Second round of changes I
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@ -126,7 +126,7 @@
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\title{Application of Optimization Algorithms for Channel Decoding}
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\subtitle{\small Midterm Presentation - 27.01.2023}
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\subtitle{\small Midterm Presentation, 27.01.2023}
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%\author{Andreas Tsouchlos}
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\author{\vspace{1.5mm} Andreas Tsouchlos}
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@ -13,10 +13,10 @@
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\item MAP rule:
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\begin{align*}
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\hat{\boldsymbol{x}}
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= \argmax_{x\in\mathbb{R}}
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= \argmax_{x\in\mathbb{R}^n}
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f_{\boldsymbol{Y}}\left( \boldsymbol{y} | \boldsymbol{x} \right)
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f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)
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= \argmax_{x\in\mathbb{R}}
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= \argmax_{x\in\mathbb{R^n}}
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e^{-L\left( \boldsymbol{y} | \boldsymbol{x}\right)}
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f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)
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\end{align*}
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@ -24,23 +24,34 @@
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\begin{align*}
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f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)
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= \frac{1}{\left| \mathcal{C}\left( \boldsymbol{H} \right) \right| }
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\sum_{c \in \mathcal{C}\left( \boldsymbol{H} \right) }
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\sum_{\boldsymbol{c} \in \mathcal{C}\left( \boldsymbol{H} \right) }
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\delta\left( \boldsymbol{x} - \left( -1 \right)^{\boldsymbol{c}} \right)
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\approx \frac{1}{Z} e^{-\gamma h\left( x \right) }
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\approx \frac{1}{Z} e^{-\gamma h\left( \boldsymbol{x} \right) }
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\end{align*}
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\item Code constraint polynomial:
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\begin{align*}
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h\left( \boldsymbol{x} \right) =
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\underbrace{\sum_{j=1}^{n} \left( x_j^2 - 1 \right)^2}_{\text{Bipolar constraint}}
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+ \underbrace{\sum_{i=1}^{m} \left[ \left(
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\prod_{j\in\mathcal{A}\left( i \right)} x_j\right) -1 \right]^2}
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_{\text{Parity constraint}},
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\hspace{5mm}\mathcal{A}\left( i \right) \equiv \left\{
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j | j\in \mathcal{J},
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\boldsymbol{H}_{j,i} = 1
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\right\},
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i \in \mathcal{I}
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\end{align*}
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\begin{minipage}[c]{0.56\textwidth}
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\raggedright
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\begin{align*}
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h\left( \boldsymbol{x} \right) =
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\underbrace{\sum_{j=1}^{n} \left( x_j^2 - 1 \right)^2}_{\text{Bipolar
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constraint}}
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+ \underbrace{\sum_{i=1}^{m} \left[ \left(
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\prod_{j\in\mathcal{A}\left( i \right)} x_j\right) -1 \right]^2}
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_{\text{Parity constraint}},
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\end{align*}
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\end{minipage}%
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\begin{minipage}[c]{0.4\textwidth}
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\raggedleft
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\begin{flalign*}
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\mathcal{I} &\equiv \left\{\text{``Set of all variable nodes''}\right\} &\\
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\mathcal{J} &\equiv \left\{\text{``Set of all check nodes''}\right\} &\\
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\mathcal{A}\left( i \right) &\equiv \left\{j | j\in \mathcal{J},
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\boldsymbol{H}_{j,i} = 1
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\right\}, i \in \mathcal{I}&\\
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\end{flalign*}
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\end{minipage}
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\hfill
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\end{itemize}
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\end{frame}
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@ -63,8 +74,8 @@
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\item Code proximal operator \cite{proximal_algorithms}:
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\begin{align*}
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\text{prox}_{\gamma h} \left( \boldsymbol{x} \right) &\equiv
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\argmin_{\boldsymbol{z}\in\mathbb{R}} \left(
