Poisson models and mean-squared error for correlator estimator of time delay

A method for modeling large errors in correlation-based time-delay estimation is developed in terms of level-crossing probabilities. The level-crossing interpretation for peak ambiguity leads directly to an exact expression for the probability of large error involving the hazard function associated with the level-crossing process. Two models for the distribution of the error over the level-crossing time yield approximations to the mean-square error (MSE) that involve the low-order (<4) finite-dimensional distributions of the associated level-crossing process. Application of an inhomogeneous Poisson model for the level crossings reduces the form of the approximations to a weighted sum of the Cramer-Rao lower bound and the second moment of a function of the level-crossing intensity over time. Explicit expressions for the large error probability and the MSE approximations are obtained under a Gaussian model for the correlator statistics. Results of computer simulation are presented that indicate the accuracy of the approximations