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latex/presentations/midterm/presentation.tex
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@ -126,7 +126,7 @@
\title{Application of Optimization Algorithms for Channel Decoding} \title{Application of Optimization Algorithms for Channel Decoding}
\subtitle{\small Midterm Presentation - 27.01.2023} \subtitle{\small Midterm Presentation, 27.01.2023}
%\author{Andreas Tsouchlos} %\author{Andreas Tsouchlos}
\author{\vspace{1.5mm} Andreas Tsouchlos} \author{\vspace{1.5mm} Andreas Tsouchlos}

44
latex/presentations/midterm/sections/decoding_algorithms.tex
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@ -13,10 +13,10 @@
\item MAP rule: \item MAP rule:
\begin{align*} \begin{align*}
\hat{\boldsymbol{x}} \hat{\boldsymbol{x}}
= \argmax_{x\in\mathbb{R}} = \argmax_{x\in\mathbb{R}^n}
f_{\boldsymbol{Y}}\left( \boldsymbol{y} | \boldsymbol{x} \right) f_{\boldsymbol{Y}}\left( \boldsymbol{y} | \boldsymbol{x} \right)
f_{\boldsymbol{X}}\left( \boldsymbol{x} \right) f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)
= \argmax_{x\in\mathbb{R}} = \argmax_{x\in\mathbb{R^n}}
e^{-L\left( \boldsymbol{y} | \boldsymbol{x}\right)} e^{-L\left( \boldsymbol{y} | \boldsymbol{x}\right)}
f_{\boldsymbol{X}}\left( \boldsymbol{x} \right) f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)
\end{align*} \end{align*}
@ -24,23 +24,34 @@
\begin{align*} \begin{align*}
f_{\boldsymbol{X}}\left( \boldsymbol{x} \right) f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)
= \frac{1}{\left| \mathcal{C}\left( \boldsymbol{H} \right) \right| } = \frac{1}{\left| \mathcal{C}\left( \boldsymbol{H} \right) \right| }
\sum_{c \in \mathcal{C}\left( \boldsymbol{H} \right) } \sum_{\boldsymbol{c} \in \mathcal{C}\left( \boldsymbol{H} \right) }
\delta\left( \boldsymbol{x} - \left( -1 \right)^{\boldsymbol{c}} \right) \delta\left( \boldsymbol{x} - \left( -1 \right)^{\boldsymbol{c}} \right)
\approx \frac{1}{Z} e^{-\gamma h\left( x \right) } \approx \frac{1}{Z} e^{-\gamma h\left( \boldsymbol{x} \right) }
\end{align*} \end{align*}
\item Code constraint polynomial: \item Code constraint polynomial:
\begin{minipage}[c]{0.56\textwidth}
\raggedright
\begin{align*} \begin{align*}
h\left( \boldsymbol{x} \right) = h\left( \boldsymbol{x} \right) =
\underbrace{\sum_{j=1}^{n} \left( x_j^2 - 1 \right)^2}_{\text{Bipolar constraint}} \underbrace{\sum_{j=1}^{n} \left( x_j^2 - 1 \right)^2}_{\text{Bipolar
constraint}}
+ \underbrace{\sum_{i=1}^{m} \left[ \left( + \underbrace{\sum_{i=1}^{m} \left[ \left(
\prod_{j\in\mathcal{A}\left( i \right)} x_j\right) -1 \right]^2} \prod_{j\in\mathcal{A}\left( i \right)} x_j\right) -1 \right]^2}
_{\text{Parity constraint}}, _{\text{Parity constraint}},
\hspace{5mm}\mathcal{A}\left( i \right) \equiv \left\{
j | j\in \mathcal{J},
\boldsymbol{H}_{j,i} = 1
\right\},
i \in \mathcal{I}
\end{align*} \end{align*}
\end{minipage}%
\begin{minipage}[c]{0.4\textwidth}
\raggedleft
\begin{flalign*}
\mathcal{I} &\equiv \left\{\text{``Set of all variable nodes''}\right\} &\\
\mathcal{J} &\equiv \left\{\text{``Set of all check nodes''}\right\} &\\
\mathcal{A}\left( i \right) &\equiv \left\{j | j\in \mathcal{J},
\boldsymbol{H}_{j,i} = 1
\right\}, i \in \mathcal{I}&\\
\end{flalign*}
\end{minipage}
\hfill
\end{itemize} \end{itemize}
\end{frame} \end{frame}
@ -63,8 +74,8 @@
\item Code proximal operator \cite{proximal_algorithms}: \item Code proximal operator \cite{proximal_algorithms}:
\begin{align*} \begin{align*}
