From 6939c6bcc59f6fe5993a6fb05fae17758031e573 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Mon, 23 Jan 2023 20:07:40 +0100 Subject: [PATCH] Added footnotes mentioning used code; Added Hybrid algorithm --- .../midterm/sections/decoding_algorithms.tex | 2 +- .../midterm/sections/examination_results.tex | 415 ++++++++++++------ .../sections/theoretical_background.tex | 4 +- 3 files changed, 284 insertions(+), 137 deletions(-) diff --git a/latex/presentations/midterm/sections/decoding_algorithms.tex b/latex/presentations/midterm/sections/decoding_algorithms.tex index 48e44ad..406bbef 100644 --- a/latex/presentations/midterm/sections/decoding_algorithms.tex +++ b/latex/presentations/midterm/sections/decoding_algorithms.tex @@ -58,7 +58,7 @@ \left( \boldsymbol{y} | \boldsymbol{x} \right) \right) \end{align*} \note{Notational difference between $f$ and $f_X$ or $f_Y$} - \item Code proximal operator: + \item Code proximal operator \cite{proximal_algorithms}: \begin{align*} \text{prox}_{\gamma h} \left( \boldsymbol{x} \right) &\equiv arg\min_{\boldsymbol{z}\in\mathbb{R}} \left( diff --git a/latex/presentations/midterm/sections/examination_results.tex b/latex/presentations/midterm/sections/examination_results.tex index 2b3148e..6074599 100644 --- a/latex/presentations/midterm/sections/examination_results.tex +++ b/latex/presentations/midterm/sections/examination_results.tex @@ -3,14 +3,17 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Proximal Decoder}% -\label{sub:Ex Proximal Decoder} +\subsection{Proximal Decoding}% +\label{sub:Ex Proximal Decoding} \begin{frame}[t] - \frametitle{Proximal Decoder: Examination Results} + \frametitle{Proximal Decoding: Bit Error Rate and Performance} \begin{itemize} - \item AWGN Channel - (3,6) regular LDPC Code with $n=204, k=102$: - \vspace{2mm} + \item Comparison of simulation + \footnote{(3,6) regular LDPC Code with $n=204, k=102$ + \cite[Code: 204.33.484]{mackay_enc}} + with results of Wadayama and Takabe + \begin{figure}[H] \centering @@ -62,71 +65,90 @@ \item $\mathcal{O}\left(n \right) $ time complexity - same as BP; Only multiplication and addition necessary \cite{proximal_paper} \item Measured Performance: Between $\SI{0.5}{\mega\bit / \second}$ and - $\SI{2.5}{\mega\bit / \second}$ - Intel Core i7-7700HQ @ 2.80GHz\\ - ($\sim \SI{10}{\second}$ for the shown plot) - \todo{Use the shown bitrate, or half? - ($n_{iterations} \cdot n$ or $n_{iterations} \cdot k$?)} + $\SI{2.5}{\mega\bit / \second}$ - Intel Core i7-7700HQ @ 2.80GHz% +% \\ ($\sim \SI{10}{\second}$ for the shown plot) \end{itemize} + \vspace{3mm} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}[t] - \frametitle{Proximal Decoder: Choice of $\gamma$} + \frametitle{Proximal Decoding: Choice of $\gamma$} + \setcounter{footnote}{0} - \begin{figure}[H] - \centering - - \begin{subfigure}[c]{0.5\textwidth} - \centering - - \begin{tikzpicture}[scale=0.52] - \begin{semilogyaxis}[xlabel={SNR}, ylabel={BER}, - grid=both, grid style={line width=.1pt}, - legend style={at={(0.05,0.05)},anchor=south west}, - ymin=3e-7, ymax=1.5,] - \foreach \gamma in {0.01, 0.05, 0.15}{ - \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{\gamma}] {res/2d_ber_fer_dfr_20433484_proximal.csv}; - \legend{\gamma} - } - \legend{$\gamma=0.01$, $\gamma=0.05$, $\gamma=0.15$} - \end{semilogyaxis} - \end{tikzpicture} - \end{subfigure}% - \begin{subfigure}[c]{0.5\textwidth} - \centering - - \begin{tikzpicture}[scale=0.7] - \begin{axis}[view={75}{60}, - zmode=log, - xlabel={SNR}, - ylabel={$\gamma$}, - zlabel={BER},] - \addplot3[surf, mesh/rows=17, mesh/cols=14, colormap/viridis] table [col sep=comma, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv}; - \addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $} - \addplot3[red, line width=1.5] table[col sep=comma, discard if