orm limit H(Y) = 1, regardless of the input probability (q). The case E = 0.5 corresponds to a maximum uncertainty in the output symbols, i.e., according to Eq. (12.13):p(y1)=q+-2eq= 0.5 = p(y2), which is independent of the input probability distribution. It is correct to call this communication channel useless, because the output symbols are 100% uncorrelated with the input symbols. Second, we observe from the result in Eq. (12.15) that the equivocation H(Y|X) = f(e) is independent of the input probability distribution. This was expected, since equivocation defines the uncertainty of Y given the knowledge of X. It represents a measure of the channel's noise, i.e., H(Y|X) = f(e). In the limit E 0, we have H(Y|X) 0, which means that the communication channel (CC) is ideal or noiseless. The transition matrix of the ideal or noiseless channel is defined in Eq. (12.3). In this case, Eqs. (12.14)-(12.17) show that all entropy measures are equal, i.e., H(X) = H(Y) = H(X, Y) = H(X;Y) = f(q). Third, we consider the mutual information H(X; Y) of the binary symmetric channel, which is defined in Eq. (12.17). Figure 12.7 shows plots of H(Y; X) as a function of the input probability parameter q and the channel noise parameter E for E = 0 to E = 0.35 (noting that the same curves are obtained with E 1-E). The input source entropy H(X) is also shown for reference. We observe from Fig. 12.7 that the mutual information H(X; Y) is never greater than the input source entropy H(X). This is expected, since, by definition, H(X;) Y) = H(X) - H(X|Y) < H(X), Eq. (12.11), with the upper bound a result of the fact that H(X|Y) > 0. When the channel is noiseless or ideal (E = 0), the mutual information equals the source entropy and, as we have seen earlier, all entropy measures are equal: 12.3 Channel entropy and mutual information Probability q Figure 12.7 Mutual information H(X;) of binary symmetric channel, as a function of the probability q = p(x1) and channel noise parameter E. The input source entropy H(X) is also shown. H(X)=H(Y)=H(X,