Geometric-View-Based Evaluation of Generalized Marcum Q-Function

This paper presents a geometric-view-based evaluation for the generalized Marcum Q-function Q_m(a,b), which is interpreted as the probability that a random Gaussian point whose mean vector has an Euclidean norm a≥, lies outside of an N-dimensional sphere/hypersphere of radius b≥0, centered on the origin point O, where N=2m≥3. By using the geometric symmetry of the sphere/hypersphere and the symmetry property of the joint Gaussian probability density function as well as a proper selection for the center point of the transformed spherical/hyperspherical coordinates, a direct, simpler evaluation approach is presented which leads to several new, compact, finite-integral representations of Q_m(a,b). The new representations are given in an unified form in the sense that they are valid for both an integer m and a non-integer m. Their application to error performance analysis of digital communications is illustrated.