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.gitignore
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.gitignore
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build/
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.latexmkrc
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.latexmkrc
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$pdflatex="pdflatex -shell-escape -interaction=nonstopmode -synctex=1 %O %S";
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$out_dir = 'build';
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common.tex
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common.tex
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% Author: Andreas Tsouchlos
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%
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% Collection of useful commands and definitions
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%
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% ||====================================================================||
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% || WARNING ||
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% ||====================================================================||
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% || ||
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% || The following packages have to be included before using this file: ||
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% || amsmath ||
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% || pgfplots ||
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% || caption ||
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% || listings ||
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% ||====================================================================||
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Math Symbols %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\DeclareMathOperator*{\argmin}{\arg\!\min}
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\DeclareMathOperator*{\argmax}{\arg\!\max}
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\DeclareMathOperator\sign{sign}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Graphics & Plotting %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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% Filters for Pgfplots
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% Source: https://tex.stackexchange.com/a/58563 (modified)
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%
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\pgfplotsset{
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discard if/.style 2 args={
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x filter/.append code={
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\edef\tempa{\thisrow{#1}}
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\edef\tempb{#2}
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\ifx\tempa\tempb
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\def\pgfmathresult{inf}
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\fi
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}
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},
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discard if not/.style 2 args={
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x filter/.append code={
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\edef\tempa{\thisrow{#1}}
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\edef\tempb{#2}
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\ifx\tempa\tempb
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\else
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\def\pgfmathresult{inf}
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\fi
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}
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},
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discard if gt/.style 2 args={
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x filter/.append code={
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\edef\tempa{\thisrow{#1}}
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\edef\tempb{#2}
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\ifdim\tempa pt > \tempb pt
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\def\pgfmathresult{inf}
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\fi
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}
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},
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discard if lt/.style 2 args={
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x filter/.append code={
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\edef\tempa{\thisrow{#1}}
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\edef\tempb{#2}
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\ifdim\tempa pt < \tempb pt
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\def\pgfmathresult{inf}
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\fi
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}
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}
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}
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%
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% Styles
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%
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\pgfplotsset{
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fancy marks/.style={
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% TODO
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}
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}
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%
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% Colors
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%
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\definecolor{kit-green100}{rgb}{0,.59,.51}
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\definecolor{kit-green70}{rgb}{.3,.71,.65}
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\definecolor{kit-green50}{rgb}{.50,.79,.75}
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\definecolor{kit-green30}{rgb}{.69,.87,.85}
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\definecolor{kit-green15}{rgb}{.85,.93,.93}
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\definecolor{KITgreen}{rgb}{0,.59,.51}
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\definecolor{KITpalegreen}{RGB}{130,190,60}
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\colorlet{kit-maigreen100}{KITpalegreen}
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\colorlet{kit-maigreen70}{KITpalegreen!70}
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\colorlet{kit-maigreen50}{KITpalegreen!50}
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\colorlet{kit-maigreen30}{KITpalegreen!30}
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\colorlet{kit-maigreen15}{KITpalegreen!15}
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\definecolor{KITblue}{rgb}{.27,.39,.66}
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\definecolor{kit-blue100}{rgb}{.27,.39,.67}
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\definecolor{kit-blue70}{rgb}{.49,.57,.76}
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\definecolor{kit-blue50}{rgb}{.64,.69,.83}
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\definecolor{kit-blue30}{rgb}{.78,.82,.9}
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\definecolor{kit-blue15}{rgb}{.89,.91,.95}
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\definecolor{KITyellow}{rgb}{.98,.89,0}
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\definecolor{kit-yellow100}{cmyk}{0,.05,1,0}
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\definecolor{kit-yellow70}{cmyk}{0,.035,.7,0}
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\definecolor{kit-yellow50}{cmyk}{0,.025,.5,0}
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\definecolor{kit-yellow30}{cmyk}{0,.015,.3,0}
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\definecolor{kit-yellow15}{cmyk}{0,.0075,.15,0}
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\definecolor{KITorange}{rgb}{.87,.60,.10}
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\definecolor{kit-orange100}{cmyk}{0,.45,1,0}
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\definecolor{kit-orange70}{cmyk}{0,.315,.7,0}
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\definecolor{kit-orange50}{cmyk}{0,.225,.5,0}
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\definecolor{kit-orange30}{cmyk}{0,.135,.3,0}
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\definecolor{kit-orange15}{cmyk}{0,.0675,.15,0}
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\definecolor{KITred}{rgb}{.63,.13,.13}
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\definecolor{kit-red100}{cmyk}{.25,1,1,0}
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\definecolor{kit-red70}{cmyk}{.175,.7,.7,0}
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\definecolor{kit-red50}{cmyk}{.125,.5,.5,0}
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\definecolor{kit-red30}{cmyk}{.075,.3,.3,0}
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\definecolor{kit-red15}{cmyk}{.0375,.15,.15,0}
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\definecolor{KITpurple}{RGB}{160,0,120}
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\colorlet{kit-purple100}{KITpurple}
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\colorlet{kit-purple70}{KITpurple!70}
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\colorlet{kit-purple50}{KITpurple!50}
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\colorlet{kit-purple30}{KITpurple!30}
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\colorlet{kit-purple15}{KITpurple!15}
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\definecolor{KITcyanblue}{RGB}{80,170,230}
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\colorlet{kit-cyanblue100}{KITcyanblue}
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\colorlet{kit-cyanblue70}{KITcyanblue!70}
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\colorlet{kit-cyanblue50}{KITcyanblue!50}
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\colorlet{kit-cyanblue30}{KITcyanblue!30}
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\colorlet{kit-cyanblue15}{KITcyanblue!15}
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%
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% Color Schemes
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%
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% Define colormaps
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\pgfplotsset{
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colormap={mako}{
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rgb=(0.18195582, 0.11955283, 0.23136943)
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rgb=(0.25307401, 0.23772973, 0.48316271)
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rgb=(0.21607792, 0.39736958, 0.61948028)
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rgb=(0.20344718, 0.56074869, 0.65649508)
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rgb=(0.25187832, 0.71827158, 0.67872193)
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rgb=(0.54578602, 0.8544913, 0.69848331)
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},
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colormap={rocket}{
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rgb=(0.20973515, 0.09747934, 0.24238489)
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rgb=(0.43860848, 0.12177004, 0.34119475)
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rgb=(0.67824099, 0.09192342, 0.3504148)
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rgb=(0.8833417, 0.19830556, 0.26014181)
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rgb=(0.95381595, 0.46373781, 0.31769923)
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rgb=(0.96516917, 0.70776351, 0.5606593)
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},
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colormap={cividis}{
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rgb=(0.130669, 0.231458, 0.43284)
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rgb=(0.298421, 0.332247, 0.423973)
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rgb=(0.42512, 0.431334, 0.447692)
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rgb=(0.555393, 0.537807, 0.471147)
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rgb=(0.695985, 0.648334, 0.440072)
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rgb=(0.849223, 0.771947, 0.359729)
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},
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colormap={cel}{
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color=(KITred!90!black);
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color=(kit-blue100);
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color=(kit-green70);
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color=(kit-yellow70!80!kit-orange70);
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},
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}
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% Define cycle lists
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\pgfplotscreateplotcyclelist{mako}{%
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[samples of colormap={4} of mako]%
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}
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\pgfplotscreateplotcyclelist{rocket}{%
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[samples of colormap={4} of rocket]%
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}
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\pgfplotscreateplotcyclelist{cividis}{%
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[samples of colormap={4} of cividis]%
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}
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\pgfplotscreateplotcyclelist{viridis}{%
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[samples of colormap={4} of viridis]%
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}
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\pgfplotscreateplotcyclelist{cel}{%
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[samples of colormap={4} of cel]%
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}
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% Define individual colors
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\makeatletter
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\def\extractcolormapcolor#1#2{%
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\expandafter\pgfplotscolormapaccess\expandafter[\pgfplotspointmetatransformedrange]%
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[1.0]%
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{#2}%
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{\pgfkeysvalueof{/pgfplots/colormap name}}%
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\def\pgfplots@loc@TMPb{\pgfutil@definecolor{#1}{\csname pgfpl@cm@\pgfkeysvalueof{/pgfplots/colormap name}@colspace\endcsname}}%
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\expandafter\pgfplots@loc@TMPb\expandafter{\pgfmathresult}%
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}%
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\def\getcolorbyvalue#1{
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\csname pgfpl@cm@\pgfkeysvalueof{/pgfplots/colormap name}@colspace\endcsname
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}
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\makeatother
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\def\setschemecolors{
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\extractcolormapcolor{scol0}{0}
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\extractcolormapcolor{scol1}{333}
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\extractcolormapcolor{scol2}{666}
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\extractcolormapcolor{scol3}{1000}
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}
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% Define color schemes
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\pgfplotsset{
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/pgfplots/colorscheme/mako/.style={
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colormap name={mako},
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cycle list name={mako},
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/utils/exec={\setschemecolors},
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},
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/pgfplots/colorscheme/rocket/.style={
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colormap name={rocket},
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cycle list name={rocket},
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/utils/exec={\setschemecolors},
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},
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/pgfplots/colorscheme/cividis/.style={
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colormap name={cividis},
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cycle list name={cividis},
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/utils/exec={\setschemecolors},
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},
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/pgfplots/colorscheme/viridis/.style={
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colormap name={viridis},
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cycle list name={viridis},
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/utils/exec={\setschemecolors},
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},
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/pgfplots/colorscheme/cel/.style={
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colormap name={cel},
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cycle list name={cel},
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/utils/exec={\setschemecolors},
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},
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}
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814
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\documentclass[journal]{IEEEtran}
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\usepackage{amsmath,amsfonts}
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\usepackage{algorithmic}
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\usepackage{algorithm}
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\usepackage{array}
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\usepackage[caption=false,font=normalsize,labelfont=sf,textfont=sf]{subfig}
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\usepackage{textcomp}
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\usepackage{stfloats}
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\usepackage{url}
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\usepackage{verbatim}
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\usepackage{graphicx}
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\usepackage{cite}
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\usepackage[dvipsnames]{xcolor}
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\usepackage{float}
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\usepackage{tikz}
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\usepackage{pgfplots}
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\usetikzlibrary{spy, arrows.meta,arrows}
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\pgfplotsset{compat=newest}
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\usepgfplotslibrary{colormaps}
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\usepackage{listings}
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\usepackage{lipsum}
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\usepackage{dsfont}
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\usepackage{ulem}
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\usepackage{siunitx}
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\hyphenation{op-tical net-works semi-conduc-tor IEEE-Xplore}
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Inputs & Global Options
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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\input{common.tex}
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\pgfplotsset{colorscheme/cel}
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Title, Header, Footer, etc.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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\begin{document}
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\title{A Note on Improving Proximal Decoding for Linear Block Codes}
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\author{Andreas Tsouchlos, Holger Jäkel, and Laurent Schmalen\\
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Communications Engineering Lab (CEL), Karlsruhe Institute of Technology (KIT)\\
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Hertzstr. 16, 76187 Karlsruhe, Germany, Email: \texttt{\{first.last\}@kit.edu}}
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% TODO
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\markboth{Journal of \LaTeX\ Class Files,~Vol.~14, No.~8, August~2021}%
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{Shell \MakeLowercase{\textit{et al.}}: A Sample Article Using IEEEtran.cls
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for IEEE Journals}
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\maketitle
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Abstract & Index Terms
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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\begin{abstract}
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In this paper, the proximal decoding algorithm is considered within the
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context of \textit{additive white Gaussian noise} (AWGN) channels.
