Modeling, Estimation, and Bounds for Precision Two-way Time Transfer and Ranging

Over-the-air precision two-way time transfer and ranging (PTTR) uses the signals exchanged between two RF transceivers to estimate the range between them and the offset between their clocks. Our paper makes three significant contributions to the estimation and position, navigation, and timing literature. First, we introduce a propagation channel model characterized by its amplitude (attenuation), phase shift, and propagation delay. This three-parameter formulation is quite practical for modeling signal exchange over line-of-sight channels with arbitrary additive wide-sense stationary noise that also captures electronic effects. Second, we derive the Cramer-Rao Lower Bound (CRLB) for this three parameter channel model. We show that there is a canonical definition of phase shift for which the CRLB is a diagonal matrix. The derived CRLB depends on simple properties of the signal and noise power spectral density (PSD) related to whitened bandwidth and whitened center frequency as well as the signal to noise ratio. Third, we derive a maximum likelihood estimator (MLE) for the amplitude, phase shift, and delay given the reference waveform, received waveform samples, and noise PSD. A novel aspect of the derived MLE is that it is implemented entirely in the frequency domain, which makes the MLE more computationally efficient than a typical time-domain implementation. Using an example reference waveform, we show that the derived MLE achieves an empirical covariance close to the CRLB for sufficiently high signal to noise ratios. The estimation results of our paper are directly relevant to over-the-air PTTR applications operating in direct-path, line-of-sight propagation environments. The generality of the waveform and noise formulations allow our estimator and bounds to be applied to arbitrary PTTR ranging waveforms in colored interference environments. We present a system architecture utilizing this estimator that enables PTTR on existing wireless networks with no changes to the communications protocol or radio hardware or software.