From 311013716421f8f0ecedddadd1c222de8cb0ba81 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Mon, 19 Dec 2022 14:16:12 +0100
Subject: [PATCH] Changed variable names

---
 .../sections/theoretical_background.tex       | 20 +++++++++++++------
 1 file changed, 14 insertions(+), 6 deletions(-)

diff --git a/latex/presentations/midterm/sections/theoretical_background.tex b/latex/presentations/midterm/sections/theoretical_background.tex
index b058c53..8ea98a8 100644
--- a/latex/presentations/midterm/sections/theoretical_background.tex
+++ b/latex/presentations/midterm/sections/theoretical_background.tex
@@ -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}