\text{prox}_{\gamma h} \left( \boldsymbol{x} \right) &\equiv \text{prox}_{\gamma h} \left( \boldsymbol{x} \right) &\equiv
\argmin_{\boldsymbol{z}\in\mathbb{R}} \left( \argmin_{\boldsymbol{t}\in\mathbb{R}^n} \left(
\gamma h\left( \boldsymbol{z} \right) + \frac{1}{2} \lVert \boldsymbol{z} \gamma h\left( \boldsymbol{t} \right) + \frac{1}{2} \lVert \boldsymbol{t}
- \boldsymbol{x} \rVert^2 \right)\\ - \boldsymbol{x} \rVert^2 \right)\\
&\approx \boldsymbol{x} - \gamma \nabla h\left( \boldsymbol{x} \right), &\approx \boldsymbol{x} - \gamma \nabla h\left( \boldsymbol{x} \right),
\hspace{5mm} \gamma \text{ small} \hspace{5mm} \gamma \text{ small}
@ -77,7 +88,8 @@
\hspace{10mm} \text{``Gradient descent step''}\\ \hspace{10mm} \text{``Gradient descent step''}\\
\boldsymbol{s} &\leftarrow \boldsymbol{r} \boldsymbol{s} &\leftarrow \boldsymbol{r}
- \gamma \nabla h\left( \boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r}
\right) \hspace{29mm} \text{``Code proximal step''} \right), \hspace{9mm} \gamma > 0
\hspace{10mm} \text{``Code proximal step''}
\end{align*} \end{align*}
\end{itemize} \end{itemize}
\end{frame} \end{frame}
@ -87,7 +99,7 @@
\begin{frame}[t, fragile] \begin{frame}[t, fragile]
\frametitle{Proximal Decoding: Algorithm} \frametitle{Proximal Decoding: Algorithm}
\begin{itemize} \begin{itemize}
\item Resulting iterative decoding algorithm \cite{proximal_paper}: \item Iterative decoding algorithm \cite{proximal_paper}:
\end{itemize} \end{itemize}
\vspace{2mm} \vspace{2mm}
@ -121,12 +133,12 @@ return $\boldsymbol{\hat{c}}$
\boldsymbol{c} : \lambda_{\boldsymbol{c}} \ge 0, \boldsymbol{c} : \lambda_{\boldsymbol{c}} \ge 0,
\sum_{\boldsymbol{c}\in\mathcal{C}}\lambda_{\boldsymbol{c}} = 1 \sum_{\boldsymbol{c}\in\mathcal{C}}\lambda_{\boldsymbol{c}} = 1
\right\}, \right\},
\hspace{5mm} \lambda_{\boldsymbol{c}} \in \mathbb{R} \hspace{5mm} \lambda_{\boldsymbol{c}} \in \mathbb{R}_{\ge 0}
\end{align*} \end{align*}
\item Cost function: \item Cost function:
\begin{align*} \begin{align*}
\sum_{i=1}^{n} \gamma_i c_i, \sum_{i=1}^{n} \gamma_i c_i,
\hspace{5mm}\gamma_i = \log\left( \hspace{5mm}\gamma_i = \ln\left(
\frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right) \frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right)
\end{align*} \end{align*}
\item LP formulation of ML decoding: \item LP formulation of ML decoding:

81
latex/presentations/midterm/sections/examination_results.tex
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@ -1,5 +1,5 @@
\section{Examination Results}% \section{Analysis}%
\label{sec:Examination Results} \label{sec:Analysis}
\subsection{Proximal Decoding}% \subsection{Proximal Decoding}%
@ -11,10 +11,10 @@
\frametitle{Proximal Decoding: Bit Error Rate and Performance} \frametitle{Proximal Decoding: Bit Error Rate and Performance}
\vspace*{-0.5cm} \vspace*{-0.5cm}
\begin{itemize} \begin{itemize}
\item Comparison of simulation \item Comparison of simulation%
\footnote{(3,6) regular LDPC code with $n=204, k=102$ \footnote{(3,6) regular LDPC code with $n=204, k=102$
\cite[\text{204.33.484}]{mackay_enc}} \cite[\text{204.33.484}]{mackay_enc}}
with results of Wadayama et al. with results of Wadayama et al. \cite{proximal_paper}
\end{itemize} \end{itemize}
\begin{figure}[H] \begin{figure}[H]
@ -78,9 +78,9 @@
\vspace*{-0.5cm} \vspace*{-0.5cm}
\begin{itemize} \begin{itemize}
\item $\mathcal{O}\left(n \right) $ time complexity - same as BP; \item $\mathcal{O}\left(n \right) $ time complexity - same as BP;
only multiplication and addition necessary \cite{proximal_paper} only multiplication and addition necessary
