hapter 11. To recall briefly, a communication channel is a transmission means for encoded information. Its constituents are an originator source (generating message symbols), an encoder, a transmitter, a physical transmission pipe, a receiver, a decoder, and a recipient source (restituting message symbols). The two sources (originator and recipient) may be discrete or continuous. The encoding and decoding scheme may include not only symbol-to- codeword conversion and the reverse, but also data compression and error correction, which we will not be concerned with in this chapter. Here, we shall consider binary channels. These include two binary sources X = {x1, x2} as the originator and Y = (y1,12) as the recipient, with associated probability distributions p(xj) and p(yi), i = 1, 2. The events are discrete (as opposed to continuous); hence binary channels are one elementary variant of discrete channels. A binary channel can be represented schematically through the diagram shown in Fig. 12.1. Each of the four paths corresponds to the event that, given a symbol Xi from originator source X, a symbol yj is output from the recipient source Y. The conditional probability of this event is p(yj|xi). 12.1 Binary symmetric channel p(y1x2) =E Originator Recipient source source Noisy channel Figure 12.1 Binary symmetric channel showing the originator source X = {x1,x2}, the recipient source Y = {y1,y2}, the conditional probabilities p(yj|xi), and the corresponding transition matrix P(Y)X). In the ideal case of a noiseless channel, the two sources are 100% correlated,¹ i.e., (12.1) Kronecker symbol. In this case, the channel is ideal indeed, since the transmission is 100% deterministic, which causes no corruption of information. The set of conditional probabilities p(y)|xj) associated with the channel defines a j x i transition matrix. This matrix, noted P(Y)X), can be written as2 (12.2) 1 Since there is no reason to have a one-to-one correspondence between the symbols or indices, we can also have p(yj|xi) = 1 for i # j and p