Added first draft of 'channel coding with LPDC codes' section

latex/thesis/chapters/theoretical_background.tex

@@ -44,34 +44,6 @@ Lastly, the optimization methods utilized are described.
 
 \tikzstyle{box} = [rectangle, minimum width=1.5cm, minimum height=0.7cm,
                           rounded corners=0.1cm, text centered, draw=black, fill=KITgreen!80]
-\begin{figure}[htpb]
-    \centering
-
-    \begin{tikzpicture}[scale=1, transform shape]
-        \node (in) {$\boldsymbol{c}$};
-        \node[box, right=0.5cm of in] (bpskmap) {Mapper};
-        \node[right=1.5cm of bpskmap,
-              draw, circle, inner sep=0pt, minimum size=0.5cm] (add) {$+$};
-        \node[below=0.5cm of add] (noise) {$\boldsymbol{z}$};
-        \node[box, right=1.5cm of add] (decoder) {Decoder};
-        \node[box, right=1.5cm of decoder] (demapper) {Demapper};
-        \node[right=0.5cm of demapper] (out) {$\boldsymbol{\hat{c}}$};
-
-        \node at ($(bpskmap.east)!0.5!(add.west) + (0,0.3cm)$) {$\boldsymbol{x}$};
-        \node at ($(add.east)!0.5!(decoder.west) + (0,0.3cm)$) {$\boldsymbol{y}$};
-        \node at ($(decoder.east)!0.5!(demapper.west) + (0,0.3cm)$) {$\boldsymbol{\hat{x}}$};
-
-        \draw[->] (in) -- (bpskmap);
-        \draw[->] (bpskmap) -- (add);
-        \draw[->] (add) -- (decoder);
-        \draw[->] (noise) -- (add);
-        \draw[->] (decoder) -- (demapper);
-        \draw[->] (demapper) -- (out);
-    \end{tikzpicture}
-
-    \caption{Overview of notation}
-    \label{fig:notation}
-\end{figure}
 
 \todo{Note about $\tilde{\boldsymbol{c}}$ (and maybe $\tilde{\boldsymbol{x}}$?)}
 
@@ -80,11 +52,101 @@ Lastly, the optimization methods utilized are described.
 \section{Channel Coding with LDPC Codes}
 \label{sec:theo:Channel Coding with LDPC Codes}
 