\item Measured Performance: $\sim\SI{10000}{}$ frames/s \item Measured performance: $\sim\SI{10000}{}$ frames/s
- Intel Core i7-7700HQ @ 2.80GHz; $n=204$ on Intel Core i7-7700HQ @ 2.80GHz; $n=204$
\end{itemize} \end{itemize}
\vspace{3mm} \vspace{3mm}
\end{frame} \end{frame}
@ -92,7 +92,7 @@
\setcounter{footnote}{0} \setcounter{footnote}{0}
\begin{itemize} \begin{itemize}
\item Simulation \item Simulation%
\footnote{(3,6) regular LDPC code with $n=204, k=102$ \footnote{(3,6) regular LDPC code with $n=204, k=102$
\cite[\text{204.33.484}]{mackay_enc}} \cite[\text{204.33.484}]{mackay_enc}}
results for different values of $\gamma$ results for different values of $\gamma$
@ -392,7 +392,7 @@
\setcounter{footnote}{0} \setcounter{footnote}{0}
\begin{itemize} \begin{itemize}
\item Analysis of simulated \item Analysis of simulated%
\footnote{(3,6) regular LDPC code with $n=204, k=102$ \footnote{(3,6) regular LDPC code with $n=204, k=102$
\cite[\text{204.33.484}]{mackay_enc}} \cite[\text{204.33.484}]{mackay_enc}}
BER and FER BER and FER
@ -532,8 +532,8 @@ return $\boldsymbol{\hat{c}}$
table [col sep=comma, x=k, y=grad_h_1] table [col sep=comma, x=k, y=grad_h_1]
{res/proximal/comp_bch_7_4_combined.csv}; {res/proximal/comp_bch_7_4_combined.csv};
\addlegendentry{est} \addlegendentry{est}
\addlegendentry{$\nabla L \left[ 2 \right] $} \addlegendentry{$\left(\nabla L \right)_2$}
\addlegendentry{$\nabla h \left[ 2 \right] $} \addlegendentry{$\left(\nabla h \right)_2 $}
\end{axis} \end{axis}
\end{tikzpicture}\\ \end{tikzpicture}\\
\begin{tikzpicture}[scale = 0.35] \begin{tikzpicture}[scale = 0.35]
@ -556,8 +556,8 @@ return $\boldsymbol{\hat{c}}$
table [col sep=comma, x=k, y=grad_h_2] table [col sep=comma, x=k, y=grad_h_2]
{res/proximal/comp_bch_7_4_combined.csv}; {res/proximal/comp_bch_7_4_combined.csv};
\addlegendentry{est} \addlegendentry{est}
\addlegendentry{$\nabla L \left[ 3 \right] $} \addlegendentry{$\left(\nabla L \right)_3$}
\addlegendentry{$\nabla h \left[ 3 \right] $} \addlegendentry{$\left(\nabla h \right)_3 $}
\end{axis} \end{axis}
\end{tikzpicture}\\ \end{tikzpicture}\\
\begin{tikzpicture}[scale = 0.35] \begin{tikzpicture}[scale = 0.35]
@ -580,8 +580,8 @@ return $\boldsymbol{\hat{c}}$
table [col sep=comma, x=k, y=grad_h_3] table [col sep=comma, x=k, y=grad_h_3]
{res/proximal/comp_bch_7_4_combined.csv}; {res/proximal/comp_bch_7_4_combined.csv};
\addlegendentry{est} \addlegendentry{est}
\addlegendentry{$\nabla L \left[ 4 \right] $} \addlegendentry{$\left(\nabla L \right)_4$}
\addlegendentry{$\nabla h \left[ 4 \right] $} \addlegendentry{$\left(\nabla h \right)_4 $}
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\end{minipage}% \end{minipage}%
@ -608,8 +608,8 @@ return $\boldsymbol{\hat{c}}$
table [col sep=comma, x=k, y=grad_h_0] table [col sep=comma, x=k, y=grad_h_0]
{res/proximal/comp_bch_7_4_combined.csv}; {res/proximal/comp_bch_7_4_combined.csv};
\addlegendentry{est} \addlegendentry{est}
\addlegendentry{$\nabla L \left[ 1 \right] $} \addlegendentry{$\left(\nabla L \right)_1$}
\addlegendentry{$\nabla h \left[ 1 \right] $} \addlegendentry{$\left(\nabla h \right)_1 $}
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\end{minipage}% \end{minipage}%
@ -636,8 +636,8 @@ return $\boldsymbol{\hat{c}}$
table [col sep=comma, x=k, y=grad_h_4] table [col sep=comma, x=k, y=grad_h_4]
{res/proximal/comp_bch_7_4_combined.csv}; {res/proximal/comp_bch_7_4_combined.csv};
\addlegendentry{est} \addlegendentry{est}