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\gamma h\left( \boldsymbol{z} \right) + \frac{1}{2} \lVert \boldsymbol{z}
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\argmin_{\boldsymbol{t}\in\mathbb{R}^n} \left(
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\gamma h\left( \boldsymbol{t} \right) + \frac{1}{2} \lVert \boldsymbol{t}
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- \boldsymbol{x} \rVert^2 \right)\\
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&\approx \boldsymbol{x} - \gamma \nabla h\left( \boldsymbol{x} \right),
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\hspace{5mm} \gamma \text{ small}
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@ -77,7 +88,8 @@
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\hspace{10mm} \text{``Gradient descent step''}\\
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\boldsymbol{s} &\leftarrow \boldsymbol{r}
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- \gamma \nabla h\left( \boldsymbol{r}
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\right) \hspace{29mm} \text{``Code proximal step''}
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\right), \hspace{9mm} \gamma > 0
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\hspace{10mm} \text{``Code proximal step''}
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\end{align*}
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\end{itemize}
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\end{frame}
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@ -87,7 +99,7 @@
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\begin{frame}[t, fragile]
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\frametitle{Proximal Decoding: Algorithm}
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\begin{itemize}
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\item Resulting iterative decoding algorithm \cite{proximal_paper}:
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\item Iterative decoding algorithm \cite{proximal_paper}:
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\end{itemize}
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\vspace{2mm}
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@ -121,12 +133,12 @@ return $\boldsymbol{\hat{c}}$
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\boldsymbol{c} : \lambda_{\boldsymbol{c}} \ge 0,
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\sum_{\boldsymbol{c}\in\mathcal{C}}\lambda_{\boldsymbol{c}} = 1
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\right\},
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\hspace{5mm} \lambda_{\boldsymbol{c}} \in \mathbb{R}
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\hspace{5mm} \lambda_{\boldsymbol{c}} \in \mathbb{R}_{\ge 0}
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\end{align*}
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\item Cost function:
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\begin{align*}
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\sum_{i=1}^{n} \gamma_i c_i,
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\hspace{5mm}\gamma_i = \log\left(
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\hspace{5mm}\gamma_i = \ln\left(
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\frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right)
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\end{align*}
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\item LP formulation of ML decoding:
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@ -1,5 +1,5 @@
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\section{Examination Results}%
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\label{sec:Examination Results}
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\section{Analysis}%
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\label{sec:Analysis}
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\subsection{Proximal Decoding}%
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@ -11,10 +11,10 @@
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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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\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.
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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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@ -78,9 +78,9 @@
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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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- Intel Core i7-7700HQ @ 2.80GHz; $n=204$
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only multiplication and addition necessary
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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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@ -92,7 +92,7 @@
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\setcounter{footnote}{0}
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\begin{itemize}
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\item Simulation
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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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@ -392,7 +392,7 @@
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\setcounter{footnote}{0}
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\begin{itemize}
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\item Analysis of simulated
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\item Analysis of simulated%
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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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BER and FER
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@ -532,8 +532,8 @@ return $\boldsymbol{\hat{c}}$
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table [col sep=comma, x=k, y=grad_h_1]
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{res/proximal/comp_bch_7_4_combined.csv};
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\addlegendentry{est}
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\addlegendentry{$\nabla L \left[ 2 \right] $}
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\addlegendentry{$\nabla h \left[ 2 \right] $}
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\addlegendentry{$\left(\nabla L \right)_2$}
