diff --git a/latex/thesis/chapters/comparison.tex b/latex/thesis/chapters/comparison.tex index 0d0c468..d70e964 100644 --- a/latex/thesis/chapters/comparison.tex +++ b/latex/thesis/chapters/comparison.tex @@ -270,6 +270,11 @@ to the constraints never being quite satisfied. With \ac{LP} decoding using \ac{ADMM}, the constraints are fulfilled for each parity check individualy after each iteration of the decoding process. +It should be noted that while in this thesis proximal decoding was +examined with respect to its performance in \ac{AWGN} channels, in +\cite{proximal_paper} it is presented as a method applicable to non-trivial +channel models such as \ac{LDPC}-coded massive \ac{MIMO} channels, perhaps +broadening its usefulness beyond what is shown here. The timing requirements of the decoding algorithms are visualized in figure \ref{fig:comp:time}. diff --git a/latex/thesis/chapters/conclusion.tex b/latex/thesis/chapters/conclusion.tex index 54a7992..82e5576 100644 --- a/latex/thesis/chapters/conclusion.tex +++ b/latex/thesis/chapters/conclusion.tex @@ -1,8 +1,55 @@ \chapter{Conclusion}% \label{chapter:conclusion} -\begin{itemize} - \item Summary of results - \item Future work -\end{itemize} +In the context of this thesis, two decoding algorithms were considered: +proximal decoding and \ac{LP} decoding using \ac{ADMM}. +The two algorithms were first analyzed individually, before comparing them +based on simulation results as well as their theoretical structure. + +For proximal decoding, the effect of each parameter on the behavior of the +decoder was examined, leading to an approach to choosing the value of each +of the parameters. +The convergence properties of the algorithm were investigated in the context +of the relatively high decoding failure rate, to derive an approach to correct +possible wrong componets of the estimate. +Based on this approach, an improvement over proximal decoding was suggested, +leading to a decoding gain of up to $\SI{1}{dB}$, depending on the code and +the parameters considered. + +For \ac{LP} decoding using \ac{ADMM}, the circumstances brought about via the +relaxation while formulating the \ac{LP} decoding problem were first explored. +The decomposable nature arising from the relocation of the constraints into +the objective function itself was recognized as the major driver in enabling +the efficent implementation of the decoding algorithm. +Based on simulation results, general guidelines for choosing each parameter +were again derived. +The decoding performance, in form of the \ac{FER}, of the algorithm was +analyzed, observing that \ac{LP} decoding using \ac{ADMM} nearly reaches that +of \ac{BP}, staying within approximately $\SI{0.5}{dB}$ depending on the code +in question. + +Finally, strong parallells were discovered with regard to the theoretical +structure of the two algorithms, both in the constitution of their respective +objective functions as in the iterative approaches used to minimize them. +One difference noted was the approximate nature of the minimization in the +case of proximal decoding, leading to the constraints never being truly +satisfied. +In conjunction with the alternating minimization with respect to the same +variable leading to oscillatory behavior, this was identified as the +root cause of its comparatively worse decoding performance. +Furthermore, both algorithms were expressed as message passing algorithms, +justifying their similar computational performance. + +While the modified proximal decoding algorithm presented in section +\ref{sec:prox:Improved Implementation} shows some promising results, further +investigation is required to determine how different choices of parameters +affect the decoding performance. +Additionally, a more mathematically rigorous foundation for determining the +potentially wrong components of the estimate is desirable. +Another area benefiting from future work is the