Transition points in the capacity-achieving distribution for the peak-power limited AWGN and free-space optical intensity channels

The capacity-achieving input distribution for many channels like the additive white Gaussian noise (AWGN) channel and the free-space optical intensity (FSOI) channel under the 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 a 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. To this end, 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. For the real AWGN channel case, we show that for the 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. For the FSOI channel, we give transition points where binary gives way to ternary, and in some cases where ternary gives way to quaternary, in the presence of the peak-power constraint and with or without the average-power constraint.