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9
latex/presentations/final/presentation.tex
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@ -300,12 +300,13 @@
\input{sections/proximal_decoding.tex} \input{sections/proximal_decoding.tex}
\input{sections/lp_dec_using_admm.tex} \input{sections/lp_dec_using_admm.tex}
\input{sections/comparison.tex} \input{sections/comparison.tex}
\input{sections/conclusion.tex}
\input{sections/question_slide.tex} \input{sections/question_slide.tex}
\begin{frame}[allowframebreaks] % \begin{frame}[allowframebreaks]
\frametitle{Bibliography} % \frametitle{Bibliography}
\printbibliography[heading=none] % \printbibliography[heading=none]
\end{frame} % \end{frame}
\input{sections/appendix.tex} \input{sections/appendix.tex}

82
latex/presentations/final/sections/appendix.tex
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@ -27,7 +27,7 @@
ylabel={$\gamma$}, ylabel={$\gamma$},
zlabel={BER}, zlabel={BER},
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
legend pos=outer north east,] legend pos=outer north east,]
\addplot3[surf, \addplot3[surf,
@ -50,10 +50,10 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$ \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$
\cite[\text{96.3.965}]{mackay_enc}} \citereference{Mac23, 96.3.965}}
\end{subfigure}% \end{subfigure}%
\begin{subfigure}[t]{0.33\textwidth} \begin{subfigure}[t]{0.33\textwidth}
\centering \centering
@ -65,7 +65,7 @@
ylabel={$\gamma$}, ylabel={$\gamma$},
zlabel={BER}, zlabel={BER},
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
legend pos=outer north east,] legend pos=outer north east,]
\addplot3[surf, \addplot3[surf,
@ -88,10 +88,10 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$ \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$
\cite[\text{204.33.484}]{mackay_enc}} \citereference{Mac23, 204.33.484}}
\end{subfigure}% \end{subfigure}%
\begin{subfigure}[t]{0.33\textwidth} \begin{subfigure}[t]{0.33\textwidth}
\centering \centering
@ -103,7 +103,7 @@
ylabel={$\gamma$}, ylabel={$\gamma$},
zlabel={BER}, zlabel={BER},
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
legend pos=outer north east,] legend pos=outer north east,]
\addplot3[surf, \addplot3[surf,
@ -126,12 +126,14 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$ \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$
\cite[\text{408.33.844}]{mackay_enc}} \citereference{Mac23, 408.33.844}}
\end{subfigure}% \end{subfigure}%
\vspace*{-2mm}
\begin{subfigure}[t]{0.33\textwidth} \begin{subfigure}[t]{0.33\textwidth}
\centering \centering
@ -142,7 +144,7 @@
ylabel={$\gamma$}, ylabel={$\gamma$},
zlabel={BER}, zlabel={BER},
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
legend pos=outer north east,] legend pos=outer north east,]
\addplot3[surf, \addplot3[surf,
@ -165,7 +167,7 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{BCH code with $n=31, k=26$} \caption{BCH code with $n=31, k=26$}
\end{subfigure}% \end{subfigure}%
@ -179,7 +181,7 @@
ylabel={$\gamma$}, ylabel={$\gamma$},
zlabel={BER}, zlabel={BER},
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
legend pos=outer north east,] legend pos=outer north east,]
\addplot3[surf, \addplot3[surf,
@ -202,10 +204,10 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$ \caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$
\cite[\text{204.55.187}]{mackay_enc}} \citereference{Mac23, 204.55.187}}
\end{subfigure}% \end{subfigure}%
\begin{subfigure}[t]{0.33\textwidth} \begin{subfigure}[t]{0.33\textwidth}
\centering \centering
@ -217,7 +219,7 @@
ylabel={$\gamma$}, ylabel={$\gamma$},
zlabel={BER}, zlabel={BER},
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
legend pos=outer north east,] legend pos=outer north east,]
\addplot3[surf, \addplot3[surf,
@ -240,10 +242,10 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{LDPC code (progressive edge growth construction) with $n=504, k=252$ \caption{LDPC code (progressive edge growth construction) with $n=504, k=252$
\cite[\text{PEGReg252x504}]{mackay_enc}} \citereference{Mac23, PEGReg252x504}}
\end{subfigure}% \end{subfigure}%
\end{figure} \end{figure}
\end{minipage}% \end{minipage}%
@ -271,6 +273,11 @@
\end{tikzpicture} \end{tikzpicture}
\end{figure} \end{figure}
\end{minipage} \end{minipage}
\vspace*{-1mm}
\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame} \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -296,7 +303,7 @@
xlabel={$\rho$}, ylabel={Avg. \# iter.}, xlabel={$\rho$}, ylabel={Avg. \# iter.},
ymode=log, ymode=log,
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
] ]
\addplot[ForestGreen, line width=1pt] \addplot[ForestGreen, line width=1pt]
table [col sep=comma, x=rho, y=k_avg, table [col sep=comma, x=rho, y=k_avg,
@ -313,10 +320,10 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$ \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$
\cite[\text{96.3.965}]{mackay_enc}} \citereference{Mac23, 96.3.965}}
\end{subfigure}% \end{subfigure}%
\begin{subfigure}[t]{0.33\textwidth} \begin{subfigure}[t]{0.33\textwidth}
\centering \centering
@ -327,7 +334,7 @@
xlabel={$\rho$}, ylabel={Avg. \# iter.}, xlabel={$\rho$}, ylabel={Avg. \# iter.},
ymode=log, ymode=log,
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
] ]
\addplot[ForestGreen, line width=1pt] \addplot[ForestGreen, line width=1pt]
table [col sep=comma, x=rho, y=k_avg, table [col sep=comma, x=rho, y=k_avg,
@ -344,10 +351,10 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$ \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$
\cite[\text{204.33.484}]{mackay_enc}} \citereference{Mac23, 204.33.484}}
\end{subfigure}% \end{subfigure}%
\begin{subfigure}[t]{0.33\textwidth} \begin{subfigure}[t]{0.33\textwidth}
\centering \centering
@ -358,7 +365,7 @@
xlabel={$\rho$}, ylabel={Avg. \# iter.}, xlabel={$\rho$}, ylabel={Avg. \# iter.},
ymode=log, ymode=log,
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
] ]
\addplot[ForestGreen, line width=1pt] \addplot[ForestGreen, line width=1pt]
table [col sep=comma, x=rho, y=k_avg, table [col sep=comma, x=rho, y=k_avg,
@ -375,12 +382,14 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$ \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$
\cite[\text{408.33.844}]{mackay_enc}} \citereference{Mac23, 408.33.844}}
\end{subfigure}% \end{subfigure}%
\vspace*{-2mm}
