New exponential-type integral representations of the generalized marcum Q-function of real-order with applications

摘要

This article derives new “exponential-type” contour and finite-range integral representations for the generalized M-th order Marcum Q-function QM(α, β) when its real order M>0 is not necessarily an integer. These new forms have both computational and analytical utilities, and are very attractive for computing the statistical expectations of all three functions of the form QM(a√γ, b√γ) and QM(a√γ, b) with respect to the probablility density function of γ random variable. This feature is not possible with the existing trigonometric integral representations for QM(α, β) due to the presence of cross-product terms. We also show that all known exponential-type integral representations for QM(α, β) discovered by Helstom [2], Simon [15], Tellambura et. al. [9] and Annamalai et. al. [10] can be obtained from our contour integral via appropriate variable substutions. Several applications of our novel integral representations of QM(α, β) are also provided such as the evaluation of the receiver operating characteristics (ROC) and the partial area under the ROC curves of diversity-enabled energy detectors, and unified error probability analyses of coherent, differentially coherent and noncoherent binary and quaternary digital modulations in a myriad of fading environments.