Applying Hankel’s Expansion for Performance Analysis in Double-Nakagami (Generalized- $K$ ) Fading Channels

The double-scattering fading models have found applications in many transmission scenarios. In this paper, approximate probability density functions (PDFs) are proposed for double-Nakagami fading and Gamma–Gamma fading, based on Hankel’s expansion of the modified Bessel function of the second kind. The proposed PDF is not only consistent with the exact PDF when the difference between the two shape parameters is equal to an integer plus one half, but also accurately approximates the distribution when the shape parameters are both greater than or equal to 2. The proposed approximate PDF has been compared with the power series expansion and the mixture Gamma model. It is found that the proposed PDF is in simple form and no parameters need to be tabulated. Moreover, the proposed PDF does not require an increasing number of terms to maintain precision with increasing diversity order of the reception scheme. Closed-form outage probability, ergodic capacity, and symbol error rate expressions have been derived using the proposed approximate PDFs for equal-gain combining, selection combining, and two-stage reception schemes. The proposed approximations are also leveraged for performance analysis of space-time block codes and free-space optical links. The precision of the proposed approximations is validated by comparing with the simulation results.