A brief introduction to detection problems and illustration

Abstract

In this paper, some basics concepts in statistical approachesfor detection of signals embedded in noise are brieflyreviewed. Though most material presented in the first part ofthe paper is well documented in the litterature (seereferences), some effort is made to provide a self-containedand concise introduction to the subject. The importance andrelevance of the likelihood ratio is highlighted from aBayesian formulation of the problem; the optimal (in the senseof maximized detection probability for given false alarm rates)Neyman Pearson test and likelihood ratio test are discussed inthe general framework of test performance studies. Therefore,the usefullness of the ROCs (Receiver OperatingCharacteristics) is illustrated on simple examples. When someunknown parameters must be taken into account, extension of theprevious approaches are mentionned, with some emphasis put onthe GLRT (generalized likelihood ratio test). Most resultsintroduced in this tutorial are then applied and discussed inthe framework of the extra-solar planet detection problem. Aobservation model involving Poisson random variables isstudied. A thorough study in performed, and the Gaussianasymptotics are discussed in the case where all parameters ofthe model are assumed. In a more realistic situation where someparameters must be estimated, the GLRT is derived and itsperformances are evaluated by Monte Carlo simulation.