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Changed variable names

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This commit is contained in:
Andreas Tsouchlos 2022-12-19 14:16:12 +01:00
parent fab8ab64ac
commit 3110137164
1 changed files with 14 additions and 6 deletions
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20
latex/presentations/midterm/sections/theoretical_background.tex
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@ -20,7 +20,8 @@
\begin{frame}[t]
\frametitle{Presumptions: Channel \& Modulation}
\tikzstyle{mapper} = [rectangle, minimum width=1.5cm, rounded corners=0.1cm, minimum height=0.7cm, text centered, draw=black, fill=KITgreen!80]
\tikzstyle{mapper} = [rectangle, minimum width=1.5cm, minimum height=0.7cm,
rounded corners=0.1cm, text centered, draw=black, fill=KITgreen!80]
\begin{figure}[htpb]
\centering
@ -43,7 +44,8 @@
\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\}
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*}
@ -68,16 +70,22 @@
\begin{itemize}
\item Codeword Polytope:
\begin{align*}
\text{poly}\left( \mathcal{C} \right) = \left\{ \sum_{\boldsymbol{y}\in\mathcal{C}} \lambda_{\boldsymbol{y}} \boldsymbol{y} : \lambda_{\boldsymbol{y}} \ge 0, \sum_{\boldsymbol{y}\in\mathcal{C}}\lambda_{\boldsymbol{y}} = 1 \right\}, \hspace{5mm} \lambda_{\boldsymbol{y}} \in \mathbb{R}
\text{poly}\left( \mathcal{C} \right) =
\left\{
\sum_{\boldsymbol{c}\in\mathcal{C}} \lambda_{\boldsymbol{c}} \boldsymbol{c}
: \lambda_{\boldsymbol{c}} \ge 0,
\sum_{\boldsymbol{c}\in\mathcal{C}}\lambda_{\boldsymbol{c}} = 1
\right\},
\hspace{5mm} \lambda_{\boldsymbol{c}} \in \mathbb{R}
\end{align*}
\item Cost Function:
\begin{align*}
\gamma_i = \log\left( \frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right)
\gamma_i = \log\left( \frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right),
\hspace{5mm} i = \left\{ 1, \ldots, n \right\}
\end{align*}
\todo{Why is ``the cost of decoding $\hat{y} = 1$'' a valid choice for an overall cost function?}
\item LP Formulation:
\begin{align*}
&\text{minimize } \sum_{i=1}^{n} \gamma_i f_i, \hspace{5mm} f_i = \sum_{\boldsymbol{y}} \lambda_{\boldsymbol{y}}y_i\\
&\text{minimize } \sum_{i=1}^{n} \gamma_i f_i \\
&\text{subject to } \boldsymbol{f}\in\text{poly}\left( \mathcal{C} \right)
\end{align*}
\end{itemize}