From 5989c336217c07d3ba01c49fd691e39b52ae986b Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Tue, 11 Apr 2023 20:52:25 +0200 Subject: [PATCH] Fixed spelling and grammatical errors in theory chapter --- .../chapters/theoretical_background.tex | 36 +++++++++---------- 1 file changed, 18 insertions(+), 18 deletions(-) diff --git a/latex/thesis/chapters/theoretical_background.tex b/latex/thesis/chapters/theoretical_background.tex index b8d10e3..75801a5 100644 --- a/latex/thesis/chapters/theoretical_background.tex +++ b/latex/thesis/chapters/theoretical_background.tex @@ -3,7 +3,7 @@ In this chapter, the theoretical background necessary to understand this work is given. -First, the used notation is clarified. +First, the notation used is clarified. The physical aspects are detailed - the used modulation scheme and channel model. A short introduction to channel coding with binary linear codes and especially \ac{LDPC} codes is given. @@ -33,7 +33,7 @@ of indexed variables:% x_{\left[ m:n \right] } &:= \left\{ x_m, x_{m+1}, \ldots, x_{n-1}, x_n \right\} .\end{align*} % -In order to designate elemen-twise operations, in particular the\textit{Hadamard product} +In order to designate elemen-twise operations, in particular the \textit{Hadamard product} and the \textit{Hadamard power}, the operator $\circ$ will be used:% % \begin{alignat*}{3} @@ -82,7 +82,7 @@ Encoding the information using \textit{binary linear codes} is one way of conducting this process, whereby \textit{data words} are mapped onto longer \textit{codewords}, which carry redundant information. \Ac{LDPC} codes have become especially popular, since they are able to -reach arbitrarily small probabilities of error at coderates up to the capacity +reach arbitrarily small probabilities of error at code rates up to the capacity of the channel \cite[Sec. II.B.]{mackay_rediscovery} while having a structure that allows for very efficient decoding. @@ -252,13 +252,13 @@ Figure \ref{fig:theo:tanner_graph} shows the tanner graph for the \label{fig:theo:tanner_graph} \end{figure}% % -\noindent \acp{CN} and \acp{VN}, and by extention the rows and columns of +\noindent \acp{CN} and \acp{VN}, and by extension the rows and columns of $\boldsymbol{H}$, are indexed with the variables $j$ and $i$. The sets of all \acp{CN} and all \acp{VN} are denoted by $\mathcal{J} := \left[ 1:m \right]$ and $\mathcal{I} := \left[ 1:n \right]$, respectively. -The \textit{neighbourhood} of the $j$th \ac{CN}, i.e., the set of all adjacent \acp{VN}, +The \textit{neighborhood} of the $j$th \ac{CN}, i.e., the set of all adjacent \acp{VN}, is denoted by $N_c\left( j \right)$. -The neighbourhood of the $i$th \ac{VN} is denoted by $N_v\left( i \right)$. +The neighborhood of the $i$th \ac{VN} is denoted by $N_v\left( i \right)$. For the code depicted in figure \ref{fig:theo:tanner_graph}, for example, $N_c\left( 1 \right) = \left\{ 1, 3, 5, 7 \right\}$ and $N_v\left( 3 \right) = \left\{ 1, 2 \right\}$. @@ -282,7 +282,7 @@ the use of \ac{BP} impractical for applications where a very low \ac{BER} is desired \cite[Sec. 15.3]{ryan_lin_2009}. Another popular decoding method for \ac{LDPC} codes is the \textit{min-sum algorithm}. -This is a simplification of \ac{BP} using an approximation of the the +This is a simplification of \ac{BP} using an approximation of the non-linear $\tanh$ function to improve the computational performance. @@ -478,13 +478,13 @@ and minimizing $g$ using the proximal operator % Since $g$ is minimized with the proximal operator and is thus not required to be differentiable, it can be used to encode the constraints of the problem -(e.g., in the form of an \textit{indicator funcion}, as mentioned in +(e.g., in the form of an \textit{indicator function}, as