M-APSK Constellation Optimization in the Presence of Phase Reference Error

Considering the analytical issue of
${M}$
-ary amplitude phase-shift keying design in the presence of a phase reference error (PRE), we present a systematic optimization formulation to first minimize the error floor of the optimum detector, which decides on the number of signal points on each ring for all possible numbers of rings. Then with a closed-form minimum symbol error probability formulation, we are able to determine the optimal number of rings and the corresponding ring radii for specific PRE variance and average signal-to-noise ratio. This approach involves purely numerical computation without time-consuming simulations, and works well for all
$M\geq 16 $
.