New representations and bounds for the generalized Marcum Q-function via a geometric approach, and an application

  • Authors:

  • Rong Li;Pooi Yuen Kam;Hua Fu

  • Affiliations:

  • Department of Electrical and Computer Engineering, National University of Singapore, Singapore;Department of Electrical and Computer Engineering, National University of Singapore, Singapore;Department of Electrical and Computer Engineering, National University of Singapore, Singapore

  • Venue:
  • IEEE Transactions on Communications
  • Year:
  • 2010

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Abstract

The generalized Marcum Q-function of order m, Qm(a, b), is interpreted geometrically as the probability of a 2m-dimensional, real, Gaussian random vector z2m, whose mean vector has a Frobenius norm of a, lying outside of a hyperball BO,b2m of 2m dimensions, with radius b, and centered at the origin O. Based on this new geometric view, some new representations and closed-form bounds are derived for Qm(a, b). For the case that m is an odd multiple of 0.5, a new closed-form representation is derived, which involves only simple exponential and erfc functions. For the case that m is an integer, a pair of new, finite-integral representations for Qm(a, b) is derived. Some generic exponential bounds and erfc bounds are also derived by computing the probability of z2m lying outside of various bounding geometrical shapes whose surfaces tightly enclose, or are tightly enclosed by the surface of BO, b2m. These bounding shapes consist of an arbitrarily large number of parts. As their closeness of fit with BO, b2m improves, our generic bounds approach the exact value of Qm(a, b). The function Qm(a, b) is proved to be an increasing function of its order when 2m is a positive integer. Thus, Qm+0.5(a, b) and Qm-0.5(a, b) can be used as tight upper and lower bounds, respectively, on Qm(a, b). Their average is a good approximation to Qm(a, b). An application of our new representations and bounds is also given.