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An analysis of the convergence behavior of the algorithm shows that it is an
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inherent property of proximal decoding to enter an
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oscillating behavior of the estimate after a number of iterations.
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Due to this oscillation, frame errors arising during decoding can often
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be attributed to only a few remaining wrongly decoded components.
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In this paper, an improvement of the algorithm is proposed by appending an
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additional step, in which these erroneous components are attempted to be
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corrected.
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An empirical rule is suggested, with which the components most likely needing
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correction can be determined.
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Using this insight and performing a subsequent ``ML-in-the-list'' decoding,
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a gain of up to approximately 1 dB is achieved compared to proximal decoding,
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depending on the parameters chosen and the code considered.
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\end{abstract}
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\begin{IEEEkeywords}
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Optimization-based decoding, Proximal decoding, ML-in-the-list.
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\end{IEEEkeywords}
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Content
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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\IEEEPARstart{C}{hannel} coding using binary linear codes is a way of enhancing
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the reliability of data by detecting and correcting any errors that may occur
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during its transmission or storage.
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One class of binary linear codes, \textit{low-density parity-check} (LDPC)
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||||||
|
codes, has become especially popular due to its ability to reach arbitrarily
|
||||||
|
small probabilities of error at code rates up to the capacity of the channel
|
||||||
|
\cite{mackay99}, while retaining a structure that allows for very efficient
|
||||||
|
decoding.
|
||||||
|
While the established decoders for LDPC codes, such as belief propagation (BP)
|
||||||
|
and the min-sum algorithm, offer good decoding performance, they are generally
|
||||||
|
not optimal and exhibit an error floor for high
|
||||||
|
\textit{signal-to-noise ratios} (SNRs) \cite{channel_codes_book}, making them
|
||||||
|
unsuitable for applications with extreme reliability requirements.
|
||||||
|
|
||||||
|
Optimization based decoding algorithms are an entirely different way of
|
||||||
|
approaching the decoding problem.
|
||||||
|
A number of different such algorithms have been introduced.
|
||||||
|
The field of \textit{linear programming} (LP) decoding \cite{feldman_paper},
|
||||||
|
for example, represents one class of such algorithms, based on a reformulation
|
||||||
|
of the \textit{maximum likelihood} (ML) decoding problem as a linear program.
|
||||||
|
Many different optimization algorithms can be used to solve the resulting
|
||||||
|
problem \cite{interior_point_decoding, ADMM, adaptive_lp_decoding}.
|
||||||
|
Recently, proximal decoding for LDPC codes was presented by
|
||||||
|
Wadayama et al. \cite{proximal_paper}.
|
||||||
|
It is a novel approach and relies on a non-convex optimization formulation
|
||||||
|
of the \textit{maximum a posteriori} (MAP) decoding problem.
|
||||||
|
|
||||||
|
The aim of this work is to improve upon the performance of proximal decoding by
|
||||||
|
first presenting an examination of the algorithm's behavior and then suggesting
|
||||||
|
an approach to mitigate some of its flaws.
|
||||||
|
This analysis is performed within the context of
|
||||||
|
\textit{additive white Gaussian noise} (AWGN) channels.
|
||||||
|
It is first observed that, while the algorithm initially moves the estimate in
|
||||||
|
the right direction, in the final steps of the decoding process convergence to
|
||||||
|
the correct codeword is often not achieved.
|
||||||
|
Furthermore, it is suggested that the reason for this behavior is the nature
|
||||||
|
of the decoding algorithm itself, comprising two separate gradient descent
|
||||||
|
steps working adversarially.
|
||||||
|
|
||||||
|
A method to mitigate this effect is proposed by appending an additional step
|
||||||
|
to the decoding process.
|
||||||
|
In this additional step, the components of the estimate with the highest
|
||||||
|
probability of being erroneous are identified.
|
||||||
|
New codewords are then generated, over which an ``ML-in-the-list''
|
||||||
|
\cite{ml_in_the_list} decoding is performed.
|
||||||
|
A process to conduct this identification is proposed in this paper.
|
||||||
|
Using the improved algorithm, a gain of up to
|
||||||
|
approximately 1 dB can be achieved compared to proximal decoding, depending on
|
||||||
|
the parameters chosen and the code considered.
|
||||||
|
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\section{Preliminaries}
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\subsection{Notation}
|
||||||
|
|
||||||
|
When considering binary linear codes, data words are mapped onto
|
||||||
|
codewords, the lengths of which are denoted by $k \in \mathbb{N}$
|
||||||
|
and $n \in \mathbb{N}$, respectively, with $k \le n$.
|
||||||
|
The set of codewords $\mathcal{C} \subset \mathbb{F}_2^n$ of a binary linear
|
||||||
|
code can be represented using the parity-check matrix
|
||||||
|
$\boldsymbol{H} \in \mathbb{F}_2^{m \times n} $, where $m$ represents the
|
||||||
|
number of parity-checks:
|
||||||
|
%
|
||||||
|
\begin{align*}
|
||||||
|
\mathcal{C} := \left\{ \boldsymbol{c} \in \mathbb{F}_2^n :
|
||||||
|
\boldsymbol{H}\boldsymbol{c}^\text{T} = \boldsymbol{0} \right\}
|
||||||
|
\end{align*}
|
||||||
|
%
|
||||||
|
|
||||||
|
The check nodes $j \in \mathcal{J}:=\left\{1, \ldots, m\right\}$ each correspond
|
||||||
|
to a parity check, i.e., row of $\boldsymbol{H}$.
|
||||||
|
The variable nodes $i \in \mathcal{I}:=\left\{1, \ldots, n\right\}$ correspond
|
||||||
|
to the components of a codeword being subjected to a parity check, i.e., to
|
||||||
|
columns of $\boldsymbol{H}$.
|
||||||
|
The neighborhood of a parity check $j$, i.e., the set of indices of components
|
||||||
|
relevant for the according parity check, is denoted by
|
||||||
|
$N_c(j) := \left\{i \mid i \in \mathcal{I}, \boldsymbol{H}_{j,i} = 1 \right\},
|
||||||
|
\hspace{2mm} j \in \mathcal{J}$.
|
||||||
|
|
||||||
|
In order to transmit a codeword $\boldsymbol{c} \in \mathbb{F}_2^n$, it is
|
||||||
|
mapped onto a \textit{binary phase shift keying} (BPSK) symbol via
|
||||||
|
$\boldsymbol{x} = 1 - 2\boldsymbol{c}$, with
|
||||||
|
$ \boldsymbol{x} \in \left\{-1, 1\right\}^n$, which is then transmitted over an
|
||||||
|
AWGN channel.
|
||||||
|
The received vector $\boldsymbol{y} \in \mathbb{R}^n$ is decoded to obtain an
|
||||||
|
estimate of the transmitted codeword, denoted as
|
||||||
|
$\hat{\boldsymbol{c}} \in \mathbb{F}_2^n$.
|
||||||
|
A distinction is made between $\boldsymbol{x} \in \left\{-1, 1\right\}^n$
|
||||||
|
and $\tilde{\boldsymbol{x}} \in \mathbb{R}^n$,
|
||||||
|
the former denoting the BPSK symbol physically transmitted over the channel and
|
||||||
|
the latter being used as a variable during the optimization process.
|
||||||
|
The posterior probability of having transmitted $\boldsymbol{x}$ when receiving
|
||||||
|
$\boldsymbol{y}$ is expressed as a \textit{probability mass function} (PMF)
|
||||||
|
$p_{\boldsymbol{X}\mid\boldsymbol{Y}}(\boldsymbol{x} \mid \boldsymbol{y})$.
