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The mutual information obtained on the symbol xi when the symbol yj is received is given by:

[2.47]

where i(xi, yj) represents the decrease in uncertainty on xi due to the receipt of yj.

The average value of the mutual information, or the amount of information I(X, Y) transmitted through the channel is:

[2.48]

or:

[2.49]

Hence:

[2.50]

Given relationships [2.36] and [2.37], we get:

[2.51]

[2.52]

[2.53]

Special cases.

 – Noiseless channel: X and Y symbols are linked, so:I(X, Y)=H(X)=H(Y)

 – Channel with maximum power noise: X and Y symbols are independent, therefore:I(X, Y) = 0