The calculation of the probability of detection and the generalized Marcum Q-function

Abstract

A highly reliable, accurate, and efficient method of calculating the probability of detection, P/sub N/(X,Y), for N incoherently integrated samples, where X is the constant received signal-to-noise ratio of a single pulse and Y is the normalized threshold level, is presented. The useful range of parameters easily exceeds most needs. On a VAX/11 computer with double precision calculations, better than 13-place absolute accuracy is normally achieved. There is a gradual loss of accuracy with increasing parameter values. For example, for N=10/sup 9/, and with both NX and Y near 10/sup 7/, the accuracy can drop to ten places. The function P/sub N/(X,Y) can be equated to the generalized Marcum Q-function, Q/sub m/( alpha , beta ). The corresponding limits on alpha and beta are roughly 4500 for the 13-place accuracy and 60000 for ultimate (INTEGER*4) limit.