Second-order statistics and discrete-time detection modeling for partially saturated processes

The basic problem in underwater detection is formulated under the general assumptions of partially saturated propagation of narrow-band and acoustic signals. Expressions for the joint probability density function (PDF) of ρ, the short-time average root-mean-square pressure at the receiver are obtained. This joint PDF is a general result reducing to the PDF’s for the fully saturated and the unsaturated cases for limiting values of the appropriate variables. Subsequently, defining detection as occurring whenever ρ exceeds a specified threshold level ρ0 and, using the above results, the upcrossing and downcrossing statistics of the envelope process are studied. Closed form expressions for the probability mass functions (PMF’s) of the interarrival time (time between two successive detections) and holding time (time between an upcrossing and the first subsequent downcrossing) are obtained. Results using our partially saturated detection model reduce, in limiting cases, to results already obtained in the literature for fully saturated and unsaturated propagation.