This algorithm provides a method for non-linear multiple target tracking that does not require association of targets. This is done by recursive Bayesian estimation of the density corresponding to the expected number of targets in each measurable set -- the Probability Hypothesis Density (PHD). Efficient Monte Carlo estimation is achieved by giving this density the role of the single target state probability density in the conventional particle filter. The problem setup for our algorithm includes (1) a bounded region of interest containing a changing number of targets, (2) independent observations each accompanied by estimates of false alarm probability and the probability that the observation represents something new, (3) an estimate of the Poisson rate at which targets leave the region of interest. The prototype application of this filter is to aid in short range acoustic contact detection and alertment for submarine systems. The filter uses as input passive acoustic detections from a fully automated process, which generates a large numbers of valid and false detections. The filter does not require specific target classification. Although the mathematical theory of Probability Hypothesis Density estimation has been developed in the context of modern Random Set Theory, our development relies on elementary methods instead. The principal tools are conditioning on the expected number of targets and identification of the PHD with the density for the proposition that at least one target is present.