Reorganized proximal analysis results and continued writing

latex/thesis/chapters/proximal_decoding.tex

@@ -322,8 +322,21 @@ $[-\eta, \eta]$ individually.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Simulation Results}%
+\section{Analysis and Simulation Results}%
-\label{sec:prox:Simulation Results}
+\label{sec:prox:Analysis and Simulation Results}
+
+In this section, the general behaviour of the proximal decoding algorithm is
+analyzed.
+The impact of the parameters $\gamma$, as well as $\omega$, $K$ and $\eta$ is
+examined.
+The decoding performance is assessed on the basis of the \ac{BER} and the
+\ac{FER} as well as the \textit{decoding failure rate} - the rate at which
+the algorithm produces erroneous results.
+The convergence properties are reviewed and related to the decoding
+performance.
+Finally, the computational performance is examined on a theoretical basis
+as well as on the basis of the implementation completed in the context of this
+work.
 
 All simulation results presented hereafter are based on Monte Carlo
 simulations.
@@ -333,9 +346,18 @@ stated.
 \todo{Mention number of datapoints from which each graph was created for
 non ber and fer curves}
 
+
+\subsection{Choice of Parameters}
+
+First, the effect of the parameter $\gamma$ is investigated.
 Figure \ref{fig:prox:results} shows a comparison of the decoding performance
 of the proximal decoding algorithm as presented by Wadayama et al. in
 \cite{proximal_paper} and the implementation realized for this work.
+\noindent The \ac{BER} curves for three different choices of the parameter
+$\gamma$ are shown, as well as the \ac{BER} curve resulting from decoding
+using \ac{BP}, as a reference.
+The results from Wadayama et al. are shown with solid lines,
+while the newly generated ones are shown with dashed lines.
 
 \begin{figure}[H]
     \centering
@@ -344,9 +366,9 @@ of the proximal decoding algorithm as presented by Wadayama et al. in
         \begin{axis}[grid=both, grid style={line width=.1pt},
                      xlabel={$E_b / N_0$ (dB)}, ylabel={BER},
                      ymode=log,
-                     legend style={at={(0.5,-0.55)},anchor=south},
+                     legend style={at={(0.5,-0.7)},anchor=south},
-                     width=0.75\textwidth,
+                     width=0.6\textwidth,
-                     height=0.5625\textwidth,
+                     height=0.45\textwidth,
                      ymax=1.2, ymin=0.8e-4,
                      xtick={1, 2, ..., 5},
                      xmin=0.9, xmax=5.6,
@@ -388,24 +410,37 @@ of the proximal decoding algorithm as presented by Wadayama et al. in
         \end{axis}
     \end{tikzpicture}
     
-    \caption{Simulation results\protect\footnotemark{} for $\omega = 0.05, K=100$}
+    \caption{Comparison of datapoints from Wadayama et al. with own simulation results%
+        \protect\footnotemark{}}
     \label{fig:prox:results}
 \end{figure}
 %
-\footnotetext{(3,6) regular LDPC code with n = 204, k = 102 \cite[204.33.484]{mackay_enc}}%
+\footnotetext{(3,6) regular LDPC code with $n = 204$, $k = 102$
+    \cite[\text{204.33.484}]{mackay_enc}; $\omega = 0.05, K=200, \eta=1.5$
+}%
 %
-
+\noindent It is noticeable that for a moderately chosen value of $\gamma$
-Looking at the graph in figure \ref{fig:prox:results} one might notice that for
+($\gamma = 0.05$) the decoding performance is better than for low
-a moderately chosen value of $\gamma$ ($\gamma = 0.05$) the decoding
+($\gamma = 0.01$) or high ($\gamma = 0.15$) values.
-performance is better than for low ($\gamma = 0.01$) or high
-($\gamma = 0.15$) values.
 The question arises if there is some optimal value maximazing the decoding
 performance, especially since the decoding performance seems to dramatically
 depend on $\gamma$.
 To better understand how $\gamma$ and the decoding performance are
 related, figure \ref{fig:prox:results} was recreated, but with a considerably
-larger selection of values for $\gamma$ (figure \ref{fig:prox:results_3d}).%
+larger selection of values for $\gamma$.
-%
+In this new graph, shown in figure \ref{fig:prox:results_3d}, instead of
+stacking the \ac{BER} curves on top of one another in the same plot, the
+visualization is extended to three dimensions.
+The previously shown results are highlighted.
+
+Evidently, while the decoding performance does depend on the value of
+$\gamma$, there is no single optimal value offering optimal performance, but
+rather a certain interval in which it stays largely unchanged.
+When examining a number of different codes (figure
+\ref{fig:prox:results_3d_multiple}), it is apparent that while the exact
+landscape of the graph depends on the code, the general behaviour is the same
+in each case.
+
 \begin{figure}[H]
     \centering
 
