0090-6778 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TCOMM.2017.2688396, IEEE Transactions on Communications 1 Generalized MGF of Beckmann Fading with Applications to Wireless Communications Performance Analysis Juan P. Pe˜ na-Mart´ ın, Juan M. Romero-Jerez, Senior Member, IEEE, F. J. Lopez-Martinez, Member, IEEE Abstract—The Beckmann distribution is a general multipath fading model for the received radio signal in the presence of a large number of scatterers, which can thence be modeled as a complex Gaussian random variable where both the in- phase and quadrature components have arbitrary mean and variance. However, the complicated nature of this distribution has prevented its widespread use and relatively few analytical results have been reported for this otherwise useful fading model. In this paper, we derive a closed-form expression for the generalized moment-generating function (MGF) of the signal-to-noise ratio (SNR) of Beckmann fading, which permits to circumvent the inherent analytical complexity of this model. This is a new and useful result, as it is key for evaluating several important performance metrics of different wireless communication systems and also permits to readily compute the moments of the output SNR. Thus, we obtain simple exact expressions for the energy detection performance in Beckmann fading channels, both in terms of the receiver operating characteristic (ROC) curve and of the area under ROC curve. We also analyze the outage probability in interference limited systems affected by Beckmann fading, as well as the outage probability of secrecy capacity in wiretap Beckmann fading channels. Monte Carlo simulations have been performed to validate the derived expressions. Index Terms—Beckmann fading, Maximal Ratio Combining (MRC), Square-Law Combining (SLC), Energy Detection, Re- ceiver Operating Characteristic (ROC), Secrecy Capacity. I. I NTRODUCTION Because of the presence of multiple scatterers, the radio signal in wireless environments is built from the superposi- tion of a number of individual waves, each with a certain amplitude and phase. Thus, the complex baseband signal (or, equivalently, field) of a wireless channel can be expressed as v  Re jΦ = n  i=1 A i e jφi , (1) where R is the amplitude and Φ the phase of the resulting signal, and A i and φ i denotes, respectively, the amplitudes and phases of the individual components. By assuming a sufficiently large number of paths, and by virtue of the Central Limit Theorem [1], the received signal can be modeled as a complex Gaussian random variable (RV), which can thence be J. P. Pe˜ na-Mart´ ın and J. M. Romero-Jerez are with the Department of Electronic Technology, F. J. L´ opez-Mart´ ınez is with the Department of Communications Engineering, E.T.S.I. Telecomunicaci´ on, University of M´ alaga, 29071 M´ alaga, Spain (e-mail: jppena@uma.es, romero@dte.uma.es, fjlopezm@ic.uma.es). The material in this paper was submitted, in part, to the IEEE 85th Vehicular Technology Conference. written as v = X + jY . This topic was originally addressed by Beckmann [2, 3] in its more general form by assuming arbitrary mean and variance for the real and imaginary parts of v, i.e., X ∼N (μ x ,σ 2 x ) and Y ∼N (μ y ,σ 2 y ), being X and Y independent 1 . This corresponds to the most accurate way to characterize the scattering of electromagnetic waves from rough surfaces [5], on which the distribution of the received signal envelope R = |v| is that of the modulus of a complex Gaussian RV. The Beckmann distribution includes the most popular clas- sical fading models used in practice, such as the Rician [6], Hoyt (Nakagami-q) [7] and Rayleigh distributions, as particular cases. Unlike other state-of-the-art envelope fading models [8, 9], the effect of imbalances in the line-of-sight (LOS) and non-LOS (NLOS) components is considered at the same time. Thus, the Beckmann fading model effectively captures the correlation between the amplitudes and phases of each ray component in (1) [10]. Besides, it allows for modeling LOS propagation conditions with a Hoyt-distributed diffuse component, which accurately fit field measurements in different scenarios [10–12]. However, the distribution of the signal envelope R has a very complicated form, being its chief probability functions, namely probability density function (PDF) and cumulative density function (CDF), unavailable in closed-form [4, 5, 13]. This fact has hindered the performance evaluation of wireless communication systems operating under this otherwise intuitive and physically-justified fading model. For this reason, and despite remarkable efforts have been made in order to analyze different performance metrics such as capacity, error probability, level crossing statistics and outage probability under Beckmann fading [14–16], there are still many communication-theoretic open problems which remain unexplored when Beckmann fading is considered. The contribution of this paper is two-fold: we derive a closed-form expression for the generalized moment generating function (MGF) of the signal-to-noise ratio (SNR) under Beckmann fading, which is given in terms of elementary functions and from which the moments of the output SNR can be readily obtained. We then illustrate the applicability of the generalized MGF in three different scenarios of interest, and for which the performance under Beckmann fading has been 1 As argued in [4], the assumption of independence does not cause any loss of generality. Should X and Y be correlated, there exists a linear transformation (equivalent to rotating the axis a certain angle ϕ) which yields a pair of uncorrelated Gaussian RVs X ′ and Y ′ with non-zero mean and non-identical variances. The symbol ∼ means statistically distributed as.