diff --git a/compression.tex b/compression.tex new file mode 100644 index 0000000..e69de29 diff --git a/correction.tex b/correction.tex new file mode 100644 index 0000000..e69de29 diff --git a/crypto.tex b/crypto.tex new file mode 100644 index 0000000..e69de29 diff --git a/entropy.bib b/entropy.bib new file mode 100644 index 0000000..e73b5b9 --- /dev/null +++ b/entropy.bib @@ -0,0 +1,35 @@ + @misc{ enwiki:shannon-hartley, + author = "{Wikipedia contributors}", + title = "Shannon–Hartley theorem --- {Wikipedia}{,} The Free Encyclopedia", + year = "2025", + howpublished = "\url{https://en.wikipedia.org/w/index.php?title=Shannon%E2%80%93Hartley_theorem&oldid=1316080633}", + note = "[Online; accessed 29-October-2025]" + } + @misc{ enwiki:noisy-channel, + author = "{Wikipedia contributors}", + title = "Noisy-channel coding theorem --- {Wikipedia}{,} The Free Encyclopedia", + year = "2025", + howpublished = "\url{https://en.wikipedia.org/w/index.php?title=Noisy-channel_coding_theorem&oldid=1285893870}", + note = "[Online; accessed 29-October-2025]" + } + @misc{ enwiki:source-coding, + author = "{Wikipedia contributors}", + title = "Shannon's source coding theorem --- {Wikipedia}{,} The Free Encyclopedia", + year = "2025", + howpublished = "\url{https://en.wikipedia.org/w/index.php?title=Shannon%27s_source_coding_theorem&oldid=1301398440}", + note = "[Online; accessed 29-October-2025]" + } + @misc{ dewiki:nyquist-shannon, + author = "Wikipedia", + title = "Nyquist-Shannon-Abtasttheorem --- Wikipedia{,} die freie Enzyklopädie", + year = "2025", + url = "\url{https://de.wikipedia.org/w/index.php?title=Nyquist-Shannon-Abtasttheorem&oldid=255540066}", + note = "[Online; Stand 29. Oktober 2025]" + } + @misc{ enwiki:information-content, + author = "{Wikipedia contributors}", + title = "Information content --- {Wikipedia}{,} The Free Encyclopedia", + year = "2025", + howpublished = "\url{https://en.wikipedia.org/w/index.php?title=Information_content&oldid=1313862600}", + note = "[Online; accessed 29-October-2025]" + } diff --git a/entropy.tex b/entropy.tex index 2586608..5908164 100644 --- a/entropy.tex +++ b/entropy.tex @@ -10,6 +10,8 @@ \usepackage{amsmath} \PassOptionsToPackage{hyphens}{url} \usepackage{hyperref} % allows urls to follow line breaks of text +\usepackage[style=ieee, backend=biber, maxnames=1, minnames=1]{biblatex} +\addbibresource{entropy.bib} @@ -147,7 +149,10 @@ as a cost function in machine learning % relation to hamming distance and efficient codes \subsection{Noisy communication channels} -Given a model of +The noisy channel coding theorem was stated by \textit{Claude Shannon} in 1948, but first rigorous proof was +provided in 1954 by Amiel Feinstein. +It is of foundational to information theory, stating that given a noisy channel with capacity $C$ +and information transmitted at $R$ \cite{enwiki:shannon-hartley} \begin{figure}[H] \begin{tikzpicture} @@ -173,4 +178,31 @@ Given a model of \label{fig:noisy-channel} \end{figure} +\begin{figure}[H] +\begin{tikzpicture} + \def\boxw{2.5cm} + \def\n{4} + \pgfmathsetmacro{\gap}{(\textwidth - \n*\boxw)/(\n-1)} + \node (S) at (0,0) [draw, align=center, text width=\boxw] {Information Source}; + \node (S0) at (\boxw + \gap,1) [draw, circle] {0}; + \node (S1) at (\boxw + \gap,-1) [draw, circle] {1}; + \node (D0) at ({2*(\boxw + \gap)},1) [draw, circle] {0}; + \node (D1) at ({2*(\boxw + \gap)},-1) [draw, circle] {1}; + \node (D) at ({3*(\boxw + \gap)},0) [draw, align=center, text width=\boxw] {Destination}; + + \draw[->] (S) -- (S0); + \draw[->] (S) -- (S1); + + \draw[->,dashed] (S0) -- (D0) node[midway, above] {$p$}; + \draw[->,dashed] (S0) -- (D1) node[pos=0.8, above] {$1-p$}; + \draw[->,dashed] (S1) -- (D0) node[pos= 0.2, above] {$1-p$}; + \draw[->,dashed] (S1) -- (D1) node[midway, below] {$p$}; + + \draw[->] (D0) -- (D); + \draw[->] (D1) -- (D); +\end{tikzpicture} +\caption{Model of a binary symmetric channel} +\label{fig:binary-channel} +\end{figure} + \end{document} \ No newline at end of file