1941 lines
90 KiB
TeX
1941 lines
90 KiB
TeX
\section{Analysis}%
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\label{sec:Analysis}
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\subsection{Proximal Decoding}%
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\label{sub:Ex Proximal Decoding}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{Proximal Decoding: Bit Error Rate and Performance}
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\vspace*{-0.5cm}
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\begin{itemize}
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\item Comparison of simulation%
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\footnote{(3,6) regular LDPC code with $n=204, k=102$
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\cite[\text{204.33.484}]{mackay_enc}}
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with results of Wadayama et al. \cite{proximal_paper}
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\end{itemize}
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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grid=both, grid style={line width=.1pt},
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xlabel={$E_b / N_0$ (dB)}, ylabel={BER},
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ymode=log,
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legend pos=outer north east,
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width=0.45\textwidth,
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height=0.3375\textwidth,
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ymax=1.2, ymin=0.8e-4,
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xtick={1, 2, ..., 5},
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xmin=0.9, xmax=5.6,
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legend columns=2,
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]
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\addplot [ForestGreen, mark=triangle, dashed, line width=1pt]
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table [x=SNR, y=BER, col sep=comma,
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discard if not={gamma}{0.15},
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discard if gt={SNR}{5.5},]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addlegendentry{$\gamma = 0.15$}
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\addplot [ForestGreen, mark=*, line width=1pt]
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table [x=SNR, y=gamma_0_15, col sep=comma] {res/ber_paper.csv};
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\addlegendentry{$\gamma = 0.15$ (Wadayama et al.)}
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\addplot [NavyBlue, mark=triangle, dashed, line width=1pt]
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table [x=SNR, y=BER, col sep=comma,
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discard if not={gamma}{0.01},
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discard if gt={SNR}{5.5},]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addlegendentry{$\gamma = 0.01$}
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\addplot [NavyBlue, mark=*, line width=1pt]
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table [x=SNR, y=gamma_0_01, col sep=comma] {res/ber_paper.csv};
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\addlegendentry{$\gamma = 0.01$ (Wadayama et al.)}
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\addplot [RedOrange, mark=triangle, dashed, line width=1pt]
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table [x=SNR, y=BER, col sep=comma,
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discard if not={gamma}{0.05},
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discard if gt={SNR}{5.5},]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addlegendentry{$\gamma = 0.05$}
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\addplot [RedOrange, mark=*, line width=1pt]
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table [x=SNR, y=gamma_0_05, col sep=comma] {res/ber_paper.csv};
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\addlegendentry{$\gamma = 0.05$ (Wadayama et al.)}
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\addplot [RoyalPurple, mark=*, line width=1pt]
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table [x=SNR, y=BP, col sep=comma] {res/ber_paper.csv};
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\addlegendentry{BP}
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\end{axis}
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\end{tikzpicture}
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\caption{Simulation results for $\omega = 0.05, K=100$}
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\label{fig:sim_results_prox}
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\end{figure}
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\vspace*{-0.5cm}
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\begin{itemize}
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\item $\mathcal{O}\left(n \right) $ time complexity - same as BP;
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only multiplication and addition necessary \cite{proximal_paper}
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\item Measured performance: $\sim\SI{10000}{}$ frames/s
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on Intel Core i7-7700HQ @ 2.80GHz; $n=204$
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\end{itemize}
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\vspace{3mm}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t]
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\frametitle{Proximal Decoding: Choice of $\gamma$}
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\setcounter{footnote}{0}
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\begin{itemize}
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\item Simulation%
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\footnote{(3,6) regular LDPC code with $n=204, k=102$
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\cite[\text{204.33.484}]{mackay_enc}}
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results for different values of $\gamma$
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\end{itemize}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[c]{0.5\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{semilogyaxis}[xlabel={$E_b / N_0$ (dB)}, ylabel={BER},
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grid=both, grid style={line width=.1pt},
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width=0.8\textwidth,
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height=0.6\textwidth,
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legend style={at={(0.05,0.05)},anchor=south west},
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ymin=3e-7, ymax=1.5,]
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\addplot [ForestGreen, mark=*] table [x=SNR, y=BER,
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col sep=comma, discard if not={gamma}{0.15}]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addlegendentry{$\gamma = 0.15$}
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\addplot [NavyBlue, mark=*] table [x=SNR, y=BER,
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col sep=comma, discard if not={gamma}{0.01}]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addlegendentry{$\gamma = 0.01$}
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\addplot [RedOrange, mark=*] table [x=SNR, y=BER,
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col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addlegendentry{$\gamma = 0.05$}
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\end{semilogyaxis}
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\end{tikzpicture}
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\end{subfigure}%
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\begin{subfigure}[c]{0.5\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[view={75}{30},
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zmode=log,
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xlabel={$E_b / N_0$ (dB)},
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ylabel={$\gamma$},
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zlabel={BER},
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width=0.8\textwidth,
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height=0.6\textwidth,
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legend pos=outer north east,]
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\addplot3[surf,
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mesh/rows=17, mesh/cols=14,
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colormap/viridis] table [col sep=comma,
