Added supplementary slides for choice of gamma, mu and rho

latex/thesis/bibliography.bib

@@ -76,7 +76,7 @@
 @mastersthesis{yanxia_lu_thesis,
     author      = {Lu, Yanxia},
     title       = {Realization of Channel Decoding Using Optimization Techniques},
-    year        = {2023},
+    year        = {2022},
     type        = {Bachelor's Thesis},
     institution = {KIT},
 }

latex/thesis/chapters/comparison.tex

@@ -87,7 +87,7 @@ return $\boldsymbol{s}$
                 + \underbrace{\sum\nolimits_{j\in\mathcal{J}} g_j\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}} \right) }
                     _{\text{Constraints}} \\
             \text{subject to}\hspace{5mm} &
-                \tilde{\boldsymbol{c}} \in \mathbb{R}^n
+                \tilde{\boldsymbol{c}} \in \left[ 0, 1 \right]^n
         \end{align*}
     
         \begin{genericAlgorithm}[caption={}, label={},

latex/thesis/chapters/theoretical_background.tex

@@ -21,7 +21,7 @@ For example:%
 %
 \begin{align*}
     x \in \left\{ -1, 1 \right\} &\to \tilde{x} \in \mathbb{R}\\
-    c \in \mathbb{F}_2           &\to \tilde{c} \in \left[ 0, 1 \right]
+    c \in \mathbb{F}_2           &\to \tilde{c} \in \left[ 0, 1 \right] \subseteq \mathbb{R}
 .\end{align*}
 %
 Additionally, a shorthand notation will be used to denote series of indices and series
@@ -29,9 +29,10 @@ of indexed variables:%
 %
 \begin{align*}
     \left[ m:n \right]      &:= \left\{ m, m+1, \ldots, n-1, n \right\},
-        \hspace{5mm} m,n\in\mathbb{Z}\\
+        \hspace{5mm} m < n, \hspace{2mm} m,n\in\mathbb{Z}\\
     x_{\left[ m:n \right] } &:= \left\{ x_m, x_{m+1}, \ldots, x_{n-1}, x_n \right\} 
 .\end{align*}
+\todo{Not really slicing. How should it be denoted?}
 %
 In order to designate elemen-twise operations, in particular the \textit{Hadamard product}
 and the \textit{Hadamard power}, the operator $\circ$ will be used:%
@@ -50,16 +51,17 @@ and the \textit{Hadamard power}, the operator $\circ$ will be used:%
 \section{Preliminaries: Channel Model and Modulation}
 \label{sec:theo:Preliminaries: Channel Model and Modulation}
 
-In order to transmit a bit-word $\boldsymbol{c}$ of length $n$ over a channel,
-it has to be mapped onto a symbol $\boldsymbol{x}$ that can be physically
-transmitted.
+In order to transmit a bit-word $\boldsymbol{c} \in \mathbb{F}_2^n$ of length
+$n$ over a channel, it has to be mapped onto a symbol
+$\boldsymbol{x} \in \mathbb{R}^n$ that can be physically transmitted.
 This is known as modulation. The modulation scheme chosen here is \ac{BPSK}:%
 %
 \begin{align*}
-    \boldsymbol{x} = \left( -1 \right)^{\boldsymbol{c}}
+    \boldsymbol{x} = 1 - 2\boldsymbol{c}
 .\end{align*}
 %
-The symbol that reaches the receiver, $\boldsymbol{y}$, is distorted by the channel.
+The transmitted symbol is distorted by the channel and denoted by
+$\boldsymbol{y} \in \mathbb{R}^n$.
 This distortion is described by the channel model, which in the context of
 this thesis is chosen to be \ac{AWGN}:%
 %
@@ -87,7 +89,7 @@ of the channel \cite[Sec. II.B.]{mackay_rediscovery} while having a structure
 that allows for very efficient decoding.
 
 The lengths of the data words and codewords are denoted by $k\in\mathbb{N}$
-and $n\in\mathbb{N}$, respectively.
+and $n\in\mathbb{N}$, respectively, with $k \le n$.
 The set of codewords $\mathcal{C} \subset \mathbb{F}_2^n$ of a binary
 linear code can be represented using the \textit{parity-check matrix}
 $\boldsymbol{H} \in \mathbb{F}_2^{m\times n}$, where $m$ represents
@@ -103,14 +105,14 @@ $\boldsymbol{c} \in \mathbb{F}_2^n$ using the \textit{generator matrix}
 $\boldsymbol{G} \in \mathbb{F}_2^{k\times n}$:%
 %
 \begin{align*}
-    \boldsymbol{c} = \boldsymbol{u}^\text{T}\boldsymbol{G}
+    \boldsymbol{c} = \boldsymbol{u}\boldsymbol{G}
 .\end{align*}
 %
 
 After obtaining a codeword from a data word, it is transmitted over a channel
 as described in section \ref{sec:theo:Preliminaries: Channel Model and Modulation}.
 The received signal $\boldsymbol{y}$ is then decoded to obtain
-an estimate of the transmitted codeword, $\hat{\boldsymbol{c}}$.
+an estimate of the transmitted codeword, denoted as $\hat{\boldsymbol{c}}$.
 Finally, the encoding procedure is reversed and an estimate of the originally
 sent data word, $\hat{\boldsymbol{u}}$, is produced.
 The methods examined in this work are all based on \textit{soft-decision} decoding,
@@ -170,7 +172,17 @@ criterion:%
         \right)
 .\end{align*}%
 %
-
+The \ac{MAP}- and \ac{ML}-criteria are closely connected through
+\textit{Bayes' theorem}:%
+%
+\begin{align*}
+    \argmax_{c\in\mathcal{C}} p_{\boldsymbol{C} \mid \boldsymbol{Y}}
+        \left( \boldsymbol{c} \mid \boldsymbol{y} \right)
+    = TODO
+.\end{align*}
+%
+This has the consequence that if the probability \ldots, the two criteria are
+equivalent.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -472,8 +484,8 @@ and minimizing $g$ using the proximal operator
 \cite[Sec. 4.2]{proximal_algorithms}:%
 %
 \begin{align*}
-    \boldsymbol{x} \leftarrow \boldsymbol{x} - \lambda \nabla f\left( \boldsymbol{x} \right) \\
-    \boldsymbol{x} \leftarrow \textbf{prox}_{\lambda g} \left( \boldsymbol{x} \right) 
+    \boldsymbol{x} &\leftarrow \boldsymbol{x} - \lambda \nabla f\left( \boldsymbol{x} \right) \\
+    \boldsymbol{x} &\leftarrow \textbf{prox}_{\lambda g} \left( \boldsymbol{x} \right) 
 ,\end{align*}
 %
 Since $g$ is minimized with the proximal operator and is thus not required