mentioned in \cite[Sec. 1.2]{proximal_algorithms}). The \ac{ADMM} is another optimization method. In this thesis it will be used to solve a \textit{linear program}, which is a special type of convex optimization problem, where the objective function -is linear and the constraints consist of linear equalities and inequalities. +is linear, and the constraints consist of linear equalities and inequalities. Generally, any linear program can be expressed in \textit{standard form}% \footnote{The inequality $\boldsymbol{x} \ge \boldsymbol{0}$ is to be interpreted componentwise.} @@ -499,7 +499,7 @@ interpreted componentwise.} \label{eq:theo:admm_standard} \end{alignat}% % -A technique called \textit{lagrangian relaxation} \cite[Sec. 11.4]{intro_to_lin_opt_book} +A technique called \textit{Lagrangian relaxation} \cite[Sec. 11.4]{intro_to_lin_opt_book} can then be applied. First, some of the constraints are moved into the objective function itself and weights $\boldsymbol{\lambda}$ are introduced. A new, relaxed problem @@ -515,7 +515,7 @@ is then formulated as \label{eq:theo:admm_relaxed} \end{align}% % -the new objective function being the \textit{lagrangian}% +the new objective function being the \textit{Lagrangian}% % \begin{align*} \mathcal{L}\left( \boldsymbol{x}, \boldsymbol{\lambda} \right) @@ -525,10 +525,10 @@ the new objective function being the \textit{lagrangian}% .\end{align*}% % This problem is not directly equivalent to the original one, as the -solution now depends on the choice of the \textit{lagrange multipliers} +solution now depends on the choice of the \textit{Lagrange multipliers} $\boldsymbol{\lambda}$. Interestingly, however, for this particular class of problems, -the minimum of the objective function (herafter called \textit{optimal objective}) +the minimum of the objective function (hereafter called \textit{optimal objective}) of the relaxed problem (\ref{eq:theo:admm_relaxed}) is a lower bound for the optimal objective of the original problem (\ref{eq:theo:admm_standard}) \cite[Sec. 4.1]{intro_to_lin_opt_book}:% @@ -599,7 +599,7 @@ $g_i: \mathbb{R}^{n_i} \rightarrow \mathbb{R}$, i.e., $g\left( \boldsymbol{x} \right) = \sum_{i=1}^{N} g_i \left( \boldsymbol{x}_i \right)$, where $\boldsymbol{x}_i,\hspace{1mm} i\in [1:N]$ are subvectors of -$\boldsymbol{x}$, the lagrangian is as well: +$\boldsymbol{x}$, the Lagrangian is as well: % \begin{align*} \text{minimize }\hspace{5mm} & \sum_{i=1}^{N} g_i\left( \boldsymbol{x}_i \right) \\ @@ -638,18 +638,18 @@ This modified version of dual ascent is called \textit{dual decomposition}: .\end{align*} % -The \ac{ADMM} works the same way as dual decomposition. -It only differs in the use of an \textit{augmented lagrangian} +\ac{ADMM} works the same way as dual decomposition. +It only differs in the use of an \textit{augmented Lagrangian} $\mathcal{L}_\mu\left( \boldsymbol{x}_{[1:N]}, \boldsymbol{\lambda} \right)$ in order to strengthen the convergence properties. -The augmented lagrangian extends the ordinary one with an additional penalty term +The augmented Lagrangian extends the ordinary one with an additional penalty term with the penaly parameter $\mu$: % \begin{align*} \mathcal{L}_\mu \left( \boldsymbol{x}_{[1:N]}, \boldsymbol{\lambda} \right) = \underbrace{\sum_{i=1}^{N} g_i\left( \boldsymbol{x_i} \right) + \boldsymbol{\lambda}^\text{T}\left( \boldsymbol{b} - - \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x}_i \right)}_{\text{Ordinary lagrangian}} + - \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x}_i \right)}_{\text{Ordinary Lagrangian}} + \underbrace{\frac{\mu}{2}\left\Vert \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x}_i - \boldsymbol{b} \right\Vert_2^2}_{\text{Penalty term}}, \hspace{5mm} \mu > 0