From 4ae8f6660315e69d18d2801c399ae35df4e22993 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Mon, 30 Mar 2026 23:19:11 +0200
Subject: [PATCH] [thesis] Add BP text and error floor figure

---
 src/thesis/acronyms.tex                | 15 ++++
 src/thesis/chapters/2_fundamentals.tex | 94 +++++++++++++++++++++++---
 2 files changed, 99 insertions(+), 10 deletions(-)

diff --git a/src/thesis/acronyms.tex b/src/thesis/acronyms.tex
index 6606381..f1c96a4 100644
--- a/src/thesis/acronyms.tex
+++ b/src/thesis/acronyms.tex
@@ -8,6 +8,21 @@
     long=belief propagation
 }
 
+\DeclareAcronym{nms}{
+    short=NMS,
+    long=normalized min-sum
+}
+
+\DeclareAcronym{spa}{
+    short=SPA,
+    long=sum-product algorithm
+}
+
+\DeclareAcronym{llr}{
+    short=LLR,
+    long=log-likelihood ratio
+}
+
 \DeclareAcronym{sc}{
     short=SC,
     long=spatially coupled
diff --git a/src/thesis/chapters/2_fundamentals.tex b/src/thesis/chapters/2_fundamentals.tex
index 5673373..597906a 100644
--- a/src/thesis/chapters/2_fundamentals.tex
+++ b/src/thesis/chapters/2_fundamentals.tex
@@ -308,17 +308,79 @@ 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.
 
+% TODO: Make this look better
 \begin{figure}[t]
     \centering
 
     \begin{tikzpicture}
         \begin{axis}[
-                domain=-5:5,
-                width=\figwidth,
-                height=\figheight,
+                width=12cm,
+                height=9cm,
+                xlabel={$E_b/N_0$ (dB)},
+                ylabel={\ac{ber}},
+                xmin=0, xmax=6,
+                ymin=1e-9, ymax=1,
+                ticks=none,
+                % y tick label={},
+                ymode=log,
+                grid=both,
+                grid style={line width=0.2pt, draw=gray!30},
+                major grid style={line width=0.4pt, draw=gray!50},
+                legend pos=north east,
+                legend cell align={left},
             ]
-            \addplot+[mark=none, line width=1pt]
-            {x^2};
+
+            \addplot+[mark=none, solid, thick, smooth] coordinates {
+                (0.0, 1.2e-1)
+                (0.3, 1.1e-1)
+                (0.5, 9e-2)
+                (0.7, 5e-2)
+                (0.8, 2e-2)
+                (0.9, 5e-3)
+                (1.0, 8e-4)
+                (1.1, 1e-4)
+                (1.2, 1.5e-5)
+                (1.3, 3e-6)
+                (1.4, 5e-7)
+                (1.5, 8e-8)
+                (1.6, 2e-8)
+                (1.8, 8e-9)
+                (2.0, 5e-9)
+                (2.5, 3e-9)
+                (3.0, 2e-9)
+            };
+            \addlegendentry{Regular LDPC-BC}
+
+            \addplot+[mark=none, solid, thick, smooth] coordinates {
+                (0.0, 1.5e-1)
+                (0.3, 1.4e-1)
+                (0.5, 1.2e-1)
+                (0.6, 1.0e-1)
+                (0.7, 6e-2)
+                (0.8, 1e-2)
+                (0.85, 2e-3)
+                (0.9, 2e-4)
+                (0.95, 2e-5)
+                (1.0, 1.5e-6)
+                (1.05, 1e-7)
+                (1.1, 1e-8)
+                (1.2, 3e-9)
+                (1.5, 1.5e-9)
+                (2.0, 1e-9)
+                (2.5, 8e-10)
+                (3.0, 6e-10)
+            };
+            \addlegendentry{Irregular LDPC-BC}
+
+            \draw[red, thick, rounded corners=12pt]
+            (axis cs:0.55, 2e-3) rectangle (axis cs:1.55, 5e-5);
+            \node[red, font=\small\bfseries, anchor=west] at (axis
+            cs:1.6, 4e-4) {Waterfall};
+
+            \draw[red, thick, rounded corners=12pt]
+            (axis cs:1.6, 8e-9) rectangle (axis cs:3.2, 4e-10);
+            \node[red, font=\small\bfseries, anchor=west] at (axis
+            cs:3.3, 2e-9) {Error floor};
         \end{axis}
     \end{tikzpicture}
 
@@ -330,7 +392,7 @@ qualitative performance characteristic of an \ac{ldpc} code
     \label{fig:ldpc-perf}
 \end{figure}
 
-Broadly, there are two kinds of \ac{LDPC} codes, \textit{regular} and
+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
@@ -364,13 +426,25 @@ waterfall region \cite[Intro.]{costello_spatially_2014}.
 
 \subsection{Belief Propagation}
 
-\red{[key points (sub-optimal but good enough, low complexity, \ldots)]} \\
-\red{[top-level overview (iterative algorithm that approximates \ldots)]}
+% 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}.
+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]}
 
 \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