From 773f3b110943d717637c238938d4c5ae5b9d4435 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Wed, 15 Feb 2023 18:08:11 +0100
Subject: [PATCH] Added introduction to theoretical background; minor changes
 to decoding techniques

---
 latex/thesis/chapters/decoding_techniques.tex |  5 +--
 .../chapters/theoretical_background.tex       | 34 ++++++++++++-------
 2 files changed, 25 insertions(+), 14 deletions(-)

diff --git a/latex/thesis/chapters/decoding_techniques.tex b/latex/thesis/chapters/decoding_techniques.tex
index 8488bef..9df74dd 100644
--- a/latex/thesis/chapters/decoding_techniques.tex
+++ b/latex/thesis/chapters/decoding_techniques.tex
@@ -40,7 +40,7 @@ $f$ has to be minimized under certain constraints:%
 
 In contrast to the established message-passing decoding algorithms,
 the viewpoint then changes from observing the decoding process in its
-tanner graph based representation (as shown in figure \ref{fig:dec:tanner})
+tanner graph representation (as shown in figure \ref{fig:dec:tanner})
 into a spacial representation, where the codewords are some of the edges
 of a hypercube and the goal is to find that point $\boldsymbol{x}$,
 \todo{$\boldsymbol{x}$? Or some other variable?}
@@ -255,7 +255,8 @@ immediately approximalted by a gradient-descent step:%
     \hspace{5mm} \gamma \text{ small}
 .\end{align*}%
 %
-The second step thus becomes%
+The second step thus becomes \todo{Write the formulation optimization problem properly
+    (as shown in the introductory section)}%
 %
 \begin{align*}
     \boldsymbol{s} \leftarrow \boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right),
diff --git a/latex/thesis/chapters/theoretical_background.tex b/latex/thesis/chapters/theoretical_background.tex
index 54cec97..de3bf23 100644
--- a/latex/thesis/chapters/theoretical_background.tex
+++ b/latex/thesis/chapters/theoretical_background.tex
@@ -1,15 +1,14 @@
 \chapter{Theoretical Background}%
 \label{chapter:theoretical_background}
 
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Preliminaries: Channel Model and Modulation}
-\label{sec:theo:Preliminaries: Channel Model and Modulation}
-
-\begin{itemize}
-    \item \Ac{AWGN}
-    \item \Ac{BPSK}
-\end{itemize}
+In this chapter, the theoretical background necessary to understand this
+work is given.
+First, the used notation is clarified.
+The physical aspects are detailed - the used modulation scheme and channel model.
+A short introduction of channel coding with binary linear codes and especially
+\ac{LDPC} codes is given.
+The established methods of decoding LPDC codes are briefly explained.
+Lastly, the optimization methods utilized are described.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -21,8 +20,8 @@
 %
 
 \begin{itemize}
-    \item General remarks on notation (matrices, PDF, etc.)
-    \item \sout{Diagram from midterm presentation}
+    \item General remarks on notation (matrices, \ldots)
+    \item Probabilistic quantities (random variables, \acp{PDF}, \ldots)
 \end{itemize}
 
 %
@@ -59,6 +58,17 @@
     \caption{Overview of notation}
     \label{fig:notation}
 \end{figure}
+\todo{Move figure to 'Channel and Modulation'}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Preliminaries: Channel Model and Modulation}
+\label{sec:theo:Preliminaries: Channel Model and Modulation}
+
+\begin{itemize}
+    \item \Ac{AWGN}
+    \item \Ac{BPSK}
+\end{itemize}
+
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -68,7 +78,7 @@
 \begin{itemize}
     \item Introduction
     \item Binary linear codes
-    \item \Ac{LDPC} codes
+    \item \Ac{LDPC} codes (especially $i$, $j$, parity check matrix $\boldsymbol{H}$, etc.)
 \end{itemize}