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Andreas Tsouchlos 2023-01-25 22:01:01 +01:00
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2
latex/presentations/midterm/presentation.tex
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@ -121,7 +121,7 @@
\title{Application of Optimization Algorithms for Channel Decoding} \title{Application of Optimization Algorithms for Channel Decoding}
\subtitle{\small Midterm Presentation} \subtitle{\small Midterm Presentation - 27.01.2023}
%\author{Andreas Tsouchlos} %\author{Andreas Tsouchlos}
\author{\vspace{1.5mm} Andreas Tsouchlos} \author{\vspace{1.5mm} Andreas Tsouchlos}

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latex/presentations/midterm/sections/decoding_algorithms.tex
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@ -116,7 +116,7 @@ Output $\boldsymbol{\hat{x}}$
\begin{minipage}[c]{0.6\linewidth} \begin{minipage}[c]{0.6\linewidth}
\begin{itemize} \begin{itemize}
\item Codeword Polytope: \item Codeword polytope:
\begin{align*} \begin{align*}
\text{poly}\left( \mathcal{C} \right) = \text{poly}\left( \mathcal{C} \right) =
\left\{ \left\{
@ -126,13 +126,13 @@ Output $\boldsymbol{\hat{x}}$
\right\}, \right\},
\hspace{5mm} \lambda_{\boldsymbol{c}} \in \mathbb{R} \hspace{5mm} \lambda_{\boldsymbol{c}} \in \mathbb{R}
\end{align*} \end{align*}
\item Cost Function: \item Cost function:
\begin{align*} \begin{align*}
\sum_{i=1}^{n} \gamma_i c_i, \sum_{i=1}^{n} \gamma_i c_i,
\hspace{5mm}\gamma_i = \log\left( \hspace{5mm}\gamma_i = \log\left(
\frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right) \frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right)
\end{align*} \end{align*}
\item LP Formulation of ML Decoding: \item LP formulation of ML decoding:
\begin{align*} \begin{align*}
&\text{minimize } \sum_{i=1}^{n} \gamma_i f_i \\ &\text{minimize } \sum_{i=1}^{n} \gamma_i f_i \\
&\text{subject to } \boldsymbol{f}\in\text{poly}\left( \mathcal{C} \right) &\text{subject to } \boldsymbol{f}\in\text{poly}\left( \mathcal{C} \right)

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latex/presentations/midterm/sections/examination_results.tex
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@ -67,8 +67,8 @@
\end{figure} \end{figure}
\item $\mathcal{O}\left(n \right) $ time complexity - same as BP; \item $\mathcal{O}\left(n \right) $ time complexity - same as BP;
Only multiplication and addition necessary \cite{proximal_paper} only multiplication and addition necessary \cite{proximal_paper}
\item Measured Performance: $\sim\SI{10000}{frames / \second}$ \item Measured Performance: $\sim\SI{10000}{}$ frames/s
- Intel Core i7-7700HQ @ 2.80GHz; $n=204$ - Intel Core i7-7700HQ @ 2.80GHz; $n=204$
\end{itemize} \end{itemize}
\vspace{3mm} \vspace{3mm}
@ -81,8 +81,8 @@
\setcounter{footnote}{0} \setcounter{footnote}{0}
\begin{itemize} \begin{itemize}
\item Comparison of simulation \item Analysis of simulation
\footnote{(3,6) regular LDPC Code with $n=204, k=102$ \footnote{(3,6) regular LDPC code with $n=204, k=102$
\cite[\text{204.33.484}]{mackay_enc}} \cite[\text{204.33.484}]{mackay_enc}}
results for different values of $\gamma$ results for different values of $\gamma$
\end{itemize} \end{itemize}
@ -367,8 +367,8 @@
\setcounter{footnote}{0} \setcounter{footnote}{0}
\begin{itemize} \begin{itemize}
\item Comparison of simulated \item Analysis of simulated
\footnote{(3,6) regular LDPC Code with $n=204, k=102$ \footnote{(3,6) regular LDPC code with $n=204, k=102$
