an.tsouchlos/ba-thesis
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases 1 Wiki Activity

Reworked introduction to decoding using optimization

Browse Source
This commit is contained in:
Andreas Tsouchlos 2023-02-19 11:02:33 +01:00
parent da7162e0c9
commit 51bec2b5a7
1 changed files with 15 additions and 16 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

31
latex/thesis/chapters/decoding_techniques.tex
View File

@ -30,25 +30,24 @@ the \ac{ML} decoding problem:%
f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c} \right) f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c} \right)
.\end{align*}% .\end{align*}%
% %
\todo{Note about these generally being the same thing, when the a priori probability
is uniformly distributed}%
\todo{Here the two problems are written in terms of $\hat{\boldsymbol{c}}$; below MAP
decoding is applied in terms of $\hat{\boldsymbol{x}}$. Is that a problem?}%
The goal is to arrive at a formulation, where a certain objective function The goal is to arrive at a formulation, where a certain objective function
$f$ has to be minimized under certain constraints:% $f$ has to be minimized under certain constraints:%
% %
\begin{align*} \begin{align*}
\text{minimize } f\left( \boldsymbol{x} \right)\\ \text{minimize } f\left( \boldsymbol{c} \right)\\
\text{subject to \ldots} \text{subject to $\boldsymbol{c} \in D$}
.\end{align*} ,\end{align*}%
%
where $D$ is the domain of values attainable for $c$ and represents the
constraints.
In contrast to the established message-passing decoding algorithms, In contrast to the established message-passing decoding algorithms,
the viewpoint then changes from observing the decoding process in its the viewpoint then changes from observing the decoding process in its
tanner graph representation (as shown in figure \ref{fig:dec:tanner}) tanner graph representation (as shown in figure \ref{fig:dec:tanner})
to a spacial representation, where the codewords are some of the edges to a spacial representation (figure \ref{fig:dec:spacial}),
of a hypercube and the goal is to find that point $\boldsymbol{x}$, where the codewords are some of the edges of a hypercube.
\todo{$\boldsymbol{x}$? Or some other variable?} The goal is to find that point $\boldsymbol{c}$,
which minimizes the objective function $f$ (as shown in figure \ref{fig:dec:spacial}). which minimizes the objective function $f$.
% %
% Figure showing decoding space % Figure showing decoding space
@ -143,11 +142,11 @@ which minimizes the objective function $f$ (as shown in figure \ref{fig:dec:spac
\node[color=KITblue, right=0cm of c110] {$\left( 1, 1, 0 \right) $}; \node[color=KITblue, right=0cm of c110] {$\left( 1, 1, 0 \right) $};
\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $}; \node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
% x % c
\node[color=KITgreen, fill=KITgreen, \node[color=KITgreen, fill=KITgreen,
draw, circle, inner sep=0pt, minimum size=4pt] (f) at (0.9, 0.7, 1) {}; draw, circle, inner sep=0pt, minimum size=4pt] (c) at (0.9, 0.7, 1) {};
\node[color=KITgreen, right=0cm of f] {$\boldsymbol{x}$}; \node[color=KITgreen, right=0cm of c] {$\boldsymbol{c}$};
\end{tikzpicture} \end{tikzpicture}
\caption{Spacial representation of a single parity-check code} \caption{Spacial representation of a single parity-check code}
@ -164,7 +163,6 @@ which minimizes the objective function $f$ (as shown in figure \ref{fig:dec:spac
\label{sec:dec:LP Decoding} \label{sec:dec:LP Decoding}
\Ac{LP} decoding is a subject area introduced by Feldman et al. \Ac{LP} decoding is a subject area introduced by Feldman et al.
\todo{Space before citation?}
\cite{feldman_paper}. They reframe the decoding problem as an \cite{feldman_paper}. They reframe the decoding problem as an
\textit{integer linear program} and subsequently present two relaxations into \textit{integer linear program} and subsequently present two relaxations into
\textit{linear programs}, one representing a formulation of exact \ac{LP} \textit{linear programs}, one representing a formulation of exact \ac{LP}
@ -179,7 +177,8 @@ making the \ac{ML} and \ac{MAP} decoding problems equivalent.}%
% %
\begin{align} \begin{align}
\hat{\boldsymbol{c}} = \argmax_{\boldsymbol{c} \in \mathcal{C}} \hat{\boldsymbol{c}} = \argmax_{\boldsymbol{c} \in \mathcal{C}}
f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c} \right)% f_{\boldsymbol{Y} \mid \boldsymbol{C}}
\left( \boldsymbol{y} \mid \boldsymbol{c} \right)%
\label{eq:lp:ml} \label{eq:lp:ml}
.\end{align}% .\end{align}%
% %