# \(\alpha\)-Fluctuating Nakagami-\(m\) Fading Model for Wireless Communications
## 5 Simulation and Results' Analysis
### Numerical Computation Discussion  
Comparing the derived results with those for the underlying Fluctuating Nakagami-\(m\) model, it is clear that the computational complexity is of the same order. Specifically, both models yield results in terms of either Meijer G-functions or Fox H-functions. Moreover, the expressions for ABER and capacity are provided in terms of a single univariate Fox H-function. In contrast, the classical approach for \(\alpha\)-variate fading channels typically relies on Diophantine approximation of \(\alpha\), resulting in ABER and capacity expressions represented as series of Meijer G-functions (see, for example, [40; 51]). The number of summands in such series is determined by the approximation, which can be quite large for non-integer \(\alpha\), thereby significantly increasing the overall computational complexity. Moreover, results for related models (see, e.g., [29; 37; 38]) are given in terms of multivariate hypergeometric-type functions (such as Meijer G-functions, Fox H-functions, or Kampe de Feriet functions), which are considerably more complex. While univariate hypergeometric-type functions are implemented in most modern Computer Algebra Systems (CASs), their multivariate counterparts are generally not. However, some numerical routines exist for evaluating such functions (see, e.g., [52; 53; 54; 55]). It must also be noted that even existing CAS implementations of univariate functions present certain issues--such as significant computational slowdown for non-integer \(\alpha\), or divergence for large values of \(\alpha\). To address these drawbacks, as noted in the remarks following the proof of Theorem 1, the derived results can be implemented using univariate contour integrals in the complex domain (see, for example, expression (14) for the MGF). The remaining question is whether this implementation is computationally more efficient than brute-force numerical integration. For further analysis, one restricts attention to ABER calculation for coherent modulation, specifically QPSK. To this end, ABER will be evaluated in two ways: using numerical integration of (22) (denoted as \(\mathbb{P}^{num}er\)), and using the derived closed-form expression (24), represented via a Mellin-Barnes integral (denoted as \(\overline{\mathbb{P}}^{f}er\)). Both integrals were computed numerically using Wolfram Mathematica. In the latter case, the integration contour \(\mathcal{L}\) was chosen as \(\mathcal{L}:\left(-\frac{\delta m}{4}-i\delta;-\frac{\delta m}{4}+i\delta\right)\) (see the explanation following Theorem 1 for justification), where \(\delta\in\mathbb{R}\) was adjusted to ensure at least 5-digit accuracy while accelerating the computation. The computation times for these two approaches are denoted as \(t_{num}\) and \(t_{cf}\), respectively. Thus, the following numerical performance metrics can be defined:  
* \(\epsilon_{er}=\left|\frac{\overline{\mathbb{P}}_{er}^{num}-\overline{\mathbb{ P}}_{er}^{f}}{\overline{\mathbb{P}}_{er}^{num}}\right|\)--relative error of the obtained solution for the ABER;
* \(\epsilon_{t}=\dfrac{t_{num}}{t_{cf}}\)--computational time gain/loss of the proposed solution relative to numerical evaluation. A solution is considered computationally efficient if \(\epsilon_{t}>1\) while \(\epsilon_{er}<\varepsilon\), where \(\varepsilon\) denotes the required accuracy threshold. The results for various values of \(\alpha\) (with \(m=2\) and \(m_{s}=3\)) are presented in Figures 9 and 10. It should be noted that the average SNR range was upper-bounded (at 20 dB), since in the high-SNR regime, a highly efficient approximation (derived in Corollary 3) becomes applicable. This approximation offers dramatically lower computational complexity while maintaining excellent accuracy. The results clearly demonstrate the high accuracy of the closed-form solution across the low-to-moderate SNR range.