Carrier and Clock Recovery in (Turbo-)Coded Systems: Cramér-Rao Bound and Synchronizer Performance

In this paper, we derive the Cramér-Rao bound (CRB) for joint carrier phase, carrier frequency, and timing estimation from a noisy linearly modulated signal with encoded data symbols. We obtain a closed-form expression for the CRB in terms of the marginal a posteriori probabilities of the coded symbols, allowing efficient numerical evaluation of the CRB for a wide range of coded systems by means of the BCJR algorithm. Simulation results are presented for a rate 1/2 turbo code combined with QPSK mapping. We point out that the synchronization parameters for the coded system are essentially decoupled. We find that, at the normal (i.e., low) operating SNR of the turbo-coded system, the true CRB for coded transmission is (i) essentially the same as the modified CRB and (ii) considerably smaller than the true CRB for uncoded transmission. Comparison of actual synchronizer performance with the CRB for turbo-coded QPSK reveals that a “code-aware” soft-decision-directed synchronizer can perform very closely to this CRB, whereas “code-unaware” estimators such as the conventional non-data-aided algorithm are substantially worse; when operating on coded signals, the performance of the latter synchronizers is still limited by the CRB for uncoded transmission.