not={gamma}{0.05}, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv}; - \addlegendentry{$\gamma = 0.05$} - \addplot3[blue, line width=1.5] table[col sep=comma, discard if not={gamma}{0.01}, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv}; - \addlegendentry{$\gamma = 0.01$} - \addplot3[brown, line width=1.5] table[col sep=comma, discard if not={gamma}{0.15}, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv}; - \addlegendentry{$\gamma = 0.15$} - \end{axis} - \end{tikzpicture} - \end{subfigure} + \begin{itemize} + \item Comparison of simulation + \footnote{(3,6) regular LDPC Code with $n=204, k=102$ + \cite[Code: 204.33.484]{mackay_enc}} + results for different values of $\gamma$ + \end{itemize} - \caption{BER for $\omega = 0.05, K=100$} - \label{fig:ber_3d} - \end{figure} + \begin{figure}[H] + \centering + + \begin{subfigure}[c]{0.5\textwidth} + \centering + \begin{tikzpicture}[scale=0.52] + \begin{semilogyaxis}[xlabel={SNR}, ylabel={BER}, + grid=both, grid style={line width=.1pt}, + legend style={at={(0.05,0.05)},anchor=south west}, + ymin=3e-7, ymax=1.5,] + \foreach \gamma in {0.01, 0.05, 0.15}{ + \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{\gamma}] {res/2d_ber_fer_dfr_20433484_proximal.csv}; + \legend{\gamma} + } + \legend{$\gamma=0.01$, $\gamma=0.05$, $\gamma=0.15$} + \end{semilogyaxis} + \end{tikzpicture} + \end{subfigure}% + \begin{subfigure}[c]{0.5\textwidth} + \centering + + \begin{tikzpicture}[scale=0.55] + \begin{axis}[view={75}{60}, + zmode=log, + xlabel={SNR}, + ylabel={$\gamma$}, + zlabel={BER},] + \addplot3[surf, mesh/rows=17, mesh/cols=14, colormap/viridis] table [col sep=comma, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv}; + \addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $} + \addplot3[red, line width=1.5] table[col sep=comma, discard if not={gamma}{0.05}, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv}; + \addlegendentry{$\gamma = 0.05$} + \addplot3[blue, line width=1.5] table[col sep=comma, discard if not={gamma}{0.01}, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv}; + \addlegendentry{$\gamma = 0.01$} + \addplot3[brown, line width=1.5] table[col sep=comma, discard if not={gamma}{0.15}, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv}; + \addlegendentry{$\gamma = 0.15$} + \end{axis} + \end{tikzpicture} + \end{subfigure} + + \caption{BER for $\omega = 0.05, K=100$} + \label{fig:ber_3d} + \end{figure} + + \begin{itemize} + \item Not great benefit in finding the optimal value for $\gamma$ + \end{itemize} + \vspace{3mm} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}[t, fragile] - \frametitle{Proximal Decoder: Frame Error Rate} + \frametitle{Proximal Decoding: Frame Error Rate} + \setcounter{footnote}{0} + + \begin{itemize} + \item Comparison of simulated + \footnote{(3,6) regular LDPC Code with $n=204, k=102$ + \cite[Code: 204.33.484]{mackay_enc}} + BER and FER + \end{itemize} \begin{minipage}{.4\textwidth} \centering - + \begin{algorithm}[caption={}, label={}, basicstyle=\fontsize{7.5}{9.5}\selectfont ] @@ -143,11 +165,11 @@ Output $\boldsymbol{\hat{x}}$ \end{minipage}% \begin{minipage}{.6\textwidth} \centering - \begin{figure}[H] + \vspace*{-8mm} \centering - \begin{tikzpicture}[scale=0.45] + \begin{tikzpicture}[scale=0.42] \begin{axis}[ grid=both, xlabel={SNR}, ylabel={BER}, @@ -165,8 +187,8 @@ Output $\boldsymbol{\hat{x}}$ {res/2d_ber_fer_dfr_20433484_proximal.csv}; \addlegendentry{$\gamma = 0.05$} \end{axis} - \end{tikzpicture}\\ - \begin{tikzpicture}[scale=0.45] + \end{tikzpicture} + \begin{tikzpicture}[scale=0.42] \begin{axis}[ grid=both, xlabel={SNR}, ylabel={FER}, @@ -184,8 +206,8 @@ Output $\boldsymbol{\hat{x}}$ {res/2d_ber_fer_dfr_20433484_proximal.csv}; \addlegendentry{$\gamma = 0.05$} \end{axis} - \end{tikzpicture} - \begin{tikzpicture}[scale=0.45] + \end{tikzpicture}\\ + \begin{tikzpicture}[scale=0.42] \begin{axis}[ grid=both, xlabel={SNR}, ylabel={Decoding Failure Rate}, @@ -212,84 +234,209 @@ Output $\boldsymbol{\hat{x}}$ \end{frame} -\begin{frame}[t] - \frametitle{title} - \begin{figure}[H] +\newcommand{\tikzmarknew}[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)$)$) + node [align=center, text width=2cm, pos=0.5, anchor=west] {#4}; + \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{tikzpicture}[scale=0.45] - \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, + + \begin{algorithm}[caption={}, label={}, + basicstyle=\fontsize{6.5}{7.5}\selectfont ] -% \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.15}] -% {res/2d_ber_fer_dfr_20433484.csv}; -% \addlegendentry{$\gamma = 0.15$} -% \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.01}] -% {res/2d_ber_fer_dfr_20433484.csv}; -% \addlegendentry{$\gamma = 0.01$} - \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}] - {res/2d_ber_fer_dfr_20433484_proximal.csv}; - \addlegendentry{proximal} - \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}] - {res/2d_ber_fer_dfr_20433484_hybrid.csv}; - \addlegendentry{hybrid prox. \& ML} - \end{axis} - \end{tikzpicture}\\ - \begin{tikzpicture}[scale=0.45] - \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}{0.15}] -% {res/2d_ber_fer_dfr_20433484.csv}; -% \addlegendentry{$\gamma = 0.15$} -% \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}] -% {res/2d_ber_fer_dfr_20433484.csv}; -% \addlegendentry{$\gamma = 0.01$} - \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}] - {res/2d_ber_fer_dfr_20433484_proximal.csv}; - \addlegendentry{proximal} - \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}] - {res/2d_ber_fer_dfr_20433484_hybrid.csv}; - \addlegendentry{hybrid prox. \& ML} - \end{axis} - \end{tikzpicture} - \begin{tikzpicture}[scale=0.45] - \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}{0.15}] -% {res/2d_ber_fer_dfr_20433484.csv}; -% \addlegendentry{$\gamma = 0.15$} -% \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.01}] -% {res/2d_ber_fer_dfr_20433484.csv}; -% \addlegendentry{$\gamma = 0.01$} - \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}] - {res/2d_ber_fer_dfr_20433484_proximal.csv}; - \addlegendentry{proximal} - \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}] - {res/2d_ber_fer_dfr_20433484_hybrid.csv}; - \addlegendentry{hybrid prox. \& ML} - \end{axis} - \end{tikzpicture} - - \caption{Simulation results for $\gamma = 0.05, \omega = 0.05, K=200$} - \label{fig:simulation_results_hybrid} - \end{figure} - +$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$$\hspace{4.185cm}\tikzmarknew{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 $\tikzmarknew{prox-end}$ +Find $N$ most probably wrong bits $\hspace{2cm}\tikzmarknew{ml-start}$ +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 $d\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right)$ $\tikzmarknew{ml-end}$ + \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}{0.15}] + % {res/2d_ber_fer_dfr_20433484.csv}; + % \addlegendentry{$\gamma = 0.15$} + % \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.01}] + % {res/2d_ber_fer_dfr_20433484.csv}; + % \addlegendentry{$\gamma = 0.01$} + \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}] + {res/2d_ber_fer_dfr_20433484_proximal.csv}; + \addlegendentry{proximal} + \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}] + {res/2d_ber_fer_dfr_20433484_hybrid.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}{0.15}] + % {res/2d_ber_fer_dfr_20433484.csv}; + % \addlegendentry{$\gamma = 0.15$} + % \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}] + % {res/2d_ber_fer_dfr_20433484.csv}; + % \addlegendentry{$\gamma = 0.01$} + \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}] + {res/2d_ber_fer_dfr_20433484_proximal.csv}; + \addlegendentry{proximal} + \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}] + {res/2d_ber_fer_dfr_20433484_hybrid.