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\addlegendentry{$\left(\nabla h \right)_2 $}
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\end{axis}
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\end{tikzpicture}\\
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\begin{tikzpicture}[scale = 0.35]
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@ -556,8 +556,8 @@ return $\boldsymbol{\hat{c}}$
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table [col sep=comma, x=k, y=grad_h_2]
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{res/proximal/comp_bch_7_4_combined.csv};
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\addlegendentry{est}
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\addlegendentry{$\nabla L \left[ 3 \right] $}
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\addlegendentry{$\nabla h \left[ 3 \right] $}
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\addlegendentry{$\left(\nabla L \right)_3$}
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\addlegendentry{$\left(\nabla h \right)_3 $}
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\end{axis}
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\end{tikzpicture}\\
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\begin{tikzpicture}[scale = 0.35]
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@ -580,8 +580,8 @@ return $\boldsymbol{\hat{c}}$
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table [col sep=comma, x=k, y=grad_h_3]
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{res/proximal/comp_bch_7_4_combined.csv};
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\addlegendentry{est}
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\addlegendentry{$\nabla L \left[ 4 \right] $}
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\addlegendentry{$\nabla h \left[ 4 \right] $}
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\addlegendentry{$\left(\nabla L \right)_4$}
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\addlegendentry{$\left(\nabla h \right)_4 $}
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\end{axis}
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\end{tikzpicture}
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\end{minipage}%
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@ -608,8 +608,8 @@ return $\boldsymbol{\hat{c}}$
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table [col sep=comma, x=k, y=grad_h_0]
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{res/proximal/comp_bch_7_4_combined.csv};
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\addlegendentry{est}
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\addlegendentry{$\nabla L \left[ 1 \right] $}
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\addlegendentry{$\nabla h \left[ 1 \right] $}
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\addlegendentry{$\left(\nabla L \right)_1$}
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\addlegendentry{$\left(\nabla h \right)_1 $}
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\end{axis}
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\end{tikzpicture}
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\end{minipage}%
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@ -636,8 +636,8 @@ return $\boldsymbol{\hat{c}}$
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table [col sep=comma, x=k, y=grad_h_4]
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{res/proximal/comp_bch_7_4_combined.csv};
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\addlegendentry{est}
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\addlegendentry{$\nabla L \left[ 5 \right] $}
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\addlegendentry{$\nabla h \left[ 5 \right] $}
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\addlegendentry{$\left(\nabla L \right)_5$}
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\addlegendentry{$\left(\nabla h \right)_5 $}
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\end{axis}
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\end{tikzpicture}\\
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\begin{tikzpicture}[scale = 0.35]
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@ -660,8 +660,8 @@ return $\boldsymbol{\hat{c}}$
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table [col sep=comma, x=k, y=grad_h_5]
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{res/proximal/comp_bch_7_4_combined.csv};
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\addlegendentry{est}
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\addlegendentry{$\nabla L \left[ 6 \right] $}
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\addlegendentry{$\nabla h \left[ 6 \right] $}
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\addlegendentry{$\left(\nabla L \right)_6$}
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\addlegendentry{$\left(\nabla h \right)_6 $}
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\end{axis}
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\end{tikzpicture}\\
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\begin{tikzpicture}[scale = 0.35]
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@ -684,14 +684,14 @@ return $\boldsymbol{\hat{c}}$
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table [col sep=comma, x=k, y=grad_h_6]
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{res/proximal/comp_bch_7_4_combined.csv};
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\addlegendentry{est}
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\addlegendentry{$\nabla L \left[ 7 \right] $}
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\addlegendentry{$\nabla h \left[ 7 \right] $}
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\addlegendentry{$\left(\nabla L \right)_7$}
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\addlegendentry{$\left(\nabla h \right)_7 $}
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\end{axis}
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\end{tikzpicture}
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\end{minipage}
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\caption{Internal variables of proximal decoder
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as a function of the iteration ($n=7$)\footnotemark}
