Nonlinear Filters for Bayesian Estimation of Pulse Arrival Time in Additive White Gaussian Noise

Bayesian estimation of the arrival time of a single pulse is considered in the presence of a white noise. The employed model is a stochastic process consisted of a randomly delayed causal pulse and an additive white Gaussian noise, where the prior density of the delay is known. The history of this stochastic process is given at every point in time and the problem is to obtain the conditional expectation of an arbitrary function of the arrival time. The paper adopts a stochastic differential equation approach to develop nonlinear filters to efficiently compute this estimation. The filtering problem is resolved for a number of pulse shapes which allow for a finite-dimensional solution. These pulse shapes include step, exponential, and rectangular functions, as well as a piecewise constant function which well approximates a broad class of waveforms. The application of this filtering problem in real-time pulse arrival detection is discussed. The task of this operation is to report the event of pulse arrival as soon after occurrence as possible. The performance of nonlinear filtering is numerically verified for this application.