Generic exponential bounds on the generalized Marcum Q-function via the geometric approach

Abstract

The generalized Marcum Q-function, Qm(a, b), can be interpreted geometrically as the probability of a 2m-dimensional, real, Gaussian random vector Z2m, whose mean vector has a Frobenius norm of a, lying outside of a hyperball double-struck B signO,b 2m], of 2m dimensions, with radius b, and centered at the origin O. Based on this geometric view, we propose some new generic exponential bounds on Qm(a, b) for the case where m is an integer. These generic exponential bounds are obtained by computing the probability of Z2m lying outside of some bounding geometrical shapes whose surfaces tightly enclose, or are tightly enclosed by the surface of double-struck B sign O,b 2m]. The bounding geometrical shapes used in the derivation consist of an arbitrarily large number of parts. As their closeness of fit with double-struck B sign O,b 2m improves, the generic exponential bounds obtained approach the exact value of Qm(a, b). These generic exponential bounds only involve the exponential function, and thus, are easy to handle in analytical computations. Our numerical results show that when evaluated with a few terms, these generic exponential bounds are much tighter than the existing exponential bounds in the literature for a wide range of arguments. For the case of a > b, our generic upper exponential bound is the first upper exponential bound on Qm(a, b). © 2007 IEEE.