Asymptotically Exact Approximations for the Symmetric Difference of Generalized Marcum $Q$-Functions

In this paper, we derive two simple and asymptotically exact approximations for the function defined as ΔQ
_m
(a, b)
₌
Δ Q
_m
(a, b) - Q
_m
(b, a). The generalized Marcum Q-function Q
_m
(a, b) appears in many scenarios in communications in this particular form and is referred to as the symmetric difference of generalized Marcum Q-functions or the difference of generalized Marcum Q-functions with reversed arguments. We show that the symmetric difference of Marcum Q-functions can be expressed in terms of a single Gaussian Q-function for large and even moderate values of the arguments a and b. A second approximation for ΔQ
_m
(a, b) is also given in terms of the exponential function. We illustrate the applicability of these new approximations in different scenarios: 1) statistical characterization of Hoyt fading; 2) performance analysis of communication systems; 3) level crossing statistics of a sampled Rayleigh envelope; and 4) asymptotic approximation of the Rice I
_e
-function.