\begin{subfigure}[t]{0.33\textwidth} \begin{subfigure}[t]{0.33\textwidth}
\centering \centering
@ -390,7 +399,7 @@
xlabel={$\rho$}, ylabel={Avg. \# iter.}, xlabel={$\rho$}, ylabel={Avg. \# iter.},
ymode=log, ymode=log,
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
] ]
\addplot[ForestGreen, line width=1pt] \addplot[ForestGreen, line width=1pt]
table [col sep=comma, x=rho, y=k_avg, table [col sep=comma, x=rho, y=k_avg,
@ -407,7 +416,7 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{BCH code with $n=31, k=26$} \caption{BCH code with $n=31, k=26$}
\end{subfigure}% \end{subfigure}%
@ -420,7 +429,7 @@
xlabel={$\rho$}, ylabel={Avg. \# iter.}, xlabel={$\rho$}, ylabel={Avg. \# iter.},
ymode=log, ymode=log,
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
] ]
\addplot[ForestGreen, line width=1pt] \addplot[ForestGreen, line width=1pt]
table [col sep=comma, x=rho, y=k_avg, table [col sep=comma, x=rho, y=k_avg,
@ -437,10 +446,10 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$ \caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$
\cite[\text{204.55.187}]{mackay_enc}} \citereference{Mac23, 204.55.187}}
\end{subfigure}% \end{subfigure}%
\begin{subfigure}[t]{0.33\textwidth} \begin{subfigure}[t]{0.33\textwidth}
\centering \centering
@ -451,7 +460,7 @@
xlabel={$\rho$}, ylabel={Avg. \# iter.}, xlabel={$\rho$}, ylabel={Avg. \# iter.},
ymode=log, ymode=log,
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
] ]
\addplot[ForestGreen, line width=1pt] \addplot[ForestGreen, line width=1pt]
table [col sep=comma, x=rho, y=k_avg, table [col sep=comma, x=rho, y=k_avg,
@ -468,10 +477,10 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{LDPC code (progressive edge growth construction) with $n=504, k=252$ \caption{LDPC code (progressive edge growth construction) with $n=504, k=252$
\cite[\text{PEGReg252x504}]{mackay_enc}} \citereference{Mac23, PEGReg252x504}}
\end{subfigure}% \end{subfigure}%
\end{figure} \end{figure}
\end{minipage}% \end{minipage}%
@ -500,5 +509,10 @@
\end{tikzpicture} \end{tikzpicture}
\end{figure} \end{figure}
\end{minipage} \end{minipage}
\vspace*{-1mm}
\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame} \end{frame}

118
latex/presentations/final/sections/comparison.tex
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@ -11,8 +11,10 @@
\begin{frame}[t] \begin{frame}[t]
\frametitle{Comparison: Convergence Behavior} \frametitle{Comparison: Convergence Behavior}
\vspace*{-3mm}
\begin{itemize} \begin{itemize}
\item (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc} \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484}
\end{itemize} \end{itemize}
\begin{figure}[H] \begin{figure}[H]
@ -105,6 +107,14 @@
\begin{itemize} \begin{itemize}
\item Minimum number of iterations independant of SNR \item Minimum number of iterations independant of SNR
\end{itemize} \end{itemize}
\bigskip
\smallskip
\smallskip
\smallskip
\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame} \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -114,11 +124,11 @@
\vspace*{-6.5mm} \vspace*{-6.5mm}
\begin{itemize} \begin{itemize}
\item The points shown are from the following codes: BCH $\left( 31, 11 \right)$; \item Codes: BCH $\left( 31, 11 \right)$; BCH $\left( 31, 26 \right)$;
BCH $\left( 31, 26 \right)$; \\ \citereference{Mac23, 96.3.965; 204.33.484; 204.55.187; 408.33.844; PEGReg252x504}
\cite[\text{96.3.965; 204.33.484; 204.55.187; 408.33.844; PEGReg252x504}]{mackay_enc}
\item Measured performance: $\sim\SI{10000}{}$ frames/s \item Measured performance: $\sim\SI{10000}{}$ frames/s
on Intel Core i7-7700HQ @ 2.80GHz; $n=204$ on Intel Core i7-7700HQ @ 2.80GHz; $n=204$
\item Both algorithms are $\mathcal{O}\left( n \right)$ on average
\end{itemize} \end{itemize}
\begin{figure}[H] \begin{figure}[H]
@ -130,8 +140,8 @@
xlabel={$n$}, ylabel={time per frame (s)}, xlabel={$n$}, ylabel={time per frame (s)},
legend style={at={(0.05,0.6)},anchor=south west}, legend style={at={(0.05,0.6)},anchor=south west},
legend cell align={left}, legend cell align={left},
width=0.45\textwidth, width=0.43\textwidth,
height=0.3375\textwidth, height=0.32\textwidth,
] ]
\addplot[RedOrange, only marks, mark=square*] \addplot[RedOrange, only marks, mark=square*]
table [col sep=comma, x=n, y=spf] table [col sep=comma, x=n, y=spf]
@ -151,10 +161,12 @@
\end{tikzpicture} \end{tikzpicture}
\end{figure} \end{figure}
\begin{itemize} \bigskip
\item Both algorithms are $\mathcal{O}\left( n \right)$ on average \smallskip
\item LP decoding implementation significantly faster \smallskip
\end{itemize}
\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame} \end{frame}
@ -165,7 +177,7 @@
\vspace{-0.5cm} \vspace{-0.5cm}
\begin{itemize} \begin{itemize}
\item (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc} \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484}
\end{itemize} \end{itemize}
\begin{figure}[H] \begin{figure}[H]
@ -179,8 +191,8 @@
legend columns=2, legend columns=2,
legend style={at={(0.5,-0.45)},anchor=south}, legend style={at={(0.5,-0.45)},anchor=south},
ymax=1.5, ymin=3e-8, ymax=1.5, ymin=3e-8,
width=0.48\textwidth, width=0.45\textwidth,
height=0.35\textwidth, height=0.33\textwidth,
] ]
\addplot[RedOrange, mark=*, line width=1pt] \addplot[RedOrange, mark=*, line width=1pt]
table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}] table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
@ -208,8 +220,8 @@
legend columns=2, legend columns=2,
legend style={at={(0.5,-0.45)},anchor=south}, legend style={at={(0.5,-0.45)},anchor=south},
ymax=1.5, ymin=3e-8, ymax=1.5, ymin=3e-8,
width=0.48\textwidth, width=0.45\textwidth,
height=0.35\textwidth, height=0.33\textwidth,
] ]
\addplot[RedOrange, mark=*, solid, line width=1pt] \addplot[RedOrange, mark=*, solid, line width=1pt]
@ -256,6 +268,13 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\end{figure} \end{figure}
\smallskip
\smallskip
\smallskip
\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame} \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -318,7 +337,7 @@
\centering \centering
\begin{figure}[H] \begin{figure}[H]
\vspace*{-0.7cm} \vspace*{-0.8cm}
\centering \centering
\begin{subfigure}[t]{0.33\textwidth} \begin{subfigure}[t]{0.33\textwidth}
@ -331,7 +350,7 @@
ymode=log, ymode=log,
ymax=1.5, ymin=8e-5, ymax=1.5, ymin=8e-5,
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
] ]