\cite[\text{204.33.484}]{mackay_enc}} \cite[\text{204.33.484}]{mackay_enc}}
BER and FER BER and FER
\end{itemize} \end{itemize}
@ -660,8 +660,8 @@ Output $\boldsymbol{\hat{x}}$
\setcounter{footnote}{0} \setcounter{footnote}{0}
\begin{itemize} \begin{itemize}
\item For larger $n$, the Gradient itself starts to oscillate \item For larger $n$, the gradient itself starts to oscillate
\item The Amplitude of the oscillation seems to be highly correlated \item The amplitude of the oscillation seems to be highly correlated
with the probability of a bit error with the probability of a bit error
\end{itemize} \end{itemize}
@ -721,12 +721,12 @@ Output $\boldsymbol{\hat{x}}$
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\caption{Corellation between bit error and amplitude of oscillation} \caption{Correlation between bit error and amplitude of oscillation}
\end{subfigure} \end{subfigure}
\end{figure} \end{figure}
\footnotetext{A single decoding is shown, using a (3,6) regular LDPC Code \footnotetext{A single decoding is shown, using a (3,6) regular LDPC code
with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc}; with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc};
$\gamma = 0.05, \omega = 0.05, E_b / N_0 = \SI{5}{dB}$} $\gamma = 0.05, \omega = 0.05, E_b / N_0 = \SI{5}{dB}$}
\end{frame} \end{frame}
@ -1364,11 +1364,11 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{x}}_n\text{ with lowest }d
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\caption{Average error for $\SI{500000}{decodings}, \caption{Average error for $\SI{500000}{}$ decodings,$
\omega = 0.05, \gamma = 0.05, K=200$\footnotemark} \omega = 0.05, \gamma = 0.05, K=200$\footnotemark}
\end{figure} \end{figure}
\footnotetext{Simulation performed with (3,6) regular LDPC Code with $n=204, k=102$ \footnotetext{Simulation performed with (3,6) regular LDPC code with $n=204, k=102$
\cite[Code: 204.33.484]{mackay_enc}} \cite[Code: 204.33.484]{mackay_enc}}
\begin{itemize} \begin{itemize}

6
latex/presentations/midterm/sections/forthcoming_examination.tex
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@ -1,5 +1,5 @@
\section{Forthcoming Examination}% \section{Forthcoming Examinations}%
\label{sec:Forthcoming Examination} \label{sec:Forthcoming Examinations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -7,7 +7,7 @@
\label{sub:LP Decoding} \label{sub:LP Decoding}
\begin{frame}[t] \begin{frame}[t]
\frametitle{Forthcoming Examination: LP Decoding} \frametitle{Forthcoming Examinations: LP Decoding}
\begin{itemize} \begin{itemize}
\item Test the (Alternating Direction Method of Multipliers) ADMM \item Test the (Alternating Direction Method of Multipliers) ADMM

31
latex/presentations/midterm/sections/theoretical_background.tex
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@ -11,8 +11,8 @@
\begin{itemize} \begin{itemize}
\item The general [ML] decoding problem for linear codes and the general problem \item The general [ML] decoding problem for linear codes and the general problem
of finding the weights of a linear code are both NP-complete. \cite{ml_np_hard_proof} of finding the weights of a linear code are both NP-complete. \cite{ml_np_hard_proof}
\item The iterative message–passing algorithms preffered in practice do not guarantee \item The iterative message–passing algorithms preferred in practice do not guarantee
optimality and may fail to decode correctly when the graph contains cycles optimality and may fail to decode correctly when the graph contains cycles.