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addlegendentry{$\gamma = \left[ 0\text{:}0.01\text{:}0.16 \right] $}
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\addplot3[NavyBlue, line width=1.5] table [col sep=comma,
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discard if not={gamma}{0.01},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addlegendentry{$\gamma = 0.01$}
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\addplot3[RedOrange, line width=1.5] table [col sep=comma,
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discard if not={gamma}{0.05},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addlegendentry{$\gamma = 0.05$}
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\addplot3[ForestGreen, line width=1.5] table [col sep=comma,
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discard if not={gamma}{0.15},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addlegendentry{$\gamma = 0.15$}
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\end{axis}
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\end{tikzpicture}
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\end{subfigure}
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\caption{BER for $\omega = 0.05, K=100$}
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\label{fig:ber_3d}
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\end{figure}
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\begin{itemize}
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\item Not great benefit in finding the optimal value for $\gamma$
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\end{itemize}
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\vspace{3mm}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\begin{frame}[t]
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% \frametitle{Proximal Decoding: Choice of $\gamma$}
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%
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% \begin{minipage}[c]{0.9\textwidth}
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% \centering
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%
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% \begin{figure}[H]
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% \vspace*{-0.5cm}
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% \hspace*{-0.7cm}
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% \centering
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% \begin{subfigure}[c]{0.33\textwidth}
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% \centering
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% \begin{tikzpicture}[scale=0.4]
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% \begin{axis}[view={75}{30},
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% zmode=log,
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% xlabel={$E_b / N_0$ (dB)},
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% ylabel={$\gamma$},
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% zlabel={BER},]
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% \addplot3[surf,
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% mesh/rows=17, mesh/cols=10,
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% colormap/viridis] table [col sep=comma,
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_963965.csv};
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% \addplot3[RedOrange, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.05},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_963965.csv};
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% \addplot3[NavyBlue, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.01},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_963965.csv};
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% \addplot3[ForestGreen, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.15},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_963965.csv};
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% \end{axis}
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% \end{tikzpicture}
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% \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$ \cite[\text{96.3.965}]{mackay_enc}}
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% \end{subfigure}%
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% \begin{subfigure}[c]{0.33\textwidth}
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% \centering
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% \begin{tikzpicture}[scale=0.4]
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% \begin{axis}[view={75}{30},
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% zmode=log,
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% xlabel={$E_b/N_0$ (dB)},
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% ylabel={$\gamma$},
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% zlabel={BER},]
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% \addplot3[surf,
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% mesh/rows=17, mesh/cols=14,
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% colormap/viridis] table [col sep=comma,
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_20433484.csv};
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% \addplot3[RedOrange, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.05},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_20433484.csv};
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% \addplot3[NavyBlue, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.01},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_20433484.csv};
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% \addplot3[ForestGreen, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.15},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_20433484.csv};
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% \end{axis}
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% \end{tikzpicture}
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% \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc}}
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% \end{subfigure}%
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% \begin{subfigure}[c]{0.33\textwidth}
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% \centering
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% \begin{tikzpicture}[scale=0.4]
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% \begin{axis}[view={75}{30},
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% zmode=log,
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% xlabel={$E_b / N_0$ (dB)},
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% ylabel={$\gamma$},
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% zlabel={BER},]
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% \addplot3[surf,
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% mesh/rows=17, mesh/cols=10,
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% colormap/viridis] table [col sep=comma,
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_40833844.csv};
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% \addplot3[RedOrange, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.05},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_40833844.csv};
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% \addplot3[NavyBlue, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.01},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_40833844.csv};
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% \addplot3[ForestGreen, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.15},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_40833844.csv};
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% \end{axis}
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% \end{tikzpicture}
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% \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$ \cite[\text{408.33.844}]{mackay_enc}}
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% \end{subfigure}
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% \hspace*{-0.7cm}
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% \begin{subfigure}[c]{0.33\textwidth}
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% \centering
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% \begin{tikzpicture}[scale=0.4]
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% \begin{axis}[view={75}{30},
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% zmode=log,
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% xlabel={$E_b / N_0$ (dB)},
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% ylabel={$\gamma$},
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% zlabel={BER},]