|
||||||
|
Likewise, the likelihood of receiving $\boldsymbol{y}$ upon transmitting
|
||||||
|
$\boldsymbol{x}$ is expressed as a \textit{probability density function} (PDF)
|
||||||
|
$f_{\boldsymbol{Y}\mid\boldsymbol{X}}(\boldsymbol{y} \mid \boldsymbol{x})$.
|
||||||
|
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\subsection{Proximal Decoding}
|
||||||
|
|
||||||
|
Proximal decoding was proposed by Wadayama et al. as a novel formulation
|
||||||
|
of optimization-based decoding \cite{proximal_paper}.
|
||||||
|
With proximal decoding, the proximal gradient method \cite{proximal_algorithms}
|
||||||
|
is used to solve a non-convex optimization formulation of the MAP decoding
|
||||||
|
problem.
|
||||||
|
|
||||||
|
When making the equal probability assumption for all codewords, MAP and ML
|
||||||
|
decoding are equivalent and, specifically for AWGN channels, correspond to a
|
||||||
|
nearest-neighbor decision.
|
||||||
|
For this reason, decoding can be done using a figure of merit that describes
|
||||||
|
the distance from a given vector to a codeword.
|
||||||
|
One such expression, formulated under the assumption of BPSK, is the
|
||||||
|
\textit{code-constraint polynomial} \cite{proximal_paper}
|
||||||
|
%
|
||||||
|
\begin{align*}
|
||||||
|
h\left( \tilde{\boldsymbol{x}} \right) =
|
||||||
|
\underbrace{\sum_{i=1}^{n}
|
||||||
|
\left( \tilde{x_i}^2-1 \right) ^2}_{\text{Bipolar constraint}}
|
||||||
|
+ \underbrace{\sum_{j=1}^{m} \left[
|
||||||
|
\left( \prod_{i\in N_c \left( j \right) } \tilde{x_i} \right)
|
||||||
|
-1 \right] ^2}_{\text{Parity constraint}}
|
||||||
|
.\end{align*}%
|
||||||
|
%
|
||||||
|
Its intent is to penalize vectors far from a codeword and favor those close
|
||||||
|
to one.
|
||||||
|
It comprises two terms: one representing the bipolar constraint
|
||||||
|
and one representing the parity constraint, incorporating all of the
|
||||||
|
information regarding the code.
|
||||||
|
|
||||||
|
The channel model can be considered using the negative log-likelihood
|
||||||
|
%
|
||||||
|
\begin{align*}
|
||||||
|
L \left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) = -\ln\left(
|
||||||
|
f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}\left(
|
||||||
|
\boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) \right)
|
||||||
|
.\end{align*}
|
||||||
|
%
|
||||||
|
The information about the channel and the code are consolidated in the objective
|
||||||
|
function \cite{proximal_paper}
|
||||||
|
%
|
||||||
|
\begin{align*}
|
||||||
|
g\left( \tilde{\boldsymbol{x}} \right)
|
||||||
|
= L\left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)
|
||||||
|
+ \gamma h\left( \tilde{\boldsymbol{x}} \right),
|
||||||
|
\hspace{5mm} \gamma > 0%
|
||||||
|
.\end{align*}
|
||||||
|
%
|
||||||
|
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:
|
||||||
|
%
|
||||||
|
\begin{alignat}{3}
|
||||||
|
\boldsymbol{r} &\leftarrow \boldsymbol{s}
|
||||||
|
- \omega \left( \boldsymbol{s} - \boldsymbol{y} \right)
|
||||||
|
\hspace{5mm }&&\omega > 0 \label{eq:r_update}\\
|
||||||
|
\boldsymbol{s} &\leftarrow \boldsymbol{r}
|
||||||
|
- \gamma \nabla h\left( \boldsymbol{r} \right),
|
||||||
|
\hspace{5mm} &&\gamma > 0 \label{eq:s_update}
|
||||||
|
.\end{alignat}
|
||||||
|
%
|
||||||
|
An equation for determining $\nabla h(\boldsymbol{r})$ is given in
|
||||||
|
\cite{proximal_paper}.
|
||||||
|
It should be noted that the variables $\boldsymbol{r}$ and $\boldsymbol{s}$
|
||||||
|
really represent $\tilde{\boldsymbol{x}}$ during different
|
||||||
|
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
|
||||||
|
$\Pi_\eta : \mathbb{R}^n \rightarrow \left[-\eta, \eta\right]^n$, 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
|
||||||
|
presented in algorithm \ref{alg:proximal_decoding}.
|
||||||
|
|
||||||
|
\begin{algorithm}
|
||||||
|
\caption{Proximal decoding algorithm for an AWGN channel.}
|
||||||
|
\label{alg:proximal_decoding}
|
||||||
|
|
||||||
|
\begin{algorithmic}
|
||||||
|
\STATE $\boldsymbol{s} \leftarrow \boldsymbol{0}$
|
||||||
|
\STATE \textbf{for} $K$ iterations \textbf{do}
|
||||||
|
\STATE \hspace{5mm} $\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \left( \boldsymbol{s} - \boldsymbol{y} \right) $
|
||||||
|
\STATE \hspace{5mm} $\boldsymbol{s} \leftarrow \Pi_\eta \left(\boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right) \right)$
|
||||||
|
\STATE \hspace{5mm} $\boldsymbol{\hat{c}} \leftarrow \mathds{1} \left\{ \text{sign}\left( \boldsymbol{s} \right) = -1 \right\}$
|
||||||
|
\STATE \hspace{5mm} \textbf{if} $\boldsymbol{H}\boldsymbol{\hat{c}} = \boldsymbol{0}$ \textbf{do}
|
||||||
|
\STATE \hspace{10mm} \textbf{return} $\boldsymbol{\hat{c}}$
|
||||||
|
\STATE \hspace{5mm} \textbf{end if}
|
||||||
|
\STATE \textbf{end for}
|
||||||
|
\STATE \textbf{return} $\boldsymbol{\hat{c}}$
|
||||||
|
\end{algorithmic}
|
||||||
|
\end{algorithm}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\section{Improved algorithm}
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\subsection{Analysis of Convergence Behavior}
|
||||||
|
|
||||||
|
In figure \ref{fig:fer vs ber}, the \textit{frame error rate} (FER),
|
||||||
|
\textit{bit error rate} (BER) and \textit{decoding failure rate} (DFR) of
|
||||||
|
proximal decoding are shown for an LDPC code with $n=204$ and $k=102$
|
||||||
|
\cite[204.33.484]{mackay}.
|
||||||
|
A decoding failure is defined as a decoding operation, the result of which is
|
||||||
|
not a valid codeword, i.e., as non-convergence of the algorithm.
|
||||||
|
The parameters chosen for this simulation are $\gamma=0.05, \omega=0.05,
|
||||||
|
\eta=1.5$ and $K=200$.
|
||||||
|
They were determined to offer the best performance in a preliminary examination,
|
||||||
|
where the effect of changing multiple parameters was simulated over a wide
|
||||||
|
range of values.
|
||||||
|
It is apparent that the DFR completely dominates the FER after a certain SNR.
|
||||||
|
This means that most frame errors are not due to the algorithm converging
|
||||||
|
to the wrong codeword, but due to the algorithm not converging at all.
|
||||||
|
|
||||||
|
As proximal decoding is an optimization-based decoding method, one possible
|
||||||
|
explanation for this effect might be that during the decoding process convergence
|
||||||
|
on the final codeword is often not achieved, although the estimate is moving in
|
||||||
|
the right general direction.
|
||||||
|
This would suggest that most frame errors occur due to only a few incorrectly
|
||||||
|
decoded bits.
|
||||||
|
|
||||||
|
\pgfplotsset{
|
||||||
|
FERPlot/.style={
|
||||||
|
line width=1pt,
|
||||||
|
densely dashed,
|
||||||
|
mark=triangle,
|
||||||
|
fancy marks
|
||||||
|
},
|
||||||
|
BERPlot/.style={
|
||||||
|
line width=1pt,
|
||||||
|
mark=*,
|
||||||
|
fancy marks,
|
||||||
|
},
|
||||||
|
DFRPlot/.style={
|
||||||
|
only marks,
|
||||||
|
mark=square*,
|
||||||
|
fancy marks,
|
||||||
|
}}
|
||||||
|
|
||||||
|
\begin{figure}[h]
|
||||||
|
\centering
|
||||||
|
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\begin{axis}[
|
||||||
|
grid=both,
|
||||||
|
xlabel={$E_\text{b} / N_0$ (dB)}, ylabel={},
|
||||||
|
ymode=log,
|
||||||
|
ymax=1.5, ymin=2e-7,
|
||||||
|
width=0.45\textwidth,
|
||||||
|
height=0.3375\textwidth,
|
||||||
|
legend pos = south west,
|
||||||
|
]
|
||||||
|
\addplot+[FERPlot]
|
||||||
|
table [x=SNR, y=FER, col sep=comma,
|
||||||
|
discard if not={gamma}{0.05},
|
||||||
|
discard if gt={SNR}{9}]
|
||||||
|
{res/proximal_ber_fer_dfr_20433484.csv};
|
||||||
|
\addlegendentry{FER}
|
||||||
|
\addplot+[DFRPlot]
|
||||||
|
table [x=SNR, y=DFR, col sep=comma,
|
||||||
|
discard if not={gamma}{0.05},
|
||||||
|
discard if gt={SNR}{9}]
|
||||||
|
{res/proximal_ber_fer_dfr_20433484.csv};
|
||||||
|
\addlegendentry{DFR}
|
||||||
|
\addplot+[BERPlot]
|
||||||
|
table [x=SNR, y=BER, col sep=comma,
|
||||||
|
discard if not={gamma}{0.05},
|
||||||
|
discard if gt={SNR}{7.5}]
|
||||||
|
{res/proximal_ber_fer_dfr_20433484.csv};
|
||||||
|
\addlegendentry{BER}
|
||||||
|
\end{axis}
|
||||||
|
\end{tikzpicture}
|
||||||
|
|
||||||
|
\caption{FER, DFR, and BER for $\left( 3, 6 \right)$-regular LDPC code with
|
||||||
|
$n=204, k=102$ \cite[\text{204.33.484}]{mackay}.
