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Rewrote introduction and conclusion

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Andreas Tsouchlos 2023-04-24 12:29:01 +02:00
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18
latex/thesis/chapters/conclusion.tex
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@ -1,4 +1,4 @@
\chapter{Conclusion}% \chapter{Conclusion and Outlook}%
\label{chapter:conclusion} \label{chapter:conclusion}
In the context of this thesis, two decoding algorithms were considered: In the context of this thesis, two decoding algorithms were considered:
@ -11,16 +11,16 @@ decoder was examined, leading to an approach to choosing the value of each
of the parameters. of the parameters.
The convergence properties of the algorithm were investigated in the context The convergence properties of the algorithm were investigated in the context
of the relatively high decoding failure rate, to derive an approach to correct of the relatively high decoding failure rate, to derive an approach to correct
possible wrong componets of the estimate. possible wrong components of the estimate.
Based on this approach, an improvement over proximal decoding was suggested, 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 leading to a decoding gain of up to $\SI{1}{dB}$, depending on the code and
the parameters considered. the parameters considered.
For \ac{LP} decoding using \ac{ADMM}, the circumstances brought about via the For \ac{LP} decoding using \ac{ADMM}, the circumstances brought about by the
relaxation while formulating the \ac{LP} decoding problem were first explored. \ac{LP} relaxation were first explored.
The decomposable nature arising from the relocation of the constraints into The decomposable nature arising from the relocation of the constraints into
the objective function itself was recognized as the major driver in enabling the objective function itself was recognized as the major driver in enabling
the efficent implementation of the decoding algorithm. an efficient implementation of the decoding algorithm.
Based on simulation results, general guidelines for choosing each parameter Based on simulation results, general guidelines for choosing each parameter
were again derived. were again derived.
The decoding performance, in form of the \ac{FER}, of the algorithm was The decoding performance, in form of the \ac{FER}, of the algorithm was
@ -28,15 +28,15 @@ 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 of \ac{BP}, staying within approximately $\SI{0.5}{dB}$ depending on the code
in question. in question.
Finally, strong parallells were discovered with regard to the theoretical Finally, strong parallels were discovered with regard to the theoretical
structure of the two algorithms, both in the constitution of their respective structure of the two algorithms, both in the constitution of their respective
objective functions as in the iterative approaches used to minimize them. objective functions as in the iterative approaches used to minimize them.
One difference noted was the approximate nature of the minimization in the One difference noted was the approximate nature of the minimization in the
case of proximal decoding, leading to the constraints never being truly case of proximal decoding, leading to the constraints never being truly
satisfied. satisfied.
In conjunction with the alternating minimization with respect to the same In conjunction with the alternating minimization with respect to the same
variable leading to oscillatory behavior, this was identified as the variable, leading to oscillatory behavior, this was identified as
root cause of its comparatively worse decoding performance. a possible cause of its comparatively worse decoding performance.
Furthermore, both algorithms were expressed as message passing algorithms, Furthermore, both algorithms were expressed as message passing algorithms,
justifying their similar computational performance. justifying their similar computational performance.
@ -46,7 +46,7 @@ investigation is required to determine how different choices of parameters
affect the decoding performance. affect the decoding performance.
Additionally, a more mathematically rigorous foundation for determining the Additionally, a more mathematically rigorous foundation for determining the
potentially wrong components of the estimate is desirable. potentially wrong components of the estimate is desirable.
Another area benefiting from future work is the expantion of the \ac{ADMM} Another area benefiting from future work is the expansion of the \ac{ADMM}
based \ac{LP} decoder into a decoder approximating \ac{ML} performance, based \ac{LP} decoder into a decoder approximating \ac{ML} performance,
using \textit{adaptive \ac{LP} decoding}. using \textit{adaptive \ac{LP} decoding}.
With this method, the successive addition of redundant parity checks is used With this method, the successive addition of redundant parity checks is used

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latex/thesis/chapters/introduction.tex
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@ -19,7 +19,7 @@ linear codes was conducted in Feldman's 2003 Ph.D. thesis and subsequent paper,
establishing the field of \ac{LP} decoding \cite{feldman_thesis}, \cite{feldman_paper}. establishing the field of \ac{LP} decoding \cite{feldman_thesis}, \cite{feldman_paper}.
There, the \ac{ML} decoding problem is approximated by a \textit{linear program}, There, the \ac{ML} decoding problem is approximated by a \textit{linear program},
a linear, convex optimization problem, which can subsequently be solved using a linear, convex optimization problem, which can subsequently be solved using
a number of different algorithms \cite{alp}, \cite{interior_point}, several different algorithms \cite{alp}, \cite{interior_point},
\cite{original_admm}, \cite{pdd}. \cite{original_admm}, \cite{pdd}.
More recently, novel approaches such as \textit{proximal decoding} have been More recently, novel approaches such as \textit{proximal decoding} have been
introduced. Proximal decoding is based on a non-convex optimization formulation introduced. Proximal decoding is based on a non-convex optimization formulation
@ -40,4 +40,6 @@ the existing literature.
Specifically, the proximal decoding algorithm and \ac{LP} decoding using Specifically, the proximal decoding algorithm and \ac{LP} decoding using
the \ac{ADMM} \cite{original_admm} are explored within the context of the \ac{ADMM} \cite{original_admm} are explored within the context of
\ac{BPSK} modulated \ac{AWGN} channels. \ac{BPSK} modulated \ac{AWGN} channels.
Implementations of both decoding methods are produced, and based on simulation
results from those implementations the algorithms are examined and compared.