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% \addplot3[surf,
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% mesh/rows=17, mesh/cols=10,
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% colormap/viridis] table [col sep=comma,
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
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% \addplot3[RedOrange, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.05},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
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% \addplot3[NavyBlue, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.01},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
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% \addplot3[ForestGreen, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.15},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
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% \end{axis}
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% \end{tikzpicture}
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% \caption{BCH code with $n=31, k=26$\\[2\baselineskip]}
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% \end{subfigure}%
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% \begin{subfigure}[c]{0.33\textwidth}
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% \centering
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% \begin{tikzpicture}[scale=0.4]
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% \begin{axis}[view={75}{30},
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% zmode=log,
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% xlabel={$E_b / N_0$ (dB)},
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% ylabel={$\gamma$},
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% zlabel={BER},]
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% \addplot3[surf,
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% mesh/rows=17, mesh/cols=10,
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% colormap/viridis] table [col sep=comma,
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_20455187.csv};
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% \addplot3[RedOrange, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.05},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_20455187.csv};
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% \addplot3[NavyBlue, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.01},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_20455187.csv};
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% \addplot3[ForestGreen, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.15},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_20455187.csv};
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% \end{axis}
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% \end{tikzpicture}
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% \caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$ \cite[\text{204.55.187}]{mackay_enc}}
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% \end{subfigure}%
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% \begin{subfigure}[c]{0.33\textwidth}
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% \centering
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% \begin{tikzpicture}[scale=0.4]
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% \begin{axis}[view={75}{30},
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% zmode=log,
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% xlabel={$E_b / N_0$ (dB)},
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% ylabel={$\gamma$},
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% zlabel={BER},]
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% \addplot3[surf,
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% mesh/rows=17, mesh/cols=10,
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% colormap/viridis] table [col sep=comma,
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
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% \addplot3[RedOrange, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.05},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
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% \addplot3[NavyBlue, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.01},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
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% \addplot3[ForestGreen, line width=1.5] table[col sep=comma,
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% discard if not={gamma}{0.15},
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% x=SNR, y=gamma, z=BER]
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% {res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
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% \end{axis}
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% \end{tikzpicture}
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% \caption{LDPC code (Progressive Edge Growth Construction) with $n=504, k=252$ \cite[\text{PEGReg252x504}]{mackay_enc}}
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% \end{subfigure}%
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% \end{figure}
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% \end{minipage}%
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% \begin{minipage}[c]{0.1\textwidth}
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% \centering
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%
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% \begin{figure}[H]
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% \vspace*{-1.5cm}
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% \hspace*{-0.75cm}
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% \centering
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% \begin{tikzpicture}[scale=0.7]
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% \begin{axis}[hide axis,
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% xmin=10, xmax=50,
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% ymin=0, ymax=0.4,
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% legend style={draw=white!15!black,legend cell align=left}]
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% \addlegendimage{surf, colormap/viridis}
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% \addlegendentry{$\gamma = \left[ 0\text{ : }0.01\text{ : }0.16 \right] $};
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% \addlegendimage{NavyBlue, line width=1.5pt}
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% \addlegendentry{$\gamma = 0.01$};
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% \addlegendimage{RedOrange, line width=1.5pt}
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% \addlegendentry{$\gamma = 0.05$};
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% \addlegendimage{ForestGreen, line width=1.5pt}
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% \addlegendentry{$\gamma = 0.15$};
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% \end{axis}
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% \end{tikzpicture}
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% \end{figure}
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% \end{minipage}
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%\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[t, fragile]
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\frametitle{Proximal Decoding: Frame Error Rate}
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\setcounter{footnote}{0}
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\begin{itemize}
|
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\item Analysis of simulated%
|
|
\footnote{(3,6) regular LDPC code with $n=204, k=102$
|
|
\cite[\text{204.33.484}]{mackay_enc}}
|
|
BER and FER
|
|
\end{itemize}
|
|
|
|
\begin{minipage}{.4\textwidth}
|
|
\centering
|
|
|
|
\begin{figure}[htpb]
|
|
\centering
|
|
|
|
\begin{algorithm}[caption={}, label={}]
|
|
$\boldsymbol{s} \leftarrow \boldsymbol{0}$
|
|
for $K$ iterations do
|
|
$\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \nabla L \left( \boldsymbol{y} \mid \boldsymbol{s} \right) $
|
|
$\boldsymbol{s} \leftarrow \boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right) $
|
|
$\boldsymbol{\hat{x}} \leftarrow \text{sign}\left( \boldsymbol{s} \right) $
|
|
if $\boldsymbol{H}\boldsymbol{\hat{c}} = \boldsymbol{0}$ do
|
|
return $\boldsymbol{\hat{c}}$
|
|
end if
|
|
end for
|
|
return $\boldsymbol{\hat{c}}$
|
|
\end{algorithm}
|
|
|
|
\vspace*{-5mm}
|
|
|
|
\caption{Proximal decoding algorithm \cite{proximal_paper}}
|
|
\end{figure}
|
|
\end{minipage}%
|
|
\begin{minipage}{.6\textwidth}
|
|
\centering
|
|
\begin{figure}[H]
|
|
\vspace*{-9mm}
|
|
\centering
|
|
|
|
