Wrote begining of LP decoding theory

latex/thesis/abbreviations.tex

@@ -40,16 +40,6 @@
     long  = frame error rate
 }
 
-\DeclareAcronym{LP}{
-    short = LP,
-    long  = linear programming
-}
-
-\DeclareAcronym{LDPC}{
-    short = LDPC,
-    long  = low-density parity-check
-}
-
 %
 % M
 %
@@ -64,6 +54,20 @@
     long  = maximum likelihood
 }
 
+%
+%L
+%
+
+\DeclareAcronym{LP}{
+    short = LP,
+    long  = linear programming
+}
+
+\DeclareAcronym{LDPC}{
+    short = LDPC,
+    long  = low-density parity-check
+}
+
 %
 % P
 %

latex/thesis/chapters/decoding_techniques.tex

@@ -25,11 +25,15 @@ the \ac{ML} decoding problem:%
 %
 \begin{align*}
     \hat{\boldsymbol{c}}_{\text{\ac{MAP}}} &= \argmax_{c \in \mathcal{C}}
-        f_{\boldsymbol{X} \mid \boldsymbol{Y}} \left( \boldsymbol{x} \mid \boldsymbol{y} \right)\\
+        f_{\boldsymbol{C} \mid \boldsymbol{Y}} \left( \boldsymbol{c} \mid \boldsymbol{y} \right)\\
     \hat{\boldsymbol{c}}_{\text{\ac{ML}}} &= \argmax_{c \in \mathcal{C}}
-        f_{\boldsymbol{Y} \mid \boldsymbol{X}} \left( \boldsymbol{y} \mid \boldsymbol{x} \right) 
-.\end{align*}
+        f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c} \right) 
+.\end{align*}%
 %
+\todo{Note about these generally being the same thing, when the a priori probability
+is uniformly distributed}%
+\todo{Here the two problems are written in terms of $\hat{\boldsymbol{c}}$; below MAP
+decoding is applied in terms of $\hat{\boldsymbol{x}}$. Is that a problem?}%
 The goal is to arrive at a formulation, where a certain objective function
 $f$ has to be minimized under certain constraints:%
 %
@@ -41,7 +45,7 @@ $f$ has to be minimized under certain constraints:%
 In contrast to the established message-passing decoding algorithms,
 the viewpoint then changes from observing the decoding process in its
 tanner graph representation (as shown in figure \ref{fig:dec:tanner})
-into a spacial representation, where the codewords are some of the edges
+to a spacial representation, where the codewords are some of the edges
 of a hypercube and the goal is to find that point $\boldsymbol{x}$,
 \todo{$\boldsymbol{x}$? Or some other variable?}
 which minimizes the objective function $f$ (as shown in figure \ref{fig:dec:spacial}).
@@ -127,6 +131,8 @@ which minimizes the objective function $f$ (as shown in figure \ref{fig:dec:spac
         \caption{Spacial representation of a single parity-check code}
         \label{fig:dec:spacial}
     \end{subfigure}%
+
+    \caption{Different representations of the decoding problem}
 \end{figure}
 
 
@@ -135,6 +141,52 @@ which minimizes the objective function $f$ (as shown in figure \ref{fig:dec:spac
 \section{LP Decoding using ADMM}%
 \label{sec:dec:LP Decoding using ADMM}
 
