\documentclass[12pt]{article}
% version = 1.00 of latexdemo.tex 2011 Feb 01

% Thomas D. Schneider, Ph.D.
% toms@alum.mit.edu
% toms@alum.mit.edu (permanent)
% https://alum.mit.edu/www/toms (permanent)

\usepackage{html}

\begin{document}

\bibliographystyle{unsrt}  % define bibliography style

\begin{center}
\textbf{\large Shannon's Fundamental Equations}
\\
by Tom Schneider
\\
2011 Feb 01
\end{center}

\vspace{12pt}

Here's an equation from Shannon's famous 1948 paper \cite{Shannon1948}:
\begin{equation}
H = - \sum_{i=1}^M p_i \log_2 p_i
\;\;\;\;\;\mbox{bits per second}
\label{eqn.H}
\end{equation}
where $p_i$ are the probabilities of symbols in a message. 
Shannon used that equation for the \emph{uncertainty}
to define information as the decrease of uncertainty
at the receiver
from before receiving the message to after:
\begin{equation}
R = H_{\mbox{before}}
  - H_{\mbox{after}}
\;\;\;\;\;\mbox{bits per second}
\label{eqn.R}
\end{equation}

Then in 1949 Shannon used a beautiful geometric method
to derive the channel capacity equation \cite{Shannon1949}:
\begin{equation}
C = W \log_2 \left( \frac{P}{N} + 1\right)
\;\;\;\;\;\mbox{bits per second}
\label{eqn.C}
\end{equation}
where
$W$ is the bandwidth of the communication,
$P$ is the power received
$N$ is the noise at the receiver.

Finally, Shannon related the information $R$
(equation \ref{eqn.R})
to the channel capacity
(equation \ref{eqn.C})
in the famous channel capacity theorem.
This theorem says that $R$ cannot exceed $C$.
But if
$R \le C$
then one may have as few errors in the communication as desired.
Even Shannon found this surprising
\cite{Shannon1949}.

\bibliography{latexdemo.bib} % show the bibliography

\end{document}
