[thesis] Top-align figures; add content outline; write sc intro

src/thesis/acronyms.tex

@@ -37,3 +37,18 @@
     short=CN,
     long=chek node
 }
+
+\DeclareAcronym{ber}{
+    short=BER,
+    long=bit error rate
+}
+
+\DeclareAcronym{fer}{
+    short=FER,
+    long=frame error rate
+}
+
+\DeclareAcronym{awgn}{
+    short=AWGN,
+    long=additive white Gaussian noise
+}

src/thesis/chapters/2_fundamentals.tex

@@ -140,7 +140,7 @@ Finally, we differentiate between \textit{soft-decision} decoding, where
 $\bm{y} \in \mathbb{R}^n$, and \textit{hard-decision} decoding, where
 $\bm{y} \in \mathbb{F}_2^n$ \cite[Sec.~1.5.1.3]{ryan_channel_2009}.
 %
-\begin{figure}[h]
+\begin{figure}[t]
     \centering
 
     \tikzset{
@@ -194,8 +194,8 @@ This would quickly render decoding intractable as we increase the block length.
 We can get around this problem by constructing $\bm{H}$ in such a
 manner that the number of nonzero entries grows less than quadratically, e.g.,
 only linearly.
-This is exactly the motivation behind \ac{ldpc} codes \cite[Ch.
+This is exactly the motivation behind \ac{ldpc} codes
-1]{gallager_low_1960}.
+\cite[Ch.~1]{gallager_low_1960}.
 
 %
 % Tanner Graph, VNs and CNs
@@ -217,7 +217,7 @@ the \acp{vn} that make up the corresponding parity check
 Figure \ref{PCM and Tanner graph of the Hamming code} shows this
 construction for the [7,4,3]-Hamming code.
 %
-\begin{figure}[H]
+\begin{figure}[t]
     \centering
 
     \begin{align*}
@@ -296,43 +296,95 @@ and that of a check node $j$ as
 $\mathcal{N}_\text{C} (j) = \left\{ j \in \mathcal{J} : \bm{H}_{j,i}
 = 1 \right\}$.
 
-% TODO: Do we need any of these?
+%
-% \red{
+% Error floor and waterfall regions
-%     \begin{itemize}
+%
-%         \item Cycles (? - Only if needed later)
+
-%         \item Regular vs irregular (? - only if needed later)
+We typically evaluate the performance of LDPC codes using the
-%     \end{itemize}
+\ac{ber} or the \ac{fer} (a \textit{frame} referes to one whole
-% }
+transmitted block in this context).
+Considering an \ac{awgn} channel, \autoref{fig:ldpc-perf} shows a
+qualitative performance characteristic of an \ac{ldpc} code
+\cite[Fig.~1]{costello_spatially_2014}. We talk of the
+\textit{waterfall} and the \textit{error floor} regions.
+
+\begin{figure}[t]
+    \centering
+
+    \begin{tikzpicture}
+        \begin{axis}[
+                domain=-5:5,
+                width=\figwidth,
+                height=\figheight,
+            ]
+            \addplot+[mark=none, line width=1pt]
+            {x^2};
+        \end{axis}
+    \end{tikzpicture}
+
+    \caption{
+        Qualitative performance characteristic of \ac{ldpc} code
+        in an \ac{awgn} channel. Adapted from
+        \cite[Fig.~1]{costello_spatially_2014}.
+    }
+    \label{fig:ldpc-perf}
+\end{figure}
+
+Broadly, there are two kinds of \ac{LDPC} codes, \textit{regular} and
+\textit{irregular}.
+Regular codes are characterized by the fact that the weights, i.e.,
+the numbers of ones, of their rows and columns are constant
+\cite[Sec.~5.1.1]{ryan_channel_2009}.
+Already during their introduction, regular \ac{ldpc} codes where shown to have
+a minimum distance scaling linearly with the block length $n$ for
+large values \cite[Ch.~2,~Theorem~1]{gallager_low_1960},
+which leads to them not exhibiting an error floor under \ac{ml} decoding.
+Irregular codes, on the other hand, generally do exhibit an error floor,
+their redeming quality being the ability to reach near-capacity
+performance in the waterfall region \cite[Intro.]{costello_spatially_2014}.
 
 \subsection{Spatially-Coupled LDPC Codes}
 
 A relatively recent development in the world of \ac{ldpc} codes is
-that of \ac{sc}-\ac{ldpc} codes.\\
+that of \ac{sc}-\ac{ldpc} codes.
-\red{[a bit more history (developed by \ldots, developed from \ldots,
+Their key feature is that they combine the best properties of regular
-\ldots)]}\\
+and irregular codes.
-\red{[core concept]}
+They have a minimum distance that grows linearly with $n$, promising
+good error floor behavior, and capacity approaching
+iterative decoding behavior, promising good performance in the
+waterfall region \cite[Intro.]{costello_spatially_2014}.
 
 \red{
     \begin{itemize}
-        \item Tanner graph + PCM
+        \item Core construction idea
-        \item Key benefits and reasoning behind them
+        \item Tanner graph \cite[Fig.~3]{costello_spatially_2014} +
-        \item Cite \cite{costello_spatially_2014} \cite{hassan_fully_2016}
+            PCM \cite[Eq. 1]{hassan_fully_2016}
     \end{itemize}
 }
 
 \subsection{Belief Propagation}
 
-\red{[short intro]} \\
 \red{[key points (sub-optimal but good enough, low complexity, \ldots)]} \\
 \red{[top-level overview (iterative algorithm that approximates \ldots)]}
 
 \red{
     \begin{itemize}
+        \item Thanks to their sparsity, an efficient iterative
+            decoder exists for LDPC codes, called \ac{bp}.
         \item SPA and NMS algorithms
             % TODO: Would it be better to split this into a separate section?
         \item Sliding-window decoding of SC-LDPC codes
-        \item Cite \cite{ryan_channel_2009} \cite{hassan_fully_2016}
+            \cite{costello_spatially_2014} \cite{hassan_fully_2016}
-            \cite{costello_spatially_2014}
+            \begin{itemize}
+                \item Windowed decoding
+                \item The core property of SC-LDPC decoders is the
+                    passing of reliability information (in the form
+                    of LLRs, i.e., soft information) from one window
+                    to the next.
+                    This way, the highly reliable information from
+                    the initial windows is passed on to subsequent
+                    windows \cite{costello_spatially_2014}.
+            \end{itemize}
     \end{itemize}
 }
 

src/thesis/main.tex

@@ -28,6 +28,8 @@
 %
 
 \newcommand{\red}[1]{\textcolor{red}{#1}}
+\newcommand{\figwidth}{10cm}
+\newcommand{\figheight}{7.5cm}
 
 %
 %