+\Ac{LP} decoding is a subject area introduced by Feldman et al.
+\cite{feldman_paper}. They reframed the decoding problem as an
+\textit{integer linear program} and subsequently presented a relaxation into
+a \textit{linear program}, lifting the integer requirement.
+The optimization method used to solve this problem that is examined in this
+work is the \ac{ADMM}.
+\todo{With or without 'the'?}
+\todo{Why chose ADMM?}
+
+Feldman at al. begin by looking at the \ac{ML} decoding problem%
+\footnote{They assume that all codewords are equally likely to be transmitted,
+making the \ac{ML} and \ac{MAP} decoding problems essentially equivalent}%
+\todo{Dot after footnote?}%
+%
+\begin{align*}
+    \hat{\boldsymbol{x}} = \argmax_{\boldsymbol{x} \in
+        \left\{ \left( -1 \right)^{\boldsymbol{c}}
+            \text{ : } \boldsymbol{c} \in \mathcal{C} \right\} }
+        f_{\boldsymbol{Y} \mid \boldsymbol{X}} \left( \boldsymbol{y} \mid \boldsymbol{x} \right) 
+.\end{align*}
+%
+\todo{Define $\mathcal{X}$ as $\left\{ \left( -1 \right)
+    ^{\boldsymbol{c}} : \boldsymbol{c}\in \mathcal{C} \right\} $?}%
+They suggest that maximizing the likelihood
+$f_{\boldsymbol{Y} \mid \boldsymbol{X}}\left( \boldsymbol{y} \mid \boldsymbol{x} \right)$
+is equivalent to minimizing the negative log-likelihood.
+
+\ldots
+
+Based on this, they propose their cost function%
+\footnote{In this context, \textit{cost function} and \textit{objective function}
+mean the same thing}
+for the \ac{LP} decoding problem:%
+%
+\begin{align*}
+    \sum_{i=1}^{n} \gamma_i c_i,
+        \hspace{5mm} \gamma_i = \ln\left(
+            \frac{f_{\boldsymbol{Y} | \boldsymbol{C}}
+                \left( Y_i = y_i  \mid C_i = 0 \right) }
+            {f_{\boldsymbol{Y} | \boldsymbol{C}}
+                \left( Y_i = y_i | C_i = 1 \right) } \right) \\
+.\end{align*}
+%
+
+The 
+
 \begin{itemize}
     \item Equivalent \ac{ML} optimization problem
     \item \Ac{LP} relaxation
@@ -305,6 +357,7 @@ The components of the gradient of the code-constraint polynomial can be computed
                 - \prod_{j\in\mathcal{A}\left( i \right) }x_j \right)
 .\end{align*}%
 \todo{Only multiplication?}%
+\todo{$x_k$: $k$ or some other indexing variable?}%
 %
 In the case of \ac{AWGN}, the likelihood
 $f_{\boldsymbol{Y} \mid \boldsymbol{X}}\left( \boldsymbol{y} \mid \boldsymbol{x} \right)$

latex/thesis/chapters/theoretical_background.tex

@@ -24,6 +24,20 @@ Lastly, the optimization methods utilized are described.
     \item Probabilistic quantities (random variables, \acp{PDF}, \ldots)
 \end{itemize}
 
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Preliminaries: Channel Model and Modulation}
+\label{sec:theo:Preliminaries: Channel Model and Modulation}
+
+%
+% TODOs
+%
+
+\begin{itemize}
+    \item \Ac{AWGN}
+    \item \Ac{BPSK}
+\end{itemize}
+
 %
 % Figure showing notation for entire coding / decoding process
 %
@@ -58,17 +72,6 @@ Lastly, the optimization methods utilized are described.
     \caption{Overview of notation}
     \label{fig:notation}
 \end{figure}
-\todo{Move figure to 'Channel and Modulation'}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Preliminaries: Channel Model and Modulation}
-\label{sec:theo:Preliminaries: Channel Model and Modulation}
-
-\begin{itemize}
-    \item \Ac{AWGN}
-    \item \Ac{BPSK}
-\end{itemize}
-
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

latex/thesis/thesis.tex

@@ -122,12 +122,12 @@
     %   - Results summary
     %
     % 2. Theoretical Background
+    %   2.2 Notation
+    %       - General remarks on notation (matrices, ...)
+    %       - Probabilistic quantities(random variables, PDFs, ...)
     %   2.1 Preliminaries: Channel Model and Modulation
     %       - AWGN
     %       - BPSK
-    %   2.2 Notation
-    %       - General remarks on notation (matrices, PDF, etc.)
-    %       - Diagram from midterm presentation
     %   2.3 Channel Coding with LDPC Codes
     %       - Introduction
     %       - Binary linear codes