\addplot[RedOrange, line width=1pt, mark=*, solid] \addplot[RedOrange, line width=1pt, mark=*, solid]
@ -351,10 +370,10 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$ \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$
\cite[\text{96.3.965}]{mackay_enc}} \citereference{Mac23, 96.33.965}}
\end{subfigure}% \end{subfigure}%
\begin{subfigure}[t]{0.33\textwidth} \begin{subfigure}[t]{0.33\textwidth}
\centering \centering
@ -366,7 +385,7 @@
ymode=log, ymode=log,
ymax=1.5, ymin=8e-5, ymax=1.5, ymin=8e-5,
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
] ]
\addplot[RedOrange, line width=1pt, mark=*] \addplot[RedOrange, line width=1pt, mark=*]
@ -390,10 +409,10 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$ \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$
\cite[\text{204.33.484}]{mackay_enc}} \citereference{Mac23, 204.33.484}}
\end{subfigure}% \end{subfigure}%
\begin{subfigure}[t]{0.33\textwidth} \begin{subfigure}[t]{0.33\textwidth}
\centering \centering
@ -405,7 +424,7 @@
ymode=log, ymode=log,
ymax=1.5, ymin=8e-5, ymax=1.5, ymin=8e-5,
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
] ]
\addplot[RedOrange, line width=1pt, mark=*] \addplot[RedOrange, line width=1pt, mark=*]
@ -421,12 +440,14 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$ \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$
\cite[\text{408.33.844}]{mackay_enc}} \citereference{Mac23, 408.33.844}}
\end{subfigure}% \end{subfigure}%
\vspace*{-2mm}
\begin{subfigure}[t]{0.33\textwidth} \begin{subfigure}[t]{0.33\textwidth}
\centering \centering
@ -437,7 +458,7 @@
ymode=log, ymode=log,
ymax=1.5, ymin=8e-5, ymax=1.5, ymin=8e-5,
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
] ]
\addplot[RedOrange, line width=1pt, mark=*] \addplot[RedOrange, line width=1pt, mark=*]
@ -459,7 +480,7 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{BCH code with $n=31, k=26$} \caption{BCH code with $n=31, k=26$}
\end{subfigure}% \end{subfigure}%
@ -473,7 +494,7 @@
ymode=log, ymode=log,
ymax=1.5, ymin=8e-5, ymax=1.5, ymin=8e-5,
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
] ]
\addplot[RedOrange, line width=1pt, mark=*] \addplot[RedOrange, line width=1pt, mark=*]
@ -488,10 +509,10 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$ \caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$
\cite[\text{204.55.187}]{mackay_enc}} \citereference{Mac23, 204.55.187}}
\end{subfigure}% \end{subfigure}%
\begin{subfigure}[t]{0.33\textwidth} \begin{subfigure}[t]{0.33\textwidth}
\centering \centering
@ -503,7 +524,7 @@
ymode=log, ymode=log,
ymax=1.5, ymin=8e-5, ymax=1.5, ymin=8e-5,
width=1.2\textwidth, width=1.2\textwidth,
height=0.85\textwidth, height=0.825\textwidth,
] ]
\addplot[RedOrange, line width=1pt, mark=*] \addplot[RedOrange, line width=1pt, mark=*]
@ -519,10 +540,10 @@
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\vspace*{-1mm} \vspace*{-2mm}
\caption{LDPC code (progressive edge growth construction) with $n=504, k=252$ \caption{LDPC code (progressive edge growth construction) with $n=504, k=252$
\cite[\text{PEGReg252x504}]{mackay_enc}} \citereference{Mac23, PEGReg252x504}}
\end{subfigure}% \end{subfigure}%
\end{figure} \end{figure}
\end{minipage}% \end{minipage}%
@ -548,19 +569,17 @@
\addlegendimage{NavyBlue, line width=1pt, mark=*} \addlegendimage{NavyBlue, line width=1pt, mark=*}
\addlegendentry{ADMM} \addlegendentry{ADMM}
% \addlegendimage{RoyalPurple, line width=1pt, mark=*}
% \addlegendentry{BP}
\addlegendimage{Black, line width=1pt, mark=*} \addlegendimage{Black, line width=1pt, mark=*}
\addlegendentry{ML} \addlegendentry{ML}
% \addlegendimage{PineGreen, line width=1pt, mark=triangle*, solid}
% \addlegendentry{ML}
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\end{figure} \end{figure}
\end{minipage} \end{minipage}
\vspace*{-1mm}
\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame} \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -700,24 +719,3 @@ return $\tilde{\boldsymbol{c}}$
\end{figure}% \end{figure}%
\end{frame} \end{frame}
\begin{frame}[t]
\frametitle{Conclusion}
\begin{itemize}
\item Analysis of the general behavior of the two decoding algorithms
\begin{itemize}
\item Parameter choice based on decoding performance and time complexity
\item Verification of theoretical considerations with simulation results
\end{itemize}
\item Suggestion for improvement of proximal decoding
\begin{itemize}
\item Addition of "ML-in-the-List" step
\item Up to $\sim \SI{1}{dB}$ gain under certain conditions
\end{itemize}
\item Comparison
\begin{itemize}
\item based on simulation results
\item based on their theoretical structure
\end{itemize}
\end{itemize}
\end{frame}

26
latex/presentations/final/sections/conclusion.tex Normal file
View File

@ -0,0 +1,26 @@
\section{Conclusion and Outlook}%
\label{sec:Comparison}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{Conclusion}
\begin{itemize}
\item Analysis of the general behavior of the two decoding algorithms
\begin{itemize}
\item Parameter choice based on decoding performance and time complexity
\item Verification of theoretical considerations with simulation results
\end{itemize}
\item Suggestion for improvement of proximal decoding
\begin{itemize}
\item Addition of "ML-in-the-List" step
\item Up to $\sim \SI{1}{dB}$ gain under certain conditions
\end{itemize}
\item Comparison
\begin{itemize}
\item based on simulation results
\item based on their theoretical structure
\end{itemize}
\end{itemize}
\end{frame}

163
latex/presentations/final/sections/lp_dec_using_admm.tex
View File

@ -7,7 +7,9 @@
\label{sub:LP Decoding using ADMM} \label{sub:LP Decoding using ADMM}
\begin{frame}[t] \begin{frame}[t]
\frametitle{LP Decoding \cite{feldman_paper}} \frametitle{LP Decoding \citereference{FWK05}}
\vspace*{-3mm}
\begin{itemize} \begin{itemize}
\item General reformulation of ML decoding as a linear program (LP) \item General reformulation of ML decoding as a linear program (LP)
@ -36,6 +38,13 @@
\end{align*} \end{align*}
\item Goal: relaxation of constraints to make a practical solution to the problem feasible \item Goal: relaxation of constraints to make a practical solution to the problem feasible
\end{itemize} \end{itemize}
\bigskip
\bigskip
\addreference{FWK05}{J. Feldman; M.J. Wainwright; D.R. Karger: \emph{Using linear programming
to Decode Binary linear codes}.