\cite{ldpc_conv} \cite{ldpc_conv}
\item The standard message-passing algorithms used for decoding [LDPC and turbo codes] \item The standard message-passing algorithms used for decoding [LDPC and turbo codes]
are often difficult to analyze. \cite{feldman_thesis} are often difficult to analyze. \cite{feldman_thesis}
@ -48,7 +48,7 @@
\begin{itemize} \begin{itemize}
\item Examination of ``Proximal Decoding'' \item Examination of ``Proximal Decoding''
\item Examination of ``Iterative Point Decoding'' \item Examination of ``Interior Point Decoding''
\end{itemize} \end{itemize}
\end{frame} \end{frame}
@ -66,14 +66,14 @@
\centering \centering
\begin{tikzpicture}[scale=1, transform shape] \begin{tikzpicture}[scale=1, transform shape]
\node (in) {$c\left[ k \right] $}; \node (in) {$\boldsymbol{c}$};
\node[mapper, right=0.5cm of in] (bpskmap) {Mapper}; \node[mapper, right=0.5cm of in] (bpskmap) {Mapper};
\node[right=1.5cm of bpskmap, \node[right=1.5cm of bpskmap,
draw, circle, inner sep=0pt, minimum size=0.5cm] (add) {$+$}; draw, circle, inner sep=0pt, minimum size=0.5cm] (add) {$+$};
\node[right=0.5cm of add] (out) {$y\left[ k \right] $}; \node[right=0.5cm of add] (out) {$\boldsymbol{y}$};
\node[below=0.5cm of add] (noise) {$n\left[ k \right] $}; \node[below=0.5cm of add] (noise) {$\boldsymbol{z}$};
\node at ($(bpskmap.east)!0.5!(add.west) + (0,0.3cm)$) {$x\left[ k \right] $}; \node at ($(bpskmap.east)!0.5!(add.west) + (0,0.3cm)$) {$\boldsymbol{x}$};
\draw[->] (in) -- (bpskmap); \draw[->] (in) -- (bpskmap);
\draw[->] (bpskmap) -- (add); \draw[->] (bpskmap) -- (add);
@ -83,22 +83,21 @@
\end{figure} \end{figure}
\begin{itemize} \begin{itemize}
\item All simulations are performed with BPSK Modulation: \item All simulations are performed with BPSK modulation:
\begin{align*} \begin{align*}
x\left[ k \right] = \left( -1 \right)^{c\left[ k \right] }, \boldsymbol{x} = \left( -1 \right)^{\boldsymbol{c}},
\hspace{5mm} \boldsymbol{c} \in \mathbb{F}_2^n, \hspace{5mm} \boldsymbol{c} \in \mathbb{F}_2^n
\hspace{2mm} k\in \left\{ 1, \ldots, n \right\}
\end{align*} \end{align*}
\item The used channel model is AWGN: \item The used channel model is AWGN:
\begin{align*} \begin{align*}
\boldsymbol{y} = \boldsymbol{x} + \boldsymbol{n}, \boldsymbol{y} = \boldsymbol{x} + \boldsymbol{z},
\hspace{5mm}\boldsymbol{n}\sim \mathcal{N} \hspace{5mm}\boldsymbol{z}\sim \mathcal{N}
\left(0,\frac{1}{2}\left(\frac{k}{n}\frac{E_b}{N_0}\right)^{-1}\right), \left(0,\frac{1}{2}\left(\frac{k}{n}\frac{E_b}{N_0}\right)^{-1}\right),
\hspace{2mm} \boldsymbol{y}, \boldsymbol{n} \in \mathbb{R}^n \hspace{2mm} \boldsymbol{y}, \boldsymbol{z} \in \mathbb{R}^n
\end{align*} \end{align*}
\item All-zeros assumption: \item All-zeros assumption:
\begin{align*} \begin{align*}
\boldsymbol{c} = 0 \boldsymbol{c} = \boldsymbol{0}
\end{align*} \end{align*}
\end{itemize} \end{itemize}
\end{frame} \end{frame}
@ -113,7 +112,7 @@
\begin{minipage}[c]{0.6\linewidth} \begin{minipage}[c]{0.6\linewidth}
\begin{itemize} \begin{itemize}
\item Reormulate decoding problem as optimization problem \item Reformulate decoding problem as optimization problem
\begin{itemize} \begin{itemize}
\item Establish objective function \item Establish objective function
\item Establish constraints \item Establish constraints