(yilxj) =0 for i=j. Thus, a perfect binary symmetric channel can equivalently have either of the transition matrixes (see definition in text): Since in the binary system, bit parity (which of the 1 or 0 received bits represents the actual 0 in a given code) is a matter of convention, the noiseless channel can be seen as having an identity transition matrix. According to the transition-matrix representation, one can also define for any source Z = X, Y the prob- ability vector P (Z) = [p(z1), p (z2)] to obtain the matrix-vector relation P (Y) = P (Y|X) P (X), which stems from the property of conditional probabilities: p(yj) = p(yj|xi)p(xi). Channel entropy To recall from Chapter 1, the properties of the transition-matrix elements are twofold: (a) p(yjlx1)p(x1)+p(yjlx2)p(x2)=p(yj); (b) p(y1lxi)+p(y2lxi)=1(column elements add to unity). The channel is said to be symmetric if its transition matrix P(Y|X) is symmetric, i.e., [P(Y|X)] [P(Y|X)] or the matrix elements are invariant by permutation of row and column indices, i.e., ,TP(Y) = P(Y|X). In the case of the ideal or noiseless channel, we have, according to Eqs. (12.1) and (12.2): P(Y)X) (12.3) which is the identity matrix. In the general, or nonideal, case, where the channel is corrupted by noise, we have p(yilxi) = 1 - E and, hence, p(yj#ilxi) = E, where E is a positive real number satisfying 0<8<1. The smaller E, the closer the channel is to ideal or noiseless. The degree of noise "corruption" is, thus, defined by the parameter E, which, as we shall see later in this chapter, defines the symbol error probability. According to the definition in Eq. (4.2), in the most general case the transition matrix of the binary symmetric channel takes the form: (12.4) A limiting case is given by the parameter value E = 0.5. In this case, the elements of the transition matrix in Eq. (12.4) are all equal to p(y; /xi) = 0.5. Thus, given the knowledge of the originator symbol Xi (namely, x1 or x2), there is an equal probability that the recipient will output any of the symbols yj 