-\begin{itemize}
+Channel coding describes the process of adding redundancy to information
-    \item Introduction
+transmitted over a channel in order to detect and correct any errors
-    \item Binary linear codes
+that may occur during the transmission.
-    \item \Ac{LDPC} codes (especially $i$, $j$, parity check matrix $\boldsymbol{H}$, $N\left( j  \right) $ \& $N\left( i \right) $, etc.)
+Encoding the information using \textit{binary linear codes} is one way of
-\end{itemize}
+conducting this process, whereby \textit{data words} are mapped onto longer
+\textit{codewords}, which carry redundant information.
+It can be shown that as the length of the encoded data words becomes greater,
+the theoretically achievable error-correcting capabilities of the code become
+better, asymptotically approaching the capacity of the channel.
+For this reason, \ac{LDPC} codes have become especially popular, given their
+low memory requirements even for very large codes.
+
+The lengths of the data words and codewords are denoted by $k$ and $n$,
+respectively.
+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
+the number of parity-checks:%
+%
+\begin{align*}
+    \mathcal{C} := \left\{ \boldsymbol{c} \in \mathbb{F}_2^n :
+            \boldsymbol{H}\boldsymbol{c}^\text{T} = \boldsymbol{0} \right\}
+.\end{align*}
+%
+A data word $\boldsymbol{u} \in \mathbb{F}_2^k$ can be mapped onto a codword
+$\boldsymbol{c} \in \mathbb{F}_2^n$ using the \textit{generator matrix}
+$\boldsymbol{G} \in \mathbb{F}_2^{k\times n}$:%
+%
+\begin{align*}
+    \boldsymbol{c} = \boldsymbol{u}\boldsymbol{G}
+.\end{align*}
+%
+
+After obtaining a codeword from a data word, it is transmitted over a channel,
+as shown in figure \ref{fig:theo:channel_overview}.
+Using the selected modulation scheme, $\boldsymbol{c}$ is mapped onto
+$\boldsymbol{x}$.
+The channel distorts $\boldsymbol{x}$ into $\boldsymbol{y}$, which is what
+reaches the receiver.
+The received signal $\boldsymbol{y}$ is then decoded at the receiver to obtain
+an estimate of the transmitted codeword, $\hat{\boldsymbol{c}}$.
+Finally, the encoding procedure is reversed and an estimate for the originally
+sent data word is obtained.
+
+\begin{figure}[htpb]
+    \centering
+
+    \begin{tikzpicture}[scale=1, transform shape]
+        \node (c) {$\boldsymbol{c}$};
+        \node[box, right=0.5cm of c] (bpskmap) {Mapper};
+        \node[right=1.5cm of bpskmap,
+              draw, circle, inner sep=0pt, minimum size=0.5cm] (add) {$+$};
+        \node[box, right=1.5cm of add] (decoder) {Decoder};
+        \node[box, right=1.5cm of decoder] (demapper) {Demapper};
+        \node[right=0.5cm of demapper] (out) {$\boldsymbol{\hat{c}}$};
+
+        \node (x) at ($(bpskmap.east)!0.5!(add.west) + (0,0.3cm)$) {$\boldsymbol{x}$};
+        \node (y) at ($(add.east)!0.5!(decoder.west) + (0,0.3cm)$) {$\boldsymbol{y}$};
+        \node (x_hat) at ($(decoder.east)!0.5!(demapper.west) + (0,0.3cm)$)
+            {$\boldsymbol{\hat{x}}$};
+        \node[below=0.5cm of add] (z) {$\boldsymbol{z}$};
+
+        \draw[->] (c) -- (bpskmap);
+        \draw[->] (bpskmap) -- (add);
+        \draw[->] (add) -- (decoder);
+        \draw[->] (z) -- (add);
+        \draw[->] (decoder) -- (demapper);
+        \draw[->] (demapper) -- (out);
+
+        \coordinate (top_left)      at ($(x.north west) + (-0.1cm, 0.1cm)$);
+        \coordinate (top_right)     at ($(y.north east) + (+0.1cm, 0.1cm)$);
+        \coordinate (bottom_center) at ($(z.south)      + (0cm,   -0.1cm)$);
+        \draw[dashed] (top_left) -- (top_right) |- (bottom_center) -| cycle;
+        \node[below=0.25cm of z] (text) {Channel};
+    \end{tikzpicture}
+
+    \caption{Overview of codeword transmission}
+    \label{fig:theo:channel_overview}
+\end{figure}
+
+The decoding process itself is generally based either on the \ac{MAP} or the \ac{ML}
+criterion:%
+%
+\begin{align*}
+    \hat{\boldsymbol{c}}_{\text{\ac{MAP}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}}
+    p_{\boldsymbol{C} \mid \boldsymbol{Y}} \left(\boldsymbol{c} \mid \boldsymbol{y}
+        \right)
+    \hat{\boldsymbol{c}}_{\text{\ac{ML}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}}
+    f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c}
+        \right)
+.\end{align*}%
+%
+The methods examined in this work are all based on \textit{soft-decision} decoding,
+i.e., $\boldsymbol{y}$ is considered to be in $\mathbb{R}^n$ and no preliminary decision
+is made by a demodulator.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -94,6 +156,7 @@ Lastly, the optimization methods utilized are described.
 \begin{itemize}
     \item Introduction to message passing
     \item Overview of \ac{BP} algorithm
+    \item \Ac{LDPC} codes (especially $i$, $j$, parity check matrix $\boldsymbol{H}$, $N\left( j  \right) $ \& $N\left( i \right) $, etc.)
 \end{itemize}