Generic Exponential Bounds on the Generalized Marcum Q-Function via the Geometric Approach

The generalized Marcum Q-function, Q_m (a,b), can be interpreted geometrically as the probability of a 2m- dimensional, real, Gaussian random vector z_2m, whose mean vector has a Frobenius norm of a, lying outside of a hyperball B^2m _O,b 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 Q_m (a, b) for the case where m is an integer. These generic exponential bounds are obtained by computing the probability of z_2m lying outside of some bounding geometrical shapes whose surfaces tightly enclose, or are tightly enclosed by the surface of B^2m _O,b ldr The bounding geometrical shapes used in the derivation consist of an arbitrarily large number of parts. As their closeness of fit with B^2m _O,b improves, the generic exponential bounds obtained approach the exact value of Q_m (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 Q_m (a,b).