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

This article derives new “exponential-type” contour and finite-range integral representations for the generalized M-th order Marcum Q-function Q_M(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 Q_M(a√γ, b√γ) and Q_M(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 Q_M(α, β) due to the presence of cross-product terms. We also show that all known exponential-type integral representations for Q_M(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 Q_M(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.