On the Approximation of the Generalized-K PDF by a Gamma PDF Using the Moment Matching Method

Using the Nakagami probability density function (PDF) to model multipath fading and the Gamma PDF to model shadowing, in a wireless channel, has led to a closed-form expression for the composite fading PDF, known as the generalized-.fi" PDF (also called Gamma-Gamma PDF). However, further derivations have shown that the cumulative distribution function (CDF) and the characteristic function of the generalized-K PDF contain special functions that are involved to handle. In this paper, an approximation of the generalized-K PDF by the familiar gamma PDF is introduced. The parameters of the approximating Gamma PDF are computed using the moment matching method. The accuracy of this approximation in the lower and upper tail regions is enhanced by adjusting the parameters of the approximating gamma distribution in each region. The CDF and the complementary CDF plots show that this approximation is sufficiently accurate for both integer and non-integer practical values of the multipath fading and shadowing parameters. The region-wise approximation obtained by the adjusted moment matching method is used to well- approximate the PDF of the sum of identically and independent generalized-K random variables. Applications of the obtained results arise in distributed antenna systems (DASs), cooperative relay networks, radar, and sonar systems.