Optimum Pilot Sequences for Data-Aided Synchronization

We compute the pilot sequences that minimize the Cramér-Rao bound(CRB) for carrier phase offset, carrier frequency offset, and symbol timing offset estimation. If we require the pilot sequence to be drawn from a discrete set of constellation points, then the alternating sequence is the CRB-minimizing sequence. If we relax the constellation constraint and apply a total energy constraint, we obtain three different sequences that minimize the CRB for each of the synchronization parameters. We show that the maximum-likelihood (ML) estimator achieves the CRB for any of the sequences presented. We also explore the impact of using the pilot symbols in a preamble with unknown data symbols immediately following. The interference caused by the unknown data symbols reduces the performance of the ML estimator beyond an SNR threshold.