\begin{tikzpicture}[scale=0.7]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$E_b / N_0$ (dB)}, ylabel={BER},
|
|
ymode=log,
|
|
legend style={at={(0.05,0.05)},anchor=south west},
|
|
ymax=1.5, ymin=3e-7,
|
|
width=0.7\textwidth,
|
|
height=0.6\textwidth,
|
|
]
|
|
\addplot [ForestGreen, mark=*]
|
|
table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.15$}
|
|
\addplot [NavyBlue, mark=*]
|
|
table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.01$}
|
|
\addplot [RedOrange, mark=*]
|
|
table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.05$}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\begin{tikzpicture}[scale=0.7]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
|
|
ymode=log,
|
|
legend style={at={(0.05,0.05)},anchor=south west},
|
|
ymax=1.5, ymin=3e-7,
|
|
width=0.7\textwidth,
|
|
height=0.6\textwidth,
|
|
]
|
|
\addplot [ForestGreen, mark=*]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.15$}
|
|
\addplot [NavyBlue, mark=*]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.01$}
|
|
\addplot [RedOrange, mark=*]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.05$}
|
|
\end{axis}
|
|
\end{tikzpicture}\\
|
|
\begin{tikzpicture}[scale=0.7]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$E_b / N_0$ (dB)}, ylabel={Decoding Failure Rate},
|
|
ymode=log,
|
|
legend style={at={(0.05,0.05)},anchor=south west},
|
|
ymax=1.5, ymin=3e-7,
|
|
width=0.7\textwidth,
|
|
height=0.6\textwidth,
|
|
]
|
|
\addplot [ForestGreen, mark=*]
|
|
table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.15$}
|
|
\addplot [NavyBlue, mark=*]
|
|
table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.01$}
|
|
\addplot [RedOrange, mark=*]
|
|
table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addlegendentry{$\gamma = 0.05$}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{Simulation results for $\omega = 0.05, K=100$}
|
|
\label{fig:simulation_results_ber_fer_dfr}
|
|
\end{figure}
|
|
\end{minipage}
|
|
\end{frame}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{frame}[t]
|
|
\frametitle{Proximal Decoding: Oscillation of Estimate}
|
|
\setcounter{footnote}{0}
|
|
|
|
\begin{itemize}
|
|
\item $\nabla L \left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) $
|
|
and $\nabla h \left( \tilde{\boldsymbol{x}} \right) $ generally end up
|
|
in an equilibrium
|
|
\end{itemize}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
\begin{minipage}[c]{0.25\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}[scale = 0.7]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=1.2\textwidth,
|
|
height=0.45\textwidth,
|
|
scale only axis,
|
|
xtick={0, 50, ..., 200},
|
|
xticklabels={0, 25, ..., 100},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_1]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_1]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_1]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_2$}
|
|
\addlegendentry{$\left(\nabla h \right)_2 $}
|
|
\end{axis}
|
|
\end{tikzpicture}\\
|
|
\begin{tikzpicture}[scale = 0.7]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=1.2\textwidth,
|
|
height=0.45\textwidth,
|
|
scale only axis,
|
|
xtick={0, 50, ..., 200},
|
|
xticklabels={0, 25, ..., 100},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_2]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_2]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_2]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_3$}
|
|
\addlegendentry{$\left(\nabla h \right)_3 $}
|
|
\end{axis}
|
|
\end{tikzpicture}\\
|
|
\begin{tikzpicture}[scale = 0.7]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=1.2\textwidth,
|
|
height=0.45\textwidth,
|
|
scale only axis,
|
|
xtick={0, 50, ..., 200},
|
|
xticklabels={0, 25, ..., 100},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_3]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_3]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_3]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_4$}
|
|
\addlegendentry{$\left(\nabla h \right)_4 $}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{minipage}%
|
|
\begin{minipage}[c]{0.5\textwidth}
|
|
\vspace*{-1cm}
|
|
\centering
|
|
|
|
\begin{tikzpicture}[, spy using outlines={circle, magnification=6,
|
|
connect spies}]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=0.8\textwidth,
|
|
height=0.35\textwidth,
|
|
scale only axis,
|
|
xtick={0, 50, ..., 200},
|
|
xticklabels={0, 25, ..., 100},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_0]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_0]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_0]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_1$}
|
|
\addlegendentry{$\left(\nabla h \right)_1 $}
|
|
|
|
\coordinate (spypoint) at (axis cs:100,1.12);
|
|
\coordinate (magnifyglass) at (axis cs:180,2.75);
|
|
\end{axis}
|
|
\spy [black, size=2cm] on (spypoint)
|
|
in node[fill=white] at (magnifyglass);
|
|
\end{tikzpicture}
|
|
\end{minipage}%
|
|
\begin{minipage}[c]{0.25\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}[scale = 0.7]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=1.2\textwidth,
|
|
height=0.45\textwidth,
|
|
scale only axis,
|
|
xtick={0, 50, ..., 200},
|
|
xticklabels={0, 25, ..., 100},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_4]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_4]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_4]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_5$}
|
|
\addlegendentry{$\left(\nabla h \right)_5 $}
|
|
\end{axis}
|
|
\end{tikzpicture}\\
|
|
\begin{tikzpicture}[scale = 0.7]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=1.2\textwidth,
|
|
height=0.45\textwidth,
|
|
scale only axis,
|
|
xtick={0, 50, ..., 200},
|
|
xticklabels={0, 25, ..., 100},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_5]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_5]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_5]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_6$}
|
|
\addlegendentry{$\left(\nabla h \right)_6 $}
|
|
\end{axis}
|
|
\end{tikzpicture}\\
|
|
\begin{tikzpicture}[scale = 0.7]
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=1.2\textwidth,
|
|
height=0.45\textwidth,
|
|
scale only axis,
|
|
xtick={0, 50, ..., 200},
|
|
xticklabels={0, 25, ..., 100},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_6]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_6]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_6]
|
|
{res/proximal/comp_bch_7_4_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_7$}
|
|
\addlegendentry{$\left(\nabla h \right)_7 $}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{minipage}
|
|
|
|
\caption{Internal variables of proximal decoder
|
|
as a function of the number of iterations ($n=7$)\footnotemark}
|
|
|
|
\footnotetext{A single decoding is shown, using the BCH$\left( 7,4 \right) $ code;
|
|
$\gamma = 0.05, \omega = 0.05, E_b / N_0 = \SI{5}{dB}$}
|
|
\end{figure}
|
|
\end{frame}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{frame}[t]
|
|
\frametitle{Proximal Decoding: Visualization of Gradients}
|
|
\setcounter{footnote}{0}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\begin{subfigure}[c]{0.5\textwidth}
|
|
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[xmin = -1.25, xmax=1.25,
|
|
ymin = -1.25, ymax=1.25,
|
|
xlabel={$x_1$}, ylabel={$x_2$},
|
|
width=0.9\textwidth,
|
|
height=0.8\textwidth,
|
|
grid=major, grid style={dotted},
|
|
view={0}{90}]
|
|
\addplot3[point meta=\thisrow{grad_norm},
|
|
point meta min=1,
|
|
point meta max=3,
|
|
quiver={u=\thisrow{grad_0},
|
|
v=\thisrow{grad_1},
|
|
scale arrows=.05,
|
|
every arrow/.append style={%
|
|
line width=.3+\pgfplotspointmetatransformed/1000,
|
|
-{Latex[length=0pt 5,width=0pt 3]}
|
|
},
|
|
},
|
|
quiver/colored = {mapped color},
|
|
colormap/rocket,
|
|
-stealth,
|
|
]
|
|
table[col sep=comma] {res/proximal/2d_grad_L.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{$\nabla L \left(\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)$
|
|
for a repetition code with $n=2$\footnotemark}
|
|
\end{subfigure}%
|
|
\begin{subfigure}[c]{0.5\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[xmin = -1.25, xmax=1.25,
|
|
ymin = -1.25, ymax=1.25,
|
|
width=0.9\textwidth,
|
|
height=0.8\textwidth,
|
|
xlabel={$x_1$}, ylabel={$x_2$},
|
|
grid=major, grid style={dotted},
|
|
view={0}{90}]
|
|
\addplot3[point meta=\thisrow{grad_norm},
|
|
point meta min=1,
|
|
point meta max=4,
|
|
quiver={u=\thisrow{grad_0},
|
|
v=\thisrow{grad_1},
|
|
scale arrows=.03,
|
|
every arrow/.append style={%
|
|
line width=.3+\pgfplotspointmetatransformed/1000,
|
|
-{Latex[length=0pt 5,width=0pt 3]}
|
|
},
|
|
},
|
|
quiver/colored = {mapped color},
|
|
colormap/rocket,
|
|
-stealth,
|
|
]
|
|
