109 lines
3.4 KiB
TeX
109 lines
3.4 KiB
TeX
\chapter{Theoretical Background}%
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\label{chapter:theoretical_background}
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In this chapter, the theoretical background necessary to understand this
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work is given.
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First, the used notation is clarified.
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The physical aspects are detailed - the used modulation scheme and channel model.
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A short introduction of channel coding with binary linear codes and especially
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\ac{LDPC} codes is given.
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The established methods of decoding LPDC codes are briefly explained.
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Lastly, the optimization methods utilized are described.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Notation}
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\label{sec:theo:Notation}
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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 remarks on notation (matrices, \ldots)
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\item Probabilistic quantities (random variables, \acp{PDF}, \ldots)
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\end{itemize}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Preliminaries: Channel Model and Modulation}
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\label{sec:theo:Preliminaries: Channel Model and Modulation}
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%
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% TODOs
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%
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\begin{itemize}
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\item \Ac{AWGN}
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\item \Ac{BPSK}
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\end{itemize}
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%
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% Figure showing notation for entire coding / decoding process
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%
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\tikzstyle{box} = [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[box, 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) {$\boldsymbol{z}$};
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\node[box, right=1.5cm of add] (decoder) {Decoder};
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\node[box, 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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\caption{Overview of notation}
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\label{fig:notation}
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\end{figure}
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\todo{Note about $\tilde{\boldsymbol{c}}$ (and maybe $\tilde{\boldsymbol{x}}$?)}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Channel Coding with LDPC Codes}
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\label{sec:theo:Channel Coding with LDPC Codes}
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\begin{itemize}
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\item Introduction
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\item Binary linear codes
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\item \Ac{LDPC} codes (especially $i$, $j$, parity check matrix $\boldsymbol{H}$, $N\left( j \right) $ \& $N\left( i \right) $, etc.)
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\end{itemize}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Decoding LDPC Codes using Belief Propagation}
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\label{sec:theo:Decoding LDPC Codes using Belief Propagation}
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\begin{itemize}
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\item Introduction to message passing
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\item Overview of \ac{BP} algorithm
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\end{itemize}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Optimization Methods}
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\label{sec:theo:Optimization Methods}
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\begin{itemize}
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\item \Ac{ADMM}
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\item Proximal gradient method
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\end{itemize}
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