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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}
.\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*}
\left[ m:n \right] &:= \left\{ m, m+1, \ldots, n-1, n \right\},
\hspace{5mm} m < n, \hspace{2mm} m,n\in\mathbb{Z}
.\end{align*}
\todo{Not really slicing. How should it be denoted?}
%
In order to designate elemen-twise operations, in particular the \textit{Hadamard product}
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}
.\end{align*}
%
The transmitted symbol is distorted by the channel and denoted by
The transmitted symbol is distorted by the channel and denoted as
$\boldsymbol{y} \in \mathbb{R}^n$.
This distortion is described by the channel model, which in the context of
this thesis is chosen to be \ac{AWGN}:%
@ -170,17 +169,23 @@ criterion:%
\right)
.\end{align*}%
%
The \ac{MAP}- and \ac{ML}-criteria are closely connected through
\textit{Bayes' theorem}:%
The two criteria are closely connected through Bayes' theorem and are equivalent
when the prior probability of transmitting a codeword is the same for all
codewords:
%
\begin{align*}
\argmax_{c\in\mathcal{C}} p_{\boldsymbol{C} \mid \boldsymbol{Y}}
\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*}
%
This has the consequence that if the probability \ldots, the two criteria are
equivalent.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%