\addlegendentry{$\nabla L \left[ 5 \right] $} \addlegendentry{$\left(\nabla L \right)_5$}
\addlegendentry{$\nabla h \left[ 5 \right] $} \addlegendentry{$\left(\nabla h \right)_5 $}
\end{axis} \end{axis}
\end{tikzpicture}\\ \end{tikzpicture}\\
\begin{tikzpicture}[scale = 0.35] \begin{tikzpicture}[scale = 0.35]
@ -660,8 +660,8 @@ return $\boldsymbol{\hat{c}}$
table [col sep=comma, x=k, y=grad_h_5] table [col sep=comma, x=k, y=grad_h_5]
{res/proximal/comp_bch_7_4_combined.csv}; {res/proximal/comp_bch_7_4_combined.csv};
\addlegendentry{est} \addlegendentry{est}
\addlegendentry{$\nabla L \left[ 6 \right] $} \addlegendentry{$\left(\nabla L \right)_6$}
\addlegendentry{$\nabla h \left[ 6 \right] $} \addlegendentry{$\left(\nabla h \right)_6 $}
\end{axis} \end{axis}
\end{tikzpicture}\\ \end{tikzpicture}\\
\begin{tikzpicture}[scale = 0.35] \begin{tikzpicture}[scale = 0.35]
@ -684,14 +684,14 @@ return $\boldsymbol{\hat{c}}$
table [col sep=comma, x=k, y=grad_h_6] table [col sep=comma, x=k, y=grad_h_6]
{res/proximal/comp_bch_7_4_combined.csv}; {res/proximal/comp_bch_7_4_combined.csv};
\addlegendentry{est} \addlegendentry{est}
\addlegendentry{$\nabla L \left[ 7 \right] $} \addlegendentry{$\left(\nabla L \right)_7$}
\addlegendentry{$\nabla h \left[ 7 \right] $} \addlegendentry{$\left(\nabla h \right)_7 $}
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\end{minipage} \end{minipage}
\caption{Internal variables of proximal decoder \caption{Internal variables of proximal decoder
as a function of the iteration ($n=7$)\footnotemark} as a function of the number of iterations ($n=7$)\footnotemark}
\footnotetext{A single decoding is shown, using the BCH$\left( 7,4 \right) $ code; \footnotetext{A single decoding is shown, using the BCH$\left( 7,4 \right) $ code;
$\gamma = 0.05, \omega = 0.05, E_b / N_0 = \SI{5}{dB}$} $\gamma = 0.05, \omega = 0.05, E_b / N_0 = \SI{5}{dB}$}
@ -1018,7 +1018,7 @@ $\textcolor{KITblue}{\text{Compute }d_H\left( \boldsymbol{ \tilde{c}}_n, \boldsy
$\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d_H\left( \boldsymbol{ \tilde{c}}_n, \boldsymbol{\hat{c}} \right)}$ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d_H\left( \boldsymbol{ \tilde{c}}_n, \boldsymbol{\hat{c}} \right)}$
\end{algorithm} \end{algorithm}
\caption{Hybrid proximal \& ML decoding algorithm} \caption{Improved proximal decoding algorithm}
\end{figure} \end{figure}
\end{minipage} \end{minipage}
\end{frame} \end{frame}
@ -1029,8 +1029,8 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
\frametitle{Proximal Decoding: Improvement using ``ML-on-List''} \frametitle{Proximal Decoding: Improvement using ``ML-on-List''}
\begin{itemize} \begin{itemize}
\item Comparison of proximal \& hybrid proximal-ML (correction of $N = \SI{12}{\bit}$) \item Comparison of proximal \& improved (correction of $N = \SI{12}{\bit}$)
decoding simulation decoding simulation%
\footnote{(3,6) regular LDPC code with $n=204, k=102$ \footnote{(3,6) regular LDPC code with $n=204, k=102$
\cite[Code: 204.33.484]{mackay_enc}} \cite[Code: 204.33.484]{mackay_enc}}
results results
@ -1150,13 +1150,13 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
\addlegendentry{proximal, $\gamma = 0.05$} \addlegendentry{proximal, $\gamma = 0.05$}
\addlegendimage{Emerald, mark=triangle, densely dashed} \addlegendimage{Emerald, mark=triangle, densely dashed}
\addlegendentry{hybrid, $\gamma = 0.15$} \addlegendentry{improved, $\gamma = 0.15$}
\addlegendimage{RoyalPurple, mark=triangle, densely dashed} \addlegendimage{RoyalPurple, mark=triangle, densely dashed}
