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Rewrote Decoding Techniques introduction

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Andreas Tsouchlos 2023-02-19 17:39:40 +01:00
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latex/thesis/chapters/decoding_techniques.tex
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@ -3,8 +3,10 @@
In this chapter, the decoding techniques examined in this work are detailed. In this chapter, the decoding techniques examined in this work are detailed.
First, an overview of the general methodology of using optimization methods First, an overview of the general methodology of using optimization methods
for channel decoding is given. Afterwards, the specific decoding techniques for channel decoding is given.
themselves are explained. Then, the field of \ac{LP} decoding and an \ac{ADMM}-based \ac{LP} decoding
algorithm are introduced.
Finally, the \textit{proximal decoding} algorithm is presented.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -46,7 +48,7 @@ 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 spatial representation (figure \ref{fig:dec:spatial}), to a spatial representation (figure \ref{fig:dec:spatial}),
where the codewords are some of the edges of a hypercube. where the codewords are some of the edges of a hypercube.
The goal is to find that point $\boldsymbol{c}$, The goal is to find the point $\boldsymbol{c}$,
which minimizes the objective function $f$. which minimizes the objective function $f$.
% %
@ -256,12 +258,12 @@ the transfer matrix would be $\boldsymbol{T}_j =
0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
\end{bmatrix} $ (example taken from \cite[Sec. II, A]{efficient_lp_dec_admm})} \end{bmatrix} $ (example taken from \cite[Sec. II, A]{efficient_lp_dec_admm}).}
(i.e. the relevant components of $\boldsymbol{c}$ for parity-check $j$) (i.e. the relevant components of $\boldsymbol{c}$ for parity-check $j$)
and $\mathcal{P}_{d}$ is the \textit{check polytope}, the convex hull of all and $\mathcal{P}_{d}$ is the \textit{check polytope}, the convex hull of all
binary vectors of length $d$ with even parity% binary vectors of length $d$ with even parity%
\footnote{Essentially $\mathcal{P}_{d_j}$ is the set of vectors that satisfy \footnote{Essentially $\mathcal{P}_{d_j}$ is the set of vectors that satisfy
parity-check $j$, but extended to continuous domain.}% parity-check $j$, but extended to the continuous domain.}%
. .
In figure \ref{fig:dec:poly}, the two relaxations are compared for an In figure \ref{fig:dec:poly}, the two relaxations are compared for an