|
||||||
|
Parameters used for simulation: $\gamma =0.05,\omega = 0.05,
|
||||||
|
\eta = 1.5, K=200$.
|
||||||
|
}
|
||||||
|
\label{fig:fer vs ber}
|
||||||
|
\end{figure}%
|
||||||
|
%
|
||||||
|
An approach for lowering the FER might then be to append an ``ML-in-the-list''
|
||||||
|
\cite{ml_in_the_list} step to the decoding process shown in algorithm
|
||||||
|
\ref{alg:proximal_decoding}.
|
||||||
|
This step would consist of determining the $N \in \mathbb{N}$ most probably
|
||||||
|
wrong bits, finding all variations of the current estimate with those bits
|
||||||
|
modified, and performing ML decoding on this list.
|
||||||
|
|
||||||
|
This approach crucially relies on identifying the most probably wrong bits.
|
||||||
|
Therefore, the convergence properties of proximal decoding are investigated.
|
||||||
|
Considering equations (\ref{eq:s_update}) and (\ref{eq:r_update}), figure
|
||||||
|
\ref{fig:grad} shows the two gradients along which the minimization is
|
||||||
|
performed for a repetition code with $n=2$.
|
||||||
|
It is apparent that a net movement will result as long as the two gradients
|
||||||
|
have a common component.
|
||||||
|
As soon as this common component is exhausted, they will work in opposing
|
||||||
|
directions and an oscillation of the estimate will take place.
|
||||||
|
This behavior matches the conjecture that the reason for the high DFR is a
|
||||||
|
failure to converge to the correct codeword in the final steps of the
|
||||||
|
optimization process.%
|
||||||
|
%
|
||||||
|
\begin{figure}[h]
|
||||||
|
\centering
|
||||||
|
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\begin{axis}[xmin = -1.25, xmax=1.25,
|
||||||
|
ymin = -1.25, ymax=1.25,
|
||||||
|
xlabel={$\tilde{x}_1$},
|
||||||
|
ylabel={$\tilde{x}_2$},
|
||||||
|
y label style={at={(axis description cs:-0.06,0.5)},anchor=south},
|
||||||
|
width=0.45\textwidth,
|
||||||
|
height=0.3375\textwidth,
|
||||||
|
grid=major, grid style={dotted},
|
||||||
|
view={0}{90}]
|
||||||
|
\addplot3[point meta=\thisrow{grad_norm},
|
||||||
|
point meta min=1,
|
||||||
|
point meta max=2.5,
|
||||||
|
quiver={u=\thisrow{grad_0},
|
||||||
|
v=\thisrow{grad_1},
|
||||||
|
scale arrows=.05,
|
||||||
|
every arrow/.append style={%
|
||||||
|
line width=.3
|
||||||
|
+\pgfplotspointmetatransformed/1000,
|
||||||
|
-{Latex[length=0pt 5,width=0pt 3]}
|
||||||
|
},
|
||||||
|
},
|
||||||
|
quiver/colored = {mapped color},
|
||||||
|
-stealth,
|
||||||
|
]
|
||||||
|
table[col sep=comma] {res/2d_grad_L.csv};
|
||||||
|
\end{axis}
|
||||||
|
\begin{axis}[hide axis,
|
||||||
|
width=0.45\textwidth,
|
||||||
|
height=0.3375\textwidth,
|
||||||
|
xmin=10, xmax=50,
|
||||||
|
ymin=0, ymax=0.4,
|
||||||
|
legend style={draw=white!15!black,
|
||||||
|
legend cell align=left,
|
||||||
|
empty legend,
|
||||||
|
at={(0.9775,0.97)},anchor=north east}]
|
||||||
|
\addlegendimage{mark=none}
|
||||||
|
\addlegendentry{
|
||||||
|
$\nabla L\left(\boldsymbol{y}
|
||||||
|
\mid \tilde{\boldsymbol{x}}\right)$
|
||||||
|
};
|
||||||
|
\end{axis}
|
||||||
|
\end{tikzpicture}
|
||||||
|
|
||||||
|
\vspace{3mm}
|
||||||
|
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\begin{axis}[xmin = -1.25, xmax=1.25,
|
||||||
|
ymin = -1.25, ymax=1.25,
|
||||||
|
width=0.45\textwidth,
|
||||||
|
height=0.3375\textwidth,
|
||||||
|
xlabel={$\tilde{x}_1$},
|
||||||
|
ylabel={$\tilde{x}_2$},
|
||||||
|
y label style={at={(axis description cs:-0.06,0.5)},anchor=south},
|
||||||
|
grid=major, grid style={dotted},
|
||||||
|
view={0}{90}]
|
||||||
|
\addplot3[point meta=\thisrow{grad_norm},
|
||||||
|
point meta min=1,
|
||||||
|
point meta max=7,
|
||||||
|
quiver={u=\thisrow{grad_0},
|
||||||
|
v=\thisrow{grad_1},
|
||||||
|
scale arrows=.03,
|
||||||
|
every arrow/.append style={%
|
||||||
|
line width=.5
|
||||||
|
+\pgfplotspointmetatransformed/1000,
|
||||||
|
-{Latex[length=0pt 5,width=0pt 3]}
|
||||||
|
},
|
||||||
|
},
|
||||||
|
quiver/colored = {mapped color},
|
||||||
|
-stealth,
|
||||||
|
]
|
||||||
|
table[col sep=comma] {res/2d_grad_h.csv};
|
||||||
|
\end{axis}
|
||||||
|
\begin{axis}[hide axis,
|
||||||
|
width=0.45\textwidth,
|
||||||
|
height=0.3375\textwidth,
|
||||||
|
xmin=10, xmax=50,
|
||||||
|
ymin=0, ymax=0.4,
|
||||||
|
legend style={draw=white!15!black,
|
||||||
|
legend cell align=left,
|
||||||
|
empty legend,
|
||||||
|
at={(0.9775,0.97)},anchor=north east}]
|
||||||
|
\addlegendimage{mark=none}
|
||||||
|
\addlegendentry{$\nabla h\left(\tilde{\boldsymbol{x}}\right)$};
|
||||||
|
\end{axis}
|
||||||
|
\end{tikzpicture}
|
||||||
|
|
||||||
|
\caption{Gradients
|
||||||
|
$\nabla L\left(\boldsymbol{y} \mid \tilde{\boldsymbol{x}}\right)$
|
||||||
|
and $\nabla h \left( \tilde{\boldsymbol{x}} \right)$ for a repetition
|
||||||
|
code with $n=2$.
|
||||||
|
Shown for $\boldsymbol{y} = \begin{bmatrix} -0.5 & 0.8 \end{bmatrix}$.
|
||||||
|
}
|
||||||
|
\label{fig:grad}
|
||||||
|
\end{figure}%
|
||||||
|
%
|
||||||
|
In figure \ref{fig:prox:convergence_large_n}, only component
|
||||||
|
$\left(\tilde{\boldsymbol{x}}\right)_1$ of the estimate is considered during a
|
||||||
|
decoding operation for an LDPC code with $n=204$ and $k=102$.
|
||||||
|
Two qualities may be observed.
|
||||||
|
First, the average values of the two gradients are equal, except for their sign,
|
||||||
|
leading to the aforementioned oscillation.
|
||||||
|
Second, the gradient of the code constraint polynomial itself starts to
|
||||||
|
oscillate after a certain number of iterations.
|
||||||
|
|
||||||
|
\begin{figure}[ht]
|
||||||
|
\centering
|
||||||
|
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\begin{axis}[
|
||||||
|
grid=both,
|
||||||
|
xlabel={Iterations},
|
||||||
|
width=0.45\textwidth,
|
||||||
|
height=0.3375\textwidth,
|
||||||
|
xtick={0, 100, ..., 400},
|
||||||
|
xticklabels={0, 50, ..., 200},
|
||||||
|
legend pos = south east,
|
||||||
|
]
|
||||||
|
\addplot+ [mark=none, line width=1]
|
||||||
|
table [col sep=comma, x=k, y=comb_r_s_0,
|
||||||
|
discard if gt={k}{300}]
|
||||||
|
{res/extreme_components_20433484_combined.csv};
|
||||||
|
\addplot+ [mark=none, line width=1,
|
||||||
|
discard if gt={k}{300}]
|
||||||
|
table [col sep=comma, x=k, y=grad_L_0]
|
||||||
|
{res/extreme_components_20433484_combined.csv};
|
||||||
|
\addplot+ [mark=none, line width=1]
|
||||||
|
table [col sep=comma, x=k, y=grad_h_0,
|
||||||
|
discard if gt={k}{300}]
|
||||||
|
{res/extreme_components_20433484_combined.csv};
|
||||||
|
\addlegendentry{$\left(\tilde{\boldsymbol{x}}\right)_1$}
|
||||||
|
\addlegendentry{$\left(\nabla L\right)_1$}
|
||||||
|
\addlegendentry{$\left(\nabla h\right)_1$}
|
||||||
|
\end{axis}
|
||||||
|
\end{tikzpicture}
|
||||||
|
|
||||||
|
\caption{Visualization of component $\left(\tilde{\boldsymbol{x}}\right)_1$
|
||||||
|
for a decoding operation for a (3,6) regular LDPC code with
|
||||||
|
$n=204, k=102$ \cite[\text{204.33.484}]{mackay}.