table[col sep=comma] {res/proximal/2d_grad_h.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{$\nabla h \left( \tilde{\boldsymbol{x}} \right) $ for a repetition code
|
|
with $n=2$}
|
|
\end{subfigure}%
|
|
\end{figure}
|
|
|
|
\footnotetext{In an AWGN Channel $\nabla L\left( \boldsymbol{y} \mid
|
|
\tilde{\boldsymbol{x}}\right)
|
|
\propto \left( \tilde{\boldsymbol{x}} - \boldsymbol{y} \right)$
|
|
\cite[Sec. 4.1]{proximal_paper}}
|
|
|
|
\end{frame}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{frame}[t]
|
|
\frametitle{Proximal Decoder: Oscillation of $\nabla h\left( \tilde{\boldsymbol{x}} \right) $}
|
|
\setcounter{footnote}{0}
|
|
|
|
\begin{itemize}
|
|
\item For larger $n$, the gradient itself starts to oscillate
|
|
\item The amplitude of the oscillation seems to be highly correlated
|
|
with the probability of a bit error
|
|
\end{itemize}
|
|
|
|
\begin{figure}
|
|
\centering
|
|
|
|
\begin{subfigure}[t]{0.5\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
width=0.85\textwidth,
|
|
height=0.4\textwidth,
|
|
scale only axis,
|
|
xtick={0, 100, ..., 400},
|
|
xticklabels={0, 50, ..., 200},
|
|
]
|
|
\addplot [NavyBlue, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=comb_r_s_0]
|
|
{res/proximal/extreme_components_20433484_combined.csv};
|
|
\addplot [ForestGreen, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_L_0]
|
|
{res/proximal/extreme_components_20433484_combined.csv};
|
|
\addplot [RedOrange, mark=none, line width=1]
|
|
table [col sep=comma, x=k, y=grad_h_0]
|
|
{res/proximal/extreme_components_20433484_combined.csv};
|
|
\addlegendentry{est}
|
|
\addlegendentry{$\left(\nabla L \right)_1$}
|
|
\addlegendentry{$\left(\nabla h \right)_1$}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{Internal variables of proximal decoder as a function of the iteration
|
|
($n=204$)\footnotemark}
|
|
\end{subfigure}%
|
|
\begin{subfigure}[t]{0.5\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
point meta min = -1,
|
|
point meta max = 1,
|
|
grid=both,
|
|
xlabel={$Var\left( \nabla h\left( \tilde{\boldsymbol{x}} \right) \right) $},
|
|
ylabel={Bit error (bool)},
|
|
ytick={0, 1},
|
|
width=0.85\textwidth,
|
|
height=0.4\textwidth,
|
|
scale only axis,
|
|
% colormap/viridis,
|
|
]
|
|
\addplot [RoyalPurple, only marks,]
|
|
table [col sep=comma, x=grad_h_vars, y=bit_error]
|
|
{res/proximal/extreme_components_20433484_variance.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{Correlation between bit error and amplitude of oscillation}
|
|
\end{subfigure}
|
|
\end{figure}
|
|
|
|
|
|
\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};
|
|
$\gamma = 0.05, \omega = 0.05, E_b / N_0 = \SI{5}{dB}$}
|
|
\end{frame}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%\newcommand{\tikzbracemark}[1]{\tikz[overlay,remember picture] \node (#1) {};}
|
|
%
|
|
%\newcommand*{\AddNote}[4]{%
|
|
% \begin{tikzpicture}[overlay, remember picture]
|
|
% \draw [decoration={brace,amplitude=0.5em},decorate,ultra thick]
|
|
% ($(#3)!([yshift=1.5ex]#1)!($(#3)-(0,1)$)$) --
|
|
% ($(#3)!(#2)!($(#3)-(0,1)$)$)
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% node [align=center, text width=2cm, pos=0.5, anchor=west] {#4};
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% \end{tikzpicture}
|
|
%}%
|
|
%
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%\begin{frame}[t, fragile]
|
|
% \frametitle{Proximal Decoding: Improvement using ``ML-on-List''}
|
|
% \setcounter{footnote}{0}
|
|
%
|
|
% \begin{itemize}
|
|
% \item Comparison of proximal \& hybrid-proximal-ML\\
|
|
% decoding simulation
|
|
% \footnote{(3,6) regular LDPC Code with $n=204, k=102$
|
|
% \cite[Code: 204.33.484]{mackay_enc}}
|
|
% results
|
|
% \end{itemize}
|
|
%
|
|
% \begin{minipage}{.4\textwidth}
|
|
% \centering
|
|
%
|
|
% \begin{algorithm}[caption={}, label={},
|
|
% basicstyle=\fontsize{6.5}{7.5}\selectfont
|
|
% ]
|
|
%$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$$\hspace{4.185cm}\tikzbracemark{prox-start}$
|
|
%for $k=0$ to $K-1$ do
|
|
% $\boldsymbol{r}^{\left( k+1 \right)} = \boldsymbol{s}^{(k)} - \omega \nabla L \left( \boldsymbol{s}^{(k)}; \boldsymbol{y} \right) $
|
|
% Compute $\nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right)$
|
|
% $\boldsymbol{s}^{\left( k+1 \right)} = \boldsymbol{r}^{(k+1)} - \gamma \nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right) $
|
|
% $\boldsymbol{\hat{x}} = \text{sign}\left( \boldsymbol{s}^{\left( k+1 \right) } \right) $
|
|
% If $\boldsymbol{\hat{x}}$ passes the parity check condition, output $\boldsymbol{\hat{x}}$
|
|
%end for $\tikzbracemark{prox-end}$
|
|
%Find $N$ most probably wrong bits $\hspace{2cm}\tikzbracemark{ml-start}$
|
|
%Generate variations $\boldsymbol{\tilde{x}}_n$ of $\boldsymbol{\hat{x}}$ with the $N$ bits modified
|
|
%Compute $d\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right) \forall n \in \left[ 1 : N-1 \right] $
|
|
%Output $\boldsymbol{\tilde{x}}_n$ with lowest $d\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right)$ $\tikzbracemark{ml-end}$
|
|
% \end{algorithm}
|
|
%
|
|
% \AddNote{prox-start}{prox-end}{prox-start}{\small Proximal\\Decoding}
|
|
% \AddNote{ml-start}{ml-end}{ml-start}{\small ML-on-List}
|
|
% \end{minipage}%
|
|
% \begin{minipage}{.6\textwidth}
|
|
% \centering
|
|
% \begin{figure}[H]
|
|
% \centering
|
|
% \vspace*{-12mm}
|
|
%
|
|
% \begin{tikzpicture}[scale=0.42]
|
|
% \begin{axis}[
|
|
% grid=both,
|
|
% xlabel={SNR}, ylabel={BER},
|
|
% ymode=log,
|
|
% legend style={at={(0.05,0.05)},anchor=south west},
|
|
% ymax=1.5, ymin=3e-8,
|
|
% ]
|
|
%
|
|
% \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
% \addlegendentry{proximal}
|
|
% \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
% \addlegendentry{hybrid prox. \& ML}
|
|
% \end{axis}
|
|
% \end{tikzpicture}
|
|
% \begin{tikzpicture}[scale=0.42]
|
|
% \begin{axis}[
|
|
% grid=both,
|
|
% xlabel={SNR}, ylabel={FER},
|
|
% ymode=log,
|
|
% legend style={at={(0.05,0.05)},anchor=south west},
|
|
% ymax=1.5, ymin=3e-8,
|
|
% ]
|
|
%
|
|
% \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
% \addlegendentry{proximal}
|
|
% \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
% \addlegendentry{hybrid prox. \& ML}
|
|
% \end{axis}
|
|
% \end{tikzpicture}\\
|
|
% \begin{tikzpicture}[scale=0.42]
|
|
% \begin{axis}[
|
|
% grid=both,
|
|
% xlabel={SNR}, ylabel={Decoding Failure Rate},
|
|
% ymode=log,
|
|
% legend style={at={(0.05,0.05)},anchor=south west},
|
|
% ymax=1.5, ymin=3e-8,
|
|
% ]
|
|
%
|
|
% \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
% \addlegendentry{proximal}
|
|
% \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
% \addlegendentry{hybrid prox. \& ML}
|
|
% \end{axis}
|
|
% \end{tikzpicture}
|
|
%
|
|
% \caption{Simulation results for $\gamma = 0.05, \omega = 0.05, K=200, N=12$}
|
|
% \label{fig:simulation_results_hybrid}
|
|
% \end{figure}
|
|
% \end{minipage}
|
|
%\end{frame}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{frame}[t, fragile]
|
|
\frametitle{Proximal Decoding: Improvement using \\``ML-in-the-List''}
|
|
\setcounter{footnote}{0}
|
|
|
|
\vspace*{-0.7cm}
|
|
|
|
\begin{itemize}
|
|
\item Improvement of proximal decoding by adding an ``ML-in-the-List'' step after
|
|
iterating
|
|
\end{itemize}
|
|
|
|
\vspace*{-2mm}
|
|
|
|
\begin{minipage}[t]{.48\textwidth}
|
|
\centering
|
|
|
|
\begin{figure}
|
|
\centering
|
|
|
|
\begin{algorithm}[caption={}, label={}, basicstyle=\fontsize{9.5}{11.5}\selectfont]
|
|
$\boldsymbol{s} \leftarrow \boldsymbol{0}$
|
|
for $K$ iterations do
|
|
$\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \nabla L \left( \boldsymbol{y} \mid \boldsymbol{s} \right) $
|
|
$\boldsymbol{s} \leftarrow \boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right) $
|
|
$\boldsymbol{\hat{x}} \leftarrow \text{sign}\left( \boldsymbol{s} \right) $
|
|
if $\boldsymbol{H}\boldsymbol{\hat{c}} = \boldsymbol{0}$ do
|
|
return $\boldsymbol{\hat{c}}$
|
|
end if
|
|
end for
|
|
return $\boldsymbol{\hat{c}}$
|
|
\end{algorithm}
|
|
|
|
\vspace*{-5mm}
|
|
|
|
\caption{Proximal decoding algorithm \cite{proximal_paper}}
|
|
\end{figure}
|
|
|
|
\end{minipage}%
|
|
\hfill\begin{minipage}[t]{.48\textwidth}
|
|
\centering
|
|
\begin{figure}
|
|
\centering
|
|
|
|
\begin{algorithm}[caption={}, label={}, basicstyle=\fontsize{9.5}{11.5}\selectfont]
|
|
$\boldsymbol{s} \leftarrow \boldsymbol{0}$
|
|
for $K$ iterations do
|
|
$\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \nabla L \left( \boldsymbol{y} \mid \boldsymbol{s} \right) $
|
|
$\boldsymbol{s} \leftarrow \boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right) $
|
|
$\boldsymbol{\hat{x}} \leftarrow \text{sign}\left( \boldsymbol{s} \right) $
|
|
if $\boldsymbol{H}\boldsymbol{\hat{c}} = \boldsymbol{0}$
|
|
return $\boldsymbol{\hat{c}}$
|
|
end if
|
|
end for
|
|
$\textcolor{KITblue}{\text{Find }N\text{ most probably wrong bits.}}$
|
|