@@ -416,8 +451,8 @@ larger selection of values for $\gamma$ (figure \ref{fig:prox:results_3d}).%
                      xlabel={$E_b / N_0$ (dB)},
                      ylabel={$\gamma$},
                      zlabel={BER},
-                     legend pos=outer north east,
+                     %legend pos=outer north east,
-                     %legend style={at={(0.5,-0.55)},anchor=south},
+                     legend style={at={(0.5,-0.7)},anchor=south},
                      ytick={0, 0.05, 0.1, 0.15},
                      width=0.6\textwidth,
                      height=0.45\textwidth,]
@@ -446,19 +481,21 @@ larger selection of values for $\gamma$ (figure \ref{fig:prox:results_3d}).%
         \end{axis}
     \end{tikzpicture}
 
-    \caption{BER\protect\footnotemark{} for $\omega = 0.05, K=100$}
+    \caption{Visualization of relationship between the decoding performance\protect\footnotemark{}
+        and the parameter $\gamma$}
     \label{fig:prox:results_3d}
 \end{figure}%
 %
-\footnotetext{(3,6) regular LDPC code with n = 204, k = 102 \cite[\text{204.33.484}]{mackay_enc}}%
+\footnotetext{(3,6) regular LDPC code with n = 204, k = 102 \cite[\text{204.33.484}]{mackay_enc};
+    $\omega = 0.05, K=200, \eta=1.5$
+}%
 %
-\noindent Evidently, while the performance does depend on the value of
+\noindent This indicates \todo{This is a result fit for the conclusion}
-$\gamma$, there is no single optimal value offering optimal performance, but
+that while the choice of the parameter $\gamma$ significantly
-rather a certain interval in which the performance stays largely the same.
+affects the decoding performance, there is not much benefit attainable in
-When examining a number of different codes (figure
+undertaking an extensive search for an exact optimum.
-\ref{fig:prox:results_3d_multiple}), it is apparent that while the exact
+Rather, a preliminary examination providing a rough window for $\gamma$ may
-landscape of the graph depends on the code, the general behaviour is the same
+be sufficient.
-in each case.
 
 \begin{figure}[H]
     \centering
@@ -688,12 +725,105 @@ in each case.
 A similar analysis was performed to determine the optimal values for the other
 parameters, $\omega$, $K$ and $\eta$.
 
-TODO
+Changing the parameter $\eta$ does not appear to have a significant effect on
+the decoding performance, which seems sensible considering its only purpose
+is ensuring numerical stability.
+
+\subsection{Decoding Performance}
+
+\begin{figure}[H]
+    \centering
+
+    \begin{tikzpicture}
+        \begin{axis}[grid=both,
+                     xlabel={$E_b / N_0$ (dB)}, ylabel={BER},
+                     ymode=log,
+                     width=0.48\textwidth,
+                     height=0.36\textwidth,
+                     legend style={at={(0.05,0.05)},anchor=south west},
+                     ymax=1.5, ymin=3e-7,]
+
+            \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}%
+    \hfill%
+    \begin{tikzpicture}
+        \begin{axis}[grid=both,
+                     xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
+                     ymode=log,
+                     width=0.48\textwidth,
+                     height=0.36\textwidth,
+                     legend style={at={(0.05,0.05)},anchor=south west},
+                     ymax=1.5, ymin=3e-7,]
+
+            \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}\\[1em]
+    \begin{tikzpicture}
+        \begin{axis}[grid=both,
+                     xlabel={$E_b / N_0$ (dB)}, ylabel={Decoding Failure Rate},
+                     ymode=log,
+                     width=0.48\textwidth,
+                     height=0.36\textwidth,
+                     legend style={at={(0.05,0.05)},anchor=south west},
+                     ymax=1.5, ymin=3e-7,]
+
+            \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{Cmoparison\protect\footnotemark{} of \ac{FER}, \ac{BER} and
+        decoding failure rate; $\omega = 0.05, K=100$}
+    \label{fig:prox:ber_fer_dfr}
+\end{figure}%
+%
+\footnotetext{(3,6) regular LDPC code with n = 204, k = 102 \cite[\text{204.33.484}]{mackay_enc}}%
+%
 
 Until now, only the \ac{BER} has been considered to assess the decoding
 performance.
 The \ac{FER}, however, shows considerably worse performance, as can be seen in
-figure \ref{TODO}.
+figure \ref{fig:prox:ber_fer_dfr}.
+Besides the \ac{BER} and \ac{FER} curves, the figure also shows the
+\textit{decoding failure rate}.
+This is the rate at which the iterative process produces erroneous codewords,
+i.e., the stopping criterion (line 6 of algorithm \ref{TODO}) is never
+satisfied and the maximum number of itertations $K$ is reached without
+converging to a valid codeword.
+
 One possible explanation might be found in the structure of the proxmal
 decoding algorithm \ref{TODO} itself.
 As it comprises two separate steps, one responsible for addressing the
@@ -743,6 +873,8 @@ This course of thought will be picked up in section
         \end{itemize}
 \end{itemize}
 
+\subsection{Convergence Properties}
+\subsection{Computational Performance}
 
 
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