On the discreteness of the capacity-achieving probability measure of conditional gaussian channels

A conditional Gaussian (CG) channel is a discrete-time memoryless channel such that an admissible channel input gives rise to a channel output that is Gaussian distributed with an expectation vector and a covariance matrix. In this paper, closed and bounded subset of receiver, and the channel constraint is called the bounded-input constraint is presented. The mutual information between the channel input and output with an input probability measure which satisfies the channel input constraints is maximized. An input probability measure is said to be discrete in amplitude and uniform in phase (DAUP) if the probability distribution of the amplitude is discrete with a finite number of probability mass points, and the phase is uniformly distributed