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}{0.15}] + % {res/2d_ber_fer_dfr_20433484.csv}; + % \addlegendentry{$\gamma = 0.15$} + % \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.01}] + % {res/2d_ber_fer_dfr_20433484.csv}; + % \addlegendentry{$\gamma = 0.01$} + \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}] + {res/2d_ber_fer_dfr_20433484_proximal.csv}; + \addlegendentry{proximal} + \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}] + {res/2d_ber_fer_dfr_20433484_hybrid.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] +% \frametitle{Proximal Decoding: Improvement} +% \begin{figure}[H] +% \centering +% +% \begin{tikzpicture}[scale=0.45] +% \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}{0.15}] +%% {res/2d_ber_fer_dfr_20433484.csv}; +%% \addlegendentry{$\gamma = 0.15$} +%% \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.01}] +%% {res/2d_ber_fer_dfr_20433484.csv}; +%% \addlegendentry{$\gamma = 0.01$} +% \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}] +% {res/2d_ber_fer_dfr_20433484_proximal.csv}; +% \addlegendentry{proximal} +% \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}] +% {res/2d_ber_fer_dfr_20433484_hybrid.csv}; +% \addlegendentry{hybrid prox. \& ML} +% \end{axis} +% \end{tikzpicture} +% \begin{tikzpicture}[scale=0.45] +% \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}{0.15}] +%% {res/2d_ber_fer_dfr_20433484.csv}; +%% \addlegendentry{$\gamma = 0.15$} +%% \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}] +%% {res/2d_ber_fer_dfr_20433484.csv}; +%% \addlegendentry{$\gamma = 0.01$} +% \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}] +% {res/2d_ber_fer_dfr_20433484_proximal.csv}; +% \addlegendentry{proximal} +% \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}] +% {res/2d_ber_fer_dfr_20433484_hybrid.csv}; +% \addlegendentry{hybrid prox. \& ML} +% \end{axis} +% \end{tikzpicture}\\ +% \begin{tikzpicture}[scale=0.45] +% \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}{0.15}] +%% {res/2d_ber_fer_dfr_20433484.csv}; +%% \addlegendentry{$\gamma = 0.15$} +%% \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.01}] +%% {res/2d_ber_fer_dfr_20433484.csv}; +%% \addlegendentry{$\gamma = 0.01$} +% \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}] +% {res/2d_ber_fer_dfr_20433484_proximal.csv}; +% \addlegendentry{proximal} +% \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}] +% {res/2d_ber_fer_dfr_20433484_hybrid.csv}; +% \addlegendentry{hybrid prox. \& ML} +% \end{axis} +% \end{tikzpicture} +% +% \caption{Simulation results for $\gamma = 0.05, \omega = 0.05, K=200, \SI{12}{\bit}$} +% \label{fig:simulation_results_hybrid} +% \end{figure} +% +%\end{frame} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{ADMM: Examination Results}% diff --git a/latex/presentations/midterm/sections/theoretical_background.tex b/latex/presentations/midterm/sections/theoretical_background.tex index d95223d..d683970 100644 --- a/latex/presentations/midterm/sections/theoretical_background.tex +++ b/latex/presentations/midterm/sections/theoretical_background.tex @@ -11,11 +11,11 @@ \begin{itemize} \item The general [ML] decoding problem for linear codes and the general problem of finding the weights of a linear code are both NP-complete. \cite{ml_np_hard_proof} - \item The standard message-passing algorithms used for decoding [LDPC and turbo codes] - are often difficult to analyze. \cite{feldman_thesis} \item The iterative message–passing algorithms preffered in practice do not guarantee optimality and may fail to decode correctly when the graph contains cycles \cite{ldpc_conv} + \item The standard message-passing algorithms used for decoding [LDPC and turbo codes] + are often difficult to analyze. \cite{feldman_thesis} \end{itemize} \end{frame}