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

A highly reliable, accurate, and efficient method of calculating the probability of detection, P_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=10⁹, and with both NX and Y near 10⁷, the accuracy can drop to ten places. The function P_N(X,Y ) can be equated to the generalized Marcum Q-function, Q_m(α,β). The corresponding limits on α and β are roughly 4500 for the 13-place accuracy and 60000 for ultimate (INTEGER×4) limit