Time-delay estimation for filtered Poisson processes using an EM-type algorithm

We develop a modified EM algorithm to estimate a nonrandom time shift parameter of an intensity associated with an inhomogeneous Poisson process N_t, whose points are only partially observed as a noise-contaminated output X of a linear time-invariant filter excited by a train of delta functions, a filtered Poisson process. The exact EM algorithm for computing the maximum likelihood time shift estimate generates a sequence of estimates each of which attempt to maximize a measure of similarity between the assumed shifted intensity and the conditional mean estimate of the Poisson increment dN_t. We modify the EM algorithm by using a linear approximation to this conditional mean estimate. The asymptotic performance of the modified EM algorithm is investigated by an asymptotic estimator consistency analysis. We present simulation results that show that the linearized EM algorithm converges rapidly and achieves an improvement over conventional time-delay estimation methods, such as linear matched filtering and leading edge thresholding. In these simulations our algorithm gives estimates of time delay whose mean square error virtually achieves the CR lower bound for high count rates