$\textcolor{KITblue}{\text{Generate variations } \boldsymbol{\tilde{c}}_l\text{ of }\boldsymbol{\hat{c}}\text{ with the }N\text{ bits modified.}}$
|
|
$\textcolor{KITblue}{\text{Compute }d_H\left( \boldsymbol{ \tilde{c}}_l, \boldsymbol{\hat{c}} \right) \text{ for all valid codewords } \boldsymbol{\tilde{c}}_l}$
|
|
$\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_l\text{ with lowest }d_H\left( \boldsymbol{ \tilde{c}}_l, \boldsymbol{\hat{c}} \right)}$
|
|
\end{algorithm}
|
|
|
|
\vspace*{-5mm}
|
|
|
|
\caption{Improved proximal decoding algorithm}
|
|
\end{figure}
|
|
\end{minipage}
|
|
\end{frame}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{frame}[t]
|
|
\frametitle{Proximal Decoding: Improvement using \\``ML-in-the-List''}
|
|
|
|
\vspace{-0.5cm}
|
|
|
|
\begin{itemize}
|
|
\item Comparison of proximal \& improved (correction of $N = \SI{12}{\bit}$)
|
|
decoding simulation%
|
|
\footnote{(3,6) regular LDPC code with $n=204, k=102$
|
|
\cite[Code: 204.33.484]{mackay_enc}}
|
|
results
|
|
\end{itemize}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$E_b / N_0$}, ylabel={BER},
|
|
ymode=log,
|
|
legend columns=2,
|
|
legend style={at={(0.5,-0.45)},anchor=south},
|
|
ymax=1.5, ymin=3e-8,
|
|
width=0.33\textwidth,
|
|
height=0.28\textwidth,
|
|
]
|
|
\addplot[ForestGreen, mark=*, solid]
|
|
table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addplot[Emerald, mark=triangle, densely dashed]
|
|
table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
|
|
\addplot[NavyBlue, mark=*, solid]
|
|
table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addplot[RoyalPurple, mark=triangle, densely dashed]
|
|
table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
|
|
\addplot[RedOrange, mark=*, solid]
|
|
table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addplot[red, mark=triangle, densely dashed]
|
|
table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$E_b / N_0$}, ylabel={FER},
|
|
ymode=log,
|
|
legend columns=2,
|
|
legend style={at={(0.5,-0.45)},anchor=south},
|
|
ymax=1.5, ymin=3e-8,
|
|
width=0.33\textwidth,
|
|
height=0.28\textwidth,
|
|
]
|
|
|
|
\addplot[ForestGreen, mark=*, solid]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addplot[Emerald, mark=triangle, densely dashed]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
|
|
\addplot[NavyBlue, mark=*, solid]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addplot[RoyalPurple, mark=triangle, densely dashed]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
|
|
\addplot[RedOrange, mark=*, solid]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addplot[red, mark=triangle, densely dashed]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$E_b / N_0$}, ylabel={Decoding Failure Rate},
|
|
ymode=log,
|
|
legend columns=2,
|
|
legend style={at={(0.5,-0.45)},anchor=south},
|
|
ymax=1.5, ymin=3e-8,
|
|
width=0.33\textwidth,
|
|
height=0.28\textwidth,
|
|
]
|
|
|
|
\addplot[ForestGreen, mark=*, solid]
|
|
table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addplot[Emerald, mark=triangle, densely dashed]
|
|
table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
|
|
\addplot[NavyBlue, mark=*, solid]
|
|
table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addplot[RoyalPurple, mark=triangle, densely dashed]
|
|
table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
|
|
\addplot[RedOrange, mark=*, solid]
|
|
table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
\addplot[red, mark=triangle, densely dashed]
|
|
table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\begin{tikzpicture}
|
|
\begin{axis}[hide axis,
|
|
xmin=10, xmax=50,
|
|
ymin=0, ymax=0.4,
|
|
legend columns=3,
|
|
legend style={draw=white!15!black,legend cell align=left}]
|
|
\addlegendimage{ForestGreen, mark=*, solid}
|
|
\addlegendentry{proximal, $\gamma = 0.15$}
|
|
|
|
\addlegendimage{NavyBlue, mark=*, solid}
|
|
\addlegendentry{proximal, $\gamma = 0.01$}
|
|
|
|
\addlegendimage{RedOrange, mark=*, solid}
|
|
\addlegendentry{proximal, $\gamma = 0.05$}
|
|
|
|
\addlegendimage{Emerald, mark=triangle, densely dashed}
|
|
\addlegendentry{improved, $\gamma = 0.15$}
|
|
|
|
\addlegendimage{RoyalPurple, mark=triangle, densely dashed}
|
|
\addlegendentry{improved, $\gamma = 0.01$}
|
|
|
|
\addlegendimage{red, mark=triangle, densely dashed}
|
|
\addlegendentry{improved, $\gamma = 0.05$}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{Simulation results for $\gamma = 0.05, \omega = 0.05, K=200, N=12$}
|
|
\label{fig:simulation_results_hybrid}
|
|
\end{figure}
|
|
\end{frame}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%\begin{frame}[t]
|
|
% \frametitle{Proximal Decoding: Improvement}
|
|
%
|
|
% \begin{minipage}[c]{0.9\textwidth}
|
|
% \centering
|
|
% \begin{figure}[H]
|
|
% \vspace*{-0.6cm}
|
|
% \hspace*{-0.7cm}
|
|
% \centering
|
|
% \begin{subfigure}[c]{0.33\textwidth}
|
|
% \centering
|
|
%
|
|
% \begin{tikzpicture}[scale=0.4]
|
|
% \begin{axis}[
|
|
% grid=both,
|
|
% xlabel={$E_b / N_0$}, ylabel={FER},
|
|
% ymode=log,
|
|
% legend columns=1,
|
|
% legend pos=outer north east,
|
|
% %legend columns=2,
|
|
% %legend style={at={(0.5,-0.45)},anchor=south},
|
|
% ymax=1.5, ymin=8e-5,
|
|
% ]
|
|
%
|
|
% \addplot[ForestGreen, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
% {res/proximal/2d_ber_fer_dfr_963965.csv};
|
|
% \addplot[Emerald, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
% {res/hybrid/2d_ber_fer_dfr_963965.csv};
|
|
%
|
|
% \addplot[NavyBlue, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
% {res/proximal/2d_ber_fer_dfr_963965.csv};
|
|
% \addplot[RoyalPurple, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
% {res/hybrid/2d_ber_fer_dfr_963965.csv};
|
|
%
|
|
% \addplot[RedOrange, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/proximal/2d_ber_fer_dfr_963965.csv};
|
|
% \addplot[red, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/hybrid/2d_ber_fer_dfr_963965.csv};
|
|
% \end{axis}
|
|
% \end{tikzpicture}
|
|
% \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$ \cite[\text{96.3.965}]{mackay_enc}}
|
|
% \end{subfigure}%
|
|
% \begin{subfigure}[c]{0.33\textwidth}
|
|
% \centering
|
|
% \begin{tikzpicture}[scale=0.4]
|
|
% \begin{axis}[
|
|
% grid=both,
|
|
% xlabel={$E_b / N_0$}, ylabel={FER},
|
|
% ymode=log,
|
|
% legend columns=1,
|
|
% legend pos=outer north east,
|
|
% xmin=0.5, xmax=6, xtick={1, ..., 5},
|
|
% ymax=1.5, ymin=8e-5,
|
|
% ]
|
|
%
|
|
% \addplot[ForestGreen, mark=*, solid,]
|
|
% table [x=SNR, y=FER, col sep=comma,
|
|
% discard if not={gamma}{0.15},
|
|
% discard if gt={SNR}{5.5},]
|
|
% {res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
% \addplot[Emerald, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma,
|
|
% discard if not={gamma}{0.15},
|
|
% discard if gt={SNR}{5.5},]
|
|
% {res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
%
|
|
% \addplot[NavyBlue, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma,
|
|
% discard if not={gamma}{0.01},
|
|
% discard if gt={SNR}{5.5},]
|
|
% {res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
% \addplot[RoyalPurple, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma,
|
|
% discard if not={gamma}{0.01},
|
|
% discard if gt={SNR}{5.5},]
|
|
% {res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
%
|
|
% \addplot[RedOrange, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma,
|
|
% discard if not={gamma}{0.05},
|
|
% discard if gt={SNR}{5.5},]
|
|
% {res/proximal/2d_ber_fer_dfr_20433484.csv};
|
|
% \addplot[red, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma,
|
|
% discard if not={gamma}{0.05},
|
|
% discard if gt={SNR}{5.5},]
|
|
% {res/hybrid/2d_ber_fer_dfr_20433484.csv};
|
|
% \end{axis}
|
|
% \end{tikzpicture}
|
|
% \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc}}
|
|
% \end{subfigure}%
|
|
% \begin{subfigure}[c]{0.33\textwidth}
|
|
% \centering
|
|
% \begin{tikzpicture}[scale=0.4]
|
|
% \begin{axis}[
|
|
% grid=both,
|
|
% xlabel={$E_b / N_0$}, ylabel={FER},
|
|
% ymode=log,
|
|
% legend columns=1,
|
|
% legend pos=outer north east,
|
|
% %legend columns=2,
|
|
% %legend style={at={(0.5,-0.45)},anchor=south},
|
|
% ymax=1.5, ymin=8e-5,
|
|
% ]
|
|
%
|
|
% \addplot[ForestGreen, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
% {res/proximal/2d_ber_fer_dfr_40833844.csv};
|
|
% \addplot[Emerald, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
% {res/hybrid/2d_ber_fer_dfr_40833844.csv};
|
|
%
|
|
% \addplot[NavyBlue, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
% {res/proximal/2d_ber_fer_dfr_40833844.csv};
|
|
% \addplot[RoyalPurple, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
% {res/hybrid/2d_ber_fer_dfr_40833844.csv};
|
|
%
|
|
% \addplot[RedOrange, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/proximal/2d_ber_fer_dfr_40833844.csv};
|
|
