Timing estimation for a filtered Poisson process in Gaussian noise

The problem of estimation of time shift of an inhomogeneous casually filtered Poisson process in the presence of additive Gaussian noise is discussed. Approximate expressions for the likelihood function, the MAP estimator, and the MMSE estimator that becomes increasingly accurate as the per-unit-time density of superimposed filter responses becomes small are obtained. The optimal MAP estimator takes the form of a cascade of linear and memoryless nonlinear components. For smooth point process intensities, the performance of the MAP estimator is studied via local bias and local variance. A rate distortion type lower bound on the MSE of any estimator of time delay is then derived by identification of a communications channel that accounts for the mapping from time delay to observation process. Results of numerical studies of estimator performance are presented. Based on the examples considered it is concluded: (1) the small-error MSE of the nonlinear MAP estimator can be significantly better than the small-error MSE of the optimal linear estimator: (2) the rate distortion lower bound can be significantly tighter than the Poisson limited bounds determined in previous studies