A new geometric view of the first-order marcum Q-function and some simple tight erfc-bounds

Abstract

A geometric interpretation of the first-order Marcum Q-function, Q(a,b), is introduced as the probability that a complex, Gaussian random variable with real, nonzero mean a, takes on values outside of a circular region Cb of radius b centered at the origin. This interpretation engenders a fruitful approach for deriving new representations and tight, upper/lower erfc-bounds on Q(a,b). The new representations involve finite-range integrals that facilitate analytical and numerical computations, and are simpler than similar ones in the literature. The new, simple erfc-bounds are easily obtained by using simple geometrical shapes that tightly enclose, or are tightly enclosed by the circle Cb. They involve only a few terms of erfc and exponential functions, and are close to, or even tighter than the existing bounds that involve the modified Bessel function. © 2006 IEEE.