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Reworked notation section; Added channel model and modulation section; Added decoding using BP section

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Andreas Tsouchlos 2023-03-27 11:38:03 +02:00
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9
latex/thesis/abbreviations.tex
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@ -119,6 +119,15 @@
long = probability mass function long = probability mass function
} }
%
%S
%
\DeclareAcronym{SNR}{
short = SNR,
long = signal-to-noise ratio
}
% %
% V % V
% %

2
latex/thesis/chapters/decoding_techniques.tex
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@ -72,7 +72,7 @@ which minimizes the objective function $g$.
\begin{tikzpicture}[scale=1, transform shape] \begin{tikzpicture}[scale=1, transform shape]
\node[checknode, \node[checknode,
label={[below, label distance=-0.4cm, align=center] label={[below, label distance=-0.4cm, align=center]
CN\\$\left( c_1 + c_2 + c_3 = 0 \right) $}] \acs{CN}\\$\left( c_1 + c_2 + c_3 = 0 \right) $}]
(cn) at (0, 0) {}; (cn) at (0, 0) {};
\node[variablenode, label={[above, align=center] \acs{VN}\\$\left( c_1 \right)$}] \node[variablenode, label={[above, align=center] \acs{VN}\\$\left( c_1 \right)$}]
(c1) at (-2, 2) {}; (c1) at (-2, 2) {};

289
latex/thesis/chapters/theoretical_background.tex
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@ -12,93 +12,78 @@ Lastly, the optimization methods utilized are described.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Notation} \section{General Remarks on Notation}
\label{sec:theo:Notation} \label{sec:theo:Notation}
% %
% TODOs \todo{Explain bold font $\to$ vector/matrix?}%
\todo{Explain random variables upper- and lower case and PDFs and PMFs?}%
% %
%Matrices and vectors will be depicted in bold font, matrices with upper-case
%and vectors with lower-case letters. For example:%
%%
%\begin{align*}
% \boldsymbol{H}\boldsymbol{c} &= \boldsymbol{0}
%.\end{align*}
%%
%In order to be able to distinguish between random variables and their realizations,
%random variables will be represented by upper-case and realizations by lower-case
%letters. \Acp{PDF} and \acp{PMF} will be denoted by $f$ and $p$, respectively:%
%%
%\begin{align*}
% f_{Y} &\left( y \right) := \frac{d}{dy} P\left( Y \le y \right) \\
% p_{C} &\left( c \right) := P\left( Y = y \right)
%,\end{align*}
%%
%where $P\left( . \right)$ is the probability function.
\begin{itemize} Wherever the domain of a variable is expanded, this will be indicated with a tilde.
\item General remarks on notation (matrices, \ldots) For example:%
\item Probabilistic quantities (random variables, \acp{PDF}, pdfs vs pmfs vs cdfs, \ldots) %
\end{itemize} \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]
.\end{align*}
%
Additionally a shorthand notation is used to denote series of indices and series
of indexed variables:%
%
\begin{align*}
\left[ m:n \right] &:= \left\{ m, m+1, \ldots, n-1, n \right\} \\
x_{\left[ m:n \right] } &:= \left\{ x_m, x_{m+1}, \ldots, x_{n-1}, x_n \right\}
.\end{align*}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries: Channel Model and Modulation} \section{Preliminaries: Channel Model and Modulation}
\label{sec:theo: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,
% TODOs it has to be mapped onto a symbol $\boldsymbol{x}$ that can be physically
% transmitted.
This is known as modulation. The modulation scheme chosen here is \ac{BPSK}:%
\begin{itemize}
\item \Ac{AWGN}
\item \Ac{BPSK}
\end{itemize}
%
% Figure showing notation for entire coding / decoding process
%
\tikzstyle{box} = [rectangle, minimum width=1.5cm, minimum height=0.7cm,
rounded corners=0.1cm, text centered, draw=black, fill=KITgreen!80]
\todo{Note about $\tilde{\boldsymbol{c}}$ (and maybe $\tilde{\boldsymbol{x}}$?)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Channel Coding with LDPC Codes}
\label{sec:theo:Channel Coding with LDPC Codes}
Channel coding describes the process of adding redundancy to information
transmitted over a channel in order to detect and correct any errors
that may occur during the transmission.
Encoding the information using \textit{binary linear codes} is one way of
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*} \begin{align*}
\mathcal{C} := \left\{ \boldsymbol{c} \in \mathbb{F}_2^n : \boldsymbol{x} = \left( -1 \right)^{\boldsymbol{c}}
\boldsymbol{H}\boldsymbol{c}^\text{T} = \boldsymbol{0} \right\}
.\end{align*} .\end{align*}
% %
A data word $\boldsymbol{u} \in \mathbb{F}_2^k$ can be mapped onto a codword The symbol that reaches the receiver, $\boldsymbol{y}$, is distorted by the channel.