|
||||||
|
Parameters used for simulation: $\gamma = 0.05, \omega = 0.05,
|
||||||
|
\eta = 1.5, E_b/N_0 = \SI{4}{dB}$.
|
||||||
|
}
|
||||||
|
\label{fig:prox:convergence_large_n}
|
||||||
|
\end{figure}%
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\subsection{Improvement using ``ML-in-the-list'' step}
|
||||||
|
|
||||||
|
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
|
||||||
|
$\left(\nabla h\right)_i$.
|
||||||
|
|
||||||
|
The lower the magnitude of the oscillation, the higher the probability that the
|
||||||
|
corresponding bit was not decoded correctly.
|
||||||
|
This means that this magnitude is a suitable figure of merit for determining
|
||||||
|
the probability that a given component was decoded incorrectly.%
|
||||||
|
%
|
||||||
|
\begin{figure}[H]
|
||||||
|
\centering
|
||||||
|
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\begin{axis}[
|
||||||
|
grid=both,
|
||||||
|
ylabel=$P(\hat{c}_{i'} \ne c_{i'})$,
|
||||||
|
xlabel=$i'$,
|
||||||
|
ymode=log,
|
||||||
|
width=0.45\textwidth,
|
||||||
|
height=0.3375\textwidth,
|
||||||
|
]
|
||||||
|
\addplot+ [scol1, mark=none, line width=1]
|
||||||
|
table [col sep=comma, y=p_error]{res/p_error.csv};
|
||||||
|
\end{axis}
|
||||||
|
\end{tikzpicture}
|
||||||
|
|
||||||
|
\caption{Probability that a component of the estimated codeword
|
||||||
|
$\hat{\boldsymbol{c}}\in \mathbb{F}_2^n$ is wrong for a (3,6) regular
|
||||||
|
LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay}.
|
||||||
|
The indices $i'$ are ordered such that the amplitude of oscillation of
|
||||||
|
$\left(\nabla h\right)_{i'}$ increases with $i'$.
|
||||||
|
Parameters used for simulation: $\gamma = 0.05, \omega = 0.05,
|
||||||
|
\eta = 1.5, E_b/N_0 = \SI{4}{dB}$.
|
||||||
|
Simulated with $\SI{100000000}{}$ iterations.}
|
||||||
|
\label{fig:p_error}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
The complete improved algorithm is depicted in algorithm \ref{alg:improved}.
|
||||||
|
First, the proximal decoding algorithm is applied.
|
||||||
|
If a valid codeword has been reached, i.e., if the algorithm has converged, this
|
||||||
|
is the solution returned.
|
||||||
|
Otherwise, $N \in \mathbb{N}$ components are selected based on the criterion
|
||||||
|
presented above.
|
||||||
|
Beginning with the recent estimate $\hat{\boldsymbol{c}} \in \mathbb{F}_2^n$,
|
||||||
|
all variations of words with the selected components modified are then
|
||||||
|
generated and an ``ML-in-the-list'' step is performed.
|
||||||
|
|
||||||
|
\begin{algorithm}
|
||||||
|
\caption{ML-in-the-List algorithm.}
|
||||||
|
\label{alg:ml-in-the-list}
|
||||||
|
|
||||||
|
\begin{algorithmic}
|
||||||
|
\STATE Find valid codewords under $\left(\hat{\boldsymbol{c}}_{l}\right)_{1=1}^{2^N}$
|
||||||
|
\STATE \textbf{if} no valid codewords exist
|
||||||
|
\STATE \hspace{5mm} Compute $\langle \hat{\boldsymbol{c}}_l, \hat{\boldsymbol{c}} \rangle$ for all variations $\boldsymbol{c}_l$
|
||||||
|
\STATE \textbf{else}
|
||||||
|
\STATE \hspace{5mm} Compute $\langle \hat{\boldsymbol{c}}_l, \hat{\boldsymbol{c}} \rangle$ for valid codewords
|
||||||
|
\STATE \textbf{end if}
|
||||||
|
\STATE \textbf{return} $\hat{\boldsymbol{c}}_l$ with highest $\langle \hat{\boldsymbol{c}}_l, \hat{\boldsymbol{c}} \rangle$
|
||||||
|
\end{algorithmic}
|
||||||
|
\end{algorithm}%
|
||||||
|
%
|
||||||
|
\begin{algorithm}
|
||||||
|
\caption{Improved proximal decoding algorithm.
|
||||||
|
}
|
||||||
|
\label{alg:improved}
|
||||||
|
|
||||||
|
\begin{algorithmic}
|
||||||
|
\STATE $\boldsymbol{s} \leftarrow \boldsymbol{0}$
|
||||||
|
\STATE \textbf{for} $K$ iterations \textbf{do}
|
||||||
|
\STATE \hspace{5mm} $\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \left( \boldsymbol{s} - \boldsymbol{y} \right) $
|
||||||
|
\STATE \hspace{5mm} $\boldsymbol{s} \leftarrow \Pi_\eta \left(\boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right) \right)$
|
||||||
|
\STATE \hspace{5mm} $\boldsymbol{\hat{c}} \leftarrow \mathds{1} \left\{ \text{sign}\left( \boldsymbol{s} \right) = -1 \right\}$
|
||||||
|
\STATE \hspace{10mm} \textbf{if} $\boldsymbol{H}\boldsymbol{\hat{c}} = \boldsymbol{0}$ \textbf{do}
|
||||||
|
\STATE \hspace{10mm} \textbf{return} $\boldsymbol{\hat{c}}$
|
||||||
|
\STATE \hspace{5mm} \textbf{end if}
|
||||||
|
\STATE \textbf{end for}
|
||||||
|
\STATE $\textcolor{KITblue}{\text{Estimate $N$ wrong bit indices $\mathcal{I} = \{i_1,\ldots,i_N\}$}}$
|
||||||
|
\STATE $\textcolor{KITblue}{\text{Generate candidate list $\left(\hat{\boldsymbol{c}}_{l}\right)_{l=1}^{2^N}$ by varying bits in $\mathcal{I}$}}$\vspace{1mm}
|
||||||
|
\STATE $\textcolor{KITblue}{\textbf{return ml\textunderscore in\textunderscore the\textunderscore list}\left(\left(\hat{\boldsymbol{c}}_l\right)_{1=1}^{2^N}\right)}$
|
||||||
|
\end{algorithmic}
|
||||||
|
\end{algorithm}
|
||||||
|
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\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}.
|
||||||
|
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]
|
||||||
|
\centering
|
||||||
|
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\pgfplotsset{
|
||||||
|
ProxPlot/.style={
|
||||||
|
line width=1pt,
|
||||||
|
mark=*,
|
||||||
|
fancy marks,
|
||||||
|
},
|
||||||
|
ImprPlot/.style={
|
||||||
|
line width=1pt,
|
||||||
|
mark=triangle,
|
||||||
|
densely dashed,
|
||||||
|
fancy marks,
|
||||||
|
},
|
||||||
|
}
|
||||||
|
|
||||||
|
\begin{axis}[
|
||||||
|
grid=both,
|
||||||
|
xlabel={$E_\text{b} / N_0$ (dB)},
|
||||||
|
ymode=log,
|
||||||
|
ymax=1.5, ymin=2e-7,
|
||||||
|
width=0.45\textwidth,
|
||||||
|
height=0.3375\textwidth,
|
||||||
|
legend columns=2,
|
||||||
|
legend style={draw=white!15!black,
|
||||||
|
legend cell align=left,
|
||||||
|
at={(0.5,-0.5)},anchor=south}
|
||||||
|
]
|
||||||
|
|
||||||
|
\addplot+[ProxPlot, scol1]
|
||||||
|
table [x=SNR, y=FER, col sep=comma,
|
||||||
|
discard if not={gamma}{0.05},
|
||||||
|
discard if gt={SNR}{9}]
|
||||||
|
{res/proximal_ber_fer_dfr_20433484.csv};
|
||||||
|
\addlegendentry{FER, prox. dec.};
|
||||||
|
|
||||||
|
\addplot+[ProxPlot, scol2]
|
||||||
|
table [x=SNR, y=BER, col sep=comma,
|
||||||
|
discard if not={gamma}{0.05},
|
||||||
|
discard if gt={SNR}{7.5}]
|
||||||
|
{res/proximal_ber_fer_dfr_20433484.csv};
|
||||||
|
\addlegendentry{BER, prox. dec.};
|
||||||
|
|
||||||
|
\addplot+[ImprPlot, scol1]
|
||||||
|
table [x=SNR, y=FER, col sep=comma,
|
||||||
|
discard if not={gamma}{0.05},
|
||||||
|
discard if gt={SNR}{7.5}]
|
||||||
|
{res/improved_ber_fer_dfr_20433484.csv};
|
||||||
|
\addlegendentry{FER, improved};
|
||||||
|
|
||||||
|
\addplot+[ImprPlot, scol2]
|
||||||
|
table [x=SNR, y=BER, col sep=comma,
|
||||||
|
discard if not={gamma}{0.05},
|
||||||
|
discard if gt={SNR}{6.5}]
|
||||||
|
{res/improved_ber_fer_dfr_20433484.csv};
|
||||||
|
\addlegendentry{BER, improved};
|
||||||
|
\end{axis}
|
||||||
|
\end{tikzpicture}
|
||||||
|
|
||||||
|
\caption{FER and BER of proximal decoding \cite{proximal_paper} and the
|
||||||
|
improved algorithm for a $\left( 3, 6 \right)$-regular LDPC code with
|
||||||
|
$n=204, k=102$ \cite[\text{204.33.484}]{mackay}.