IEEE Transactions on Information Theory 51.3 (2005), pp. 954–972.}
\end{frame} \end{frame}
@ -656,12 +665,16 @@
\begin{frame}[t] \begin{frame}[t]
\frametitle{LP Decoding using ADMM} \frametitle{LP Decoding using ADMM}
\vspace*{-4mm}
\begin{itemize} \begin{itemize}
\item Solution using the \textit{alternating direction method of multipliers} (ADMM) \item Solution using the \textit{alternating direction method of multipliers} (ADMM)
\citereference{$\text{Bar}^+\text{13}$}
\item Slight reformulation of the LCLP: \item Slight reformulation of the LCLP:
\begin{align*} \begin{align*}
\begin{aligned} \begin{aligned}
\text{minimize}\hspace{2mm} &\boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}} \text{minimize}\hspace{2mm} &\boldsymbol{\gamma}^\text{T}
\tilde{\boldsymbol{c}}
+ \sum_{j\in\mathcal{J}} g_j\left( \boldsymbol{z}_j \right) \\ + \sum_{j\in\mathcal{J}} g_j\left( \boldsymbol{z}_j \right) \\
\text{subject to}\hspace{2mm} & \text{subject to}\hspace{2mm} &
\boldsymbol{T}_j\tilde{\boldsymbol{c}} = \boldsymbol{z}_j, \hspace{2mm} \boldsymbol{T}_j\tilde{\boldsymbol{c}} = \boldsymbol{z}_j, \hspace{2mm}
@ -692,6 +705,12 @@
% - \boldsymbol{z}_j + \boldsymbol{u}_j\right\Vert \right) % - \boldsymbol{z}_j + \boldsymbol{u}_j\right\Vert \right)
\end{alignat*} \end{alignat*}
\end{itemize} \end{itemize}
\bigskip
\addreference{$\text{Bar}^+\text{13}$}{Siddharth Barman et al.:
\emph{Decomposition Methods for Large Scale LP Decoding}.
IEEE Transactions on Information Theory 59.12 (2013), pp. 7870–7886.}
\end{frame} \end{frame}
@ -703,12 +722,14 @@
\begin{itemize} \begin{itemize}
\item Convergence properties enhanced by over-relaxation with parameter $\rho$ \item Convergence properties enhanced by over-relaxation with parameter $\rho$
\item Simplified rules% \item Simplified rules:
\footnote{$\left( \boldsymbol{z}_j \right)_i $ is a slight abuse of notation. % \footnote{$\left( \boldsymbol{z}_j \right)_i $ is a slight abuse of notation.
What is actually meant is the component of $\boldsymbol{z}_j$ that is associated % What is actually meant is the component of $\boldsymbol{z}_j$ that is associated
with the VN $i$, i.e., $\left( \boldsymbol{T}_j^\text{T} % with the VN $i$, i.e., $\left( \boldsymbol{T}_j^\text{T}
\boldsymbol{z}_j \right)_i$.\\ % \boldsymbol{z}_j \right)_i$.\\
The same is true for $\left( \boldsymbol{u}_j \right)_i$}: % The same is true for $\left( \boldsymbol{u}_j \right)_i$}:
\vspace*{-3mm}
\begin{alignat*}{3} \begin{alignat*}{3}
\tilde{c}_i &\leftarrow \frac{1}{\left| N_v\left( i \right) \right|} \left( \tilde{c}_i &\leftarrow \frac{1}{\left| N_v\left( i \right) \right|} \left(
@ -726,12 +747,31 @@
- \boldsymbol{z}_j - \boldsymbol{z}_j
\hspace{5mm} && \forall j\in\mathcal{J} \hspace{5mm} && \forall j\in\mathcal{J}
\end{alignat*} \end{alignat*}
\vspace*{2mm}
\item The projections $\Pi_{\mathcal{P}_{d_j}}, \hspace{1mm} j\in\mathcal{J}$ \item The projections $\Pi_{\mathcal{P}_{d_j}}, \hspace{1mm} j\in\mathcal{J}$
are the main computational effort are the main computational effort
\item Many approaches exist \cite{original_admm}, \cite{efficient_lp_dec_admm}, \item Many approaches exist \citereference{$\text{Bar}^+\text{13}$},
\cite{lautern} \citereference{ZS13}, \citereference{$\text{Gen}^+\text{20}$}
\item The approach chosen here is the one described in \cite{original_admm} \item The approach chosen here is the one described in
\citereference{$\text{Bar}^+\text{13}$}
\end{itemize} \end{itemize}
\bigskip
\smallskip
\addreferences
{$\text{Bar}^+\text{13}$}{Siddharth Barman et al.:
\emph{Decomposition Methods for Large Scale LP Decoding}.
