From c1473c6eb4373f35b0d42eb5a404c157622d358d Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Sun, 19 Feb 2023 14:35:33 +0100 Subject: [PATCH] Spell checked entire decoding techniques chapter --- latex/thesis/chapters/decoding_techniques.tex | 42 +++++++++---------- 1 file changed, 21 insertions(+), 21 deletions(-) diff --git a/latex/thesis/chapters/decoding_techniques.tex b/latex/thesis/chapters/decoding_techniques.tex index ec86654..7014eeb 100644 --- a/latex/thesis/chapters/decoding_techniques.tex +++ b/latex/thesis/chapters/decoding_techniques.tex @@ -2,7 +2,7 @@ \label{chapter:decoding_techniques} In this chapter, the decoding techniques examined in this work are detailed. -First, an overview of of the general methodology of using optimization methods +First, an overview of the general methodology of using optimization methods for channel decoding is given. Afterwards, the specific decoding techniques themselves are explained. @@ -31,7 +31,7 @@ the \ac{ML} decoding problem:% .\end{align*}% % The goal is to arrive at a formulation, where a certain objective function -$f$ has to be minimized under certain constraints:% +$f$ must be minimized under certain constraints:% % \begin{align*} \text{minimize}\hspace{2mm} &f\left( \boldsymbol{c} \right)\\ @@ -44,7 +44,7 @@ constraints. In contrast to the established message-passing decoding algorithms, the viewpoint then changes from observing the decoding process in its tanner graph representation (as shown in figure \ref{fig:dec:tanner}) -to a spacial representation (figure \ref{fig:dec:spacial}), +to a spatial representation (figure \ref{fig:dec:spatial}), where the codewords are some of the edges of a hypercube. The goal is to find that point $\boldsymbol{c}$, which minimizes the objective function $f$. @@ -149,8 +149,8 @@ which minimizes the objective function $f$. \node[color=KITgreen, right=0cm of c] {$\boldsymbol{c}$}; \end{tikzpicture} - \caption{Spacial representation of a single parity-check code} - \label{fig:dec:spacial} + \caption{Spatial representation of a single parity-check code} + \label{fig:dec:spatial} \end{subfigure}% \caption{Different representations of the decoding problem} @@ -171,7 +171,7 @@ representation. To solve the resulting linear program, various optimization methods can be used. -Feldman at al. begin by looking at the \ac{ML} decoding problem% +Feldman et al. begin by looking at the \ac{ML} decoding problem% \footnote{They assume that all codewords are equally likely to be transmitted, making the \ac{ML} and \ac{MAP} decoding problems equivalent.}% % @@ -233,7 +233,7 @@ decoding, redefining the constraints in terms of the \text{codeword polytope} which represents the \textit{convex hull} of all possible codewords, i.e. the convex set of linear combinations of all codewords. However, since the number of constraints needed to characterize the codeword -polytope is exponential in the code length, this formulation is relaxed futher. +polytope is exponential in the code length, this formulation is relaxed further. By observing that each check node defines its own local single parity-check code, and thus its own \textit{local codeword polytope}, the \textit{relaxed codeword polytope} $\overline{Q}$ is defined as the intersection of all @@ -274,7 +274,7 @@ Figures \ref{fig:dec:poly:local1} and \ref{fig:dec:poly:local2} show the local codeword polytopes of each check node. Their intersection, the relaxed codeword polytope $\overline{Q}$, is shown in figure \ref{fig:dec:poly:relaxed}. -It can be seen, that the relaxed codeword polytope $\overline{Q}$ introduces +It can be seen that the relaxed codeword polytope $\overline{Q}$ introduces vertices with fractional values; these represent erroneous non-codeword solutions to the linear program and correspond to the so-called \textit{pseudocodewords} introduced in @@ -590,7 +590,7 @@ The resulting formulation of the relaxed optimization problem is the following:% \begin{itemize} \item Why ADMM? - \item Adaptive Linear Programming? + \item Adaptive linear programming? \item How ADMM is adapted to LP decoding \end{itemize} @@ -600,7 +600,7 @@ The resulting formulation of the relaxed optimization problem is the following:% \label{sec:dec:Proximal Decoding} Proximal decoding was proposed by Wadayama et. al as a novel formulation of -optimization based decoding \cite{proximal_paper}. +optimization-based decoding \cite{proximal_paper}. With this algorithm, minimization is performed using the proximal gradient method. In contrast to \ac{LP} decoding, the objective function is based on a @@ -624,7 +624,7 @@ The likelihood $f_{\boldsymbol{Y} \mid \boldsymbol{X}} \left( \boldsymbol{y} \mid \boldsymbol{x} \right) $ is a known function determined by the channel model. 