$\boldsymbol{c} \in \mathbb{F}_2^n$ using the \textit{generator matrix} The channel model used here is \ac{AWGN}:%
$\boldsymbol{G} \in \mathbb{F}_2^{k\times n}$:%
% %
\begin{align*} \begin{align*}
\boldsymbol{c} = \boldsymbol{u}\boldsymbol{G} \boldsymbol{y} = \boldsymbol{x} + \boldsymbol{z},
\hspace{5mm} \boldsymbol{z}_i \in \mathcal{N}\left( 0, \frac{\sigma^2}{2} \right),
\hspace{2mm} i \in \left[ 1:n \right]
.\end{align*} .\end{align*}
% %
This process is visualized in figure \ref{fig:theo:channel_overview}.
After obtaining a codeword from a data word, it is transmitted over a channel, \begin{figure}[H]
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 \centering
\tikzstyle{box} = [rectangle, minimum width=1.5cm, minimum height=0.7cm,
rounded corners=0.1cm, text centered, draw=black, fill=KITgreen!80]
\begin{tikzpicture}[scale=1, transform shape] \begin{tikzpicture}[scale=1, transform shape]
\node (c) {$\boldsymbol{c}$}; \node (c) {$\boldsymbol{c}$};
\node[box, right=0.5cm of c] (bpskmap) {Mapper}; \node[box, right=0.5cm of c] (bpskmap) {Mapper};
@ -128,36 +113,186 @@ sent data word is obtained.
\node[below=0.25cm of z] (text) {Channel}; \node[below=0.25cm of z] (text) {Channel};
\end{tikzpicture} \end{tikzpicture}
\caption{Overview of codeword transmission} \caption{Overview of channel and modulation}
\label{fig:theo:channel_overview} \label{fig:theo:channel_overview}
\end{figure} \end{figure}
\todo{Mapper $\to$ Modulator?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Channel Coding with LDPC Codes}
\label{sec:theo:Channel Coding with LDPC Codes}
Channel coding describes the process of adding redundancy to information
transmitted over a channel in order to detect and correct any errors
that may occur during the transmission.
Encoding the information using \textit{binary linear codes} is one way of
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.
\todo{Citation needed}
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 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, $\hat{\boldsymbol{c}}$.
Finally, the encoding procedure is reversed and an estimate of the originally
sent data word, $\hat{\boldsymbol{u}}$, is obtained.
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.
The decoding process itself is generally based either on the \ac{MAP} or the \ac{ML} The decoding process itself is generally based either on the \ac{MAP} or the \ac{ML}
criterion:% criterion:%
% %
\begin{align*} \begin{align*}
\hat{\boldsymbol{c}}_{\text{\ac{MAP}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}} \hat{\boldsymbol{c}}_{\text{\ac{MAP}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}}
p_{\boldsymbol{C} \mid \boldsymbol{Y}} \left(\boldsymbol{c} \mid \boldsymbol{y} p_{\boldsymbol{C} \mid \boldsymbol{Y}} \left(\boldsymbol{c} \mid \boldsymbol{y}
\right) \right) \\
\hat{\boldsymbol{c}}_{\text{\ac{ML}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}} \hat{\boldsymbol{c}}_{\text{\ac{ML}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}}
f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c} f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c}
\right) \right)
.\end{align*}% .\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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Decoding LDPC Codes using Belief Propagation} \section{Decoding LDPC Codes using Belief Propagation}
\label{sec:theo:Decoding LDPC Codes using Belief Propagation} \label{sec:theo:Decoding LDPC Codes using Belief Propagation}
\begin{itemize} It is often helpful to visualize codes graphically.
\item Introduction to message passing This is especially true for \ac{LDPC} codes, as the established decoding
\item Overview of \ac{BP} algorithm algorithms are \textit{message passing algorithms}, which are inherently
\item \Ac{LDPC} codes (especially $i$, $j$, parity check matrix $\boldsymbol{H}$, $N\left( j \right) $ \& $N\left( i \right) $, etc.) graph-based.
\end{itemize}
Binary linear codes with a parity-check matrix $\boldsymbol{H}$ can be
visualized using a \textit{Tanner} or \textit{factor graph}:
Each row of $\boldsymbol{H}$, which represents one parity-check, is viewed as a
\ac{CN}.
Each component of the codeword $\boldsymbol{c}$ is interpreted as a \ac{VN}.
The relationship between \acp{CN} and \acp{VN} can then be plotted by noting
which components of $\boldsymbol{c}$ are considered for which parity-check.
