## 2 System Model
### Joint Impact of Skin-Induced Attenuation and NZB Pointing Errors  
Once we obtained the expressions for skin-induced attenuation through the parameter \(h_{l,m}\) in (3) and misalignment induced fading with NZB the through the parameter \(h_{p,m}\) in (9) respectively, we can now estimate their combined impact through their product in (2), i.e., through the parameter \(h_{m}\). Indeed, since the parameter \(h_{l,m}\) is a deterministic one, by both using (2), (3), and (9) and applying a random variable (RV) transformation [60], the joint PDF of the random variable \(h_{m}\) is obtained as  
\[f_{h_{m}}(h_{m})=\frac{\psi_{m}^{2}}{\left(A_{0,m}\delta_{m}\right)^{\psi_{m}^{2 }}h_{l,m}^{\psi_{m}^{2}-1}h_{m}^{-\psi_{m}^{2}}}. \tag{10}\]  
Additionally, based on (1)-(3) the instantaneous electrical SNR, \(\gamma\), is given as [39]  
\[\gamma_{m}=\frac{\eta_{m}^{2}\exp(-\alpha(\lambda_{m})\delta_{m})h_{p,m}^{2} \overline{P}_{S}}{N_{0}}, \tag{11}\]  
where \(\overline{P}_{S}\) and \(N_{0}\) represent the signal and noise optical power spectral density (PSD), respectively.  
Since \(\mathrm{E}[h_{m}]=\int\limits_{0}^{\infty}h_{m}\,f_{h_{m}}(h_{m})dh_{m}\), by using (10), the average electrical SNR in the case of TOW links with NZB errors is obtained as [37]  
\[\mu_{m}=\frac{\eta_{m}^{2}\mathrm{E}^{2}[h_{m}]\overline{P}_{S}}{N_{0}}=\frac {\eta_{m}^{2}\psi_{m}^{2}A_{0,m}^{2}g_{m}^{2}\exp(-\alpha(\lambda_{m})\delta _{m})\overline{P}_{S}}{\left(\psi_{m}^{2}+2\right)N_{0}}, \tag{12}\]  
while for the zero boresight scenario, the expected electrical SNR, \(\mu_{m,zero}\), is given by setting in (12) that \(g_{m}=1\), while also it holds that \(\mu_{2,m}^{2}=0,\quad\mu_{y,m}^{2}=0,\quad\sigma_{x,m}^{2}=\sigma_{y,m}^{2}= \sigma_{S}^{2}\).  
More precisely, for SIMO spatial diversity FSO systems the nonzero-boresight electrical SNR could be considered as equal to the zero-boresight scenario only for the simplified SISO configuration, where \(M=1\), i.e., \(\mu_{1}=\mu_{1,zero}=\mu\). This denotes that the positioning of the laser is assumed to be performed in regard to the corresponding photodetector and thus only zero-boresight pointing errors are addressed. On the contrary, when \(M>1\), the boresight is equal zero only to one branch, while the others should also deal with nonzero boresight components. Indeed, the laser cannot be simultaneously strictly aligned with all the additional photodiodes as effectively as with the initial one. Hence, NZB pointing errors are emerged for the rest receiver apertures, where lower amount of optical power arrives at their side. Thus, the nonzero boresight consideration through (12) brings about lower electrical SNR values, i.e., \(\mu_{1}<\mu_{1,zero}\) for \(1<m\leq M\)[2]. Consequently, it becomes evident from the above that by estimating the expected value of \(h_{m}\), we conclude that the average electrical SNR in the case of TOW links depends strongly on pointing errors effect with or without non-zero boresight components. As we will more clearly see below, this is a very important outcome for the outage performance estimation as a function of the average SNR for a TOW link with pointing errors.