Generic exponential bounds and erfc-bounds on the marcum Q-function via the geometric approach

Abstract

The first-order Marcum Q-function, Q(a, b), can be interpreted geometrically as the probability that a complex, Gaussian random variable Z̃ with real mean a, takes on values outside of a circular region C O,b of radius b centered at the origin O. Bounds can thus be easily obtained by computing the probability of Z̃ lying outside of some geometrical shapes whose boundaries tightly enclose, or are tightly enclosed by the boundary of CO,b. In this paper, the bounding shapes are chosen to be a set of sectors or angular sectors of annuli to generate generic exponential bounds, and to be a set of rectangles to generate generic erfcbounds. These generic exponential bounds and erfc-bounds involve an arbitrarily large number of exponential functions and erfc functions, respectively, and are shown to approach the exact value of Q(a, b) as the number of terms involved increases. © 2006 IEEE.