Y)=H(X;Y)=f(q) = = = As the channel noise (E) increases, the mutual information is seen to decrease with 0 < H(X; Y) < H(X). When the chan- nel noise reaches the limit E 0.5, the mutual information uniformly vanishes H(X; Y) 0, regardless of the input probability (q). This is the case of the useless channel. Finally, we observe from Fig. 12.7 that the mutual information is always maximum for q = 0.5, regardless of the channel noise E. The conclusion is that for noisy binary symmetric channels, the probability distribution that maximizes mutual information is the uniform distribution. The above demonstration, based on the example of the binary symmetric channel, illustrates that in noisy channels the entropy of the output source H(Y) is not a correct measure of the information obtained from the recipient's side. This is because H(Y) also measures the uncertainty introduced by the channel noise, which is not information. Thus, noise enhances the output source uncertainty without contributing any supple- mental information according to the definition in Chapter 3. The right measure of the information available to the recipient is mutual information, i.e., the uncertainty of the output source H(Y) minus the equivocation H(Y|X). As we have seen, equivocation is the absolute measure of channel noise, and it is independent of the input probability distribution. The second important conclusion we have reached is that mutual informa- tion depends on the input probability distribution. Hence, given a noisy channel, there must exist an input probability distribution for which the channel's mutual information is maximized, meaning that the information obtained by the recipient is maximized. This Channel entropy conclusion anticipates Shannon's definition of channel capacity, which will be described in Chapter 13. Symbol error rate In this last section, we consider channels with sources X, Y having equal sizes (i.e., numbers of symbol events), and analyze the effect of channel noise in the transmission of information from ori