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Make current thesis text use CEL template

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Andreas Tsouchlos
2026-03-22 22:57:35 +01:00
parent 7703fa4023
commit c3b0c194fe
3 changed files with 51 additions and 19 deletions
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14
src/thesis/chapters/2_fundamentals.tex
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@@ -51,7 +51,7 @@ $\bm{u} \in \mathbb{F}_2^k$ of length $k \in \mathbb{N}$ (called the
A measure of the amount of introduced redundancy is the \textit{code
rate} $R = k/n$.
We call the set of all codewords $\mathcal{C}$ the \textit{code}
\cite[Section 3.1]{ryan_channel_2009}.
\cite[Sec. 3.1]{ryan_channel_2009}.
%
% d_min and the [] Notation
@@ -73,7 +73,7 @@ We define the \textit{minimum distance} of a code $\mathcal{C}$ as
\end{align*}
We can signify that a binary linear block code has information length
$k$, block length $n$ and minimum distance $d_\text{min}$ using the
notation $[n,k,d_\text{dmin}]$ \cite[Section 1.3]{macwilliams_theory_1977}.
notation $[n,k,d_\text{dmin}]$ \cite[Sec. 1.3]{macwilliams_theory_1977}.
%
% Parity checks, H, and the syndrome
@@ -88,9 +88,9 @@ additional degrees of freedom.
These conditions, called parity checks, take the form of equations
over $\mathbb{F}_2^n$, linking the individual positions of each codeword.
We can arrange the coefficients of these equations in the
\textit{parity check matrix} (\acs{pcm}) $\bm{H} \in
\textit{parity-check matrix} (\acs{pcm}) $\bm{H} \in
\mathbb{F}_2^{(n-k) \times n}$ and equivalently define the code as
\cite[Section 3.1]{ryan_channel_2009}
\cite[Sec. 3.1]{ryan_channel_2009}
\begin{align*}
\mathcal{C} = \left\{ \bm{x} \in \mathbb{F}_2^n :
\bm{H}\bm{x}^\text{T} = \bm{0} \right\}
@@ -107,7 +107,7 @@ exponentially with $n$, in contrast to keeping track of all codewords directly.
%
Figure \ref{fig:Diagram of a transmission system} visualizes the
entire communication process \cite[Section 1.1]{ryan_channel_2009}.
entire communication process \cite[Sec. 1.1]{ryan_channel_2009}.
An input message $\bm{u}\in \mathbb{F}_2^k$ is mapped onto a codeword $\bm{x}
\in \mathbb{F}_2^n$. This is passed on to a modulator, which
interacts with the physical channel.
@@ -120,7 +120,7 @@ This is done by first finding an estimate $\hat{\bm{x}}$ of the sent
codeword and undoing the encoding.
The decoding problem that we generally attempt to solve thus consists
in finding the best estimate $\hat{\bm{x}}$ given $\bm{y}$.
One approach is to use the \ac{ml} criterion \cite[Section
One approach is to use the \ac{ml} criterion \cite[Sec.
1.4]{ryan_channel_2009}
\begin{align*}
\hat{\bm{u}}_\text{ML} = \arg\max_{\bm{x} \in \mathcal{C}}
@@ -129,7 +129,7 @@ One approach is to use the \ac{ml} criterion \cite[Section
\end{align*}
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[Section 1.5.1.3]{ryan_channel_2009}.
$\bm{y} \in \mathbb{F}_2^n$ \cite[Sec. 1.5.1.3]{ryan_channel_2009}.
%
\begin{figure}[h]
\centering