 the channel noise. Further, it is assumed that: (a) The average noise amplitude is zero, i.e., (z) = 0; (b) The noise power is finite, with variance = N; (c) There exists no correlation between the noise and the input signal. These three conditions make it possible to calculate the average output power of of the Gaussian channel, as follows: =(x2)+2(x)(z) + ( (14.5) where = = P is the (average) input signal power. Consistently with the defi- nition in Eq. (14.3), the entropies of output source Y and the noise source Z are given (14.6) (14.7) respectively. The mutual information of the Gaussian channel is defined as: H(X;Y)=H(X)-H(XY): - = H(Y) - H(Y|X), (14.8) where H(Y|X) is the equivocation or conditional entropy. Given a Gaussian-distributed variable x + where X, Z are independent, the conditional probability of measuring 14.1 Gaussian channel y at the output given that x is the input is p(y/x) = p(z). Substituting this property in the definition of conditional entropy H(Y|X), we obtain: p(Zi) (14.9) p(Zi) = H(Z) = log / 2reo 2. As previously stated, the same result as in Eq. (14.9) can be obtained using the differential or continuous definition of conditional entropy, i.e., to recall from Chapter 6: (14.10) The computation of the double integral in Eq. (14.10) is elementary, albeit relatively tedious to carry out, SO we shall leave it here as a "math" exercise. We consider next the capacity of the Gaussian channel. By definition, the channel capacity C is given by the maximum of the mutual information H(X; Y) = H(Y) - H(Y|X). Using the previous results in Eqs. (14.6) and (14.7) for H(Y) and H(Y|X) = H(Z), we obtain: H(X;Y)=H(Y)-H(YX)=H(Y) = H(Z) (14.11) and using (14.12) The mutual information H(X; Y) defined in Eq. (14.12) also corresponds to the channel capacity C. This is because (given the signal power constraint P) the Gaussian distribu- tion maximizes the source entropies H(X) or H(Y), making H(X; Y) an upper bound for mutual information. Thus, we can write, equivalently: (14.13) In the above, the quan