Enable externalization; Finish writing SC-LDPC section; Add (unfinished) Tanner graph figure

src/thesis/chapters/2_fundamentals.tex

@@ -416,32 +416,154 @@ 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 Core construction idea
-        \item Tanner graph \cite[Fig.~3]{costello_spatially_2014} +
-            PCM \cite[Eq. 1]{hassan_fully_2016}
-    \end{itemize}
-}
+The essential property of \ac{sc}-\ac{ldpc} codes is that codewords
+from different \textit{spatial positions}, that would ordinarily be sent
+one after the other independently, are coupled.
+This is achieved by connecting some \acp{vn} of one spatial position to
+\acp{cn} of another, resulting in a \ac{pcm} of the form
+\cite[Eq.~1]{hassan_fully_2016}
+% TODO: Find reference and make sure notation is correct
+%
+\begin{align*}
+    \bm{H} =
+    \begin{pmatrix}
+        \bm{H}_0(1) &        &             & & \\
+        \vdots      & \ddots &             & & \\
+        \bm{H}_W(1) &        & \bm{H}_0(L) & & \\
+        & \ddots &             & & \\
+        &        & \bm{H}_W(L) & & \\
+    \end{pmatrix}
+    ,
+\end{align*}
+%
+where $W \in \mathbb{N}$ is the \textit{coupling width}.
+This construction results in a Tanner graph as depicted in
+\autoref{fig:sc-ldpc-tanner}.
+
+% TODO: Create SC-LDPC graphic
+\begin{figure}[t]
+    \centering
+
+    \tikzset{
+        VN/.style={
+            circle, fill=KITgreen, minimum width=1mm, minimum height=1mm,
+        },
+        CN/.style={
+            rectangle, fill=KITblue, minimum width=1mm, minimum height=1mm,
+        },
+    }
+
+    % TODO: Coupling over more spatial positions
+    \begin{tikzpicture}[node distance=7mm and 1cm]
+        \node[VN]                  (vn00) {};
+        \node[VN, below = of vn00] (vn01) {};
+        \node[VN, below = of vn01] (vn02) {};
+        \node[VN, below = of vn02] (vn03) {};
+        \node[VN, below = of vn03] (vn04) {};
+
+        \coordinate (temp) at ($(vn01)!0.5!(vn02)$);
+
+        \node[CN, left = of temp] (cn00) {};
+        \node[CN, below = of cn00] (cn01) {};
+
+        \draw (vn00) -- (cn00);
+        \draw (vn01) -- (cn00);
+        \draw (vn03) -- (cn00);
+        \draw (vn01) -- (cn01);
+        \draw (vn02) -- (cn01);
+        \draw (vn04) -- (cn01);
+
+        \foreach \i in {1,2,3,4} {
+            \pgfmathtruncatemacro{\prev}{\i-1}
+
+            \node[VN, right = 25mm of vn\prev 0] (vn\i0) {};
+            \node[VN, below = of vn\i0]          (vn\i1) {};
+            \node[VN, below = of vn\i1]          (vn\i2) {};
+            \node[VN, below = of vn\i2]          (vn\i3) {};
+            \node[VN, below = of vn\i3]          (vn\i4) {};
+
+            \coordinate (temp) at ($(vn\i1)!0.5!(vn\i2)$);
+
+            \node[CN, left = of temp] (cn\i0) {};
+            \node[CN, below = of cn\i0]     (cn\i1) {};
+
+            \draw (vn\i0) -- (cn\i0);
+            \draw (vn\i1) -- (cn\i0);
+            \draw (vn\i3) -- (cn\i0);
+            \draw (vn\i1) -- (cn\i1);
+            \draw (vn\i2) -- (cn\i1);
+            \draw (vn\i4) -- (cn\i1);
+        }
+
+        \foreach \i in {1,2,3,4} {
+            \pgfmathtruncatemacro{\prev}{\i-1}
+
+            \draw (vn\prev 3) -- (cn\i 0);
+            \draw (vn\prev 4) -- (cn\i 1);
+        }
+    \end{tikzpicture}
+
+    \caption{
+        Visualization of the coupling between the Tanner graphs
+        of individual spatial positions.
+    }
+    \label{fig:sc-ldpc-tanner}
+\end{figure}
+
+Note that at the first and last few spatial positions, some \acp{cn}
+have lower degrees.
+This leads to more reliable information about the
+\acp{vn} at those positions, that, as we will see, is
+later passed to subsequent spatial positions during decoding.
+This is precisely the effect that leads to the good performance of
+\ac{sc}-\ac{ldpc} codes in the waterfall region \cite{costello_spatially_2014}.
 
 \subsection{Belief Propagation}
 
 % TODO: Add exact reference
 As mentioned above, \ac{ldpc} codes are generally decoded using
 efficient iterative algorithms, something that is possilbe due to
-their sparsity \cite[\red{WHERE}?]{ryan_channel_2009}.
+their sparsity \cite[\red{WHERE?}]{ryan_channel_2009}.
 Specifically, the \ac{spa} is a general decoder that provides
 near-optimal performance across many different scenarios.
 Often, the term \ac{bp} is used to denote a whole class of variants
 of the \ac{spa}, e.g., the \ac{nms} algorithm.
-The \ac{spa} is an algorithm that approximates the marginals of the
-probability distributions of the \acp{vn} by passing
-\textit{messages} in the form of \ac{llr} values along the edges of
-the Tanner graph.
 
-\noindent\red{[SPA]} \\
-\red{[NMS]} \\
-\red{[BP for SC-LDPC codes]}
+%
+% Preliminaires (LLRs, etc.)
+%
+
+% TODO: Make this about the SPA or message passing decoders in general?
+The \ac{spa} approximates the marginals of the
+probability distributions of the \acp{vn} by passing
+\textit{messages} along the edges of the Tanner graph \red{[CITATION]}.
+The messages take the form of \acp{llr}
+%
+% TODO: Proper LLR equation
+\begin{align*}
+    L(x) = \log\left( \frac{P(X=0)}{P(X=1)} \right)
+    .%
+\end{align*}
+%
+\noindent\red{[LLRs]}
+
+%
+% SPA equations
+%
+
+\noindent\red{[SPA]}
+
+%
+% NMS equations
+%
+
+\noindent\red{[NMS]}
+
+%
+% SC-LDPC decoding
+%
+
+\noindent\red{[BP for SC-LDPC codes]}
 
 \red{
     \begin{itemize}

src/thesis/main.tex

@@ -19,7 +19,14 @@
 % ]{biblatex}
 \usepackage{todonotes}
 
-\usetikzlibrary{calc, positioning, arrows}
+\usetikzlibrary{calc, positioning, arrows, fit}
+
+\usetikzlibrary{external}
+\tikzexternalize
+
+\makeatletter
+\renewcommand{\todo}[2][]{\tikzexternaldisable\@todo[#1]{#2}\tikzexternalenable}
+\makeatother
 
 %
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