Delay estimation with a carrier modulated by a band-limited signal

Since time-delay estimation is a fundamental task in various engineering fields, several expressions for the CRB and MLE have been developed over the past decades. In all of these previous studies, a common assumption was that the wave transmission process introduced an unknown phase component, which made it impossible to exploit the phase component related to the delay from the carrier signal. However, there are practical scenarios where this unknown phase can be estimated and compensated for, enabling the utilization of the delay phase component from the carrier signal. In this context, we provide a comprehensive treatment of this scenario, including the derivation of the MLE and the associated CRB. This approach allows us to analyze the impact of each signal component (carrier frequency and baseband signal) on the achievable MSE of delay estimation relative to the SNR. It also reveals five distinct regions of operations, in contrast to the well-known three.