Partial order generalized Marcum Q-function contour integral representation

A Novel Unified Framework for Energy-Based Spectrum Sensing Analysis in the Presence of Fading

“…Though being exemplary research with a very high possible impact in various areas, it did not have any further elaboration. Moreover, independently of this work, this definition was revitalized in [41], that used a contour-integral approach proposed in [16] to analytically solve the problem of the average bit/symbol error rate calculation in the presence of fading, thus extending the results proposed in [26]. Since the approach in [16] (assumed herein) relies on a more general formulation, the Gaussian Q-function used in [26] is only a specific limiting case of a more general Marcum Q-function (used in [16] and herein); the presented results cannot be deduced from [26], and thus can be assumed as its generalization that expands the initial methodology.…”

“…Despite not being directly implemented in modern software packages (such as Matlab, Mathematica, Maple, etc. ), their computation (efficiently performed by numerical calculation of the inverse Laplace transform, for example, exhaustively discussed in [47][48][49][50]), including truncation errors, and the required number of summands and achievable computational gain (relative to the numeric integration) are frequently discussed in the literature (see, for instance, [16,28,32,51,52]).…”

“…Hence, Theorems 1 and 2 provide a solid ground for the unification of the derived quality metrics (APD and AUC) of those models. The APD and AUC expressions for the models included in Section 3.2, that can be derived directly from ( 5) and ( 6), with some exceptions, were generally reported in the literature, but in differing forms (see [8,16] for Rayleigh, [8] for Nakagami-m, [17] for Hoyt, [9,13] for η − µ, [14,30,31] for κ − µ shadowed) and hence are of limited interest, mainly for illustrative purposes and notation unification.…”

“…Proof. To prove Theorem 1, one starts by combining (1) and ( 2) and applying the contourintegral representation of the generalized Marcum Q-function given in [16]:…”

“…Numerous studies have proposed a variety of forms of representation for the ROC, AROC and AUC for several generalized channel models (as well as for the majority of simplified models) [1,[8][9][10][11][12][13][14][15][16][17], but in many cases, they do not yield closed-form expressions and are presented in terms of infinite series of hypergeometric functions of multiple variables (such as H-Fox function) (see, for instance, [14,18]), multidimensional nested infinite series (see, for instance, [19]) or parametric integrals, which are of limited usefulness for further generalization and practical implementation, or assume only integer-valued fading parameters [20], restricting their applicability. Moreover, the approaches that were used to derive those solutions were generally quite isolated and the obtained representations had dissimilar mathematical structures and notations.…”