Fixed spelling and grammatical errors in theory chapter

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