From f408b139b7236e4447e119d7941a10d3a5db8841 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Thu, 28 Dec 2023 21:31:45 +0100 Subject: [PATCH] Remove part of Conclusion; Limit lines to 80 cols; Lessen figure legend spacing --- letter.tex | 32 ++++++++++++++------------------ 1 file changed, 14 insertions(+), 18 deletions(-) diff --git a/letter.tex b/letter.tex index fa21c04..9c5c473 100644 --- a/letter.tex +++ b/letter.tex @@ -265,8 +265,8 @@ function \cite{proximal_paper} The objective function is minimized using the proximal gradient method, which amounts to iteratively performing two gradient-descent steps \cite{proximal_paper} with the given objective function and considering AWGN channels. -To this end, two helper variables, $\boldsymbol{r}$ and $\boldsymbol{s}$, are introduced, -describing the result of each of the two steps: +To this end, two helper variables, $\boldsymbol{r}$ and $\boldsymbol{s}$, are +introduced, describing the result of each of the two steps: % \begin{alignat}{3} \boldsymbol{r} &\leftarrow \boldsymbol{s} @@ -285,7 +285,8 @@ stages of the decoding process. As the gradient of the code-constraint polynomial can attain very large values in some cases, an additional step is introduced to ensure numerical stability: -every current estimate $\boldsymbol{s}$ is projected onto $\left[-\eta, \eta\right]^n$ by a projection +every current estimate $\boldsymbol{s}$ is projected onto +$\left[-\eta, \eta\right]^n$ by a projection $\Pi_\eta : \mathbb{R}^n \rightarrow \left[-\eta, \eta\right]^n$, where $\eta$ is a positive constant slightly larger than one, e.g., $\eta = 1.5$. The resulting decoding process as described in \cite{proximal_paper} is @@ -578,7 +579,8 @@ oscillate after a certain number of iterations.% Considering the magnitude of oscillation of the gradient of the code constraint polynomial, some interesting behavior may be observed. Figure \ref{fig:p_error} shows the probability that a component of the estimate -is wrong, determined through a Monte Carlo simulation, when the components of $\boldsymbol{c}$ are ordered from smallest to largest oscillation of +is wrong, determined through a Monte Carlo simulation, when the components of +$\boldsymbol{c}$ are ordered from smallest to largest oscillation of $\left(\nabla h\right)_i$. The lower the magnitude of the oscillation, the higher the probability that the @@ -666,15 +668,15 @@ generated and an ``ML-in-the-list'' step is performed. %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Simulation Results \& Discussion} -Figure \ref{fig:results} shows the FER and BER resulting from applying proximal -decoding as presented in \cite{proximal_paper} and the improved algorithm -presented here when applied to a $\left( 3,6 \right)$-regular LDPC code with $n=204$ and -$k=102$ \cite[204.33.484]{mackay}. +Figure \ref{fig:results} shows the FER and BER resulting from applying +proximal decoding as presented in \cite{proximal_paper} and the improved +algorithm presented here when applied to a $\left( 3,6 \right)$-regular LDPC +code with $n=204$ and $k=102$ \cite[204.33.484]{mackay}. The parameters chosen for the simulation are $\gamma = 0.05, \omega=0.05, \eta=1.5, K=200$. Again, these parameters were chosen,% % -\begin{figure}[H] +\begin{figure}[ht] \centering \begin{tikzpicture} @@ -703,7 +705,7 @@ Again, these parameters were chosen,% legend columns=2, legend style={draw=white!15!black, legend cell align=left, - at={(0.5,-0.5)},anchor=south} + at={(0.5,-0.44)},anchor=south} ] \addplot+[ProxPlot, scol1] @@ -772,17 +774,11 @@ Wadayama et al. \cite{proximal_paper} is introduced for AWGN channels. It relies on the fact that most errors observed in proximal decoding stem from only a few components of the estimate being wrong. These few erroneous components can mostly be corrected by appending an -additional step to the original algorithm that is only executed if the algorithm has not converged. +additional step to the original algorithm that is only executed if the +algorithm has not converged. A gain of up to $\sim\SI{1}{dB}$ can be observed, depending on the code, the parameters considered, and the SNR. -While this work serves to introduce an approach to improve proximal decoding -by appending an ``ML-in-the-list'' step, the method used to detect the most -probably wrong components of the estimate is based mainly on empirical -observation and a more mathematically rigorous foundation for determining these -components could be beneficial. - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Acknowledgements}