## 4 Numerical Results  
Using the above-extracted mathematical expressions, (19), (20), and (22), the corresponding numerical results are presented in this section using common parameter values. Thus, variations of the parameters related to the system's conditions, i.e., atmospheric turbulence, SJ, and TJ effects, with fixed receiver diversity (\(N=3\)) and DF relaying (\(M=1\)), are illustrated in Figures 2-4 for particular parameters values of the modulation scheme, i.e., OOK-RZ, 4-PPM, and 4-PAM, respectively. Specifically, binary parameter combinations were studied in all three figures, including: \(\zeta=5.03\) or \(0.59\), i.e., for weak or strong SJ effect, respectively, and whether \(\alpha=1\)\(\times\)\(10^{-8}\) or \(2\)\(\times\)\(10^{-6}\) s, with \(\beta=4\) leading to \(\sigma=5.815\)\(\times\)\(10^{-9}\) or \(1.163\)\(\times\)\(10^{-6}\) s for bit rate \(R=48\) Mbps, or \(\alpha=5\)\(\times\)\(10^{-11}\) or \(1\times\)\(10^{-8}\) s (cases with \(\sigma=1.163\)\(\times\)\(10^{-6}\) s seem to be more typical of real systems for all studied modulation schemes, according to Table I of [18], however this benchmark is too large for Gbps rates, which are treated with much smaller \(\sigma\) values. For comparison, both bit rates were studied along with an intermediate value of \(\sigma=5.815\)\(\times\)\(10^{-9}\) s.), with \(\beta=4\) leading to \(\sigma=29.075\)\(\times\)\(10^{-12}\) or \(5.815\)\(\times\)\(10^{-9}\) s for 24 Gbps, i.e., for weak or strong TJ effect, respectively [18], along with Malaga distribution parameter sets (\(a\), \(b\), \(\rho\)) = (11, 4, 1) (\(\rho=1\) corresponds to the GG sub-case.) or (25, 10, 0.75) for weak or strong turbulence conditions, respectively (corresponding to the atmosphere's index of refraction structure parameters, \(C_{n}^{2}=7.2\)\(\times\)\(10^{-15}\) m\({}^{-2/3}\) or \(1.2\)\(\times\)\(10^{-14}\) m\({}^{-2/3}\) and Rytov variance, \(\sigma_{R}^{2}=0.32\) or \(0.52\), respectively, according to [13].). In all cases, \(\Omega=0.5\) and \(b_{0}=0.25\), so that \(\Omega+2b_{0}=1\), implying that the average FSO link's optical power is normalized. The rest factors for any individual link, minutely described in [13], include a propagation distance of 1 km and operating wavelength of \(\lambda=785\) nm. Accordingly, variations of the operating system's parameters, i.e., modulation scheme format, diversity, and DF relays, are illustrated in Figures 5-7 for particular conditions, i.e., (\(a\), \(b\), \(\rho\)) = (11,4,1), \(\xi\) = 5.03, and either \(\sigma\) = 29.075 \(\times\) 10\({}^{-12}\) s for \(R\) = 48 Mbps or \(\sigma\) = 5.815 \(\times\) 10\({}^{-9}\) s for 24 Gbps with \(\alpha\) = 5 \(\times\) 10\({}^{-11}\) or 1 \(\times\) 10\({}^{-8}\) s, respectively, and \(\beta\) = 4. Similarly, binary case combinations of diversity, \(N\) = 3 or 9, DF relaying, \(M\) = 1 or 5, and formats per modulation scheme family, i.e., OOK-RZ or NRZ, 4 or 16-PPM and 4 or 16-PAM, are shown in all three figures, along with selected bit rates. Figure 4: ABER versus expected SNR for 4-PAM scheme, \(N\) = 3, \(M\) = 1, with various bitrates and spatial and time jitter conditions of (**a**) (\(a\) = 11, \(b\) = 4, \(\rho\) = 1) and (**b**) (\(a\) = 25, \(b\) = 10, \(\rho\) = 0.75). The studied curve cases are codified as in Figure 2.  
Figure 3: ABER versus expected SNR for the 4-PPM scheme, \(N\) = 3, \(M\) = 1, with various bit rates and spatial and time jitter conditions of (**a**) (\(a\) = 11, \(b\) = 4, \(\rho\) = 1) and (**b**) (\(a\) = 25, \(b\) = 10, \(\rho\) = 0.75). The studied curve cases are codified as in Figure 2.  
Figure 5: ABER versus expected SNR for (\(a=11\), \(b=4\), \(\rho=1\)), \(\xi=5.03\), OOK-RZ/NRZ schemes, and various diversity and DF multi-hop parameters, with (**a**) \(R=48\) Mbps, \(\sigma=5.815\times 10^{-9}\) s (\(\alpha=10^{-8}\) s, \(\beta=4\)) and (**b**) \(R=24\) Gbps, \(\sigma=29.075\times 10^{-12}\) s (\(\alpha=5\times 10^{-11}\) s, \(\beta=4\)).