90 lines
2.6 KiB
TeX
90 lines
2.6 KiB
TeX
\chapter{Decoding Techniques}%
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\label{chapter:decoding_techniques}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Decoding using Optimization Methods}%
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\label{sec:dec:Decoding using Optimization Methods}
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%
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% TODOs
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%
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\begin{itemize}
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\item General methodology
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\end{itemize}
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%
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% Figure showing decoding space
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%
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\begin{figure}[H]
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{245}
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\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
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% Cube
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\draw[dashed] (0, 0, 0) -- (2, 0, 0);
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\draw[dashed] (2, 0, 0) -- (2, 0, 2);
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\draw[] (2, 0, 2) -- (0, 0, 2);
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\draw[] (0, 0, 2) -- (0, 0, 0);
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\draw[] (0, 2, 0) -- (2, 2, 0);
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\draw[] (2, 2, 0) -- (2, 2, 2);
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\draw[] (2, 2, 2) -- (0, 2, 2);
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\draw[] (0, 2, 2) -- (0, 2, 0);
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\draw[] (0, 0, 0) -- (0, 2, 0);
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\draw[dashed] (2, 0, 0) -- (2, 2, 0);
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\draw[] (2, 0, 2) -- (2, 2, 2);
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\draw[] (0, 0, 2) -- (0, 2, 2);
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% Polytope Annotations
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\node[codeword] (c000) at (0, 0, 0) {};% {$\left( 0, 0, 0 \right) $};
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\node[codeword] (c101) at (2, 0, 2) {};% {$\left( 1, 0, 1 \right) $};
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\node[codeword] (c110) at (2, 2, 0) {};% {$\left( 1, 1, 0 \right) $};
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\node[codeword] (c011) at (0, 2, 2) {};% {$\left( 0, 1, 1 \right) $};
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\node[color=KITblue, right=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c101] {$\left( 1, 0, 1 \right) $};
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\node[color=KITblue, left=0cm of c110] {$\left( 1, 1, 0 \right) $};
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\node[color=KITblue, left=-0.1cm of c011] {$\left( 0, 1, 1 \right) $};
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% f
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\node[color=KITgreen, fill=KITgreen,
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draw, circle, inner sep=0pt, minimum size=4pt] (f) at (0.9, 0.7, 1) {};
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\node[color=KITgreen, right=0cm of f] {$\boldsymbol{f}$};
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\end{tikzpicture}
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\caption{Hypercube ($n=3$) and valid codewords for a single parity-check code}
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{LP Decoding using ADMM}%
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\label{sec:dec:LP Decoding using ADMM}
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\begin{itemize}
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\item Equivalent ML optimization problem
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\item LP relaxation
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\item ADMM as a solver
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\end{itemize}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Proximal Decoding}%
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\label{sec:dec:Proximal Decoding}
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\begin{itemize}
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\item Formulation of optimization problem
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\item Proximal gradient method as a solver
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\end{itemize}
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