(namely, or y2). It does not matter which originator symbol is fed into the channel, the recipient symbol output being indifferent, as in a perfect coin-flipping guess. In this case, the channel is said to be useless. Nonbinary and asymmetric discrete channels In this section, I provide a few illustrative examples of discrete communication channels, which, contrary to those described in the previous section, are asymmetric, and in some cases, nonbinary. These are referred to as the Z channel, the binary erasure channel, the noisy typewriter, and the asymmetric channel with nonoverlapping outputs. Example 12.1: Z channel This is illustrated in Fig. 12.2. The Z channel is a binary channel in which one of the two originator symbols (say, x1) is unaffected by noise and ideally converted into a unique recipient symbol (say, y1), which implies p(y1/x1) = 1 and p(y2/x1) = 0. But the channel noise affects the other symbols, giving p(y1/x2) = E and p(y2/x2) E. 12.2 Nonbinary and asymmetric discrete channels p(12)x1=0 p(12)x2)=1-8 Originator Recipient source source Noisy channel Figure 12.2 Z channel showing the originator source X = {x1,x2} the recipient source = {y1, y2}, the conditional probabilities p(yj|xi), and the corresponding transition matrix P(Y|X). The corresponding transition matrix is, therefore: (12.5) Note that stating that symbol X1 is ideally transmitted through the channel and received as symbol y1, does not mean that receiving symbol entails certainty of symbol X1 being output by the originator. Example 12.2: Binary erasure channel This is illustrated in Fig. 12.3. In the erasure channel, there exists a finite probability that the recipient source Y outputs none of the symbols from the originator source X. This is equivalent to stating that Y includes a supplemental symbol y3 = "void," which we shall call . We, thus, define the probability of Y having y3 = 0 for output as p(0/x1) = p(0/x2) = E. It is easily established that the transition matrix corresponding to input P(X) = [p(x1), I p(x2)] and output 