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Andreas Tsouchlos 2023-02-20 09:21:20 +01:00
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latex/thesis/chapters/decoding_techniques.tex
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@ -40,7 +40,7 @@ $f$ must be minimized under certain constraints:%
\text{subject to}\hspace{2mm} &\boldsymbol{c} \in D \text{subject to}\hspace{2mm} &\boldsymbol{c} \in D
,\end{align*}% ,\end{align*}%
% %
where $D$ is the domain of values attainable for $c$ and represents the where $D$ is the domain of values attainable for $\boldsymbol{c}$ and represents the
constraints. constraints.
In contrast to the established message-passing decoding algorithms, In contrast to the established message-passing decoding algorithms,
@ -214,8 +214,8 @@ decoding is the following:%
\text{subject to }\hspace{2mm} &\boldsymbol{c} \in \mathcal{C} \text{subject to }\hspace{2mm} &\boldsymbol{c} \in \mathcal{C}
.\end{align*}% .\end{align*}%
% %
\todo{$\boldsymbol{c}$ or some other variable name? e.g. $\boldsymbol{c}^{*}$. %\todo{$\boldsymbol{c}$ or some other variable name? e.g. $\boldsymbol{c}^{*}$.
Especially for the continuous variable in LP decoding} %Especially for the continuous variable in LP decoding}
As solving integer linear programs is generally NP-hard, this decoding problem As solving integer linear programs is generally NP-hard, this decoding problem
has to be approximated by one with looser constraints. has to be approximated by one with looser constraints.
@ -633,7 +633,7 @@ determined by the channel model.
The prior \ac{PDF} $f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)$ is also The prior \ac{PDF} $f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)$ is also
known as the equal probability assumption is made on known as the equal probability assumption is made on
$\mathcal{C}\left( \boldsymbol{H} \right)$. $\mathcal{C}\left( \boldsymbol{H} \right)$.
However, because in this case the considered domain is continuous, However, because the considered domain is continuous,
the prior \ac{PDF} cannot be ignored as a constant during the minimization the prior \ac{PDF} cannot be ignored as a constant during the minimization
as is often done, and has a rather unwieldy representation:% as is often done, and has a rather unwieldy representation:%
% %
@ -643,7 +643,7 @@ as is often done, and has a rather unwieldy representation:%
\sum_{c \in \mathcal{C}\left( \boldsymbol{H} \right) } \sum_{c \in \mathcal{C}\left( \boldsymbol{H} \right) }
\delta\left( \boldsymbol{x} - \left( -1 \right) ^{\boldsymbol{c}}\right) \delta\left( \boldsymbol{x} - \left( -1 \right) ^{\boldsymbol{c}}\right)
\label{eq:prox:prior_pdf} \label{eq:prox:prior_pdf}
\end{align}% .\end{align}%
% %
In order to rewrite the prior \ac{PDF} In order to rewrite the prior \ac{PDF}
$f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)$, $f_{\boldsymbol{X}}\left( \boldsymbol{x} \right)$,