Analysis of 8 point constellations - optimality and recovery

This paper proves the optimality of circular 8QAM in modulating 3 bits/symbol over an AWGN channel, using a parametrised constellation. Incorporating power amplifier's limitations, we move to a differential scheme that modulates 2 bits/symbol and analytically prove that maximum energy efficiency for this scheme can be achieved when 8PSK is the underlying constellation, and not circular 8QAM. We also discuss about non data aided carrier frequency offset estimation when circular 8QAM is used and compare the performance of modified M^th power algorithm with the generalised QPSK partitioning algorithm.