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Fixed spelling errors

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Andreas Tsouchlos 2023-04-12 23:18:43 +02:00
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latex/thesis/chapters/comparison.tex
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@ -3,7 +3,7 @@
In this chapter, proximal decoding and \ac{LP} Decoding using \ac{ADMM} are compared. In this chapter, proximal decoding and \ac{LP} Decoding using \ac{ADMM} are compared.
First the two algorithms are compared on a theoretical basis. First the two algorithms are compared on a theoretical basis.
Subsequently, their respective simulation results are examined and their Subsequently, their respective simulation results are examined, and their
differences are interpreted on the basis of their theoretical structure. differences are interpreted on the basis of their theoretical structure.
%some similarities between the proximal decoding algorithm %some similarities between the proximal decoding algorithm
@ -119,13 +119,13 @@ return $\tilde{\boldsymbol{c}}$
\end{figure}% \end{figure}%
% %
Their major differece is that while with proximal decoding the constraints Their major difference is that while with proximal decoding the constraints
are regarded in a global context, considering all parity checks at the same are regarded in a global context, considering all parity checks at the same
time, with \ac{ADMM} each parity check is time, with \ac{ADMM} each parity check is
considered separately and in a more local context (line 4 in both algorithms). considered separately and in a more local context (line 4 in both algorithms).
This difference means that while with proximal decoding the alternating This difference means that while with proximal decoding the alternating
minimization of the two parts of the objective function inevitably leads to minimization of the two parts of the objective function inevitably leads to
oscillatory behaviour (as explained in section oscillatory behavior (as explained in section
\ref{subsec:prox:conv_properties}), this is not the case with \ac{ADMM}, which \ref{subsec:prox:conv_properties}), this is not the case with \ac{ADMM}, which
partly explains the disparate decoding performance of the two methods. partly explains the disparate decoding performance of the two methods.
Furthermore, while with proximal decoding the step considering the constraints Furthermore, while with proximal decoding the step considering the constraints
@ -137,7 +137,7 @@ The contrasting treatment of the constraints (global and approximate with
proximal decoding as opposed to local and exact with \ac{LP} decoding using proximal decoding as opposed to local and exact with \ac{LP} decoding using
\ac{ADMM}) also leads to different prospects when the decoding process gets \ac{ADMM}) also leads to different prospects when the decoding process gets
stuck in a local minimum. stuck in a local minimum.
With proximal decoding this occurrs due to the approximate nature of the With proximal decoding this occurs due to the approximate nature of the
calculation, whereas with \ac{LP} decoding it occurs due to the approximate calculation, whereas with \ac{LP} decoding it occurs due to the approximate
formulation of the constraints - independent of the optimization method formulation of the constraints - independent of the optimization method
itself. itself.
@ -241,7 +241,7 @@ computed for each check node (line 6 in figure
\ref{fig:comp:message_passing:admm}), whereas \ref{fig:comp:message_passing:admm}), whereas
with proximal decoding, the same message is transmitted to all \acp{VN} with proximal decoding, the same message is transmitted to all \acp{VN}
(line 5 of figure \ref{fig:comp:message_passing:proximal}). (line 5 of figure \ref{fig:comp:message_passing:proximal}).
This means that while both algorithms have an averege time complexity of This means that while both algorithms have an average time complexity of
$\mathcal{O}\left( n \right)$, more arithmetic operations are required for the $\mathcal{O}\left( n \right)$, more arithmetic operations are required for the
\ac{ADMM} case. \ac{ADMM} case.