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as a function of the number of iterations ($n=7$)\footnotemark}
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\footnotetext{A single decoding is shown, using the BCH$\left( 7,4 \right) $ code;
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$\gamma = 0.05, \omega = 0.05, E_b / N_0 = \SI{5}{dB}$}
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@ -1018,7 +1018,7 @@ $\textcolor{KITblue}{\text{Compute }d_H\left( \boldsymbol{ \tilde{c}}_n, \boldsy
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$\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d_H\left( \boldsymbol{ \tilde{c}}_n, \boldsymbol{\hat{c}} \right)}$
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\end{algorithm}
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\caption{Hybrid proximal \& ML decoding algorithm}
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\caption{Improved proximal decoding algorithm}
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\end{figure}
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\end{minipage}
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\end{frame}
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@ -1029,8 +1029,8 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
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\frametitle{Proximal Decoding: Improvement using ``ML-on-List''}
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\begin{itemize}
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\item Comparison of proximal \& hybrid proximal-ML (correction of $N = \SI{12}{\bit}$)
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decoding simulation
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\item Comparison of proximal \& improved (correction of $N = \SI{12}{\bit}$)
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decoding simulation%
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\footnote{(3,6) regular LDPC code with $n=204, k=102$
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\cite[Code: 204.33.484]{mackay_enc}}
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results
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@ -1150,13 +1150,13 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
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\addlegendentry{proximal, $\gamma = 0.05$}
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\addlegendimage{Emerald, mark=triangle, densely dashed}
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\addlegendentry{hybrid, $\gamma = 0.15$}
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\addlegendentry{improved, $\gamma = 0.15$}
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\addlegendimage{RoyalPurple, mark=triangle, densely dashed}
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\addlegendentry{hybrid, $\gamma = 0.01$}
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\addlegendentry{improved, $\gamma = 0.01$}
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\addlegendimage{red, mark=triangle, densely dashed}
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\addlegendentry{hybrid, $\gamma = 0.05$}
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\addlegendentry{improved, $\gamma = 0.05$}
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\end{axis}
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\end{tikzpicture}
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@ -1433,19 +1433,19 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
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\addlegendentry{proximal, $\gamma = 0.15$}
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\addlegendimage{Emerald, mark=triangle, densely dashed}
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\addlegendentry{hybrid, $\gamma = 0.15$}
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\addlegendentry{improved, $\gamma = 0.15$}
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\addlegendimage{NavyBlue, mark=*, solid}
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\addlegendentry{proximal, $\gamma = 0.01$}
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\addlegendimage{RoyalPurple, mark=triangle, densely dashed}
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\addlegendentry{hybrid, $\gamma = 0.01$}
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\addlegendentry{improved, $\gamma = 0.01$}
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\addlegendimage{RedOrange, mark=*, solid}
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\addlegendentry{proximal, $\gamma = 0.05$}
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\addlegendimage{red, mark=triangle, densely dashed}
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\addlegendentry{hybrid, $\gamma = 0.05$}
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\addlegendentry{improved, $\gamma = 0.05$}
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\end{axis}
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\end{tikzpicture}
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@ -1466,7 +1466,7 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
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\begin{axis}[
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grid=both,
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xlabel={Iterations},
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ylabel={Average $\left| \boldsymbol{x}-\boldsymbol{\hat{x}} \right|$},
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ylabel={Average $\lVert \boldsymbol{c}-\boldsymbol{\hat{c}} \rVert$},
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legend pos=outer north east,
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]
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\addplot [ForestGreen, mark=none, line width=1pt]
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@ -1488,7 +1488,7 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
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\end{axis}
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\end{tikzpicture}
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\caption{Average error for $\SI{500000}{}$ decodings,$
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\caption{Average error for $\SI{500000}{}$ decodings, $
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\omega = 0.05, \gamma = 0.05, K=200$\footnotemark}
