True Cramér-Rao bound for data-aided timing and carrier synchronization with random carrier phase

The true Cramer-Rao bound (CRB) for the variance of unbiased, data-aided symbol-timing estimate has been obtained from linearly modulated waveforms through an additive white Gaussian noise channel with random carrier phase already. However, in practice, it is unusual to perform carrier synchronization prior to timing recovery. In this paper, we deduce the true CRB for data-aided time delay and frequency offset joint estimation in the presence of a random carrier phase. This new bound is analyzed and compared with the different CRBs in literatures. The agreement with the results in literatures verifies the derivation of the true CRB for data-aided timing and carrier synchronization with random carrier phase.