% \addplot[red, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/hybrid/2d_ber_fer_dfr_40833844.csv};
|
|
% \end{axis}
|
|
% \end{tikzpicture}
|
|
% \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$ \cite[\text{408.33.844}]{mackay_enc}}
|
|
% \end{subfigure}
|
|
% \hspace*{-0.7cm}
|
|
% \begin{subfigure}[c]{0.33\textwidth}
|
|
% \centering
|
|
% \begin{tikzpicture}[scale=0.4]
|
|
% \begin{axis}[
|
|
% grid=both,
|
|
% xlabel={$E_b / N_0$}, ylabel={FER},
|
|
% ymode=log,
|
|
% legend columns=1,
|
|
% legend pos=outer north east,
|
|
% %legend columns=2,
|
|
% %legend style={at={(0.5,-0.45)},anchor=south},
|
|
% ymax=1.5, ymin=8e-5,
|
|
% ]
|
|
%
|
|
% \addplot[ForestGreen, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
% {res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
|
|
% \addplot[Emerald, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
% {res/hybrid/2d_ber_fer_dfr_bch_31_26.csv};
|
|
%
|
|
% \addplot[NavyBlue, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
% {res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
|
|
% \addplot[RoyalPurple, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
% {res/hybrid/2d_ber_fer_dfr_bch_31_26.csv};
|
|
%
|
|
% \addplot[RedOrange, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
|
|
% \addplot[red, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/hybrid/2d_ber_fer_dfr_bch_31_26.csv};
|
|
% \end{axis}
|
|
% \end{tikzpicture}
|
|
% \caption{BCH code with $n=31, k=26$\\[\baselineskip]}
|
|
% \end{subfigure}%
|
|
% \begin{subfigure}[c]{0.33\textwidth}
|
|
% \centering
|
|
% \begin{tikzpicture}[scale=0.4]
|
|
% \begin{axis}[
|
|
% grid=both,
|
|
% xlabel={$E_b / N_0$}, ylabel={FER},
|
|
% ymode=log,
|
|
% legend columns=1,
|
|
% legend pos=outer north east,
|
|
% %legend columns=2,
|
|
% %legend style={at={(0.5,-0.45)},anchor=south},
|
|
% ymax=1.5, ymin=8e-5,
|
|
% ]
|
|
%
|
|
% \addplot[ForestGreen, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
% {res/proximal/2d_ber_fer_dfr_20455187.csv};
|
|
% \addplot[Emerald, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
% {res/hybrid/2d_ber_fer_dfr_20455187.csv};
|
|
%
|
|
% \addplot[NavyBlue, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
% {res/proximal/2d_ber_fer_dfr_20455187.csv};
|
|
% \addplot[RoyalPurple, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
% {res/hybrid/2d_ber_fer_dfr_20455187.csv};
|
|
%
|
|
% \addplot[RedOrange, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/proximal/2d_ber_fer_dfr_20455187.csv};
|
|
% \addplot[red, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/hybrid/2d_ber_fer_dfr_20455187.csv};
|
|
% \end{axis}
|
|
% \end{tikzpicture}
|
|
% \caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$ \cite[\text{204.55.187}]{mackay_enc}}
|
|
% \end{subfigure}%
|
|
% \begin{subfigure}[c]{0.33\textwidth}
|
|
% \centering
|
|
% \begin{tikzpicture}[scale=0.4]
|
|
% \begin{axis}[
|
|
% grid=both,
|
|
% xlabel={$E_b / N_0$}, ylabel={FER},
|
|
% ymode=log,
|
|
% legend columns=1,
|
|
% legend pos=outer north east,
|
|
% ymax=1.5, ymin=8e-5,
|
|
% ]
|
|
%
|
|
% \addplot[ForestGreen, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
% {res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
% \addplot[Emerald, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
% {res/hybrid/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
%
|
|
% \addplot[NavyBlue, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
% {res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
% \addplot[RoyalPurple, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
% {res/hybrid/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
%
|
|
% \addplot[RedOrange, mark=*, solid]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
% \addplot[red, mark=triangle, densely dashed]
|
|
% table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
% {res/hybrid/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
% \end{axis}
|
|
% \end{tikzpicture}
|
|
% \caption{LDPC code (progressive edge growth construction) with $n=504, k=252$ \cite[\text{PEGReg252x504}]{mackay_enc}}
|
|
% \end{subfigure}%
|
|
% \end{figure}
|
|
% \end{minipage}%
|
|
% \begin{minipage}[c]{0.1\textwidth}
|
|
% \centering
|
|
%
|
|
% \begin{figure}
|
|
% \vspace*{-1.5cm}
|
|
% \hspace*{-0.7cm}
|
|
% \centering
|
|
% \begin{tikzpicture}[scale=0.7]
|
|
% \begin{axis}[hide axis,
|
|
% xmin=10, xmax=50,
|
|
% ymin=0, ymax=0.4,
|
|
% legend style={draw=white!15!black,legend cell align=left}]
|
|
% \addlegendimage{ForestGreen, mark=*, solid}
|
|
% \addlegendentry{proximal, $\gamma = 0.15$}
|
|
%
|
|
% \addlegendimage{Emerald, mark=triangle, densely dashed}
|
|
% \addlegendentry{improved, $\gamma = 0.15$}
|
|
%
|
|
% \addlegendimage{NavyBlue, mark=*, solid}
|
|
% \addlegendentry{proximal, $\gamma = 0.01$}
|
|
%
|
|
% \addlegendimage{RoyalPurple, mark=triangle, densely dashed}
|
|
% \addlegendentry{improved, $\gamma = 0.01$}
|
|
%
|
|
% \addlegendimage{RedOrange, mark=*, solid}
|
|
% \addlegendentry{proximal, $\gamma = 0.05$}
|
|
%
|
|
% \addlegendimage{red, mark=triangle, densely dashed}
|
|
% \addlegendentry{improved, $\gamma = 0.05$}
|
|
% \end{axis}
|
|
% \end{tikzpicture}
|
|
%
|
|
% \end{figure}
|
|
% \end{minipage}
|
|
%\end{frame}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{frame}[t]
|
|
\frametitle{Proximal Decoding: Average Error}
|
|
\setcounter{footnote}{0}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={Iterations},
|
|
ylabel={Average $\lVert \boldsymbol{c}-\boldsymbol{\hat{c}} \rVert$},
|
|
width=0.45\textwidth,
|
|
height=0.3375\textwidth,
|
|
legend pos=outer north east,
|
|
]
|
|
\addplot [ForestGreen, mark=none, line width=1pt]
|
|
table [col sep=comma, discard if not={omega}{0.0774263682681127}, x=k, y=err]
|
|
{res/proximal/2d_avg_error_20433484_1db.csv};
|
|
\addlegendentry{$E_b / N_0 = \SI{1}{dB}$}
|
|
\addplot [NavyBlue, mark=none, line width=1pt]
|
|
table [col sep=comma, discard if not={omega}{0.0774263682681127}, x=k, y=err]
|
|
{res/proximal/2d_avg_error_20433484_3db.csv};
|
|
\addlegendentry{$E_b / N_0 = \SI{3}{dB}$}
|
|
\addplot [RedOrange, mark=none, line width=1pt]
|
|
table [col sep=comma, discard if not={omega}{0.052233450742668434}, x=k, y=err]
|
|
{res/proximal/2d_avg_error_20433484_5db.csv};
|
|
\addlegendentry{$E_b / N_0 = \SI{5}{dB}$}
|
|
\addplot [RoyalPurple, mark=none, line width=1pt]
|
|
table [col sep=comma, discard if not={omega}{0.052233450742668434}, x=k, y=err]
|
|
{res/proximal/2d_avg_error_20433484_8db.csv};
|
|
\addlegendentry{$E_b / N_0 = \SI{8}{dB}$}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{Average error for $\SI{500000}{}$ decodings, $
|
|
\omega = 0.05, \gamma = 0.05, K=200$\footnotemark}
|
|
\end{figure}
|
|
|
|
\footnotetext{Simulation performed with (3,6) regular LDPC code with $n=204, k=102$
|
|
\cite[Code: 204.33.484]{mackay_enc}}
|
|
|
|
\begin{itemize}
|
|
\item With increasing iterations, the average error asymptotically
|
|
approaches a minimum, non-zero value
|
|
\end{itemize}
|
|
\end{frame}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{frame}[t]
|
|
\frametitle{Proximal Decoding: Time Complexity}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$n$}, ylabel={time per frame (s)},
|
|
legend style={at={(0.05,0.7)},anchor=south west},
|
|
legend cell align={left},
|
|
width=0.45\textwidth,
|
|
height=0.3375\textwidth,
|
|
]
|
|
\addplot[RedOrange, only marks, mark=*]
|
|
table [col sep=comma, x=n, y=spf]
|
|
{res/proximal/fps_vs_n.csv};
|
|
\addlegendentry{proximal}
|
|
|
|
\addplot[RoyalPurple, only marks, mark=triangle*]
|
|
table [col sep=comma, x=n, y=spf]
|
|
{res/hybrid/fps_vs_n.csv};
|
|
\addlegendentry{improved ($\SI{12}{\bit}$)}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{Time complexity of proximal decoding and improved implementation\footnotemark}
|
|
\label{fig:fps_vs_n}
|
|
\end{figure}
|
|
|
|
\footnotetext{The points shown were calculated by evaluating the metadata
|
|
of BER simulation results for the following codes:
|
|
BCH $\left( 31, 11 \right)$;
|
|
BCH $\left( 31, 26 \right)$;
|
|
\cite[\text{96.3.965; 204.33.484;
|
|
204.55.187; 408.33.844; PEGReg252x504}]{mackay_enc}
|
|
}
|
|
|
|
\end{frame}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%\begin{frame}[t]
|
|
% \frametitle{Conclusion}
|
|
%
|
|
% \begin{itemize}
|
|
% \item Analysis of proximal decoding for AWGN channels:
|
|
% \begin{itemize}
|
|
% \item Error correcting performance (BER, FER, decoding failures)
|
|
% \item Computational performance ($\mathcal{O}\left( n \right) $ time complexity,
|
|
% fast implementation possible)
|
|
% \item Number of iterations independent of SNR
|
|
% \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}
|
|
% \end{itemize}
|
|
%\end{frame}
|
|
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{LP Decoding using ADMM}%
|
|
\label{sub:LP Decoding using ADMM}
