Characterizing the discrete capacity achieving distribution with peak power constraint at the transition points

The capacity-achieving input distribution for many channels like the additive white Gaussian noise (AWGN) channels under a peak power constraint is discrete with a finite number of mass points. The number of mass points is itself a variable and figuring it out is part of the optimization problem. We wish to understand the behavior of the optimal input distribution at the transition points where the number of mass points changes. We give a new set of necessary and sufficient conditions at the transition points, which offer new insights into the transition and make the computation of the optimal distribution easier. These conditions can be simplified, at least in the real channel case, by assuming the conjecture that the number of mass points increases monotonically and by at most one as the constraint on the input is relaxed. In particular, we show that for zero mean, unit variance Gaussian noise, the peak amplitude A of 1:671 and 2:786 mark the points where the binary and ternary signaling respectively are no longer optimal.