## 6 Numerical and Simulation Results  
To corroborate the derived analytical expressions in the previous sections, this section presents various numerical and simulation results under different \(\alpha\)-\(\mu\)/IGA fading channels. We adopted the Monte Carlo simulations to generate no less than 10e6 iterations. For the correlated scenarios, the simulation approaches described in [29] are considered by using a simple change of variables. In all figures, the simulation results provide a perfect matching with the numerical counterparts and confirm the availability of our theoretical derivations. In the numerical analysis, we provide some necessary methods to evaluate multivariate Fox's H-functions and multiple-fold infinite/finite series sum in order to obtain accurate results. For the former, a Python code provided in [52] has been revised to compute them, such as (42) and (47). For the two-fold infinite series sum forms in the correlated cases, such as (43) and (50), we refer to the minimum numbers of truncated terms provided in [29] to meet the given target accuracy. Noted that \(k_{min}=l_{min}=30\) is considered for various values of the nonlinear parameter (\(\alpha=0.5\), \(1.5\), \(3.5\)) because it has similar effects on system performance as the multipath parameter (\(\mu\)). Whereas for the finite series sum forms in (49), the number of terms (\(N\)) gradually increases as the average SNR (\(\overline{\gamma}\)) grows, for example, \(N=30\) as \(\overline{\gamma}=10\)dB and \(N=150\) as \(\overline{\gamma}=20\) dB Even though the number of terms (\(N\)) is so large, it takes much less time to calculate (49) than (47) in order to get the same results. Finally, we consider some comparisons and discussions about the impacts of the non-linear parameter (\(\alpha\)), the correlation coefficient (\(\rho\)), and the diversity branch (\(L\)) on the performance of different communication systems since the impacts of the multipath parameter (\(\mu\)) and the shadowing parameter (\(n\)) on the performance have been profusely discussed in the previous works, such as [29; 40; 41; 49]. In Figures 3 and 4, we plot the OP of the single-link and MRC systems as a function of \(\overline{\gamma}\) over \(\alpha\)-\(\mu\)/IGA fading channels with \(n=5\), \(\mu=2\), and \(\gamma_{th}=5\) dB, respectively. As anticipated, it can be observed that the OP of all the systems is improved when \(\overline{\gamma}\) increases. In Figure 3, some performance comparisons are given in the i.i.d. case. It is clear that the OP gets better with the increase in the number of diversity branches (\(L=1\), \(2\), and \(3\)), and the OP of MRC outperforms that of the single-link. Furthermore, the asymptotic results at the high average SNR region and the numerical results by using the mixed \(\alpha\)-\(\mu\) model are also presented, respectively. We can see that the former keeps tight with the exact results at the high average SNR level while the latter matches well with the exact results in the whole average SNR region with the truncated term \(N=12\). These results demonstrate the validity of our derivations. In Figure 4, some i.n.i.d. and correlated examples are considered. As expected, we can get some same conclusions as those in Figure 3. However, for the correlated cases, the OP of MRC degrades with the increase in the correlation coefficients (\(\rho=\rho_{N}=\rho_{G}=0\), \(0.2\), \(0.5\), and \(0.7\)). In Figures 5 and 6, we illustrate the ABEP/ASEP of different modulation schemes for various communication systems as a function of the average SNR (\(\overline{\gamma}\)) over \(\alpha\)-\(\mu\)/IGA fading channels. In Figure 5, the ABEP of DPSK for various communication systems are compared under i.i.d. fading scenarios. It can be seen that the ABEP is improved when \(\alpha\) (\(\alpha=1.5\) and \(3\)), the number of diversity branches (\(L=1\), \(2\), and \(3\)), and the average SNR (\(\overline{\gamma}=0\)\(\rightarrow\)\(20\) dB) increases, respectively.