IEEE Transactions on Information Theory 59.12 (2013), pp. 7870–7886.}
{ZS13}{Xiaojie Zhang; Paul H. Siegel: \emph{Efficient iterative LP decoding
of LDPC codes with alternating direction method of multipliers}.
2013 IEEE International Symposium on Information Theory. 2013, pp. 1501–1505.}
{$\text{Gen}^+\text{20}$}{Florian Gensheimer et al.:
\emph{A Reduced-Complexity Projection Algorithm for ADMM-Based LP Decoding}.
IEEE Transactions on Information Theory 66.8 (2020), pp. 4819–4833.}
\stopreferences
\end{frame} \end{frame}
@ -827,9 +867,11 @@
\begin{frame}[t] \begin{frame}[t]
\frametitle{LP Decoding using ADMM: Frame Error Rate} \frametitle{LP Decoding using ADMM: Frame Error Rate}
\vspace*{-6mm}
\begin{itemize} \begin{itemize}
\item ``Margulis'' LDPC code with $n = 2640$, $k = 1320$ \item ``Margulis'' LDPC code with $n = 2640$, $k = 1320$
\cite[\text{Margulis2640.1320.3}]{mackay_enc} \citereference{Mac23, Margulis2640.1320.3}
% ; $K=200, \mu = 3.3, \rho=1.9, % ; $K=200, \mu = 3.3, \rho=1.9,
% \epsilon_{\text{pri}} = 10^{-5}, \epsilon_{\text{dual}} = 10^{-5}$ % \epsilon_{\text{pri}} = 10^{-5}, \epsilon_{\text{dual}} = 10^{-5}$
\end{itemize} \end{itemize}
@ -852,50 +894,7 @@
discard if gt={SNR}{2.2}, discard if gt={SNR}{2.2},
] ]
{res/admm/fer_paper_margulis.csv}; {res/admm/fer_paper_margulis.csv};
\addlegendentry{ADMM (Barman et al.)} \addlegendentry{ADMM \citereference{$\text{Bar}^+\text{13}$}}
\addplot[RoyalPurple, line width=1pt, mark=*]
table [col sep=comma, x=SNR, y=FER, discard if gt={SNR}{2.2},]
{res/generic/fer_bp_mackay_margulis.csv};
\addlegendentry{BP (Barman et al.)}
\end{axis}
\end{tikzpicture}
\end{figure}
\begin{itemize}
\item Comparison of simulation with results from Barman et al. \cite{original_admm}
\end{itemize}
\end{frame}
\begin{frame}[t]
\frametitle{LP Decoding using ADMM: Frame Error Rate}
\begin{itemize}
\item ``Margulis'' LDPC code with $n = 2640$, $k = 1320$
\cite[\text{Margulis2640.1320.3}]{mackay_enc}
% ; $K=200, \mu = 3.3, \rho=1.9,
% \epsilon_{\text{pri}} = 10^{-5}, \epsilon_{\text{dual}} = 10^{-5}$
\end{itemize}
\begin{figure}[H]
\centering
\begin{tikzpicture}
\begin{axis}[
grid=both,
xlabel={$E_b / N_0 \left( \text{dB} \right) $}, ylabel={FER},
ymode=log,
width=0.45\textwidth,
height=0.325\textwidth,
%legend style={at={(0.03,0.04)},anchor=south west},
legend pos=outer north east,
]
\addplot[Emerald, line width=1pt, mark=*]
table [col sep=comma, x=SNR, y=FER,
discard if gt={SNR}{2.2},
]
{res/admm/fer_paper_margulis.csv};
\addlegendentry{ADMM (Barman et al.)}
\addplot[NavyBlue, mark=*, line width=1pt, densely dashed] \addplot[NavyBlue, mark=*, line width=1pt, densely dashed]
table [col sep=comma, x=SNR, y=FER,] table [col sep=comma, x=SNR, y=FER,]
{res/admm/ber_margulis264013203.csv}; {res/admm/ber_margulis264013203.csv};
@ -903,14 +902,25 @@
\addplot[RoyalPurple, line width=1pt, mark=*] \addplot[RoyalPurple, line width=1pt, mark=*]
table [col sep=comma, x=SNR, y=FER, discard if gt={SNR}{2.2},] table [col sep=comma, x=SNR, y=FER, discard if gt={SNR}{2.2},]
{res/generic/fer_bp_mackay_margulis.csv}; {res/generic/fer_bp_mackay_margulis.csv};
\addlegendentry{BP (Barman et al.)} \addlegendentry{BP \citereference{$\text{Bar}^+\text{13}$}}
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\end{figure}% \end{figure}%
\begin{itemize} \begin{itemize}
\item Comparison of simulation with results from Barman et al. \cite{original_admm} \item Comparison of simulation with results from Barman et al.
\citereference{$\text{Bar}^+\text{13}$}
\end{itemize} \end{itemize}
\bigskip
\addreferences
{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
{$\text{Bar}^+\text{13}$}{Siddharth Barman et al.:
\emph{Decomposition Methods for Large Scale LP Decoding}.
