ce entropy H(X) is maximal. No other continuous source with the same average power provides greater entropy, or average symbol uncertainty, or mean information contents. In particular, it was shown that this maximum source entropy, Hmax(X), reduces to the simple closed-form expression: (14.3) where o2 is the variance of the source PDF. By sampling the signal waveform, the originating continuous source is actually trans- formed into a discrete source. Thus the definition of discrete entropy must be used, instead of the differential or continuous definition in Eq. (14.2), but the result turns out to be strictly identical.² In the following, we will, therefore, consider the Gaussian 2 As can be shown through the following. Assume the Gaussian PDF for the source X where x is the symbol-waveform amplitude, and (x2) the symbol variance, with Ps being the average symbol or waveform power. By definition, the source entropy corresponding to discrete samples Xi which can be developed into The samples Xi can be chosen sufficiently close for the following approximations to be valid: () = 1, and = obtain Gaussian channel and Shannon-Hartley theorem channel to be discrete, which allows for considerable simplification in the entropy com- putation. However, I shall develop in footnotes or as exercises the same computations while using continuous-channel or differential entropy definitions, and show that the results are strictly identical. The conclusion from this observation is that there is no need to assume that the Gaussian channel uses a discrete alphabet or samples of a continuous source. The Gaussian channel is defined as a (discrete or continuous) communication channel that uses a Gaussian-alphabet source as the input, and has an intrinsic additive noise, also characterized by a noise source Z with Gaussian PDF. The channel noise is said to be additive, because given the input symbol X (or Xi, as sampled at time i), the output symbol y (or yi) is given by the sum y=x+z, (14.4) yi=xi+zi, where Z (or Zi) is the amplitude of