Transition points in the capacity-achieving distribution for free-space optical intensity channels

The capacity-achieving input distribution for the free-space optical intensity channels with the peak power constraint with or without the average power constraint is known to be discrete. We are interested in the structure of the capacity-achieving distribution at the transition points where the number of mass points increases. We give the necessary and sufficient conditions for the transition to occur. We consider two regimes depending on whether the average power constraint is active or not, and both the regimes have the peak power constraint. For the regime where the average power constraint is active, we give explicit conditions for the binary to ternary transition. For the regime where the average power constraint is inactive, we show that our previous work on the structure of the capacity-achieving distribution at the transition points for the Smith's case with only the peak-power constraint is applicable.