The Information Capacity of Amplitude- and Variance-Constrained Scalar Gaussian Channels

The amplitude-constrained capacity of a scalar Gaussian channel is shown to be achieved by a unique discrete random variable taking on a finite number of values. Necessary and sufficient conditions for the distribution of this random variable are obtained. These conditions permit determination of the random variable and capacity as a function of the constraint value. The capacity of the same Gaussian channel subject, additionally, to a nontrivial variance constraint is also shown to be achieved by a unique discrete random variable taking on a finite number of values. Likewise, capacity is determined as a function of both amplitudeand variance-constraint values.