expantion of the \ac{ADMM} +based \ac{LP} decoder into a decoder approximating \ac{ML} performance, +using \textit{adaptive \ac{LP} decoding}. +With this method, the successive addition of redundant parity checks is used +to mitigate the decoder becoming stuck in erroneous solutions introduced due +the relaxation of the constraints of the \ac{LP} decoding problem \cite{alp}. diff --git a/latex/thesis/chapters/discussion.tex b/latex/thesis/chapters/discussion.tex index 5c586a0..dace1ca 100644 --- a/latex/thesis/chapters/discussion.tex +++ b/latex/thesis/chapters/discussion.tex @@ -33,13 +33,6 @@ examined with respect to its performance in \ac{AWGN} channels, in channel models such as \ac{LDPC}-coded massive \ac{MIMO} channels, perhaps broadening its usefulness beyond what is shown here. -While the modified proximal decoding algorithm presented in section -\ref{sec:prox:Improved Implementation} shows some promising results, further -investigation is required to determine how different choices of parameters -affect the decoding performance. -Additionally, a more mathematically rigorous foundation for determining the -potentially wrong components of the estimate is desirable. - Another interesting approach might be the combination of proximal and \ac{LP} decoding. Performing an initial number of iterations using proximal decoding to obtain diff --git a/latex/thesis/chapters/lp_dec_using_admm.tex b/latex/thesis/chapters/lp_dec_using_admm.tex index 4a7b01e..df0b19a 100644 --- a/latex/thesis/chapters/lp_dec_using_admm.tex +++ b/latex/thesis/chapters/lp_dec_using_admm.tex @@ -1370,11 +1370,6 @@ of one another. \label{fig:admm:results} \end{figure}% % - -\footnotetext{; $K=200, \mu = 3.3, \rho=1.9, - \epsilon_{\text{pri}} = 10^{-5}, \epsilon_{\text{dual}} = 10^{-5}$ -}% -% In figure \ref{fig:admm:ber_fer}, the \ac{BER} and \ac{FER} for \ac{LP} decoding using\ac{ADMM} and \ac{BP} are shown for a (3, 6) regular \ac{LDPC} code with $n=204$. diff --git a/latex/thesis/chapters/proximal_decoding.tex b/latex/thesis/chapters/proximal_decoding.tex index 2996bd9..de3e26c 100644 --- a/latex/thesis/chapters/proximal_decoding.tex +++ b/latex/thesis/chapters/proximal_decoding.tex @@ -1195,17 +1195,10 @@ $\SI{2.80}{GHz}$ and utilizing all cores. \end{axis} \end{tikzpicture} - \caption{Timing requirements of the proximal decoding imlementation% - \protect\footnotemark{}} + \caption{Timing requirements of the proximal decoding imlementation} \label{fig:prox:time_comp} \end{figure}% % -\footnotetext{The datapoints depicted were calculated by evaluating the - metadata of \ac{FER} simulation results from the following codes: - BCH (31, 11); BCH (31, 26); \cite[\text{96.3.965; 204.33.484; 204.55.187; - 408.33.844; PEGReg252x504}]{mackay_enc} -}% -% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -1499,16 +1492,10 @@ theoretical considerations. \end{tikzpicture} \caption{Comparison of the timing requirements of the implementations of proximal - decoding and the improved algorithm\protect\footnotemark{}} + decoding and the improved algorithm} \label{fig:prox:time_complexity_comp} \end{figure}% % -\footnotetext{The datapoints depicted were calculated by evaluating the - metadata of \ac{FER} simulation results from the following codes: - BCH (31, 11); BCH (31, 26); \cite[\text{96.3.965; 204.33.484; 204.55.187; - 408.33.844; PEGReg252x504}]{mackay_enc} -}% -% In conclusion, the decoding performance of proximal decoding can be improved by appending an ML-in-the-List step when the algorithm does not produce a diff --git a/latex/thesis/thesis.tex b/latex/thesis/thesis.tex index a23a91a..20e7b82 100644 --- a/latex/thesis/thesis.tex +++ b/latex/thesis/thesis.tex @@ -218,7 +218,7 @@ \include{chapters/proximal_decoding} \include{chapters/lp_dec_using_admm} \include{chapters/comparison} - \include{chapters/discussion} +% \include{chapters/discussion} \include{chapters/conclusion} \include{chapters/appendix}