192 lines
7.3 KiB
TeX
192 lines
7.3 KiB
TeX
\section{Theoretical Background}%
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\label{sec:Theoretical Background}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Motivation}%
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\label{sub:Motivation}
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\begin{frame}[t]
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\frametitle{Motivation}
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\begin{itemize}
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\item The general [ML] decoding problem for linear codes and the general problem
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of finding the weights of a linear code are both NP-complete \cite{ml_np_hard_proof}.
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\item The iterative message–passing algorithms preferred in practice do not guarantee
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optimality and may fail to decode correctly when the graph contains cycles
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\cite{ldpc_conv}.
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\item The standard message-passing algorithms used for decoding LDPC and turbo codes
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are often difficult to analyze. \cite{feldman_thesis}
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{Previous Work}
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\begin{figure}[h]
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\centering
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\begin{subfigure}{0.33\textwidth}
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\centering
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\fbox{\includegraphics[page=1,width=.6\textwidth]{res/Bachelor_Thesis_Yanxia_Lu}}
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\end{subfigure}%
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\begin{subfigure}{0.33\textwidth}
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\centering
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\fbox{\includegraphics[page=25,width=.6\textwidth]{res/Bachelor_Thesis_Yanxia_Lu}}
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\end{subfigure}%
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\begin{subfigure}{0.33\textwidth}
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\centering
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\fbox{\includegraphics[page=60,width=.6\textwidth]{res/Bachelor_Thesis_Yanxia_Lu}}
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\end{subfigure}%
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\caption{Bachelor's Thesis by Yanxia Lu \cite{yanxia_lu_thesis}}
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\end{figure}
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\begin{itemize}
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\item Analysis of ``Proximal Decoding''
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\item Analysis of ``Interior Point Decoding''
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Presumptions}%
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\label{sub:Presumptions}
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\begin{frame}[t]
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\frametitle{Presumptions: Channel \& Modulation}
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\tikzstyle{mapper} = [rectangle, minimum width=1.5cm, minimum height=0.7cm,
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rounded corners=0.1cm, text centered, draw=black, fill=KITgreen!80]
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\begin{figure}[htpb]
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\centering
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\begin{tikzpicture}[scale=1, transform shape]
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\node (in) {$\boldsymbol{c}$};
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\node[mapper, right=0.5cm of in] (bpskmap) {Mapper};
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\node[right=1.5cm of bpskmap,
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draw, circle, inner sep=0pt, minimum size=0.5cm] (add) {$+$};
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\node[below=0.5cm of add] (noise) {$\boldsymbol{n}$};
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\node[mapper, right=1.5cm of add] (decoder) {Decoder};
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\node[mapper, right=1.5cm of decoder] (demapper) {Demapper};
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\node[right=0.5cm of demapper] (out) {$\boldsymbol{\hat{c}}$};
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\node at ($(bpskmap.east)!0.5!(add.west) + (0,0.3cm)$) {$\boldsymbol{x}$};
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\node at ($(add.east)!0.5!(decoder.west) + (0,0.3cm)$) {$\boldsymbol{y}$};
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\node at ($(decoder.east)!0.5!(demapper.west) + (0,0.3cm)$) {$\boldsymbol{\hat{x}}$};
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\draw[->] (in) -- (bpskmap);
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\draw[->] (bpskmap) -- (add);
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\draw[->] (add) -- (decoder);
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\draw[->] (noise) -- (add);
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\draw[->] (decoder) -- (demapper);
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\draw[->] (demapper) -- (out);
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\end{tikzpicture}
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\end{figure}
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\begin{itemize}
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\item All simulations are performed with BPSK:
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\begin{align*}
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\boldsymbol{x} = \left( -1 \right)^{\boldsymbol{c}},
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\hspace{5mm} \boldsymbol{c} \in \mathbb{F}_2^n,
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\hspace{2mm} \boldsymbol{x} \in \mathbb{R}^n
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\end{align*}
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\item The channel model is AWGN:
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\begin{align*}
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\boldsymbol{y} = \boldsymbol{x} + \boldsymbol{n},
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\hspace{5mm}\boldsymbol{n}\sim \mathcal{N}
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\left(0,\frac{1}{2}\left(\frac{k}{n}\frac{E_b}{N_0}\right)^{-1}\right),
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\hspace{2mm} \boldsymbol{y}, \boldsymbol{n} \in \mathbb{R}^n
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\end{align*}
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\item All-zeros assumption:
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\begin{align*}
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\boldsymbol{c} = \boldsymbol{0}
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\end{align*}
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Optimization as a Decoding Method}%
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\label{sub:Optimization as a Decoding Method}
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\begin{frame}[t]
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\frametitle{Optimization as a Decoding Method}
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\begin{minipage}[c]{0.6\linewidth}
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\begin{itemize}
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\item Reformulate decoding problem as optimization problem
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\begin{itemize}
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\item Establish objective function
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\item Establish constraints
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\end{itemize}
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\item Use optimization method to solve the new problem
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\end{itemize}
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\end{minipage}%
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\hfill%
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\begin{minipage}[c]{0.4\linewidth}
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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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% Codeword Polytope
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% \draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 0, 2);
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% \draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 2, 0);
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% \draw[line width=1pt, color=KITblue] (0, 0, 0) -- (0, 2, 2);
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% \draw[line width=1pt, color=KITblue] (2, 0, 2) -- (2, 2, 0);
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% \draw[line width=1pt, color=KITblue] (2, 0, 2) -- (0, 2, 2);
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% \draw[line width=1pt, color=KITblue] (0, 2, 2) -- (2, 2, 0);
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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) $};
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\node[color=KITblue, left=0cm of c011] {$\left( 0, 1, 1 \right) $};
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% f
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\node[color=KITgreen, fill=KITgreen,
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draw, circle, inner sep=0pt, minimum size=4pt] (f) at (0.7, 0.7, 1) {};
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\node[color=KITgreen, right=0cm of f] {$\tilde{\boldsymbol{c}}$};
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\end{tikzpicture}
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\caption{Hypercube ($n=3$) with valid codewords for single parity-check}
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\end{figure}
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\end{minipage}
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\end{frame}
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