|
||||||
|
Parameters used for simulation: $\gamma=0.05, \omega=0.05, \eta=1.5,
|
||||||
|
K=200, N=8$.
|
||||||
|
}
|
||||||
|
|
||||||
|
\label{fig:results}
|
||||||
|
\end{figure}%
|
||||||
|
%
|
||||||
|
\noindent as a preliminary examination
|
||||||
|
showed that they provide the best results for proximal decoding as well as
|
||||||
|
the improved algorithm.
|
||||||
|
All points were generated by simulating at least 100 frame errors.
|
||||||
|
The number $N$ of possibly wrong components selected was selected as $8$,
|
||||||
|
since this provides reasonable gain without requiring an unreasonable amount
|
||||||
|
of memory and computational resources.
|
||||||
|
|
||||||
|
A noticeable improvement can be observed both in the FER as well as the BER.
|
||||||
|
The gain varies significantly
|
||||||
|
with the SNR (which is to be expected, since with higher SNR values the number
|
||||||
|
of bit errors decreases, making the correction of those errors in the
|
||||||
|
``ML-in-the-list'' step more likely).
|
||||||
|
For an FER of $10^{-6}$ the gain is approximately $\SI{1}{dB}$.
|
||||||
|
Similar behavior can be observed with various other codes.
|
||||||
|
No immediate relationship between the code length and the gain was observed
|
||||||
|
during our examinations.
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\section{Conclusion}
|
||||||
|
|
||||||
|
In this paper, an improvement on proximal decoding as presented by
|
||||||
|
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.
|
||||||
|
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.
|
||||||
|
|
||||||
|
|
||||||
|
%
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
% Bibliography
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
%
|
||||||
|
|
||||||
|
|
||||||
|
\begin{thebibliography}{1}
|
||||||
|
\bibliographystyle{IEEEtran}
|
||||||
|
|
||||||
|
\bibitem{ADMM}
|
||||||
|
S. Barman, X. Liu, S. C. Draper and B. Recht, ``Decomposition Methods for Large Scale LP Decoding,'' in IEEE Transactions on Information Theory, vol. 59, no. 12, pp. 7870-7886, Dec. 2013, doi: 10.1109/TIT.2013.2281372.
|
||||||
|
|
||||||
|
\bibitem{feldman_paper}
|
||||||
|
J. Feldman, M. J. Wainwright and D. R. Karger, ``Using linear programming to Decode Binary linear codes,'' in IEEE Transactions on Information Theory, vol. 51, no. 3, pp. 954-972, March 2005, doi: 10.1109/TIT.2004.842696.
|
||||||
|
|
||||||
|
\bibitem{ml_in_the_list}
|
||||||
|
M. Geiselhart, A. Elkelesh, M. Ebada, S. Cammerer and S. t. Brink, ``Automorphism Ensemble Decoding of Reed–Muller Codes,'' in IEEE Transactions on Communications, vol. 69, no. 10, pp. 6424-6438, Oct. 2021, doi: 10.1109/TCOMM.2021.3098798.
|
||||||
|
|
||||||
|
\bibitem{mackay99}
|
||||||
|
D. J. C. MacKay, ``Good error-correcting codes based on very sparse matrices,'' in IEEE Transactions on Information Theory, vol. 45, no. 2, pp. 399-431, March 1999, doi: 10.1109/18.748992.
|
||||||
|
|
||||||
|
\bibitem{mackay}
|
||||||
|
D.J.C. MacKay, ``Encyclopedia of sparse graph codes [online],''
|
||||||
|
Available: http://www.inference.phy.cam.ac.uk/mackay/codes/data.html
|
||||||
|
|
||||||
|
\bibitem{proximal_algorithms}
|
||||||
|
N. Parikh and S. Boyd,``Proximal algorithms,'' Found. Trends Optim., vol. 1, no. 3, pp. 127–239, Jan. 2014.
|
||||||
|
|
||||||
|
\bibitem{channel_codes_book}
|
||||||
|
W. Ryan and S. Lin, Channel Codes: Classical and Modern, Cambridge, Cambridge University Press, 2009, pp. 651-670.
|
||||||
|
|
||||||
|
\bibitem{adaptive_lp_decoding}
|
||||||
|
M. H. Taghavi and P. H. Siegel, ``Adaptive Linear Programming Decoding,'' 2006 IEEE International Symposium on Information Theory, Seattle, WA, USA, 2006, pp. 1374-1378, doi: 10.1109/ISIT.2006.262071.
|
||||||
|
|
||||||
|
\bibitem{interior_point_decoding}
|
||||||
|
P. O. Vontobel, ``Interior-point algorithms for linear-programming decoding,'' 2008 Information Theory and Applications Workshop, San Diego, CA, USA, 2008, pp. 433-437, doi: 10.1109/ITA.2008.4601085.
|
||||||
|
|
||||||
|
\bibitem{proximal_paper}
|
||||||
|
T. Wadayama and S. Takabe, ``Proximal decoding for ldpc codes'' IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. advpub, 2022TAP0002, 2022.
|
||||||
|
|
||||||
|
\end{thebibliography}
|
||||||
|
|
||||||
|
|
||||||
|
\end{document}
|
||||||
901
res/2d_grad_L.csv
Normal file
901
res/2d_grad_L.csv
Normal file
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|
901
res/2d_grad_h.csv
Normal file
901
res/2d_grad_h.csv
Normal file
@ -0,0 +1,901 @@
|
|||||||
|
x_0,x_1,grad_0,grad_1,grad_norm
|
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|
1.077586206896552,-0.5603448275862069,2.492177569806063,-1.918891457214319,3.145328832413957
|
||||||
|
1.077586206896552,-0.47413793103448265,2.1275664746402096,-1.7861087477961386,2.7778991275722844
|
||||||
|
1.077586206896552,-0.38793103448275856,1.7949882631514238,-1.7378928512854164,2.4984504052312744
|
||||||
|
1.077586206896552,-0.3017241379310345,1.4944429353397055,-1.7588679835171606,2.308024321888415
|
||||||
|
1.077586206896552,-0.21551724137931028,1.2259304912050544,-1.8336583603263774,2.205721775669154
|
||||||
|
1.077586206896552,-0.12931034482758608,0.989450930747471,-1.9468881975480765,2.1838925793429116
|
||||||
|
1.077586206896552,-0.0431034482758621,0.7850042539669556,-2.0831817110172626,2.2261800735481905
|
||||||
|
1.077586206896552,0.0431034482758621,0.6125904608635072,-2.227163116568946,2.309875022711416
|
||||||
|
1.077586206896552,0.1293103448275863,0.4722095514371264,-2.363456630038132,2.410167857751749
|
||||||
|
1.077586206896552,0.2155172413793105,0.36386152568781316,-2.4766864672598308,2.503272072105596
|
||||||
|
1.077586206896552,0.3017241379310347,0.28754638361556756,-2.5514768440690476,2.5676286741954994
|
||||||
|
1.077586206896552,0.3879310344827587,0.24326412522038954,-2.5724519763007914,2.583928521649368
|
||||||
|
1.077586206896552,0.4741379310344829,0.23101475050227904,-2.524236079790069,2.5347851193865654
|
||||||
|
1.077586206896552,0.5603448275862071,0.250798259461236,-2.3914533703718885,2.4045683582738606
|
||||||
|
1.077586206896552,0.646551724137931,0.30261465209726046,-2.1587280638812576,2.1798354253136796
|
||||||
|
1.077586206896552,0.7327586206896552,0.38646392841035254,-1.810684376153183,1.8514676011228508
|
||||||
|
1.077586206896552,0.8189655172413794,0.5023460884005123,-1.3319465230226726,1.4235284095245455
|
||||||
|
1.077586206896552,0.9051724137931036,0.6502611320677394,-0.7071387203247346,0.9606688865891937
|
||||||
|
1.077586206896552,0.9913793103448278,0.8302090594120348,0.07911481610562503,0.8339701652081101
|
||||||
|
1.077586206896552,1.077586206896552,1.042189870433397,1.042189870433397,1.4738790493347687
|