\addlegendentry{hybrid, $\gamma = 0.01$} \addlegendentry{improved, $\gamma = 0.01$}
\addlegendimage{red, mark=triangle, densely dashed} \addlegendimage{red, mark=triangle, densely dashed}
\addlegendentry{hybrid, $\gamma = 0.05$} \addlegendentry{improved, $\gamma = 0.05$}
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
@ -1433,19 +1433,19 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
\addlegendentry{proximal, $\gamma = 0.15$} \addlegendentry{proximal, $\gamma = 0.15$}
\addlegendimage{Emerald, mark=triangle, densely dashed} \addlegendimage{Emerald, mark=triangle, densely dashed}
\addlegendentry{hybrid, $\gamma = 0.15$} \addlegendentry{improved, $\gamma = 0.15$}
\addlegendimage{NavyBlue, mark=*, solid} \addlegendimage{NavyBlue, mark=*, solid}
\addlegendentry{proximal, $\gamma = 0.01$} \addlegendentry{proximal, $\gamma = 0.01$}
\addlegendimage{RoyalPurple, mark=triangle, densely dashed} \addlegendimage{RoyalPurple, mark=triangle, densely dashed}
\addlegendentry{hybrid, $\gamma = 0.01$} \addlegendentry{improved, $\gamma = 0.01$}
\addlegendimage{RedOrange, mark=*, solid} \addlegendimage{RedOrange, mark=*, solid}
\addlegendentry{proximal, $\gamma = 0.05$} \addlegendentry{proximal, $\gamma = 0.05$}
\addlegendimage{red, mark=triangle, densely dashed} \addlegendimage{red, mark=triangle, densely dashed}
\addlegendentry{hybrid, $\gamma = 0.05$} \addlegendentry{improved, $\gamma = 0.05$}
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
@ -1466,7 +1466,7 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
\begin{axis}[ \begin{axis}[
grid=both, grid=both,
xlabel={Iterations}, xlabel={Iterations},
ylabel={Average $\left| \boldsymbol{x}-\boldsymbol{\hat{x}} \right|$}, ylabel={Average $\lVert \boldsymbol{c}-\boldsymbol{\hat{c}} \rVert$},
legend pos=outer north east, legend pos=outer north east,
] ]
\addplot [ForestGreen, mark=none, line width=1pt] \addplot [ForestGreen, mark=none, line width=1pt]
@ -1496,7 +1496,8 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
\cite[Code: 204.33.484]{mackay_enc}} \cite[Code: 204.33.484]{mackay_enc}}
\begin{itemize} \begin{itemize}
\item For large $k$, the average error asymptotically approaches a minimum, non-zero value \item With increasing iterations, the average error asymptotically
approaches a minimum, non-zero value
\end{itemize} \end{itemize}
\end{frame} \end{frame}
@ -1513,6 +1514,7 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
grid=both, grid=both,
xlabel={$n$}, ylabel={time per frame (s)}, xlabel={$n$}, ylabel={time per frame (s)},
legend style={at={(0.05,0.77)},anchor=south west}, legend style={at={(0.05,0.77)},anchor=south west},
legend cell align={left},
] ]
\addplot[RedOrange, only marks, mark=*] \addplot[RedOrange, only marks, mark=*]
table [col sep=comma, x=n, y=spf] table [col sep=comma, x=n, y=spf]
@ -1522,7 +1524,7 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
\addplot[RoyalPurple, only marks, mark=triangle*] \addplot[RoyalPurple, only marks, mark=triangle*]
table [col sep=comma, x=n, y=spf] table [col sep=comma, x=n, y=spf]
{res/hybrid/fps_vs_n.csv}; {res/hybrid/fps_vs_n.csv};
\addlegendentry{hybrid prox \& ML ($\SI{12}{\bit}$)} \addlegendentry{improved ($\SI{12}{\bit}$)}
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
@ -1551,13 +1553,12 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
\item Error coding performance (BER, FER, decoding failures) \item Error coding performance (BER, FER, decoding failures)