IEEE Transactions on Information Theory 59.12 (2013), pp. 7870–7886.}
\stopreferences
\end{frame} \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -918,8 +928,10 @@
\frametitle{LP Decoding using ADMM: Choice of Penalty\\ \frametitle{LP Decoding using ADMM: Choice of Penalty\\
Parameters} Parameters}
\vspace*{-5mm}
\begin{itemize} \begin{itemize}
\item (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc} \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484}
\end{itemize} \end{itemize}
\begin{figure}[H] \begin{figure}[H]
@ -933,8 +945,8 @@
grid=both, grid=both,
xlabel={$\mu$}, ylabel={FER}, xlabel={$\mu$}, ylabel={FER},
ymode=log, ymode=log,
width=0.9\textwidth, width=0.8\textwidth,
height=0.675\textwidth, height=0.6\textwidth,
] ]
\addplot[ForestGreen, line width=1pt, densely dashed, mark=*] \addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
table [col sep=comma, x=mu, y=FER, table [col sep=comma, x=mu, y=FER,
@ -959,8 +971,8 @@
grid=both, grid=both,
xlabel={$\rho$}, ylabel={FER}, xlabel={$\rho$}, ylabel={FER},
ymode=log, ymode=log,
width=0.9\textwidth, width=0.8\textwidth,
height=0.675\textwidth, height=0.6\textwidth,
] ]
\addplot[ForestGreen, line width=1pt, densely dashed, mark=*] \addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
table [col sep=comma, x=rho, y=FER, table [col sep=comma, x=rho, y=FER,
@ -1003,6 +1015,11 @@
\item Similar to $\gamma$ with proximal decoding: no clear optimum \item Similar to $\gamma$ with proximal decoding: no clear optimum
\end{itemize} \end{itemize}
\bigskip
\bigskip
\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame} \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -1010,8 +1027,10 @@
\frametitle{LP Decoding using ADMM: Choice of Penalty\\ \frametitle{LP Decoding using ADMM: Choice of Penalty\\
Parameters} Parameters}
\vspace*{-5mm}
\begin{itemize} \begin{itemize}
\item (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc} \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484}
\end{itemize} \end{itemize}
\begin{figure}[H] \begin{figure}[H]
@ -1024,8 +1043,8 @@
\begin{axis}[ \begin{axis}[
grid=both, grid=both,
xlabel={$\mu$}, ylabel={Average \# of iterations}, xlabel={$\mu$}, ylabel={Average \# of iterations},
width=0.9\textwidth, width=0.8\textwidth,
height=0.675\textwidth, height=0.6\textwidth,
] ]
\addplot[ForestGreen, line width=1pt, densely dashed, mark=*] \addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
table [col sep=comma, x=mu, y=k_avg, table [col sep=comma, x=mu, y=k_avg,
@ -1049,8 +1068,8 @@
\begin{axis}[ \begin{axis}[
grid=both, grid=both,
xlabel={$\rho$}, ylabel={Average \# of iterations}, xlabel={$\rho$}, ylabel={Average \# of iterations},
width=0.9\textwidth, width=0.8\textwidth,
height=0.675\textwidth, height=0.6\textwidth,
] ]
\addplot[ForestGreen, line width=1pt, densely dashed, mark=*] \addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
table [col sep=comma, x=rho, y=k_avg, table [col sep=comma, x=rho, y=k_avg,
@ -1092,5 +1111,13 @@
\begin{itemize} \begin{itemize}
\item For lower decoding time, choose low $\mu$ and high $\rho$ \item For lower decoding time, choose low $\mu$ and high $\rho$
\end{itemize} \end{itemize}
\bigskip
\smallskip
\smallskip
\smallskip
\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame} \end{frame}

106
latex/presentations/final/sections/proximal_decoding.tex
View File

@ -7,9 +7,9 @@
\label{sub:Decoding Algorithm} \label{sub:Decoding Algorithm}
\begin{frame}[t] \begin{frame}[t]
\frametitle{Proximal Decoding: General Idea \cite{proximal_paper}} \frametitle{Proximal Decoding: General Idea \citereference{WT22}}
\vspace*{-0.3cm} \vspace*{-0.7cm}
\begin{itemize} \begin{itemize}
\item MAP rule as continuous maximization problem: \item MAP rule as continuous maximization problem:
@ -59,6 +59,13 @@
\end{minipage} \end{minipage}
\hfill \hfill
\end{itemize} \end{itemize}
\smallskip
\smallskip
\addreference{WT22}{Tadashi Wadayama; Satoshi Takabe: \emph{Proximal Decoding for LDPC
Codes}. IEICE Transactions on Fundamentals of Electronics, Communications and Computer
Sciences advpub (2022), 2022TAP0002.}
\end{frame} \end{frame}
@ -66,6 +73,8 @@
\begin{frame}[t] \begin{frame}[t]
\frametitle{Proximal Decoding: General Idea} \frametitle{Proximal Decoding: General Idea}
\vspace*{-3mm}
\begin{itemize} \begin{itemize}
\item Objective function: \item Objective function:
\begin{align*} \begin{align*}
@ -74,7 +83,7 @@
+ \gamma h\left( \tilde{\boldsymbol{x}} \right) + \gamma h\left( \tilde{\boldsymbol{x}} \right)
\end{align*} \end{align*}
\note{Notational difference between $f$ and $f_X$ or $f_Y$} \note{Notational difference between $f$ and $f_X$ or $f_Y$}
\item Proximal operator \cite{proximal_algorithms}: \item Proximal operator \citereference{PB14}:
\begin{align*} \begin{align*}
\text{prox}_{\gamma h} \left( \tilde{\boldsymbol{x}} \right) &\equiv \text{prox}_{\gamma h} \left( \tilde{\boldsymbol{x}} \right) &\equiv
\argmin_{\boldsymbol{t}\in\mathbb{R}^n} \argmin_{\boldsymbol{t}\in\mathbb{R}^n}
@ -96,6 +105,11 @@
\hspace{10mm} \text{``Code proximal step''} \hspace{10mm} \text{``Code proximal step''}
\end{align*} \end{align*}
\end{itemize} \end{itemize}
\bigskip
\addreference{PB14}{Neal Parikh; Stephen Boyd: \emph{Proximal Algorithms}.
