7.5 Mutual information

Recall that

Mutual Information

The mutual information between two discrete random variables is denoted by $(X, Y)$ and defined as

$(X, Y)=H(X)-H(X|Y)$

Mutual information is a useful concept to measure the amount of information shared between input and output of noisychannels.

In our previous discussions it became clear that when the channel is noisy there may not be reliable communications.Therefore, the limiting factor could very well be reliability when one considers noisy channels. Claude E. Shannon in 1948changed this paradigm and stated a theorem that presents the rate (speed of communication) as the limiting factor as opposedto reliability.

This is the essence of Shannon's noisy channel coding theorem, i.e. , using only those inputs whose corresponding outputs are disjoint ( e.g. , far apart). The concept is appealing, but does not seem possible with binarychannels since the input is either zero or one. It may work if one considers a vector of binary inputs referred to as theextension channel.

$\mathrm{X\; input\; vector}=\left(\begin{array}{c}{X}_1\\ X_2\\ ⋮\\ X_n\end{array}\right)\in \overline{X}^n=\{0, 1\}^n$

$\mathrm{Y\; output\; vector}=\left(\begin{array}{c}{Y}_1\\ Y_2\\ ⋮\\ Y_n\end{array}\right)\in \overline{Y}^n=\{0, 1\}^n$

This module provides a description of the basic information necessary to understand Shannon's Noisy Channel Coding Theorem . However, for additional information on typical sequences, pleaserefer to Typical Sequences .