On the Cramer-Rao lower bound and the performance of synchronizers for (turbo) encoded systems

We derive the Cramer-Rao bound (CRB) for joint estimation of carrier phase, carrier frequency and timing from a noisy linearly modulated signal with coded 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 bound. We find that at low SNR, the CRB for coded transmission is considerably smaller than the CRB for uncoded transmission. We show that practical synchronizers that make clever use of the code properties yield a mean-square estimation error that is close to the CRB for coded transmission.