Simple tight exponential bounds on the first-order Marcum Q-function via the geometric approach

Abstract

The geometric interpretation of the first-order Marcum Q-function, Q(a, b), has been shown as the probability that a complex, Gaussian random variable Z̃ with real, nonzero mean a takes on values outside of a circular region CO,b of radius b centered at the origin O. Based on this interpretation, many new, simple, tight, upper/lower exponential bounds on Q(a, b) are easily obtained by computing the probability of Z lying outside of some simple geometrical shapes, such as circular regions, semicircular regions, sectors, and angular sectors of annuli, whose boundaries tightly enclose, or are tightly enclosed by the boundary of CO,b. The new bounds presented here only involve two exponential functions, and the best of them are in most cases much tighter than the best existing exponential bounds. In addition to these bounds, more new bounds can be obtained by using similar methods. Even some bounds in the literature can also be obtained via this geometric approach. ©2006 IEEE.