Influence of random carrier phase on true Cramer-Rao lower bound for time delay estimation

The true Cramer-Rao lower bound (CRB) for the time delay estimation has been obtained for narrowband signal with the carrier phase as a deterministic parameter already. However, the carrier phase is usually a random parameter in non-coherent receiver in the applications such as radar, sonar and communication systems. And accuracy of the time delay estimation will be affected by this nuisance carrier phase. In this paper, the true Cramer-Rao lower bound for the time delay estimation is derived and analyzed in the presence of a random carrier phase. The new bound is tighter than the one obtained under the condition that the carrier phase is not random. We show that this relation indicates the influence of the random carrier phase on the true CRB, and the penalty resulting from this random carrier phase increases severely with decreasing signal-to-noise ratio. The explanation about this influence is also given from the point of information theory. Simulations are provided to support the theoretical results.