Found. Trends Optim. 1.3 (Jan. 2014), pp. 127–239.}
\end{frame} \end{frame}
@ -103,7 +117,7 @@
\begin{frame}[t, fragile] \begin{frame}[t, fragile]
\frametitle{Proximal Decoding: Algorithm} \frametitle{Proximal Decoding: Algorithm}
\begin{itemize} \begin{itemize}
\item Iterative decoding algorithm \cite{proximal_paper}: \item Iterative decoding algorithm:
\end{itemize} \end{itemize}
\vspace{2mm} \vspace{2mm}
@ -130,8 +144,10 @@ return $\boldsymbol{\hat{c}}$
\begin{frame}[t] \begin{frame}[t]
\frametitle{Proximal Decoding: Bit Error Rate} \frametitle{Proximal Decoding: Bit Error Rate}
\vspace*{-7mm}
\begin{itemize} \begin{itemize}
\item (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc} \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484}
\end{itemize} \end{itemize}
\begin{figure}[H] \begin{figure}[H]
@ -159,7 +175,7 @@ return $\boldsymbol{\hat{c}}$
\addlegendentry{$\gamma = 0.15$} \addlegendentry{$\gamma = 0.15$}
\addplot [ForestGreen, mark=*, line width=1pt] \addplot [ForestGreen, mark=*, line width=1pt]
table [x=SNR, y=gamma_0_15, col sep=comma] {res/ber_paper.csv}; table [x=SNR, y=gamma_0_15, col sep=comma] {res/ber_paper.csv};
\addlegendentry{$\gamma = 0.15$ (Wadayama et al.)} \addlegendentry{$\gamma = 0.15$ \citereference{WT22}}
\addplot [NavyBlue, mark=triangle, dashed, line width=1pt] \addplot [NavyBlue, mark=triangle, dashed, line width=1pt]
table [x=SNR, y=BER, col sep=comma, table [x=SNR, y=BER, col sep=comma,
@ -169,7 +185,7 @@ return $\boldsymbol{\hat{c}}$
\addlegendentry{$\gamma = 0.01$} \addlegendentry{$\gamma = 0.01$}
\addplot [NavyBlue, mark=*, line width=1pt] \addplot [NavyBlue, mark=*, line width=1pt]
table [x=SNR, y=gamma_0_01, col sep=comma] {res/ber_paper.csv}; table [x=SNR, y=gamma_0_01, col sep=comma] {res/ber_paper.csv};
\addlegendentry{$\gamma = 0.01$ (Wadayama et al.)} \addlegendentry{$\gamma = 0.01$ \citereference{WT22}}
\addplot [RedOrange, mark=triangle, dashed, line width=1pt] \addplot [RedOrange, mark=triangle, dashed, line width=1pt]
table [x=SNR, y=BER, col sep=comma, table [x=SNR, y=BER, col sep=comma,
@ -179,19 +195,29 @@ return $\boldsymbol{\hat{c}}$
\addlegendentry{$\gamma = 0.05$} \addlegendentry{$\gamma = 0.05$}
\addplot [RedOrange, mark=*, line width=1pt] \addplot [RedOrange, mark=*, line width=1pt]
table [x=SNR, y=gamma_0_05, col sep=comma] {res/ber_paper.csv}; table [x=SNR, y=gamma_0_05, col sep=comma] {res/ber_paper.csv};
\addlegendentry{$\gamma = 0.05$ (Wadayama et al.)} \addlegendentry{$\gamma = 0.05$ \citereference{WT22}}
\addplot [RoyalPurple, mark=*, line width=1pt] \addplot [RoyalPurple, mark=*, line width=1pt]
table [x=SNR, y=BP, col sep=comma] {res/ber_paper.csv}; table [x=SNR, y=BP, col sep=comma] {res/ber_paper.csv};
\addlegendentry{BP} \addlegendentry{BP \citereference{WT22}}
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\end{figure} \end{figure}
\vspace*{-2mm}
\begin{itemize} \begin{itemize}
\item Comparison of simulation with results of Wadayama et al. \cite{proximal_paper} \item Comparison of simulation with results of Wadayama et al. \citereference{WT22}
\end{itemize} \end{itemize}
\bigskip
\addreference{WT22}{Tadashi Wadayama; Satoshi Takabe: \emph{Proximal Decoding for LDPC
Codes}. IEICE Transactions on Fundamentals of Electronics, Communications and Computer
Sciences advpub (2022), 2022TAP0002.}
\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}}.
\end{frame} \end{frame}
@ -199,8 +225,10 @@ return $\boldsymbol{\hat{c}}$
\begin{frame}[t] \begin{frame}[t]
\frametitle{Proximal Decoding: Choice of $\gamma$} \frametitle{Proximal Decoding: Choice of $\gamma$}
\vspace*{-5mm}
\begin{itemize} \begin{itemize}
\item (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc} \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484}
\end{itemize} \end{itemize}
\begin{figure}[H] \begin{figure}[H]
@ -273,6 +301,12 @@ return $\boldsymbol{\hat{c}}$
\begin{itemize} \begin{itemize}
\item Not great benefit in finding the optimal value for $\gamma$ \item Not great benefit in finding the optimal value for $\gamma$
\end{itemize} \end{itemize}
\bigskip
\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}}.
\end{frame} \end{frame}
@ -495,12 +529,14 @@ return $\boldsymbol{\hat{c}}$
\begin{frame}[t, fragile] \begin{frame}[t, fragile]
\frametitle{Proximal Decoding: Frame Error Rate} \frametitle{Proximal Decoding: Frame Error Rate}
\vspace*{-3mm}
\begin{itemize} \begin{itemize}
\item (3,6) regular LDPC code with $n=204$,\\ \item (3,6) regular LDPC code with $n=204$,\\
$k=102$ \cite[\text{204.33.484}]{mackay_enc} $k=102$ \citereference{Mac23, 204.33.484}
\end{itemize} \end{itemize}
\vspace*{-5mm} \vspace*{-2mm}
\begin{minipage}{.4\textwidth} \begin{minipage}{.4\textwidth}
\centering \centering
@ -521,6 +557,11 @@ end for
return $\boldsymbol{\hat{c}}$ return $\boldsymbol{\hat{c}}$
\end{algorithm} \end{algorithm}
\end{figure} \end{figure}
\vspace*{-4mm}
\hspace*{-4mm}
\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{minipage}% \end{minipage}%
\begin{minipage}{.6\textwidth} \begin{minipage}{.6\textwidth}
\centering \centering
@ -854,8 +895,8 @@ return $\boldsymbol{\hat{c}}$
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\caption{$\nabla L \left(\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)$ \caption{$\nabla L \left(\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)$%
for a repetition code with $n=2$\footnotemark} \footnotemark{} for a repetition code with $n=2$}
\end{subfigure}% \end{subfigure}%
\begin{subfigure}[c]{0.5\textwidth} \begin{subfigure}[c]{0.5\textwidth}
\centering \centering
@ -894,9 +935,7 @@ return $\boldsymbol{\hat{c}}$
\footnotetext{In an AWGN Channel $\nabla L\left( \boldsymbol{y} \mid \footnotetext{In an AWGN Channel $\nabla L\left( \boldsymbol{y} \mid
\tilde{\boldsymbol{x}}\right) \tilde{\boldsymbol{x}}\right)
\propto \left( \tilde{\boldsymbol{x}} - \boldsymbol{y} \right)$ \propto \left( \tilde{\boldsymbol{x}} - \boldsymbol{y} \right)$}
