Self-noise free second-order carrier phase synchronization of MSK-type signals

In this paper we study from a theoretical and practical perspective the problem of nondata-aided (NDA) carrier phase recovery with MSK-type modulations. As suggested in previous works, the optimal ML estimator for low SNRs is quadratic on the received data. However, self-noise limits its performance in the medium-to-high SNR interval. To overcome this limitation, we have deduced optimal second-order schemes for the whole SNR range. The proposed estimator is asymptotically efficient when the SNR is low and cancels out the self-noise at high SNRs. Closed-form expressions are obtained for the high-SNR asymptotic case and a complete discussion is provided on its applicability in closed-loop and feed-forward schemes.