Figure \ref{fig:theo:tanner_graph} shows the tanner graph for the
(7,4)-Hamming-code, which has the following parity-check matrix:%
%
\begin{align*}
\boldsymbol{H} = \begin{bmatrix}
1 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 1 & 1 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 1 & 1 & 1 & 1
\end{bmatrix}
.\end{align*}
%
%
\begin{figure}[H]
\centering
\tikzstyle{checknode} = [color=KITblue, fill=KITblue,
draw, regular polygon,regular polygon sides=4,
inner sep=0pt, minimum size=12pt]
\tikzstyle{variablenode} = [color=KITgreen, fill=KITgreen,
draw, circle, inner sep=0pt, minimum size=10pt]
\begin{tikzpicture}[scale=1, transform shape]
\node[checknode,
label={[below, label distance=-0.4cm, align=center]
\acs{CN} 1\\$\left( c_1 + c_3 + c_5 + c_7 = 0 \right) $}]
(cn1) at (-4, -1) {};
\node[checknode,
label={[below, label distance=-0.4cm, align=center]
\acs{CN} 2\\$\left( c_2 + c_3 + c_6 + c_7 = 0 \right) $}]
(cn2) at (0, -1) {};
\node[checknode,
label={[below, label distance=-0.4cm, align=center]
\acs{CN} 3\\$\left( c_4 + c_5 + c_6 + c_7 = 0 \right) $}]
(cn3) at (4, -1) {};
\node[variablenode, label={[above, align=center] \acs{VN} 1\\$c_1$}] (c1) at (-4.5, 2) {};
\node[variablenode, label={[above, align=center] \acs{VN} 2\\$c_2$}] (c2) at (-3, 2) {};
\node[variablenode, label={[above, align=center] \acs{VN} 3\\$c_3$}] (c3) at (-1.5, 2) {};
\node[variablenode, label={[above, align=center] \acs{VN} 4\\$c_4$}] (c4) at (0, 2) {};
\node[variablenode, label={[above, align=center] \acs{VN} 5\\$c_5$}] (c5) at (1.5, 2) {};
\node[variablenode, label={[above, align=center] \acs{VN} 6\\$c_6$}] (c6) at (3, 2) {};
\node[variablenode, label={[above, align=center] \acs{VN} 7\\$c_7$}] (c7) at (4.5, 2) {};
\draw (cn1) -- (c1);
\draw (cn1) -- (c3);
\draw (cn1) -- (c5);
\draw (cn1) -- (c7);
\draw (cn2) -- (c2);
\draw (cn2) -- (c3);
\draw (cn2) -- (c6);
\draw (cn2) -- (c7);
\draw (cn3) -- (c4);
\draw (cn3) -- (c5);
\draw (cn3) -- (c6);
\draw (cn3) -- (c7);
\end{tikzpicture}
\caption{Tanner graph for the (7,4)-Hamming-code}
\label{fig:theo:tanner_graph}
\end{figure}%
%
\noindent \acp{CN} and \acp{VN}, and by extention the elements of $\boldsymbol{H}$, are
indexed with the variables $j$ and $i$.
The sets of all \acp{CN} and all \acp{VN} are denoted by
$\mathcal{J} := \left[ 1:m \right]$ and $\mathcal{I} := \left[ 1:n \right]$, respectively.
The \textit{neighbourhood} of the $j$th \ac{CN}, i.e., the set of all adjacent \acp{VN},
is denoted by $N_c\left( j \right)$.
The neighbourhood of the $i$th \ac{VN} is denoted by $N_v\left( i \right)$.
For the code depicted in figure \ref{fig:theo:tanner_graph}, for example,
$N_c\left( 1 \right) = \left\{ 1, 3, 5, 7 \right\}$ and
$N_v\left( 3 \right) = \left\{ 1, 2 \right\}$.
Message passing algorithms are based on the notion of passing messages between
\acp{CN} and \acp{VN}.
\Ac{BP} is one such algorithm that is commonly used to decode \ac{LDPC} codes.
It is based on the observation that each \ac{CN} defines a single
parity-check code and each \ac{VN} defines a repetition code.
The messages transmitted between the nodes correspond to the \acp{LLR}:%
%
\begin{align*}
L_{i\to j} = \ldots
.\end{align*}
%
A number of iterations are performed, passing messages between \acp{CN} and \acp{VN}
in alternating fashion.
The bits at each \ac{VN} are then decoded based on the final values.
\ac{BP} can be shown to be equivalent to \ac{ML} decoding when the Tanner graph
is a tree, but is sub-optimal when the graph contains cycles.
This leads to generally worse performance than \ac{ML} decoding across all \acp{SNR}.
Additionally, an \textit{error floor} appears for very high \acp{SNR}, making
the use of \ac{BP} impractical for applications where a very low \ac{BER} is
desired.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -167,7 +302,7 @@ is made by a demodulator.
TODO: TODO:
\begin{itemize} \begin{itemize}
\item Intro \item Intro
\item Proximal Decoding \item Proximal gradient method
\end{itemize} \end{itemize}
\vspace{5mm} \vspace{5mm}