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Computing and Bounding the Generalized Marcum Q-Function via a Geometric Approach

The generalized Marcum Q-function, Qm(a,b), is here explained geometrically as the probability of a 2m-dimensional, real, Gaussian random vector, whose mean vector has a Frobenius norm of a, lying outside a hypersphere of 2m dimensions, with radius b, and centered at the origin. Based on this new geometric interpretation, a new closed-form representation Qm(a,b) is derived for the case where m is an odd multiple of 0.5. This representation involves only the exponential and the erfc functions, and thus is easy to handle, both numerically and analytically. For the case where m is an even multiple of 0.5, Qm+0.5(a,b) and Qm-0.5(a,b), which can be evaluated using our new representation mentioned above, are shown to be tight upper and lower bounds on Qm(a,b), respectively. They are shown in most cases to be much tighter than the existing bounds in the literature, and are valid for the entire ranges of a and b concerned. Their average is also a good approximation to Q m(a,b)