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Made polytope example figure more understandable

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Andreas Tsouchlos 2023-02-19 15:45:15 +01:00
parent 8880423d1e
commit 047e4f4009
1 changed files with 136 additions and 139 deletions
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latex/thesis/chapters/decoding_techniques.tex
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@ -270,8 +270,8 @@ Figure \ref{fig:dec:poly:exact} shows the codeword polytope
$\text{poly}\left( \mathcal{C} \right) $, i.e. the constraints for the
equivalent linear program to exact \ac{ML} decoding - only valid codewords are
feasible solutions.
Figures \ref{fig:dec:poly:local1} and \ref{fig:dec:poly:local2} show the local
codeword polytopes of each check node.
Figure \ref{fig:dec:poly:local} shows the local codeword polytope of each check
node.
Their intersection, the relaxed codeword polytope $\overline{Q}$, is shown in
figure \ref{fig:dec:poly:relaxed}.
It can be seen that the relaxed codeword polytope $\overline{Q}$ introduces
@ -302,35 +302,9 @@ The resulting formulation of the relaxed optimization problem is the following:%
% Left side - codeword polytope
%
\begin{subfigure}[c]{0.45\textwidth}
\begin{subfigure}[b]{0.49\textwidth}
\centering
\begin{subfigure}{\textwidth}
\centering
\begin{align*}
\boldsymbol{H} &=
\begin{bmatrix}
1 & 1 & 1\\
0 & 1 & 1
\end{bmatrix}\\[1em]
\mathcal{C} &= \left\{
\begin{bmatrix}
0\\
0\\
0
\end{bmatrix},
\begin{bmatrix}
0\\
1\\
1
\end{bmatrix}
\right\}
\end{align*}
\caption{Definition of the visualized code}
\label{fig:dec:poly:code_def}
\end{subfigure} \\[7em]
\begin{subfigure}{\textwidth}
\centering
@ -338,7 +312,7 @@ The resulting formulation of the relaxed optimization problem is the following:%
draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
\begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
% Cube
\coordinate (p000) at (0, 0, 0);
@ -389,17 +363,20 @@ The resulting formulation of the relaxed optimization problem is the following:%
% Right side - relaxed polytope
%
%
\begin{subfigure}[c]{0.45\textwidth}
\begin{subfigure}[b]{0.49\textwidth}
\centering
\begin{subfigure}{\textwidth}
\centering
\begin{minipage}{0.5\textwidth}
\centering
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
\begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
% Cube
\coordinate (p000) at (0, 0, 0);
@ -451,19 +428,15 @@ The resulting formulation of the relaxed optimization problem is the following:%
\node[color=KITblue, right=0cm of c110] {$\left( 1, 1, 0 \right) $};
\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
\end{tikzpicture}
\caption{Local codeword polytope of check node\\ $j=1$
$\left( c_1 + c_2 + c_3 = 0 \right)$}
\label{fig:dec:poly:local1}
\end{subfigure} \\[1em]
\begin{subfigure}{\textwidth}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
\begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
% Cube
\coordinate (p000) at (0, 0, 0);
@ -511,21 +484,41 @@ The resulting formulation of the relaxed optimization problem is the following:%
\node[color=KITblue, below=0cm of c100] {$\left( 1, 0, 0 \right) $};
\node[color=KITblue, above=0cm of c111] {$\left( 1, 1, 1 \right) $};
\end{tikzpicture}
\end{minipage}
\caption{Local codeword polytope of check node\\ $j=2$
$\left( c_2 + c_3 = 0\right)$}
\label{fig:dec:poly:local2}
\begin{tikzpicture}
\node[color=KITblue, align=center] at (-2,0)
{$j=1$\\ $\left( c_1 + c_2+ c_3 = 0 \right) $};
\node[color=KITblue, align=center] at (2,0)
{$j=2$\\ $\left(c_2 + c_3 = 0\right)$};
\end{tikzpicture}
\caption{Local codeword polytopes of the check nodes}
\label{fig:dec:poly:local}
\end{subfigure}\\[1em]
\begin{subfigure}{\textwidth}
\centering
\begin{tikzpicture}
\draw (-2, 0) -- (2, 0);
\draw (-2, 0.5) -- (-2, 0);
\draw (2, 0.5) -- (2, 0);
\node (intersection) at (0, -0.5) {Intersection};
\draw (0, 0) -- (intersection);
\draw[->] (intersection) -- (0, -1);
\end{tikzpicture}
\vspace{2mm}
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt]
\tikzstyle{pseudocodeword} = [color=KITred, fill=KITred,
draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
\begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
% Cube
\coordinate (p000) at (0, 0, 0);
@ -578,7 +571,11 @@ The resulting formulation of the relaxed optimization problem is the following:%
\end{subfigure}
\caption{Visualization of the codeword polytope and the relaxed codeword
polytope}
polytope for the code defined by the parity check matrix $\boldsymbol{H} =
\begin{bmatrix}
1 & 1 & 1\\
0 & 1 & 1
\end{bmatrix}$}
\label{fig:dec:poly}
\end{figure}%
%