||||
|
||||
\node at ++(-0.25,-0.2) (start10) {$00$}; %Variablenspalte 0
|
||||
\node at ++(-0.25,-0.7) (start20) {$01$};
|
||||
\node at ++(-0.25,-1.2) (start30) {$11$};
|
||||
\node at ++(-0.25,-1.7) (start40) {$10$};
|
||||
|
||||
\node at ++(-0.8,0.2) (var1) {$i, z_{2}$};
|
||||
\node at ++(0.0,0.6) (var1) {$z_{1}\,z_{0}$};
|
||||
|
||||
\node at ++(0.25, -0.25) (row11) {$0$}; %erste Zeile
|
||||
\node at ++(0.75, -0.25) (row12) {$1$};
|
||||
\node at ++(1.25, -0.25) (row13) {$1$};
|
||||
\node at ++(1.75, -0.25) (row14) {$0$};
|
||||
|
||||
\node at ++(0.25, -0.75) (row21) {$0$}; %zweite Zeile
|
||||
\node at ++(0.75, -0.75) (row22) {$0$};
|
||||
\node at ++(1.25, -0.75) (row23) {$0$};
|
||||
\node at ++(1.75, -0.75) (row24) {$0$};
|
||||
|
||||
\node at ++(0.25, -1.25) (row31) {$0$}; %dritte Zeile
|
||||
\node at ++(0.75, -1.25) (row32) {$0$};
|
||||
\node at ++(1.25, -1.25) (row33) {$0$};
|
||||
\node at ++(1.75, -1.25) (row34) {$0$};
|
||||
|
||||
\node at ++(0.25, -1.75) (row41) {$0$}; %vierte Zeile
|
||||
\node at ++(0.75, -1.75) (row42) {$1$};
|
||||
\node at ++(1.25, -1.75) (row43) {$1$};
|
||||
\node at ++(1.75, -1.75) (row44) {$0$};
|
||||
\end{tikzpicture}
|
||||
}\hfill
|
||||
\subfigure{
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(2,0); %obere Kante
|
||||
\draw (0,0) -- +(0,-2); %linke Kante
|
||||
\draw (1,0) -- +(0,-2); %mittlere vertikale Kante
|
||||
\draw (0,-1) -- +(2,0); %mittlere horizontale Kante
|
||||
\draw (0,-2) -- +(2,0); %untere Kante
|
||||
\draw (2,0) -- +(0,-2); %rechte Kante
|
||||
\draw (0.5,0) -- +(0,-2); %linke mittlere Kante
|
||||
\draw (1.5,0) -- +(0,-2); %rechte mittlere Kante
|
||||
\draw (0,-0.5) -- +(2,0); %obere mittlere Kante
|
||||
\draw (0,-1.5) -- +(2,0); %untere mittlere Kante
|
||||
\draw (0,0) -- +(-0.5,0.5); %schräger Strich
|
||||
|
||||
\node at ++(0.2,0.25) (start11) {$00$}; %Variablenzeile 0
|
||||
\node at ++(0.7,0.25) (start12) {$01$};
|
||||
\node at ++(1.2,0.25) (start13) {$11$};
|
||||
\node at ++(1.7,0.25) (start14) {$10$};
|
||||
|
||||
\node at ++(-0.25,-0.2) (start10) {$00$}; %Variablenspalte 0
|
||||
\node at ++(-0.25,-0.7) (start20) {$01$};
|
||||
\node at ++(-0.25,-1.2) (start30) {$11$};
|
||||
\node at ++(-0.25,-1.7) (start40) {$10$};
|
||||
|
||||
\node at ++(-0.8,0.2) (var1) {$i, z_{2}$};
|
||||
\node at ++(0.0,0.6) (var1) {$z_{1}\,z_{0}$};
|
||||
|
||||
\node at ++(0.25, -0.25) (row11) {$0$}; %erste Zeile
|
||||
\node at ++(0.75, -0.25) (row12) {$0$};
|
||||
\node at ++(1.25, -0.25) (row13) {$0$};
|
||||
\node at ++(1.75, -0.25) (row14) {$1$};
|
||||
|
||||
\node at ++(0.25, -0.75) (row21) {$0$}; %zweite Zeile
|
||||
\node at ++(0.75, -0.75) (row22) {$0$};
|
||||
\node at ++(1.25, -0.75) (row23) {$0$};
|
||||
\node at ++(1.75, -0.75) (row24) {$0$};
|
||||
|
||||
\node at ++(0.25, -1.25) (row31) {$0$}; %dritte Zeile
|
||||
\node at ++(0.75, -1.25) (row32) {$0$};
|
||||
\node at ++(1.25, -1.25) (row33) {$0$};
|
||||
\node at ++(1.75, -1.25) (row34) {$0$};
|
||||
|
||||
\node at ++(0.25, -1.75) (row41) {$0$}; %vierte Zeile
|
||||
\node at ++(0.75, -1.75) (row42) {$0$};
|
||||
\node at ++(1.25, -1.75) (row43) {$0$};
|
||||
\node at ++(1.75, -1.75) (row44) {$1$};
|
||||
\end{tikzpicture}
|
||||
}
|
||||