||||||
|
1.077586206896552,1.1637931034482758,1.2862035651318258,2.1974622268235677,2.546204989641292
|
||||||
|
1.077586206896552,1.25,1.562250143507323,3.5603076694411424,3.8879835663194453
|
||||||
|
1.1637931034482758,-1.25,7.786713528639959,-8.526122175980975,11.546760019020882
|
||||||
|
1.1637931034482758,-1.1637931034482758,7.129962534339249,-7.129962534339249,10.08328971527461
|
||||||
|
1.1637931034482758,-1.0775862068965516,6.507807054409772,-5.941375978924924,8.812008918536776
|
||||||
|
1.1637931034482758,-0.9913793103448276,5.920247088851529,-4.944986725573003,7.713768165374703
|
||||||
|
1.1637931034482758,-0.9051724137931034,5.36728263766452,-4.125418990118496,6.769549819345841
|
||||||
|
1.1637931034482758,-0.8189655172413792,4.848913700848741,-3.4672969883964067,5.961049612611979
|
||||||
|
1.1637931034482758,-0.7327586206896551,4.365140278404197,-2.9552449362417477,5.271425071394751
|
||||||
|
1.1637931034482758,-0.646551724137931,3.9159623703308855,-2.5738870494895227,4.686113083289537
|
||||||
|
1.1637931034482758,-0.5603448275862069,3.5013799766288063,-2.3078475439747423,4.193545281377965
|
||||||
|
1.1637931034482758,-0.47413793103448265,3.1213930972979607,-2.1417506355324116,3.7855238280405588
|
||||||
|
1.1637931034482758,-0.38793103448275856,2.776001732338348,-2.0602205399975393,3.4569776237883376
|
||||||
|
1.1637931034482758,-0.3017241379310345,2.4652058817499682,-2.0478814732051336,3.204849226985798
|
||||||
|
1.1637931034482758,-0.21551724137931028,2.1890055455328206,-2.0893576509901997,3.026080083561029
|
||||||
|
1.1637931034482758,-0.12931034482758608,1.9474007236869066,-2.169273289187749,2.915152857364382
|
||||||
|
1.1637931034482758,-0.0431034482758621,1.7403914162122258,-2.2722526036327855,2.8621834630821916
|
||||||
|
1.1637931034482758,0.0431034482758621,1.5679776231087774,-2.382919810160318,2.852518299367129
|
||||||
|
1.1637931034482758,0.1293103448275863,1.430159344376562,-2.4858991246053552,2.8679348332940324
|
||||||
|
1.1637931034482758,0.2155172413793105,1.3269365800155795,-2.565814762802903,2.888627024106913
|
||||||
|
1.1637931034482758,0.3017241379310347,1.2583093300258301,-2.60729094058797,2.8950489493102105
|
||||||
|
1.1637931034482758,0.3879310344827587,1.2242775944073137,-2.594951873795564,2.8692561502038934
|
||||||
|
1.1637931034482758,0.4741379310344829,1.22484137316003,-2.513421778260691,2.7959838026783492
|
||||||
|
1.1637931034482758,0.5603448275862071,1.2600006662839796,-2.3473248698183604,2.664120065519543
|
||||||
|
1.1637931034482758,0.646551724137931,1.3297554737791615,-2.0812853643035796,2.4698174806470874
|
||||||
|
1.1637931034482758,0.7327586206896552,1.4341057956455767,-1.6999274775513549,2.224053250720033
|
||||||
|
1.1637931034482758,0.8189655172413794,1.5730516318832253,-1.187875425396694,1.9711771769254673
|
||||||
|
1.1637931034482758,0.9051724137931036,1.7465929824921063,-0.5297534236746059,1.825164577890344
|
||||||
|
1.1637931034482758,0.9913793103448278,1.9547298474722208,0.28981431177990363,1.9760974449432473
|
||||||
|
1.1637931034482758,1.077586206896552,2.1974622268235677,1.2862035651318258,2.546204989641292
|
||||||
|
1.1637931034482758,1.1637931034482758,2.474790120546146,2.474790120546146,3.4998817525033066
|
||||||
|
1.1637931034482758,1.25,2.786713528639959,3.8709497621878715,4.769698559876432
|
||||||
|
1.25,-1.25,9.21875,-9.21875,13.03728127812697
|
||||||
|
1.25,-1.1637931034482758,8.526122175980975,-7.786713528639959,11.546760019020882
|
||||||
|
1.25,-1.0775862068965516,7.870652497027348,-6.562250143507318,10.247453228725888
|
||||||
|
1.25,-0.9913793103448276,7.252340963139121,-5.529984060437083,9.1201520357017
|
||||||
|
1.25,-0.9051724137931034,6.671187574316289,-4.674539495264257,8.1459230995939
|
||||||
|
1.25,-0.8189655172413792,6.127192330558858,-3.980540663823853,7.3066538054033
|
||||||
|
1.25,-0.7327586206896551,5.620355231866825,-3.4326117819508797,6.585682696422656
|
||||||
|
1.25,-0.646551724137931,5.150676278240191,-3.0153770654803393,5.96841394092694
|
||||||
|
1.25,-0.5603448275862069,4.718155469678953,-2.7134607302472418,5.442780555070663
|
||||||
|
1.25,-0.47413793103448265,4.322792806183115,-2.5114869920865956,4.999410420900515
|
||||||
|
1.25,-0.38793103448275856,3.964588287752675,-2.3940800668334075,4.631369091078224
|
||||||
|
1.25,-0.3017241379310345,3.643541914387634,-2.345864170322687,4.333413941398081
|
||||||
|
1.25,-0.21551724137931028,3.35965368608799,-2.351463518389438,4.100811330549243
|
||||||
|
1.25,-0.12931034482758608,3.112923602853745,-2.395502326868671,3.927941541728588
|
||||||
|
1.25,-0.0431034482758621,2.903351664684899,-2.4626048115953916,3.807081998975245
|
||||||
|
1.25,0.0431034482758621,2.7309378715814505,-2.537395188404609,3.7277870111604665
|
||||||
|
1.25,0.1293103448275863,2.5956822235434007,-2.6044976731313296,3.6770877518718152
|
||||||
|
1.25,0.2155172413793105,2.497584720570749,-2.648536481610562,3.640422356107945
|
||||||
|
1.25,0.3017241379310347,2.4366453626634956,-2.6541358296773137,3.6030095234076747
|
||||||
|
1.25,0.3879310344827587,2.4128641498216408,-2.6059199331665917,3.5514408489470135
|
||||||
|
1.25,0.4741379310344829,2.426241082045184,-2.4885130079134035,3.4755348910287176
|
||||||
|
1.25,0.5603448275862071,2.4767761593341264,-2.286539269752757,3.370857750716778
|
||||||
|
1.25,0.646551724137931,2.564469381688466,-1.9846229345196607,3.242719753823795
|
||||||
|
1.25,0.7327586206896552,2.6893207491082047,-1.56738821804912,3.1127402586247235
|
||||||
|
1.25,0.8189655172413794,2.851330261593341,-1.0194593361761444,3.028098677189146
|
||||||
|
1.25,0.9051724137931036,3.0504979191438766,-0.3254605047357399,3.067810635427807
|
||||||
|
1.25,0.9913793103448278,3.28682372175981,0.5299840604370849,3.3292781923774992
|
||||||
|
1.25,1.077586206896552,3.5603076694411424,1.562250143507323,3.8879835663194453
|
||||||
|
1.25,1.1637931034482758,3.8709497621878715,2.786713528639959,4.769698559876432
|
||||||
|
1.25,1.25,4.21875,4.21875,5.966213466261495
|
||||||
|
401
res/extreme_components_20433484_combined.csv
Normal file
401
res/extreme_components_20433484_combined.csv
Normal file
File diff suppressed because one or more lines are too long
17
res/improved_ber_fer_dfr_20433484.csv
Normal file
17
res/improved_ber_fer_dfr_20433484.csv
Normal file
@ -0,0 +1,17 @@
|
|||||||
|
SNR,gamma,BER,FER,DFR,num_iterations
|
||||||
|
1.0,0.05,0.09304388422035481,0.9619047619047619,0.49029126213592233,105.0
|
||||||
|
1.5,0.05,0.06962519521082769,0.8938053097345132,0.4719626168224299,113.0
|
||||||
|
2.0,0.05,0.045701357466063346,0.7769230769230769,0.43722943722943725,130.0
|
||||||
|
2.5,0.05,0.02918243320720101,0.5906432748538012,0.3713235294117647,171.0
|
||||||
|
