Data-Aided and Non-Data-Aided Maximum Likelihood SNR Estimators for CPM

Data-aided (DA) and non-data-aided (NDA) maximum likelihood (ML) estimators for the SNR of CPM in the presence of phase and frequency offset are derived and analyzed. Cramér-Rao bounds for both are obtained and compared to the simulated performance of these estimators for a full response example, CPSFK and a partial response example, GMSK. Analysis and simulations show that the performance of the DA ML estimator suffers from an unremovable bias caused by uncompensated frequency offset due to frequency offset estimation errors. As a consequence, the estimator error variance of the DA ML estimator is not able to achieve its lower bound. In contrast, the estimator error variance of the NDA ML estimator is capable of achieving its lower bound (although the lower bound for NDA ML estimator is higher than the lower bound for the DA ML estimator). This is because the NDA ML estimator is not burdened with the requirement of relying on estimates of nuisance parameters. The NDA estimator only achieves its lower bound when the observation length or true SNR are sufficiently large.