## 7 Simulations and Numerical Results  
In this part, the simulation results are obtained to analyze the effect of the detection method, atmospheric turbulence, pointing errors, relay mode, and fixed relay gain on the considered communication systems. Additionally, analytical expressions are validated by Monte Carlo simulations. The FSO link undergoes \(\mathcal{F}\)-distributed fading, in which the values of \(a\) and \(b\) can be calculated by (9) with the aperture diameter of the receiver \(D=10\) mm,\(l_{0}=5.98\) mm, \(L_{0}=0.8\) m, \(\lambda=1550\) nm, and the distance \(L\) is set to 3.5 km. We set \(\gamma_{th}=0\) dB. For strong and moderate turbulence, the refraction index structure parameter \(C_{n}^{2}\) is \(5\times 10^{-14}\) m\({}^{-2/3}\) and \(1.25\times 10^{-14}\) m\({}^{-2/3}\), respectively, which results in \(\sigma_{1}^{2}=4.0230\) and \(\sigma_{1}^{2}=1.0058\). Hence, the atmospheric turbulence parameters are derived as follows: \(a=1.4753\), \(b=4.9821\) for strong turbulence and \(a=2.5190\), \(b=5.8804\) for moderate turbulence. Figure 2 illustrates the outage probability under strong turbulence. The analytical expressions of the outage probability are based on (34) and (41). It is assumed that \(\gamma_{RF}=\gamma_{FSO}\) and \(C=1\). The outage probability decreases as the value of \(m\) changes from 1 to 6.7. For example, at 16 dB SNR in variable-gain relaying under heterodyne detection, the outage probability is \(3.31\times 10^{-2}\) with \(m=6.7\), while the outage probability increases to \(6.59\times 10^{-2}\) with \(m=1\). Moreover, it can be seen that the fixed-gain relaying offers superior performance compared with the variable-gain relaying under the IM/DD technique. For instance, at 28 dB SNR, the outage probability is \(6.58\times 10^{-2}\) for the variable-gain relaying, while the outage probability decreases to \(6.65\times 10^{-3}\) for the fixed-gain relaying. The reason is that the variable-gain relaying amplifies the noise and interference with the signal, while the relay gain is independent of the channel conditions for fixed-gain relaying. Additionally, the results also demonstrate that the system has better outage performance under HD as compared to IM/DD. For instance, at 28 dB SNR and under the situation of variable-gain relay with \(m=1\), the outage probability is \(4.54\times 10^{-3}\) under HD, while the outage probability increases to \(6.58\times 10^{-2}\) under IM/DD. Moreover, the asymptotic curves for all terms coincide with the analytical curves at high SNR values, and the simulation results match the analytical results. Figure 3 presents the outage probability performance with fixed-gain relaying under strong turbulence with \(m=1\) for various \(C\). The outage probability performance becomes worse when the value of \(C\) becomes larger (i.e., the fixed gain becomes smaller). For instance, the outage probability decreases from \(4.46\times 10^{-3}\) to \(1.72\times 10^{-3}\) when the value of \(C\) decreases from 10 to 1 (fixed gain increases). Figure 2: Outage probability performance under strong turbulence. In Figure 4, the average BER performance under strong turbulence conditions with variable-gain relaying is presented. The analytical expressions for average BER are based on (37) and (43). The figure is plotted at \(\hat{\gamma}_{FSO}=20\) dB and \(m=1\). The figure shows that CBPSK offers the best performance compared with other binary modulation schemes, and NBFSK offers the worst performance as compared with the other three modulations. For instance, at 4 dB SNR, the average BER values for the four binary modulations are 0.2217 for NBFSK, 0.1422 for DBPSK, 0.1243 for CBFSK, and 0.0745 for CBPSK. In Figure 5, the ergodic capacity performance under fixed-gain relaying is presented. The ergodic capacity expressions are based on (40) and (45). It is assumed that \(\hat{\gamma}_{RF}=10\) and \(C=1\). The figure shows that the HD method outperforms the IM/DD method.