218 lines
8.0 KiB
TeX
218 lines
8.0 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}, pdfs vs pmfs vs cdfs, \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 decoding
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\end{itemize}
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Generally, any linear program \todo{Acronym} can be expressed in \textit{standard form}%
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\footnote{The inequality $\boldsymbol{x} \ge \boldsymbol{0}$ is to be
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interpreted componentwise.}
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\cite[Sec. 1.1]{intro_to_lin_opt_book}:%
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%
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\begin{alignat}{3}
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\begin{alignedat}{3}
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\text{minimize }\hspace{2mm} && \boldsymbol{\gamma}^\text{T} \boldsymbol{x} \\
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\text{subject to }\hspace{2mm} && \boldsymbol{A}\boldsymbol{x} & = \boldsymbol{b} \\
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&& \boldsymbol{x} & \ge \boldsymbol{0}.
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\end{alignedat}
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\label{eq:theo:admm_standard}
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\end{alignat}%
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%
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A technique called \textit{lagrangian relaxation}%
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\todo{Citation needed}%
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can then be applied - some of the
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constraints are moved into the objective function itself and the weights
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$\boldsymbol{\lambda}$ are introduced. A new, relaxed problem is formulated:
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%
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\begin{align}
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\begin{aligned}
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\text{minimize }\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\boldsymbol{x}
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+ \boldsymbol{\lambda}^\text{T}\left(\boldsymbol{b}
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- \boldsymbol{A}\boldsymbol{x} \right) \\
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\text{subject to }\hspace{2mm} & \boldsymbol{x} \ge \boldsymbol{0},
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\end{aligned}
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\label{eq:theo:admm_relaxed}
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\end{align}%
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%
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the new objective function being the \textit{lagrangian}%
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%
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\begin{align*}
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\mathcal{L}\left( \boldsymbol{x}, \boldsymbol{b}, \boldsymbol{\lambda} \right)
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= \boldsymbol{\gamma}^\text{T}\boldsymbol{x}
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+ \boldsymbol{\lambda}^\text{T}\left(\boldsymbol{b}
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- \boldsymbol{A}\boldsymbol{x} \right)
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.\end{align*}%
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This problem is not directly equivalent to the original one, as the
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solution now depends on the choice of the \textit{lagrange multipliers}
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$\boldsymbol{\lambda}$.
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Interestingly, for our particular class of problems,
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the optimal objective of the relaxed problem (\ref{eq:theo:admm_relaxed}) is a lower bound for
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the optimal objective of the original problem (\ref{eq:theo:admm_standard})
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\cite[Sec. 4.1]{intro_to_lin_opt_book}:%
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%
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\begin{align*}
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\min_{\substack{\boldsymbol{x} \ge \boldsymbol{0} \\ \phantom{a}}}
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\mathcal{L}\left( \boldsymbol{x}, \boldsymbol{b}, \boldsymbol{\lambda}
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\right)
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\le
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\min_{\substack{\boldsymbol{x} \ge \boldsymbol{0} \\ \boldsymbol{A}\boldsymbol{x}
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= \boldsymbol{b}}}
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\boldsymbol{\gamma}^\text{T}\boldsymbol{x}
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.\end{align*}
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%
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Furthermore, for linear programs \textit{strong duality}
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always holds.
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\todo{Citation needed}
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This means that not only is it a lower bound, the tightest lower
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bound actually reaches the value itself:
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%
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\begin{align*}
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\max_{\boldsymbol{\lambda}} \, \min_{\boldsymbol{x} \ge \boldsymbol{0}}
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\mathcal{L}\left( \boldsymbol{x}, \boldsymbol{b}, \boldsymbol{\lambda} \right)
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= \min_{\substack{\boldsymbol{x} \ge \boldsymbol{0} \\ \boldsymbol{A}\boldsymbol{x}
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= \boldsymbol{b}}}
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\boldsymbol{\gamma}^\text{T}\boldsymbol{x}
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.\end{align*}
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%
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In other words, with the optimal choice of $\boldsymbol{\lambda}$,
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the optimal objectives of the problems (\ref{eq:theo:admm_relaxed})
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and (\ref{eq:theo:admm_standard}) have the same value.
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Thus, we can define the \textit{dual problem} as the search for the tightest lower bound:%
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%
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\begin{align}
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\text{maximize }\hspace{2mm} & \min_{\boldsymbol{x} \ge \boldsymbol{0}} \mathcal{L}
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\left( \boldsymbol{x}, \boldsymbol{b}, \boldsymbol{\lambda} \right)
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\label{eq:theo:dual}
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,\end{align}
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%
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and recover the optimal point $\boldsymbol{x}_{\text{opt}}$
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(the solution to problem (\ref{eq:theo:admm_standard}))
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from the dual optimal point $\boldsymbol{\lambda}_\text{opt}$
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(the solution to problem (\ref{eq:theo:dual}))
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by computing \cite[Sec. 2.1]{admm_distr_stats}%
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%
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\begin{align}
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\boldsymbol{x}_{\text{opt}} = \argmin_{\boldsymbol{x}}
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\mathcal{L}\left( \boldsymbol{x}, \boldsymbol{b},
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\boldsymbol{\lambda}_{\text{opt}} \right)
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\label{eq:theo:admm_obtain_primal}
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.\end{align}
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%
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The dual problem can then be solved using \textit{dual ascent}: starting with an
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initial estimate of $\boldsymbol{\lambda}$, calculate an estimate for $\boldsymbol{x}$
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using equation (\ref{eq:theo:admm_obtain_primal}); then, update $\boldsymbol{\lambda}$
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using gradient descent \cite[Sec. 2.1]{admm_distr_stats}:%
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%
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\begin{align*}
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\boldsymbol{x} &\leftarrow \argmin_{\boldsymbol{x}} \mathcal{L}\left(
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\boldsymbol{x}, \boldsymbol{b}, \boldsymbol{\lambda} \right) \\
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\boldsymbol{\lambda} &\leftarrow \boldsymbol{\lambda}
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+ \alpha\left( \boldsymbol{A}\boldsymbol{x} - \boldsymbol{b} \right),
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\hspace{5mm} \alpha > 0
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.\end{align*}
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