|
|
|
|
\begin{frame}[t]
|
|
\frametitle{LP Decoding using ADMM}
|
|
|
|
\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[RedOrange, line width=1pt]
|
|
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, only marks, mark=o, line width=1pt]
|
|
table [col sep=comma, x=SNR, y=FER,]
|
|
{res/admm/ber_margulis264013203.csv};
|
|
\addlegendentry{ADMM (Own results)}
|
|
\addplot[RoyalPurple, line width=1pt, densely dashed]
|
|
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}
|
|
|
|
\caption{Comparison of datapoints from Barman et al. with own simulation results%
|
|
\protect\footnotemark{}}
|
|
\label{fig:admm:results}
|
|
\end{figure}%
|
|
%
|
|
\footnotetext{``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{frame}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{frame}[t]
|
|
\frametitle{LP Decoding using ADMM: Choice of Penalty\\
|
|
Parameters}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\begin{subfigure}[c]{0.48\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$\mu$}, ylabel={FER},
|
|
ymode=log,
|
|
width=0.9\textwidth,
|
|
height=0.675\textwidth,
|
|
]
|
|
\addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
|
|
table [col sep=comma, x=mu, y=FER,
|
|
discard if not={SNR}{2.0},]
|
|
{res/admm/ber_2d_20433484.csv};
|
|
\addplot[RedOrange, line width=1pt, densely dashed, mark=*]
|
|
table [col sep=comma, x=mu, y=FER,
|
|
discard if not={SNR}{3.0},]
|
|
{res/admm/ber_2d_20433484.csv};
|
|
\addplot[NavyBlue, line width=1pt, densely dashed, mark=*]
|
|
table [col sep=comma, x=mu, y=FER,
|
|
discard if not={SNR}{4.0},]
|
|
{res/admm/ber_2d_20433484.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{subfigure}%
|
|
\begin{subfigure}[c]{0.48\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$\rho$}, ylabel={FER},
|
|
ymode=log,
|
|
width=0.9\textwidth,
|
|
height=0.675\textwidth,
|
|
]
|
|
\addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
|
|
table [col sep=comma, x=rho, y=FER,
|
|
discard if not={SNR}{2.0},]
|
|
{res/admm/ber_2d_20433484_rho.csv};
|
|
\addplot[RedOrange, line width=1pt, densely dashed, mark=*]
|
|
table [col sep=comma, x=rho, y=FER,
|
|
discard if not={SNR}{3.0},]
|
|
{res/admm/ber_2d_20433484_rho.csv};
|
|
\addplot[NavyBlue, line width=1pt, densely dashed, mark=*]
|
|
table [col sep=comma, x=rho, y=FER,
|
|
discard if not={SNR}{4.0},]
|
|
{res/admm/ber_2d_20433484_rho.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{subfigure}%
|
|
|
|
\begin{subfigure}[t]{\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[hide axis,
|
|
xmin=10, xmax=50,
|
|
ymin=0, ymax=0.4,
|
|
legend columns=3,
|
|
legend style={draw=white!15!black,legend cell align=left}]
|
|
|
|
\addlegendimage{ForestGreen, line width=1pt, densely dashed, mark=*}
|
|
\addlegendentry{$E_b / N_0 = \SI{2}{dB}$}
|
|
\addlegendimage{RedOrange, line width=1pt, densely dashed, mark=*}
|
|
\addlegendentry{$E_b / N_0 = \SI{3}{dB}$}
|
|
\addlegendimage{NavyBlue, line width=1pt, densely dashed, mark=*}
|
|
\addlegendentry{$E_b / N_0 = \SI{4}{dB}$}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{subfigure}
|
|
|
|
\caption{Relation between $\mu$ and $\rho$ and decoding performance%
|
|
\protect\footnotemark{}}
|
|
\end{figure}%
|
|
|
|
\footnotetext{Simulation performed with (3,6) regular LDPC code with $n=204, k=102$
|
|
\cite[Code: 204.33.484]{mackay_enc}}
|
|
|
|
\end{frame}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{frame}[t]
|
|
\frametitle{LP Decoding using ADMM: Choice of Penalty\\
|
|
Parameters}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\begin{subfigure}[c]{0.48\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$\mu$}, ylabel={Average \# of iterations},
|
|
width=0.9\textwidth,
|
|
height=0.675\textwidth,
|
|
]
|
|
\addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
|
|
table [col sep=comma, x=mu, y=k_avg,
|
|
discard if not={SNR}{2.0},]
|
|
{res/admm/mu_kavg_20433484.csv};
|
|
\addplot[RedOrange, line width=1pt, densely dashed, mark=*]
|
|
table [col sep=comma, x=mu, y=k_avg,
|
|
discard if not={SNR}{3.0},]
|
|
{res/admm/mu_kavg_20433484.csv};
|
|
\addplot[NavyBlue, line width=1pt, densely dashed, mark=*]
|
|
table [col sep=comma, x=mu, y=k_avg,
|
|
discard if not={SNR}{4.0},]
|
|
{res/admm/mu_kavg_20433484.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{subfigure}%
|
|
\begin{subfigure}[c]{0.48\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$\rho$}, ylabel={Average \# of iterations},
|
|
width=0.9\textwidth,
|
|
height=0.675\textwidth,
|
|
]
|
|
\addplot[ForestGreen, line width=1pt, densely dashed, mark=*]
|
|
table [col sep=comma, x=rho, y=k_avg,
|
|
discard if not={SNR}{2.0},]
|
|
{res/admm/rho_kavg_20433484.csv};
|
|
\addplot[RedOrange, line width=1pt, densely dashed, mark=*]
|
|
table [col sep=comma, x=rho, y=k_avg,
|
|
discard if not={SNR}{3.0},]
|
|
{res/admm/rho_kavg_20433484.csv};
|
|
\addplot[NavyBlue, line width=1pt, densely dashed, mark=*]
|
|
table [col sep=comma, x=rho, y=k_avg,
|
|
discard if not={SNR}{4.0},]
|
|
{res/admm/rho_kavg_20433484.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{subfigure}%
|
|
|
|
\begin{subfigure}[t]{\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[hide axis,
|
|
xmin=10, xmax=50,
|
|
ymin=0, ymax=0.4,
|
|
legend columns=3,
|
|
legend style={draw=white!15!black,legend cell align=left}]
|
|
|
|
\addlegendimage{ForestGreen, line width=1pt, densely dashed, mark=*}
|
|
\addlegendentry{$E_b / N_0 = \SI{2}{dB}$}
|
|
\addlegendimage{RedOrange, line width=1pt, densely dashed, mark=*}
|
|
\addlegendentry{$E_b / N_0 = \SI{3}{dB}$}
|
|
\addlegendimage{NavyBlue, line width=1pt, densely dashed, mark=*}
|
|
\addlegendentry{$E_b / N_0 = \SI{4}{dB}$}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{subfigure}
|
|
|
|
\caption{Relation between $\mu$ and $\rho$ and speed of convergence%
|
|
\protect\footnotemark{}}
|
|
\end{figure}%
|
|
|
|
\footnotetext{Simulation performed with (3,6) regular LDPC code with $n=204, k=102$
|
|
\cite[Code: 204.33.484]{mackay_enc}}
|
|
\end{frame}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{frame}[t]
|
|
\frametitle{LP Decoding using ADMM: Average Error}
|
|
|
|
\begin{figure}[H]
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
grid=both,
|
|
width=0.45\textwidth,
|
|
height=0.3375\textwidth,
|
|
legend pos=outer north east,
|
|
xlabel={Iteration}, ylabel={Average $\left\Vert \hat{\boldsymbol{c}}
|
|
- \boldsymbol{c} \right\Vert$}
|
|
]
|
|
\addplot[ForestGreen, line width=1pt]
|
|
table [col sep=comma, x=k, y=err,
|
|
discard if not={SNR}{1.0},
|
|
discard if gt={k}{100}]
|
|
{res/admm/avg_error_20433484.csv};
|
|
\addlegendentry{$E_b / N_0 = \SI{1}{dB}$}
|
|
|
|
\addplot[RedOrange, line width=1pt]
|
|
table [col sep=comma, x=k, y=err,
|
|
discard if not={SNR}{2.0},
|
|
discard if gt={k}{100}]
|
|
{res/admm/avg_error_20433484.csv};
|
|
\addlegendentry{$E_b / N_0 = \SI{2}{dB}$}
|
|
|
|
\addplot[NavyBlue, line width=1pt]
|
|
table [col sep=comma, x=k, y=err,
|
|
discard if not={SNR}{3.0},
|
|
discard if gt={k}{100}]
|
|
{res/admm/avg_error_20433484.csv};
|
|
\addlegendentry{$E_b / N_0 = \SI{3}{dB}$}
|
|
|
|
\addplot[RoyalPurple, line width=1pt]
|
|
table [col sep=comma, x=k, y=err,
|
|
discard if not={SNR}{4.0},
|
|
discard if gt={k}{100}]
|
|
{res/admm/avg_error_20433484.csv};
|
|
\addlegendentry{$E_b / N_0 = \SI{4}{dB}$}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{Average error for $\SI{100000}{}$ decodings\protect\footnotemark{}}
|
|
\end{figure}%
|
|
|
|
\footnotetext{Simulation performed with (3,6) regular LDPC code with $n=204, k=102$
|
|
\cite[Code: 204.33.484]{mackay_enc}}
|
|
\end{frame}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{frame}[t]
|
|
\frametitle{LP Decoding using ADMM: Time Complexity}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[grid=both,
|
|
xlabel={$n$}, ylabel={Time per frame (s)},
|
|
width=0.45\textwidth,
|
|
height=0.3375\textwidth,
|
|
legend style={at={(0.03,0.96)},anchor=north west},
|
|
%legend pos=outer north east,
|
|
legend cell align={left},]
|
|
|
|
% \addplot[RedOrange, only marks, mark=*]
|
|
% table [col sep=comma, x=n, y=spf]
|
|
% {res/proximal/fps_vs_n.csv};
|
|
% \addlegendentry{Proximal decoding}
|
|
\addplot[RoyalPurple, only marks, mark=triangle*]
|
|
table [col sep=comma, x=n, y=spf]
|
|
{res/admm/fps_vs_n.csv};
|
|
% \addlegendentry{LP decoding using ADMM}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{Timing requirements of the LP decoding imlementation%
|
|
\protect\footnotemark{}}
|
|
\end{figure}%
|
|
|
|
\footnotetext{The points shown were calculated by evaluating the metadata
|
|
of BER simulation results for the following codes:
|
|
BCH $\left( 31, 11 \right)$;
|
|
BCH $\left( 31, 26 \right)$;
|
|
\cite[\text{96.3.965; 204.33.484;
|
|
204.55.187; 408.33.844; PEGReg252x504}]{mackay_enc}
|
|
}
|
|
|
|
\end{frame}
|
|
|