|
||||
KV-Diagramm für $ge_{H}$
|
||||
\hfill
|
||||
KV-Diagramm für $gr_{H}$
|
||||
\end{figure}
|
||||
\begin{figure}[hbp]
|
||||
\subfigure{
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(2,0); %obere Kante
|
||||
\draw (0,0) -- +(0,-2); %linke Kante
|
||||
\draw (1,0) -- +(0,-2); %mittlere vertikale Kante
|
||||
\draw (0,-1) -- +(2,0); %mittlere horizontale Kante
|
||||
\draw (0,-2) -- +(2,0); %untere Kante
|
||||
\draw (2,0) -- +(0,-2); %rechte Kante
|
||||
\draw (0.5,0) -- +(0,-2); %linke mittlere Kante
|
||||
\draw (1.5,0) -- +(0,-2); %rechte mittlere Kante
|
||||
\draw (0,-0.5) -- +(2,0); %obere mittlere Kante
|
||||
\draw (0,-1.5) -- +(2,0); %untere mittlere Kante
|
||||
\draw (0,0) -- +(-0.5,0.5); %schräger Strich
|
||||
|
||||
\node at ++(0.2,0.25) (start11) {$00$}; %Variablenzeile 0
|
||||
\node at ++(0.7,0.25) (start12) {$01$};
|
||||
\node at ++(1.2,0.25) (start13) {$11$};
|
||||
\node at ++(1.7,0.25) (start14) {$10$};
|
||||
|
||||
\node at ++(-0.25,-0.2) (start10) {$00$}; %Variablenspalte 0
|
||||
\node at ++(-0.25,-0.7) (start20) {$01$};
|
||||
\node at ++(-0.25,-1.2) (start30) {$11$};
|
||||
\node at ++(-0.25,-1.7) (start40) {$10$};
|
||||
|
||||
\node at ++(-0.8,0.2) (var1) {$i, z_{2}$};
|
||||
\node at ++(0.0,0.6) (var1) {$z_{1}\,z_{0}$};
|
||||
|
||||
\node at ++(0.25, -0.25) (row11) {$1$}; %erste Zeile
|
||||
\node at ++(0.75, -0.25) (row12) {$1$};
|
||||
\node at ++(1.25, -0.25) (row13) {$1$};
|
||||
\node at ++(1.75, -0.25) (row14) {$1$};
|
||||
|
||||
\node at ++(0.25, -0.75) (row21) {$1$}; %zweite Zeile
|
||||
\node at ++(0.75, -0.75) (row22) {$1$};
|
||||
\node at ++(1.25, -0.75) (row23) {$0$};
|
||||
\node at ++(1.75, -0.75) (row24) {$0$};
|
||||
|
||||
\node at ++(0.25, -1.25) (row31) {$1$}; %dritte Zeile
|
||||
\node at ++(0.75, -1.25) (row32) {$1$};
|
||||
\node at ++(1.25, -1.25) (row33) {$0$};
|
||||
\node at ++(1.75, -1.25) (row34) {$0$};
|
||||
|
||||
\node at ++(0.25, -1.75) (row41) {$1$}; %vierte Zeile
|
||||
\node at ++(0.75, -1.75) (row42) {$1$};
|
||||
\node at ++(1.25, -1.75) (row43) {$1$};
|
||||
\node at ++(1.75, -1.75) (row44) {$1$};
|
||||
\end{tikzpicture}
|
||||
}\hfill
|
||||
\subfigure{
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(2,0); %obere Kante
|
||||
\draw (0,0) -- +(0,-2); %linke Kante
|
||||
\draw (1,0) -- +(0,-2); %mittlere vertikale Kante
|
||||
\draw (0,-1) -- +(2,0); %mittlere horizontale Kante
|
||||
\draw (0,-2) -- +(2,0); %untere Kante
|
||||
\draw (2,0) -- +(0,-2); %rechte Kante
|
||||
\draw (0.5,0) -- +(0,-2); %linke mittlere Kante
|
||||
\draw (1.5,0) -- +(0,-2); %rechte mittlere Kante
|
||||
\draw (0,-0.5) -- +(2,0); %obere mittlere Kante
|
||||
\draw (0,-1.5) -- +(2,0); %untere mittlere Kante
|
||||
\draw (0,0) -- +(-0.5,0.5); %schräger Strich
|
||||
|
||||
\node at ++(0.2,0.25) (start11) {$00$}; %Variablenzeile 0
|
||||
\node at ++(0.7,0.25) (start12) {$01$};
|
||||
\node at ++(1.2,0.25) (start13) {$11$};
|
||||
\node at ++(1.7,0.25) (start14) {$10$};
|
||||
|
||||
\node at ++(-0.25,-0.2) (start10) {$00$}; %Variablenspalte 0
|
||||
\node at ++(-0.25,-0.7) (start20) {$01$};
|
||||
\node at ++(-0.25,-1.2) (start30) {$11$};
|
||||
\node at ++(-0.25,-1.7) (start40) {$10$};
|
||||
|
||||
