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Added LP Decoding slide

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Andreas Tsouchlos 2022-12-19 03:36:57 +01:00
parent 7426d94324
commit 1bc78b0675
2 changed files with 83 additions and 1 deletions
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1
latex/presentations/midterm/presentation.tex
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@ -6,6 +6,7 @@
%\usepackage{amsmath}
\usepackage{float}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usepackage{subcaption}

83
latex/presentations/midterm/sections/theoretical_background.tex
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@ -18,7 +18,7 @@
\label{sub:Presumptions}
\begin{frame}[t]
\frametitle{Channel \& Modulation}
\frametitle{Presumptions: Channel \& Modulation}
\tikzstyle{mapper} = [rectangle, minimum width=1.5cm, rounded corners=0.1cm, minimum height=0.7cm, text centered, draw=black, fill=KITgreen!80]
\begin{figure}[htpb]
@ -64,6 +64,87 @@
\begin{frame}[t]
\frametitle{LP Decoding}
\begin{minipage}[c]{0.6\linewidth}
\begin{itemize}
\item Codeword Polytope:
\begin{align*}
\text{poly}\left( \mathcal{C} \right) = \left\{ \sum_{\boldsymbol{y}\in\mathcal{C}} \lambda_{\boldsymbol{y}} \boldsymbol{y} : \lambda_{\boldsymbol{y}} \ge 0, \sum_{\boldsymbol{y}\in\mathcal{C}}\lambda_{\boldsymbol{y}} = 1 \right\}, \hspace{5mm} \lambda_{\boldsymbol{y}} \in \mathbb{R}
\end{align*}
\item Cost Function:
\begin{align*}
\gamma_i = \log\left( \frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right)
\end{align*}
\item LP Formulation:
\begin{align*}
&\text{minimize } \sum_{i=1}^{n} \gamma_i f_i, \hspace{5mm} f_i = \sum_{\boldsymbol{y}} \lambda_{\boldsymbol{y}}y_i\\
&\text{subject to } \boldsymbol{f}\in\text{poly}\left( \mathcal{C} \right)
\end{align*}
\end{itemize}
\end{minipage}%
\hfill%
\begin{minipage}[c]{0.4\linewidth}
\begin{figure}[H]
\centering
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{245}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
% Cube
\draw[dashed] (0, 0, 0) -- (2, 0, 0);
\draw[dashed] (2, 0, 0) -- (2, 0, 2);
\draw[] (2, 0, 2) -- (0, 0, 2);
\draw[] (0, 0, 2) -- (0, 0, 0);
\draw[] (0, 2, 0) -- (2, 2, 0);
\draw[] (2, 2, 0) -- (2, 2, 2);
\draw[] (2, 2, 2) -- (0, 2, 2);
\draw[] (0, 2, 2) -- (0, 2, 0);
\draw[] (0, 0, 0) -- (0, 2, 0);
\draw[dashed] (2, 0, 0) -- (2, 2, 0);
\draw[] (2, 0, 2) -- (2, 2, 2);
\draw[] (0, 0, 2) -- (0, 2, 2);
% Codeword Polytope
\draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 0, 2);
\draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 2, 0);
\draw[line width=1pt, color=KITblue] (0, 0, 0) -- (0, 2, 2);
\draw[line width=1pt, color=KITblue] (2, 0, 2) -- (2, 2, 0);
\draw[line width=1pt, color=KITblue] (2, 0, 2) -- (0, 2, 2);
\draw[line width=1pt, color=KITblue] (0, 2, 2) -- (2, 2, 0);
% Polytope Annotations
\node[codeword] (c000) at (0, 0, 0) {};% {$\left( 0, 0, 0 \right) $};
\node[codeword] (c101) at (2, 0, 2) {};% {$\left( 1, 0, 1 \right) $};
\node[codeword] (c110) at (2, 2, 0) {};% {$\left( 1, 1, 0 \right) $};
\node[codeword] (c011) at (0, 2, 2) {};% {$\left( 0, 1, 1 \right) $};
\node[color=KITblue, right=0cm of c000] {$\left( 0, 0, 0 \right) $};
\node[color=KITblue, above=0cm of c101] {$\left( 1, 0, 1 \right) $};
\node[color=KITblue, left=0cm of c110] {$\left( 1, 1, 0 \right) $};
\node[color=KITblue, left=0cm of c011] {$\left( 0, 1, 1 \right) $};
% f
\node[color=KITgreen, fill=KITgreen,
draw, circle, inner sep=0pt, minimum size=4pt] (f) at (0.7, 0.7, 1) {};
\node[color=KITgreen, right=0cm of f] {$\boldsymbol{f}$};
\end{tikzpicture}
\caption{$\text{poly}\left( \mathcal{C} \right)$ for $n=3$}
\end{figure}
\end{minipage}
\end{frame}
\begin{frame}[t]
\frametitle{LP Relaxation}
\todo{TODO}
\end{frame}