s works and founding theorems. Upon completion, we will then be equipped to approach the field of quantum information theory, which represents the second part of this series of chapters. Gaussian channel Referring to Chapter 6, a continuous communications channel assumes a continuous originator source, X, whose symbol alphabet X1 , Xi can be viewed as representing time samples of a continuous, real variable X, which is associated with a continuous probability distribution function or PDF, p(x). The variable x can be conceived as representing the amplitude of some signal waveform (e.g., electromagnetic, electrical, optical, radio, acoustical). This leads one to introduce a new parameter, which is the signal power. This is the power associated with the physical waveform used to propagate the symbols through a transmission pipe. In any physical communication channel, the signal power, or more specifically the average signal power P, represents a practical Hartley established that, given a peak voltage and accuracy (noise) AS = N, the number of distinguishable pulses is m = 1 + S/N. Taking the logarithm of this result provides a measure of maximum available information: this is Hartley's law. See: http://en.wikipedia.org/wiki/Shannon%E2%80%93Hartley_theorem. 14.1 Gaussian channel constraint, which (as we shall see) must be taken into account to evaluate the channel capacity. As a well-known property, the power of a signal waveform with amplitude X is pro- portional to the square of the amplitude x2. Overlooking the proportionality constant, the corresponding average power (as averaged over all possible symbols) is, thus, given (14.1) Chapter 6 described the concept and properties of differential entropy, which is the entropy H(X) of a continuous source X. It is defined by an integral instead of a discrete sum as in the discrete channel, according to: (14.2) As a key result from this chapter, it was established that under the average-power constraint P, the Gaussian PDF is the one (and the only one) for which the sour