\cite[Sec. 4.1]{proximal_paper}}
\end{frame} \end{frame}
@ -904,9 +943,11 @@ return $\boldsymbol{\hat{c}}$
\begin{frame}[t] \begin{frame}[t]
\frametitle{Proximal Decoder: Oscillation of $\nabla h\left( \tilde{\boldsymbol{x}} \right) $} \frametitle{Proximal Decoder: Oscillation of $\nabla h\left( \tilde{\boldsymbol{x}} \right) $}
\vspace*{-3mm}
\begin{itemize} \begin{itemize}
\item Single decoding using a (3,6) regular LDPC code with $n=204, k=102$ \item Single decoding using a (3,6) regular LDPC code with $n=204, k=102$
\cite[\text{204.33.484}]{mackay_enc} \citereference{Mac23, 204.33.484}
% ; $\gamma = 0.05, \omega = 0.05, E_b / N_0 = \SI{5}{dB}$ % ; $\gamma = 0.05, \omega = 0.05, E_b / N_0 = \SI{5}{dB}$
\end{itemize} \end{itemize}
@ -969,6 +1010,12 @@ return $\boldsymbol{\hat{c}}$
\item For larger $n$, the gradient itself starts to oscillate \item For larger $n$, the gradient itself starts to oscillate
\item Dynamic range of oscillation highly correlated with probability of bit error \item Dynamic range of oscillation highly correlated with probability of bit error
\end{itemize} \end{itemize}
\bigskip
\bigskip
\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame} \end{frame}
@ -1086,7 +1133,7 @@ return $\boldsymbol{\hat{c}}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t, fragile] \begin{frame}[t, fragile]
\frametitle{Proximal Decoding: Improvement using \\``ML-in-the-List''} \frametitle{Proximal Decoding: Improvement}
\vspace*{-0.5cm} \vspace*{-0.5cm}
@ -1147,12 +1194,12 @@ $\textcolor{KITblue}{\textbf{return }\boldsymbol{\hat{c}}_l\text{ with lowest }d
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t] \begin{frame}[t]
\frametitle{Proximal Decoding: Improvement using \\``ML-in-the-List''} \frametitle{Proximal Decoding: Improvement}
\vspace{-0.5cm} \vspace{-0.6cm}
\begin{itemize} \begin{itemize}
\item (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc} \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484}
\end{itemize} \end{itemize}
\begin{figure}[H] \begin{figure}[H]
@ -1166,8 +1213,8 @@ $\textcolor{KITblue}{\textbf{return }\boldsymbol{\hat{c}}_l\text{ with lowest }d
legend columns=2, legend columns=2,
legend style={at={(0.5,-0.45)},anchor=south}, legend style={at={(0.5,-0.45)},anchor=south},
ymax=1.5, ymin=3e-8, ymax=1.5, ymin=3e-8,
width=0.48\textwidth, width=0.45\textwidth,
height=0.33\textwidth, height=0.3\textwidth,
] ]
\addplot[ForestGreen, mark=*, line width=1pt] \addplot[ForestGreen, mark=*, line width=1pt]
table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.15}] table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.15}]
@ -1199,8 +1246,8 @@ $\textcolor{KITblue}{\textbf{return }\boldsymbol{\hat{c}}_l\text{ with lowest }d
legend columns=2, legend columns=2,
legend style={at={(0.5,-0.45)},anchor=south}, legend style={at={(0.5,-0.45)},anchor=south},
ymax=1.5, ymin=3e-8, ymax=1.5, ymin=3e-8,
width=0.48\textwidth, width=0.45\textwidth,
height=0.33\textwidth, height=0.3\textwidth,
] ]
\addplot[ForestGreen, mark=*, solid, line width=1pt] \addplot[ForestGreen, mark=*, solid, line width=1pt]
@ -1256,6 +1303,11 @@ $\textcolor{KITblue}{\textbf{return }\boldsymbol{\hat{c}}_l\text{ with lowest }d
\begin{itemize} \begin{itemize}
\item Up to $\sim \SI{1}{dB}$ improvement \item Up to $\sim \SI{1}{dB}$ improvement
\end{itemize} \end{itemize}
\bigskip
\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame} \end{frame}

36
latex/presentations/final/sections/theoretical_background.tex
View File

@ -9,18 +9,37 @@
\begin{frame}[t] \begin{frame}[t]
\frametitle{Motivation} \frametitle{Motivation}
\begin{itemize} \begin{itemize}
\item The general ML decoding problem is NP-complete \cite{ml_np_hard_proof} \item General ML decoding problem NP-complete \citereference{BMT78}
\item The iterative message–passing algorithms preferred in practice do not guarantee \item Iterative message–passing algorithms preferred in practice do not guarantee
optimality when the graph contains cycles \cite{ldpc_conv} optimality when the graph contains cycles \citereference{KTP19}
\item The standard message-passing algorithms are often difficult to \item Standard message-passing algorithms difficult to analyze \citereference{FEL03}
analyze \cite{feldman_thesis}
\end{itemize} \end{itemize}
\vspace{3.5cm}
\addreferences
{BMT78}{E. Berlekamp; R. McEliece; H. van Tilborg: \emph{On the inherent intractability
of certain coding problems (Corresp.)}.
IEEE Transactions on Information Theory 24.3 (May 1978), pp. 384–386.}
{KTP19}{Banu Kabakulak; Z. Caner Taşkın; Ali Emre Pusane: \emph{Optimization–based
decoding algorithms for LDPC convolutional codes in communication systems}.
IISE Transactions 51.10 (2019), pp. 1061–1074.}
{FEL03}{Jon Feldman: \emph{Decoding error-correcting codes via linear programming}.
PhD thesis. MIT, 2003.}
\stopreferences
\end{frame} \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t] \begin{frame}[t]
\frametitle{Previous Work} \frametitle{Previous Work}
\vspace*{-5mm}
\begin{itemize}
\item Bachelor's thesis by Yanixa Lu \citereference{Lu22}
\end{itemize}
\begin{figure}[h] \begin{figure}[h]
\centering \centering
@ -39,14 +58,17 @@
\fbox{\includegraphics[page=60,width=.6\textwidth]{res/Bachelor_Thesis_Yanxia_Lu}} \fbox{\includegraphics[page=60,width=.6\textwidth]{res/Bachelor_Thesis_Yanxia_Lu}}
\end{subfigure}% \end{subfigure}%
\caption{Bachelor's Thesis by Yanxia Lu \cite{yanxia_lu_thesis}}
\end{figure} \end{figure}
\begin{itemize} \begin{itemize}
\item Analysis of ``Proximal Decoding'' \item Analysis of ``Proximal Decoding''
\item Analysis of ``Interior Point Decoding'' \item Analysis of ``Interior Point Decoding''
\end{itemize} \end{itemize}
\bigskip
\addreference{Lu22}{Yanxia Lu: \emph{Realization of Channel Decoding Using Optimization
Techniques}. Bachelor’s Thesis. KIT, 2022.}
\end{frame} \end{frame}