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