On the asymptotic performance of time-delay and Doppler estimation with a carrier modulated by a band-limited signal

Time-delay and Doppler estimation is crucial in various engineering fields, as estimating these parameters constitutes one of the key initial steps in the receiver’s operational sequence. Due to its importance, several expressions of the Cramér–Rao Bound (CRB) and Maximum Likelihood Estimation (MLE) have been derived over the years. Previous contributions started from the assumption that the transmission process introduces an unknown phase, which hindered the explicit consideration of the time-delay parameter in the carrier-phase component in theoretical derivations. However, this contribution takes into account this additional term under the assumption that such an unknown phase is inferred and compensated for. This new condition leads to the derivation of a novel MLE. Subsequently, a closed-form expression of the achievable Mean Squared Error (MSE) for the time-delay and Doppler parameters is provided for the asymptotic region, assuming the signal is band-limited. Both expressions are validated via Monte Carlo simulations. This analysis reveals five distinct regions of operation of the MLE, refining existing knowledge and providing valuable insights into time-delay estimation