\item Computational performance ($\mathcal{O}\left( n \right) $ time complexity, \item Computational performance ($\mathcal{O}\left( n \right) $ time complexity,
fast implementation possible) fast implementation possible)
\item Number of iterations required independant of SNR \item Number of iterations independent of SNR
\item Operation during iteration (oscillation of estimate)
\end{itemize} \end{itemize}
\item Suggestion for improvement of proximal decoding: \item Suggestion for improvement of proximal decoding:
\begin{itemize} \begin{itemize}
\item Addidion of ``ML-on-list'' step \item Addition of ``ML-on-list'' step
\item $\sim\SI{1}{dB}$ gain under certain conditions \item Up to $\sim\SI{1}{dB}$ gain under certain conditions
\end{itemize} \end{itemize}
\end{itemize} \end{itemize}
\end{frame} \end{frame}

8
latex/presentations/midterm/sections/forthcoming_examination.tex
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@ -1,4 +1,4 @@
\section{Forthcoming Examinations}% \section{Forthcoming Analysis}%
\label{sec:Forthcoming Examinations} \label{sec:Forthcoming Examinations}
@ -10,7 +10,7 @@
\frametitle{Forthcoming Examinations} \frametitle{Forthcoming Examinations}
\begin{itemize} \begin{itemize}
\item Test the (Alternating Direction Method of Multipliers) ADMM \item Test ADMM (Alternating Direction Method of Multipliers)
as an optimization method for LP Decoding as an optimization method for LP Decoding
\begin{itemize} \begin{itemize}
\item In LP decoding, the ML decoding problem is reduced to a linear program, \item In LP decoding, the ML decoding problem is reduced to a linear program,
@ -23,8 +23,8 @@
\end{itemize} \end{itemize}
\item Compare ADMM implementation with Proximal Decoding implementation with respect to \item Compare ADMM implementation with Proximal Decoding implementation with respect to
\begin{itemize} \begin{itemize}
\item Decoding performance (BER, FER) \item decoding performance (BER, FER)
\item Computational performance (time complexity, actual seconds per frame) \item computational performance (time complexity, actual seconds per frame)
\end{itemize} \end{itemize}
\end{itemize} \end{itemize}

13
latex/presentations/midterm/sections/theoretical_background.tex
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@ -23,7 +23,7 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t] \begin{frame}[t]
\frametitle{Previous work} \frametitle{Previous Work}
\begin{figure}[h] \begin{figure}[h]
\centering \centering
@ -47,8 +47,8 @@
\end{figure} \end{figure}
\begin{itemize} \begin{itemize}
\item Examination of ``Proximal Decoding'' \item Analysis of ``Proximal Decoding''
\item Examination of ``Interior Point Decoding'' \item Analysis of ``Interior Point Decoding''
\end{itemize} \end{itemize}
\end{frame} \end{frame}
@ -89,12 +89,13 @@
\end{figure} \end{figure}
\begin{itemize} \begin{itemize}
\item All simulations are performed with BPSK modulation: \item All simulations are performed with BPSK:
\begin{align*} \begin{align*}
\boldsymbol{x} = \left( -1 \right)^{\boldsymbol{c}}, \boldsymbol{x} = \left( -1 \right)^{\boldsymbol{c}},
\hspace{5mm} \boldsymbol{c} \in \mathbb{F}_2^n \hspace{5mm} \boldsymbol{c} \in \mathbb{F}_2^n,
\hspace{2mm} \boldsymbol{x} \in \mathbb{R}^n
\end{align*} \end{align*}
\item The used channel model is AWGN: \item The channel model is AWGN:
\begin{align*} \begin{align*}
\boldsymbol{y} = \boldsymbol{x} + \boldsymbol{z}, \boldsymbol{y} = \boldsymbol{x} + \boldsymbol{z},
\hspace{5mm}\boldsymbol{z}\sim \mathcal{N} \hspace{5mm}\boldsymbol{z}\sim \mathcal{N}