been shown, 1 albeit without demonstration, that the channel capacity takes the form C = log(20 +2 ), (13.7) where (13.8) _(1-b)f(a)-af(b) 1 A. A. Bruen and M. A. Forcinito, Cryptography, Information Theory and Error-Correction (New York: John Wiley & Sons, 2005). Note that in this reference, the transition matrix is the transposed version of the one used here. 13.1 Channel capacity b=0.5 b=0.5 Parameter a Parameter a Figure 13.1 Optimal input probability distribution nq=p(x1)=1-p(x2),corresponding to capacity C of binary channel, plotted as a function of the two transition-matrix parameters a and b (step of 0.05). The demonstration of the above result, which is elementary but far from straightfor- ward, is provided in Appendix K. This Appendix also shows that the corresponding optimal distribution is given by p(x1) = q, p(x2) = 1 - q with the parameter q defined according to: (13.9) (13.10) It is straightforward to verify that in the case a = b=1- - E (binary symmetric channel), the channel capacity defined in Eqs. (13.7)-(13.8) reduces to C=1 - f(e) and the optimal input distribution reduces to the uniform distribution p(x1) = p(x2) = q = 1/2. The case a + b = 1, which seemingly corresponds to a pole in the above definitions, is trickier to analyze. Appendix K demonstrates that the functions U, V, W, and q all continuously defined in the limit a +1 1 (or for that matter, over the full plane a, b E [0, 1]). In the limit a + b 1, it is shown that H(Y;X) = C = 0, which corresponds to the case of the useless channel. The fact that we also find, in this case, q = 1/2 (uniform input distribution) is only a consequence of the continuity of the function q. As a matter of fact, there is no optimal input distribution in useless channels, and all possible input distributions yield C = 0. Figures 13.1 and 13.2 show 2D and 3D plots of the probability q = p(x1) = 1 - p(x2) as a function of the transition-matrix parameters a, b E [0, 1] (sampled in steps of 0.05), as defined from Eqs. (13.9) and (13.10). It is seen from the f