\node at ++(-0.8,0.2) (var1) {$i, z_{2}$};
|
||||
\node at ++(0.0,0.6) (var1) {$z_{1}\,z_{0}$};
|
||||
|
||||
\node at ++(0.25, -0.25) (row11) {$0$}; %erste Zeile
|
||||
\node at ++(0.75, -0.25) (row12) {$0$};
|
||||
\node at ++(1.25, -0.25) (row13) {$0$};
|
||||
\node at ++(1.75, -0.25) (row14) {$0$};
|
||||
|
||||
\node at ++(0.25, -0.75) (row21) {$0$}; %zweite Zeile
|
||||
\node at ++(0.75, -0.75) (row22) {$1$};
|
||||
\node at ++(1.25, -0.75) (row23) {$1$};
|
||||
\node at ++(1.75, -0.75) (row24) {$0$};
|
||||
|
||||
\node at ++(0.25, -1.25) (row31) {$0$}; %dritte Zeile
|
||||
\node at ++(0.75, -1.25) (row32) {$1$};
|
||||
\node at ++(1.25, -1.25) (row33) {$1$};
|
||||
\node at ++(1.75, -1.25) (row34) {$0$};
|
||||
|
||||
\node at ++(0.25, -1.75) (row41) {$0$}; %vierte Zeile
|
||||
\node at ++(0.75, -1.75) (row42) {$0$};
|
||||
\node at ++(1.25, -1.75) (row43) {$0$};
|
||||
\node at ++(1.75, -1.75) (row44) {$0$};
|
||||
\end{tikzpicture}
|
||||
}
|
||||
|
||||
KV-Diagramm für $rt_{N}$
|
||||
\hfill
|
||||
KV-Diagramm für $ge_{N}$
|
||||
\end{figure}
|
||||
\begin{figure}[hb]
|
||||
\subfigure{
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) -- +(2,0); %obere Kante
|
||||
\draw (0,0) -- +(0,-2); %linke Kante
|
||||
\draw (1,0) -- +(0,-2); %mittlere vertikale Kante
|
||||
\draw (0,-1) -- +(2,0); %mittlere horizontale Kante
|
||||
\draw (0,-2) -- +(2,0); %untere Kante
|
||||
\draw (2,0) -- +(0,-2); %rechte Kante
|
||||
\draw (0.5,0) -- +(0,-2); %linke mittlere Kante
|
||||
\draw (1.5,0) -- +(0,-2); %rechte mittlere Kante
|
||||
\draw (0,-0.5) -- +(2,0); %obere mittlere Kante
|
||||
\draw (0,-1.5) -- +(2,0); %untere mittlere Kante
|
||||
\draw (0,0) -- +(-0.5,0.5); %schräger Strich
|
||||
|
||||
\node at ++(0.2,0.25) (start11) {$00$}; %Variablenzeile 0
|
||||
\node at ++(0.7,0.25) (start12) {$01$};
|
||||
\node at ++(1.2,0.25) (start13) {$11$};
|
||||
\node at ++(1.7,0.25) (start14) {$10$};
|
||||
|
||||
\node at ++(-0.25,-0.2) (start10) {$00$}; %Variablenspalte 0
|
||||
\node at ++(-0.25,-0.7) (start20) {$01$};
|
||||
\node at ++(-0.25,-1.2) (start30) {$11$};
|
||||
\node at ++(-0.25,-1.7) (start40) {$10$};
|
||||
|
||||
\node at ++(-0.8,0.2) (var1) {$i, z_{2}$};
|
||||
\node at ++(0.0,0.6) (var1) {$z_{1}\,z_{0}$};
|
||||
|
||||
\node at ++(0.25, -0.25) (row11) {$0$}; %erste Zeile
|
||||
\node at ++(0.75, -0.25) (row12) {$0$};
|
||||
\node at ++(1.25, -0.25) (row13) {$0$};
|
||||
\node at ++(1.75, -0.25) (row14) {$0$};
|
||||
|
||||
\node at ++(0.25, -0.75) (row21) {$0$}; %zweite Zeile
|
||||
\node at ++(0.75, -0.75) (row22) {$0$};
|
||||
\node at ++(1.25, -0.75) (row23) {$0$};
|
||||
\node at ++(1.75, -0.75) (row24) {$1$};
|
||||
|
||||
\node at ++(0.25, -1.25) (row31) {$0$}; %dritte Zeile
|
||||
\node at ++(0.75, -1.25) (row32) {$0$};
|
||||
\node at ++(1.25, -1.25) (row33) {$0$};
|
||||
\node at ++(1.75, -1.25) (row34) {$1$};
|
||||
|
||||
\node at ++(0.25, -1.75) (row41) {$0$}; %vierte Zeile
|
||||
\node at ++(0.75, -1.75) (row42) {$0$};
|
||||
\node at ++(1.25, -1.75) (row43) {$0$};
|
||||
\node at ++(1.75, -1.75) (row44) {$0$};
|
||||
\end{tikzpicture}
|
||||
}
|
||||
|
||||
KV-Diagramm für $gr_{N}$
|
||||
\end{figure}
|
||||
\end{document}
|
||||
Loading…
Reference in New Issue