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Finished introduction except proximal

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29
latex/thesis/bibliography.bib
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@ -181,3 +181,32 @@
doi={10.1109/LCOMM.2019.2911277}
}
@ARTICLE{mackay_rediscovery,
author={MacKay, D.J.C.},
journal={IEEE Transactions on Information Theory},
title={Good error-correcting codes based on very sparse matrices},
year={1999},
volume={45},
number={2},
pages={399-431},
doi={10.1109/18.748992}
}
@ARTICLE{principles_of_dig_comm,
author={Forney, G.},
year={2003},
month={01},
pages={},
title={6.451 Principles of Digital Communication II, Spring 2003}
}
@book{ryan_lin_2009,
place={Cambridge},
title={Channel Codes: Classical and Modern},
% DOI={10.1017/CBO9780511803253},
publisher={Cambridge University Press},
author={Ryan, William and Lin, Shu},
year={2009},
url={https://d1.amobbs.com/bbs_upload782111/files_35/ourdev_604508GHLFR2.pdf}
}

2
latex/thesis/chapters/decoding_techniques.tex
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@ -870,8 +870,6 @@ $\boldsymbol{z}_j$ in the previous iteration.}%
a check-node update step (lines $3$-$6$) and the $\tilde{c}_i$-updates can be understood as
a variable-node update step (lines $7$-$9$ in figure \ref{fig:lp:message_passing}).
The updates for each variable- and check-node can be perfomed in parallel.
With this interpretation it becomes clear why \ac{LP} decoding using \ac{ADMM}
is able to achieve similar computational complexity to \ac{BP}.
The main computational effort in solving the linear program then amounts to
computing the projection operation $\Pi_{\mathcal{P}_{d_j}} \left( \cdot \right) $

165
latex/thesis/chapters/theoretical_background.tex
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@ -14,27 +14,6 @@ Lastly, the optimization methods utilized are described.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{General Remarks on Notation}
\label{sec:theo:Notation}
%
\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.
Wherever the domain of a variable is expanded, this will be indicated with a tilde.
For example:%
@ -44,7 +23,7 @@ For example:%
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
Additionally, a shorthand notation will be used to denote series of indices and series
of indexed variables:%
%
\begin{align*}
@ -68,16 +47,64 @@ This is known as modulation. The modulation scheme chosen here is \ac{BPSK}:%
.\end{align*}
%
The symbol that reaches the receiver, $\boldsymbol{y}$, is distorted by the channel.
The channel model used here is \ac{AWGN}:%
This distortion is described by the channel model, which here is chosen to be \ac{AWGN}:%
%
\begin{align*}
\boldsymbol{y} = \boldsymbol{x} + \boldsymbol{z},
\hspace{5mm} \boldsymbol{z}_i \in \mathcal{N}\left( 0, \frac{\sigma^2}{2} \right),
\hspace{5mm} z_i \in \mathcal{N}\left( 0, \frac{\sigma^2}{2} \right),
\hspace{2mm} i \in \left[ 1:n \right]
.\end{align*}
%
This process is visualized in figure \ref{fig:theo:channel_overview}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.
\Ac{LDPC} codes have become especially popular, since they are able to
reach arbitrarily small probabilities of error at coderates up to the capacity
of the channel \cite[Sec. II.B.]{mackay_rediscovery} and their structure allows
for very efficient decoding.
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 process of transmitting and decoding a codeword is visualized in
figure \ref{fig:theo:channel_overview}.%
%
\begin{figure}[H]
\centering
@ -113,61 +140,13 @@ This process is visualized in figure \ref{fig:theo:channel_overview}.
\node[below=0.25cm of z] (text) {Channel};
\end{tikzpicture}
\caption{Overview of channel and modulation}
\caption{Overview of channel model and modulation}
\label{fig:theo:channel_overview}
\end{figure}
\todo{$\boldsymbol{z}$ is used to denote both the noise and the auxiliary variable for ADMM}
\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}
criterion:%
%
@ -184,8 +163,8 @@ criterion:%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Decoding LDPC Codes using Belief Propagation}
\label{sec:theo:Decoding LDPC Codes using Belief Propagation}
\section{Tanner Graphs and Belief Propagation}
\label{sec:theo:Tanner Graphs and Belief Propagation}
It is often helpful to visualize codes graphically.
This is especially true for \ac{LDPC} codes, as the established decoding
@ -200,7 +179,8 @@ 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:%
(7,4) Hamming code, which has the following parity-check matrix
\cite[Example 5.7.]{ryan_lin_2009}:%
%
\begin{align*}
\boldsymbol{H} = \begin{bmatrix}
@ -261,8 +241,8 @@ Figure \ref{fig:theo:tanner_graph} shows the tanner graph for the
\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 $i$.
\noindent \acp{CN} and \acp{VN}, and by extention the rows and columns 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},
@ -275,24 +255,17 @@ $N_v\left( 3 \right) = \left\{ 1, 2 \right\}$.
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}.
It aims to compute the posterior probabilities
$p_{C_i \mid \boldsymbol{Y}}\left(c_i = 1 | \boldsymbol{y} \right),\hspace{2mm} i\in\mathcal{I}$
\cite[Sec. III.]{mackay_rediscovery} and use them to calculate the estimate $\hat{\boldsymbol{c}}$.
For cycle-free graphs this goal is reached after a finite
number of steps and \ac{BP} is thus equivalent to \ac{MAP} decoding.
When the graph contains cycles, however, \ac{BP} only approximates the probabilities
and is sub-optimal.
This leads to generally worse performance than \ac{MAP} decoding for practical codes.
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.
desired \cite[Sec. 15.3]{ryan_lin_2009}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%