On the optimality of bit detection of certain digital modulations

In this paper, we study the optimality of bit detection for coherent M-ary phase-shift keying (PSK) and M-ary quadrature amplitude modulation (QAM), and noncoherent M-ary frequency-shift keying (FSK) signal sets. For M-PSK and M-QAM signal constellations, we employ Gray mapping, consider 8-PSK and 16-QAM signal sets as representative of the general results, and derive the log-likelihood ratio (LLR) for each bit forming the symbol. Using the LLRs, we derive the average bit-error probability (BEP) for the individual bits, and show that the decision regions and the corresponding average BEP for the case of M-PSK coincide with those obtained with the optimal symbol-based detector, whereas, in general, this is not the case for both M-QAM and M-FSK.