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\end{figure}
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@ -1496,7 +1496,8 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
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\cite[Code: 204.33.484]{mackay_enc}}
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\begin{itemize}
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\item For large $k$, the average error asymptotically approaches a minimum, non-zero value
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\item With increasing iterations, the average error asymptotically
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approaches a minimum, non-zero value
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\end{itemize}
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\end{frame}
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@ -1513,6 +1514,7 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
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grid=both,
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xlabel={$n$}, ylabel={time per frame (s)},
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legend style={at={(0.05,0.77)},anchor=south west},
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legend cell align={left},
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]
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\addplot[RedOrange, only marks, mark=*]
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table [col sep=comma, x=n, y=spf]
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@ -1522,7 +1524,7 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
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\addplot[RoyalPurple, only marks, mark=triangle*]
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table [col sep=comma, x=n, y=spf]
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{res/hybrid/fps_vs_n.csv};
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\addlegendentry{hybrid prox \& ML ($\SI{12}{\bit}$)}
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\addlegendentry{improved ($\SI{12}{\bit}$)}
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\end{axis}
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\end{tikzpicture}
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@ -1551,13 +1553,12 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{c}}_n\text{ with lowest }d
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\item Error coding performance (BER, FER, decoding failures)
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\item Computational performance ($\mathcal{O}\left( n \right) $ time complexity,
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fast implementation possible)
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\item Number of iterations required independant of SNR
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\item Operation during iteration (oscillation of estimate)
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\item Number of iterations independent of SNR
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\end{itemize}
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\item Suggestion for improvement of proximal decoding:
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\begin{itemize}
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\item Addidion of ``ML-on-list'' step
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\item $\sim\SI{1}{dB}$ gain under certain conditions
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\item Addition of ``ML-on-list'' step
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\item Up to $\sim\SI{1}{dB}$ gain under certain conditions
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\end{itemize}
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\end{itemize}
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\end{frame}
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@ -1,4 +1,4 @@
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\section{Forthcoming Examinations}%
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\section{Forthcoming Analysis}%
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\label{sec:Forthcoming Examinations}
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@ -10,7 +10,7 @@
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\frametitle{Forthcoming Examinations}
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\begin{itemize}
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\item Test the (Alternating Direction Method of Multipliers) ADMM
|
||||
\item Test ADMM (Alternating Direction Method of Multipliers)
|
||||
as an optimization method for LP Decoding
|
||||
\begin{itemize}
|
||||
\item In LP decoding, the ML decoding problem is reduced to a linear program,
|
||||
@ -23,8 +23,8 @@
|
||||
\end{itemize}
|
||||
\item Compare ADMM implementation with Proximal Decoding implementation with respect to
|
||||
\begin{itemize}
|
||||
\item Decoding performance (BER, FER)
|
||||
\item Computational performance (time complexity, actual seconds per frame)
|
||||
\item decoding performance (BER, FER)
|
||||
\item computational performance (time complexity, actual seconds per frame)
|
||||
\end{itemize}
|
||||
\end{itemize}
|
||||
|
||||
|
||||
@ -23,7 +23,7 @@
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{frame}[t]
|
||||
\frametitle{Previous work}
|
||||
\frametitle{Previous Work}
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
|
||||
@ -47,8 +47,8 @@
|
||||
\end{figure}
|
||||
|
||||
\begin{itemize}
|
||||
\item Examination of ``Proximal Decoding''
|
||||
\item Examination of ``Interior Point Decoding''
|
||||
\item Analysis of ``Proximal Decoding''
|
||||
\item Analysis of ``Interior Point Decoding''
|
||||
\end{itemize}
|
||||
\end{frame}
|
||||
|
||||
@ -89,12 +89,13 @@
|
||||
\end{figure}
|
||||
|
||||
\begin{itemize}
|
||||
\item All simulations are performed with BPSK modulation:
|
||||
\item All simulations are performed with BPSK:
|
||||
\begin{align*}
|
||||
\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*}
|
||||
\item The used channel model is AWGN:
|
||||
\item The channel model is AWGN:
|
||||
\begin{align*}
|
||||
\boldsymbol{y} = \boldsymbol{x} + \boldsymbol{z},
|
||||
\hspace{5mm}\boldsymbol{z}\sim \mathcal{N}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user