Detection of Signals with Random Moments of Appearance and Disappearance

We consider optimal detection of a signal with random moment of appearance and random length. Given the prior probability distributions of the moments of appearance and disappearance, the optimal nonsequential detection algorithm is based on a comparison of the likelihood ratio averaged over the joint distribution of these moments to some threshold. Fast algorithms generating the average likelihood ratio for discrete- and continuous-time observations are proposed. The structure of the optimal Bayesian sequential detection rule is established.