\section{Theoretical Background}%
\label{sec:Theoretical Background}


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\subsection{Motivation}%
\label{sub:Motivation}

\begin{frame}[t]
    \frametitle{Motivation}

    \todo{TODO}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Presumptions}%
\label{sub:Presumptions}

\begin{frame}[t]
    \frametitle{Channel \& Modulation}

    \tikzstyle{mapper} = [rectangle, minimum width=1.5cm, rounded corners=0.1cm, minimum height=0.7cm, text centered, draw=black, fill=KITgreen!80]
    \begin{figure}[htpb]
        \centering

        \begin{tikzpicture}[scale=1, transform shape]
            \node (in) {$c\left[ k \right] $};
            \node[mapper, right=0.5cm of in] (bpskmap) {Mapper};
            \node[right=1.5cm of bpskmap, draw, circle, inner sep=0pt, minimum size=0.5cm] (add) {$+$};
            \node[right=0.5cm of add] (out) {$y\left[ k \right] $};
            \node[below=0.5cm of add] (noise) {$n\left[ k \right] $};

            \node at ($(bpskmap.east)!0.5!(add.west) + (0,0.3cm)$) {$x\left[ k \right] $};

            \draw[->] (in) -- (bpskmap);
            \draw[->] (bpskmap) -- (add);
            \draw[->] (add) -- (out);
            \draw[->] (noise) -- (add);
        \end{tikzpicture}
    \end{figure}

    \begin{itemize}
        \item All simulations are performed with BPSK Modulation:
            \begin{align*}
                x\left[ k \right]  = \left( -1 \right)^{c\left[ k \right] }, \hspace{5mm} \boldsymbol{c} \in \mathbb{F}_2^n, \hspace{2mm} k\in \left\{ 1, \ldots, n \right\} 
            \end{align*}
        \item The used channel model is AWGN:
            \begin{align*}
                \boldsymbol{y} = \boldsymbol{x} + \boldsymbol{n},
                    \hspace{5mm}\boldsymbol{n}\sim \mathcal{N}
                        \left(0,\frac{1}{2}\left(\frac{k}{n}\frac{E_b}{N_0}\right)^{-1}\right),
                    \hspace{2mm} \boldsymbol{y}, \boldsymbol{n} \in \mathbb{R}^n
            \end{align*}
            \todo{Why $\frac{1}{2}$}
    \end{itemize}
\end{frame}


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\subsection{LP Decoding}%
\label{sub:LP Decoding}

\begin{frame}[t]
    \frametitle{LP Decoding}

    \todo{TODO}
\end{frame}