From c088a92b3b4d6a9396cd848c3c79cdf65d286ab0 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Sun, 23 Apr 2023 15:02:27 +0200
Subject: [PATCH] Wrote introduction

---
 latex/thesis/chapters/introduction.tex      | 37 +++++++++++++++------
 latex/thesis/chapters/lp_dec_using_admm.tex |  3 --
 2 files changed, 26 insertions(+), 14 deletions(-)

diff --git a/latex/thesis/chapters/introduction.tex b/latex/thesis/chapters/introduction.tex
index b259b0a..8d4644c 100644
--- a/latex/thesis/chapters/introduction.tex
+++ b/latex/thesis/chapters/introduction.tex
@@ -1,16 +1,31 @@
 \chapter{Introduction}%
 \label{chapter:introduction}
 
+Channel coding using binary linear codes is a way of enhancing the reliability
+of data by detecting and correcting any errors that may have occurred during
+transmission or storage.
+One class of binary linear codes, \ac{LDPC} codes, has become especially
+popular due to being able to reach arbitrarily small probabilities of error
+at code rates up to the capacity of the channel, while retaining a structure
+that allows for very efficient decoding.
+While the established decoders for \ac{LDPC} codes, such as \ac{BP} and the
+\textit{min-sum algorithm}, offer reasonable performance, they are suboptimal
+in most cases and exhibit a so called \textit{error floor} for high \acp{SNR},
+making them unsuitable for applications with extreme reliability requiremnts.
+Optimization based decoding algorithms are an entirely different way of approaching
+the decoding problem, in some cases coming with stronger theoretical guarantees
+and promising to alleviate the error floor issue \cite[Sec. I]{original_admm}.
 
-\begin{itemize}
-    \item Problem definition
-    \item Motivation
-        \begin{itemize}
-            \item Error floor when decoding with BP (seems to not be persent with LP decoding
-                \cite[Sec. I]{original_admm})
-            \item Strong theoretical guarantees that allow for better and better approximations
-                of ML decoding \cite[Sec. I]{original_admm}
-        \end{itemize}
-    \item Results summary
-\end{itemize}
+This thesis aims to further the analysis of optimization based decoding
+algorithms as well as verify and generalize the considerations present in
+the existing literature by considering a variety of different codes.
+Specifically, the \textit{proximal decoding} \cite{proximal_paper}
+algorithm and \ac{LP} decoding using the \ac{ADMM} \cite{original_admm} are explored.
+The two algorithms are analyzed based on their theoretical structure
+and on results of simulations conducted in the scope of this work.
+Approaches to determine the optimal value of each parameter are derived
+and the computational and decoding performance of the algorithms is examined.
+An improvement on proximal decoding is suggested, offering up to $\SI{1}{dB}$
+of gain in decoding performance, depending on the parameters chosen and the
+code considered.
 
diff --git a/latex/thesis/chapters/lp_dec_using_admm.tex b/latex/thesis/chapters/lp_dec_using_admm.tex
index df0b19a..2368ac3 100644
--- a/latex/thesis/chapters/lp_dec_using_admm.tex
+++ b/latex/thesis/chapters/lp_dec_using_admm.tex
@@ -838,9 +838,6 @@ end while
 return $\tilde{\boldsymbol{c}}$
 \end{genericAlgorithm}
 
-\todo{Projection onto $[0, 1]^n$?}
-\todo{Variable initialization}
-
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Analysis and Simulation Results}%