We consider the information transmission channel of memoryless binary symmetrical type of Figure 1.1.
Figure 1.1.Basic diagram of a digital communication
It is assumed that the signal-to-noise ratio leads to the following values of conditional probabilities of errors:
The source of binary information is considered to emit independent information with the following probabilities:
1) Calculate the source entropy H(X).
2) Calculate the entropy H(Y) at the receiver end.
3) Calculate the conditional entropy H(Y/X) (entropy of transmission error).
4) Calculate the loss of information in the transmission channel H(X/Y).
5) Deduce the average amount of information received by the recipient for each binary symbol sent I(X, Y) (mutual information).
6) Determine the channel capacity C and show that it is obtained when p1 = 0.5.
Solution of problem 1
1) By definition, we have:
then:
2) By definition, we have:
and:
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