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Minor wording changes; Added MAP and ML equivalence note

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Andreas Tsouchlos 2023-04-19 11:30:03 +02:00
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commit 894d80ca90
1 changed files with 13 additions and 8 deletions
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21
latex/thesis/chapters/theoretical_background.tex
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@ -24,13 +24,12 @@ For example:%
c \in \mathbb{F}_2 &\to \tilde{c} \in \left[ 0, 1 \right] \subseteq \mathbb{R} c \in \mathbb{F}_2 &\to \tilde{c} \in \left[ 0, 1 \right] \subseteq \mathbb{R}
.\end{align*} .\end{align*}
% %
Additionally, a shorthand notation will be used to denote a set of indices:% Additionally, a shorthand notation will be used, denoting a set of indices as%
% %
\begin{align*} \begin{align*}
\left[ m:n \right] &:= \left\{ m, m+1, \ldots, n-1, n \right\}, \left[ m:n \right] &:= \left\{ m, m+1, \ldots, n-1, n \right\},
\hspace{5mm} m < n, \hspace{2mm} m,n\in\mathbb{Z} \hspace{5mm} m < n, \hspace{2mm} m,n\in\mathbb{Z}
.\end{align*} .\end{align*}
\todo{Not really slicing. How should it be denoted?}
% %
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:% and the \textit{Hadamard power}, the operator $\circ$ will be used:%
@ -58,7 +57,7 @@ This is known as modulation. The modulation scheme chosen here is \ac{BPSK}:%
\boldsymbol{x} = 1 - 2\boldsymbol{c} \boldsymbol{x} = 1 - 2\boldsymbol{c}
.\end{align*} .\end{align*}
% %
The transmitted symbol is distorted by the channel and denoted by The transmitted symbol is distorted by the channel and denoted as
$\boldsymbol{y} \in \mathbb{R}^n$. $\boldsymbol{y} \in \mathbb{R}^n$.
This distortion is described by the channel model, which in the context of This distortion is described by the channel model, which in the context of
this thesis is chosen to be \ac{AWGN}:% this thesis is chosen to be \ac{AWGN}:%
@ -170,17 +169,23 @@ criterion:%
\right) \right)
.\end{align*}% .\end{align*}%
% %
The \ac{MAP}- and \ac{ML}-criteria are closely connected through The two criteria are closely connected through Bayes' theorem and are equivalent
\textit{Bayes' theorem}:% when the prior probability of transmitting a codeword is the same for all
codewords:
% %
\begin{align*} \begin{align*}
\argmax_{c\in\mathcal{C}} p_{\boldsymbol{C} \mid \boldsymbol{Y}} \argmax_{c\in\mathcal{C}} p_{\boldsymbol{C} \mid \boldsymbol{Y}}
\left( \boldsymbol{c} \mid \boldsymbol{y} \right) \left( \boldsymbol{c} \mid \boldsymbol{y} \right)
= TODO &= \argmax_{c\in\mathcal{C}} \frac{f_{\boldsymbol{Y} \mid \boldsymbol{C}}
\left( \boldsymbol{y} \mid \boldsymbol{c} \right) p_{\boldsymbol{C}}
\left( \boldsymbol{c} \right)}{f_{\boldsymbol{Y}}\left( \boldsymbol{y} \right) } \\
&= \argmax_{c\in\mathcal{C}} f_{\boldsymbol{Y} \mid \boldsymbol{C}}
\left( \boldsymbol{y} \mid \boldsymbol{c} \right) p_{\boldsymbol{C}}
\left( \boldsymbol{c} \right) \\
&= \argmax_{c\in\mathcal{C}}f_{\boldsymbol{Y} \mid \boldsymbol{C}}
\left( \boldsymbol{y} \mid \boldsymbol{c} \right)
.\end{align*} .\end{align*}
% %
This has the consequence that if the probability \ldots, the two criteria are
equivalent.
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