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The basic mathematical model for a communication system is the following:
:<math title="Channel model">\xrightarrow[\text{Message}]{W}
\begin{array}{ |c| }\hline \text{Encoder} \\ f_n \\ \hline\end{array} \xrightarrow[\mathrm{Encoded \atop sequence}]{X^n} \begin{array}{ |c| }\hline \text{Channel} \\ p(y|x) \\ \hline\end{array} \xrightarrow[\mathrm{Received \atop sequence}]{Y^n} \begin{array}{ |c| }\hline \text{Decoder} \\ g_n \\ \hline\end{array} \xrightarrow[\mathrm{Estimated \atop message}]{\hat W}</math>
A '''message''' ''W'' is transmitted through a noisy channel by using encoding and decoding functions. An '''encoder''' maps ''W'' into a pre-defined sequence of channel symbols of length ''n''. In its most basic model, the channel distorts each of these symbols independently of the others. The output of the channel –the received sequence– is fed into a '''decoder''' which maps the sequence into an estimate of the message. In this setting, the probability of error is defined as:
::<math> P_e = \text{Pr}\left\{ \hat{W} \neq W \right\}. </math>
'''Theorem''' (Shannon, 1948):
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