On Modeling and Pulse Phase Estimation of X-Ray Pulsars

Mathematical models are developed to characterize the X-ray pulsar signals, and the pulse phase estimation problem is addressed. The Cramér-Rao lower bound for estimation of the pulse phase is presented. Depending on employing the photon counts or direct use of the measured photon time of arrivals, two different estimation strategies are proposed and analyzed. In the first approach, utilizing the epoch folding procedure, the observed pulsar rate function on the detector is retrieved, and the pulse phase is estimated through a nonlinear least-squares fit of the empirical rate function to the known pulsar rate function. It is shown that this estimator is consistent, but not asymptotically efficient. In the second strategy, a maximum likelihood (ML) estimation problem is formulated using the probability density function of the photon time of arrivals. It is shown that the ML estimator is asymptotically efficient. Computational complexity of the proposed estimators is investigated as well. The analytical results are verified numerically via computer simulations.