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 $\mathcal{C}\left( \boldsymbol{H} \right)$. However, because in this case the considered domain is continuous, the prior \ac{PDF} cannot be ignored as a constant during the minimization @@ -653,9 +653,9 @@ the so-called \textit{code-constraint polynomial} is introduced:% The intention of this function is to provide a way to penalize vectors far from a codeword and favor those close to a codeword. In order to achieve this, the polynomial is composed of two parts: one term -representing the bibolar constraint, providing for a discrete solution of the +representing the bipolar constraint, providing for a discrete solution of the continuous optimization problem, and one term representing the parity -constraint, accomodating the role of the parity-check matrix $\boldsymbol{H}$. +constraint, accommodating the role of the parity-check matrix $\boldsymbol{H}$. The prior \ac{PDF} is then approximated using the code-constraint polynomial:% % \begin{align} @@ -666,7 +666,7 @@ The prior \ac{PDF} is then approximated using the code-constraint polynomial:% % The authors justify this approximation by arguing that for $\gamma \rightarrow \infty$, the approximation in equation -\ref{eq:prox:prior_pdf_approx} aproaches the original fuction in equation +\ref{eq:prox:prior_pdf_approx} approaches the original function in equation \ref{eq:prox:prior_pdf}. This approximation can then be plugged into equation \ref{eq:prox:vanilla_MAP} and the likelihood can be rewritten using the negative log-likelihood @@ -707,8 +707,8 @@ of \ref{eq:prox:objective_function} are considered separately: the minimization of the objective function occurs in an alternating fashion, switching between the negative log-likelihood $L\left( \boldsymbol{y} \mid \boldsymbol{x} \right) $ and the scaled -code-constaint polynomial $\gamma h\left( \boldsymbol{x} \right) $. -Two helper variables, $\boldsymbol{r}$ and $\boldsymbol{s}$ are introduced, +code-constraint polynomial $\gamma h\left( \boldsymbol{x} \right) $. +Two helper variables, $\boldsymbol{r}$ and $\boldsymbol{s}$, are introduced, describing the result of each of the two steps. The first step, minimizing the log-likelihood, is performed using gradient descent:% @@ -720,10 +720,10 @@ descent:% \label{eq:prox:step_log_likelihood} .\end{align}% % -For the second step, minimizig the scaled code-constraint polynomial, the +For the second step, minimizing the scaled code-constraint polynomial, the proximal gradient method is used and the \textit{proximal operator} of $\gamma h\left( \boldsymbol{x} \right) $ has to be computed. -It is then immediately approximalted with gradient-descent:% +It is then immediately approximated with gradient-descent:% % \begin{align*} \text{prox}_{\gamma h} \left( \boldsymbol{x} \right) &\equiv @@ -773,7 +773,7 @@ is% % Thus, the gradient of the negative log-likelihood becomes% \footnote{For the minimization, constants can be disregarded. For this reason, -it suffices to consider only the proportionality instead of the equality.}% +it suffices to consider only proportionality instead of equality.}% % \begin{align*} \nabla L \left( \boldsymbol{y} \mid \boldsymbol{x} \right) @@ -791,7 +791,7 @@ Allowing equation \ref{eq:prox:step_log_likelihood} to be rewritten as% One thing to consider during the actual decoding process, is that the gradient of the code-constraint polynomial can take on extremely large values. -In order to avoid numeric instability, an additional step is added, where all +To avoid numerical instability, an additional step is added, where all components of the current estimate are clipped to $\left[-\eta, \eta \right]$, where $\eta$ is a positive constant slightly larger than one:% % @@ -803,7 +803,7 @@ where $\eta$ is a positive constant slightly larger than one:% $\Pi_{\eta}\left( \cdot \right) $ expressing the projection onto $\left[ -\eta, \eta \right]^n$. -The iterative decoding process resulting from these considreations is shown in +The iterative decoding process resulting from these considerations is shown in figure \ref{fig:prox:alg}. \begin{figure}[H]