3.0,0.05,0.01592172301991663,0.39763779527559057,0.28450704225352114,254.0
|
||||||
|
3.5,0.05,0.006343713956170703,0.18566176470588236,0.15658914728682172,544.0
|
||||||
|
4.0,0.05,0.002085186474326746,0.06640368178829717,0.06226880394574599,1521.0
|
||||||
|
4.5,0.05,0.0005285002230367914,0.019980217606330366,0.019588828549262995,5055.0
|
||||||
|
5.0,0.05,0.00012825105681736764,0.0049208282582216805,0.004896732279647048,20525.0
|
||||||
|
5.5,0.05,2.2937044988129024e-05,0.0009664516869844794,0.0009655185599434072,104506.0
|
||||||
|
6.0,0.05,4.199412358104951e-06,0.00018215724679240927,0.00018212407157295693,554466.0
|
||||||
|
6.5,0.05,9.083774548814115e-07,3.965298533978094e-05,3.9651413042880847e-05,2547097.0
|
||||||
|
7.0,0.05,1.8621134224119705e-07,8.304542198133386e-06,8.304473233284987e-06,12162019.0
|
||||||
|
7.5,0.05,2.9553328302635507e-08,1.4127999451218143e-06,1.4127979491209494e-06,71489244.0
|
||||||
|
8.0,0.05,3.823529411764706e-09,1.8e-07,1.7999996760000583e-07,100000000.0
|
||||||
|
8.5,0.05,0.0,0.0,0.0,100000000.0
|
||||||
|
205
res/p_error.csv
Normal file
205
res/p_error.csv
Normal file
@ -0,0 +1,205 @@
|
|||||||
|
index,p_error
|
||||||
|
0,6.78e-06
|
||||||
|
1,6.68e-06
|
||||||
|
2,5.82e-06
|
||||||
|
3,1.37e-06
|
||||||
|
4,5.7e-07
|
||||||
|
5,2.9e-07
|
||||||
|
6,1.6e-07
|
||||||
|
7,1.3e-07
|
||||||
|
8,1.3e-07
|
||||||
|
9,7e-08
|
||||||
|
10,6e-08
|
||||||
|
11,7e-08
|
||||||
|
12,3e-08
|
||||||
|
13,2e-08
|
||||||
|
14,1e-08
|
||||||
|
15,5e-08
|
||||||
|
16,4e-08
|
||||||
|
17,0.0
|
||||||
|
18,1e-08
|
||||||
|
19,2e-08
|
||||||
|
20,2e-08
|
||||||
|
21,1e-08
|
||||||
|
22,2e-08
|
||||||
|
23,0.0
|
||||||
|
24,1e-08
|
||||||
|
25,5e-08
|
||||||
|
26,2e-08
|
||||||
|
27,1e-08
|
||||||
|
28,2e-08
|
||||||
|
29,0.0
|
||||||
|
30,1e-08
|
||||||
|
31,1e-08
|
||||||
|
32,1e-08
|
||||||
|
33,3e-08
|
||||||
|
34,0.0
|
||||||
|
35,1e-08
|
||||||
|
36,2e-08
|
||||||
|
37,1e-08
|
||||||
|
38,0.0
|
||||||
|
39,0.0
|
||||||
|
40,1e-08
|
||||||
|
41,1e-08
|
||||||
|
42,2e-08
|
||||||
|
43,0.0
|
||||||
|
44,1e-08
|
||||||
|
45,0.0
|
||||||
|
46,0.0
|
||||||
|
47,1e-08
|
||||||
|
48,0.0
|
||||||
|
49,0.0
|
||||||
|
50,0.0
|
||||||
|
51,1e-08
|
||||||
|
52,0.0
|
||||||
|
53,1e-08
|
||||||
|
54,0.0
|
||||||
|
55,0.0
|
||||||
|
56,0.0
|
||||||
|
57,0.0
|
||||||
|
58,0.0
|
||||||
|
59,0.0
|
||||||
|
60,1e-08
|
||||||
|
61,0.0
|
||||||
|
62,0.0
|
||||||
|
63,0.0
|
||||||
|
64,0.0
|
||||||
|
65,0.0
|
||||||
|
66,0.0
|
||||||
|
67,1e-08
|
||||||
|
68,0.0
|
||||||
|
69,2e-08
|
||||||
|
70,0.0
|
||||||
|
71,0.0
|
||||||
|
72,1e-08
|
||||||
|
73,0.0
|
||||||
|
74,0.0
|
||||||
|
75,0.0
|
||||||
|
76,0.0
|
||||||
|
77,0.0
|
||||||
|
78,0.0
|
||||||
|
79,0.0
|
||||||
|
80,0.0
|
||||||
|
81,0.0
|
||||||
|
82,0.0
|
||||||
|
83,0.0
|
||||||
|
84,0.0
|
||||||
|
85,0.0
|
||||||
|
86,0.0
|
||||||
|
87,0.0
|
||||||
|
88,0.0
|
||||||
|
89,2e-08
|
||||||
|
90,0.0
|
||||||
|
91,0.0
|
||||||
|
92,0.0
|
||||||
|
93,0.0
|
||||||
|
94,1e-08
|
||||||
|
95,0.0
|
||||||
|
96,0.0
|
||||||
|
97,0.0
|
||||||
|
98,0.0
|
||||||
|
99,0.0
|
||||||
|
100,1e-08
|
||||||
|
101,0.0
|
||||||
|
102,0.0
|
||||||
|
103,0.0
|
||||||
|
104,0.0
|
||||||
|
105,0.0
|
||||||
|
106,0.0
|
||||||
|
107,0.0
|
||||||
|
108,0.0
|
||||||
|
109,0.0
|
||||||
|
110,0.0
|
||||||
|
111,0.0
|
||||||
|
112,1e-08
|
||||||
|
113,1e-08
|
||||||
|
114,0.0
|
||||||
|
115,0.0
|
||||||
|
116,2e-08
|
||||||
|
117,0.0
|
||||||
|
118,0.0
|
||||||
|
119,0.0
|
||||||
|
120,0.0
|
||||||
|
121,0.0
|
||||||
|
122,0.0
|
||||||
|
123,1e-08
|
||||||
|
124,0.0
|
||||||
|
125,0.0
|
||||||
|
126,0.0
|
||||||
|
127,0.0
|
||||||
|
128,0.0
|
||||||
|
129,0.0
|
||||||
|
130,0.0
|
||||||
|
131,0.0
|
||||||
|
132,0.0
|
||||||
|
133,0.0
|
||||||
|
134,0.0
|
||||||
|
135,0.0
|
||||||
|
136,0.0
|
||||||
|
137,0.0
|
||||||
|
138,0.0
|
||||||
|
139,0.0
|
||||||
|
140,0.0
|
||||||
|
141,0.0
|
||||||
|
142,0.0
|
||||||
|
143,0.0
|
||||||
|
144,0.0
|
||||||
|
145,0.0
|
||||||
|
146,0.0
|
||||||
|
147,0.0
|
||||||
|
148,0.0
|
||||||
|
149,0.0
|
||||||
|
150,0.0
|
||||||
|
151,0.0
|
||||||
|
152,0.0
|
||||||
|
153,0.0
|
||||||
|
154,0.0
|
||||||
|
155,0.0
|
||||||
|
156,1e-08
|
||||||
|
157,0.0
|
||||||
|
158,1e-08
|
||||||
|
159,0.0
|
||||||
|
160,0.0
|
||||||
|
161,0.0
|
||||||
|
162,0.0
|
||||||
|
163,0.0
|
||||||
|
164,0.0
|
||||||
|
165,0.0
|
||||||
|
166,0.0
|
||||||
|
167,0.0
|
||||||
|
168,0.0
|
||||||
|
169,0.0
|
||||||
|
170,0.0
|
||||||
|
171,0.0
|
||||||
|
172,0.0
|
||||||
|
173,0.0
|
||||||
|
174,0.0
|
||||||
|
175,0.0
|
||||||
|
176,0.0
|
||||||
|
177,0.0
|
||||||
|
178,0.0
|
||||||
|
179,0.0
|
||||||
|
180,0.0
|
||||||
|
181,0.0
|
||||||
|
182,0.0
|
||||||
|
183,0.0
|
||||||
|
184,0.0
|
||||||
|
185,0.0
|
||||||
|
186,0.0
|
||||||
|
187,1e-08
|
||||||
|
188,0.0
|
||||||
|
189,0.0
|
||||||
|
190,0.0
|
||||||
|
191,0.0
|
||||||
|
192,0.0
|
||||||
|
193,0.0
|
||||||
|
194,0.0
|
||||||
|
195,0.0
|
||||||
|
196,0.0
|
||||||
|
197,0.0
|
||||||
|
198,0.0
|
||||||
|
199,0.0
|
||||||
|
200,0.0
|
||||||
|
201,0.0
|
||||||
|
202,0.0
|
||||||
|
203,1e-08
|
||||||
|
17
res/proximal_ber_fer_dfr_20433484.csv
Normal file
17
res/proximal_ber_fer_dfr_20433484.csv
Normal file
@ -0,0 +1,17 @@
|
|||||||
|
SNR,gamma,BER,FER,DFR,num_iterations
|
||||||
|
1.0,0.05,0.09869281045751634,0.9619047619047619,0.49029126213592233,105.0
|
||||||
|
1.5,0.05,0.0714538067479244,0.9099099099099099,0.47641509433962265,111.0
|
||||||
|
2.0,0.05,0.0505718954248366,0.8416666666666667,0.45701357466063347,120.0
|
||||||
|
2.5,0.05,0.029692768827276134,0.643312101910828,0.39147286821705424,157.0
|
||||||
|
3.0,0.05,0.017503699593044764,0.47641509433962265,0.3226837060702875,212.0
|
||||||
|
3.5,0.05,0.007170421360033375,0.26861702127659576,0.21174004192872117,376.0
|
||||||
|
4.0,0.05,0.0027883735974480804,0.1149032992036405,0.10306122448979592,879.0
|
||||||
|
4.5,0.05,0.001140607758254817,0.054653679653679656,0.051821446895844024,1848.0
|
||||||
|
5.0,0.05,0.000392274251548554,0.02015565755338256,0.019757433489827857,5011.0
|
||||||
|
5.5,0.05,0.00010411496574189757,0.0058772185045097465,0.0058428786301052875,17185.0
|
||||||
|
6.0,0.05,3.2864204308667e-05,0.002058158254029711,0.002053930939114166,49073.0
|
||||||
|
6.5,0.05,1.1098805067597564e-05,0.0007079869337856973,0.0007074860429114802,142658.0
|
||||||
|
7.0,0.05,2.6684330544007944e-06,0.00016917044508576608,0.00016914183128688464,597031.0
|
||||||
|
7.5,0.05,4.554000856780299e-07,2.94139917407841e-05,2.941312658332171e-05,3433740.0
|
||||||
|
8.0,0.05,8.75025641553281e-08,5.669505760554655e-06,5.669473617441322e-06,17814604.0
|
||||||
|
8.5,0.05,1.623670674453506e-08,1.0620352563949216e-06,1.062034128477234e-06,95100421.0
|
||||||
|
Loading…
Reference in New Issue
Block a user