From f433607ce632df57217e342f3a526ca643375e20 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Wed, 25 Jan 2023 22:01:01 +0100 Subject: [PATCH] First round of corrections --- latex/presentations/midterm/presentation.tex | 2 +- .../midterm/sections/decoding_algorithms.tex | 6 ++-- .../midterm/sections/examination_results.tex | 24 +++++++------- .../sections/forthcoming_examination.tex | 6 ++-- .../sections/theoretical_background.tex | 31 +++++++++---------- 5 files changed, 34 insertions(+), 35 deletions(-) diff --git a/latex/presentations/midterm/presentation.tex b/latex/presentations/midterm/presentation.tex index 6090b66..9291535 100644 --- a/latex/presentations/midterm/presentation.tex +++ b/latex/presentations/midterm/presentation.tex @@ -121,7 +121,7 @@ \title{Application of Optimization Algorithms for Channel Decoding} -\subtitle{\small Midterm Presentation} +\subtitle{\small Midterm Presentation - 27.01.2023} %\author{Andreas Tsouchlos} \author{\vspace{1.5mm} Andreas Tsouchlos} diff --git a/latex/presentations/midterm/sections/decoding_algorithms.tex b/latex/presentations/midterm/sections/decoding_algorithms.tex index 1b5d87d..2d3e823 100644 --- a/latex/presentations/midterm/sections/decoding_algorithms.tex +++ b/latex/presentations/midterm/sections/decoding_algorithms.tex @@ -116,7 +116,7 @@ Output $\boldsymbol{\hat{x}}$ \begin{minipage}[c]{0.6\linewidth} \begin{itemize} - \item Codeword Polytope: + \item Codeword polytope: \begin{align*} \text{poly}\left( \mathcal{C} \right) = \left\{ @@ -126,13 +126,13 @@ Output $\boldsymbol{\hat{x}}$ \right\}, \hspace{5mm} \lambda_{\boldsymbol{c}} \in \mathbb{R} \end{align*} - \item Cost Function: + \item Cost function: \begin{align*} \sum_{i=1}^{n} \gamma_i c_i, \hspace{5mm}\gamma_i = \log\left( \frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right) \end{align*} - \item LP Formulation of ML Decoding: + \item LP formulation of ML decoding: \begin{align*} &\text{minimize } \sum_{i=1}^{n} \gamma_i f_i \\ &\text{subject to } \boldsymbol{f}\in\text{poly}\left( \mathcal{C} \right) diff --git a/latex/presentations/midterm/sections/examination_results.tex b/latex/presentations/midterm/sections/examination_results.tex index 577197b..bf10dcc 100644 --- a/latex/presentations/midterm/sections/examination_results.tex +++ b/latex/presentations/midterm/sections/examination_results.tex @@ -67,8 +67,8 @@ \end{figure} \item $\mathcal{O}\left(n \right) $ time complexity - same as BP; - Only multiplication and addition necessary \cite{proximal_paper} - \item Measured Performance: $\sim\SI{10000}{frames / \second}$ + only multiplication and addition necessary \cite{proximal_paper} + \item Measured Performance: $\sim\SI{10000}{}$ frames/s - Intel Core i7-7700HQ @ 2.80GHz; $n=204$ \end{itemize} \vspace{3mm} @@ -81,8 +81,8 @@ \setcounter{footnote}{0} \begin{itemize} - \item Comparison of simulation - \footnote{(3,6) regular LDPC Code with $n=204, k=102$ + \item Analysis of simulation + \footnote{(3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay_enc}} results for different values of $\gamma$ \end{itemize} @@ -367,8 +367,8 @@ \setcounter{footnote}{0} \begin{itemize} - \item Comparison of simulated - \footnote{(3,6) regular LDPC Code with $n=204, k=102$ + \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} @@ -660,8 +660,8 @@ Output $\boldsymbol{\hat{x}}$ \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 + \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} @@ -721,12 +721,12 @@ Output $\boldsymbol{\hat{x}}$ \end{axis} \end{tikzpicture} - \caption{Corellation between bit error and amplitude of oscillation} + \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 + \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} @@ -1364,11 +1364,11 @@ $\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{x}}_n\text{ with lowest }d \end{axis} \end{tikzpicture} - \caption{Average error for $\SI{500000}{decodings}, + \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 $n=204, k=102$ + \footnotetext{Simulation performed with (3,6) regular LDPC code with $n=204, k=102$ \cite[Code: 204.33.484]{mackay_enc}} \begin{itemize} diff --git a/latex/presentations/midterm/sections/forthcoming_examination.tex b/latex/presentations/midterm/sections/forthcoming_examination.tex index a730667..326a053 100644 --- a/latex/presentations/midterm/sections/forthcoming_examination.tex +++ b/latex/presentations/midterm/sections/forthcoming_examination.tex @@ -1,5 +1,5 @@ -\section{Forthcoming Examination}% -\label{sec:Forthcoming Examination} +\section{Forthcoming Examinations}% +\label{sec:Forthcoming Examinations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -7,7 +7,7 @@ \label{sub:LP Decoding} \begin{frame}[t] - \frametitle{Forthcoming Examination: LP Decoding} + \frametitle{Forthcoming Examinations: LP Decoding} \begin{itemize} \item Test the (Alternating Direction Method of Multipliers) ADMM diff --git a/latex/presentations/midterm/sections/theoretical_background.tex b/latex/presentations/midterm/sections/theoretical_background.tex index 415e1b9..3b9c8e7 100644 --- a/latex/presentations/midterm/sections/theoretical_background.tex +++ b/latex/presentations/midterm/sections/theoretical_background.tex @@ -11,8 +11,8 @@ \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 iterative message–passing algorithms preffered in practice do not guarantee - optimality and may fail to decode correctly when the graph contains cycles + \item The iterative message–passing algorithms preferred 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} @@ -48,7 +48,7 @@ \begin{itemize} \item Examination of ``Proximal Decoding'' - \item Examination of ``Iterative Point Decoding'' + \item Examination of ``Interior Point Decoding'' \end{itemize} \end{frame} @@ -66,14 +66,14 @@ \centering \begin{tikzpicture}[scale=1, transform shape] - \node (in) {$c\left[ k \right] $}; + \node (in) {$\boldsymbol{c}$}; \node[mapper, right=0.5cm of in] (bpskmap) {Mapper}; \node[right=1.5cm of bpskmap, draw, circle, inner sep=0pt, minimum size=0.5cm] (add) {$+$}; - \node[right=0.5cm of add] (out) {$y\left[ k \right] $}; - \node[below=0.5cm of add] (noise) {$n\left[ k \right] $}; + \node[right=0.5cm of add] (out) {$\boldsymbol{y}$}; + \node[below=0.5cm of add] (noise) {$\boldsymbol{z}$}; - \node at ($(bpskmap.east)!0.5!(add.west) + (0,0.3cm)$) {$x\left[ k \right] $}; + \node at ($(bpskmap.east)!0.5!(add.west) + (0,0.3cm)$) {$\boldsymbol{x}$}; \draw[->] (in) -- (bpskmap); \draw[->] (bpskmap) -- (add); @@ -83,22 +83,21 @@ \end{figure} \begin{itemize} - \item All simulations are performed with BPSK Modulation: + \item All simulations are performed with BPSK modulation: \begin{align*} - x\left[ k \right] = \left( -1 \right)^{c\left[ k \right] }, - \hspace{5mm} \boldsymbol{c} \in \mathbb{F}_2^n, - \hspace{2mm} k\in \left\{ 1, \ldots, n \right\} + \boldsymbol{x} = \left( -1 \right)^{\boldsymbol{c}}, + \hspace{5mm} \boldsymbol{c} \in \mathbb{F}_2^n \end{align*} \item The used channel model is AWGN: \begin{align*} - \boldsymbol{y} = \boldsymbol{x} + \boldsymbol{n}, - \hspace{5mm}\boldsymbol{n}\sim \mathcal{N} + \boldsymbol{y} = \boldsymbol{x} + \boldsymbol{z}, + \hspace{5mm}\boldsymbol{z}\sim \mathcal{N} \left(0,\frac{1}{2}\left(\frac{k}{n}\frac{E_b}{N_0}\right)^{-1}\right), - \hspace{2mm} \boldsymbol{y}, \boldsymbol{n} \in \mathbb{R}^n + \hspace{2mm} \boldsymbol{y}, \boldsymbol{z} \in \mathbb{R}^n \end{align*} \item All-zeros assumption: \begin{align*} - \boldsymbol{c} = 0 + \boldsymbol{c} = \boldsymbol{0} \end{align*} \end{itemize} \end{frame} @@ -113,7 +112,7 @@ \begin{minipage}[c]{0.6\linewidth} \begin{itemize} - \item Reormulate decoding problem as optimization problem + \item Reformulate decoding problem as optimization problem \begin{itemize} \item Establish objective function \item Establish constraints