## 2 System and Channel Model
### Generalized Pointing Errors Model  
In order to accurately describe the stochastic nature of the pointing errors effect including the boresight component, the general and realistic Beckmann distribution statistical distribution model was used. The PDF of the \(n\)-th radial displacement at the receiver \(R_{n}\), according to the Beckmann model which takes into consideration the influence of beam width, detector size, the different jitter for elevation and horizontal displacement as well as the effect of pointing errors with boresight, is obtained as described in [22; 28; 69]  
\[ \begin{split} f_{R_{n}}(R_{n})&=\frac{R_{n}}{2\pi v_{x,n}v_{y,n}}\\ &\times\int_{0}^{2\pi}exp\left(-\frac{(R_{x}\cos\xi_{n}-\mu_{x,n} )^{2}}{2\sigma_{x,n}^{2}}-\frac{\left(R_{x}\sin\xi_{n}-\mu_{y,n}\right)^{2}}{ 2\sigma_{y,n}^{2}}\right)d\xi_{n},\end{split}  \tag{11}\]  
where \(\zeta_{n}\) is the divergence angle describing the increase in the beam radius at the receiver with a distance \(z_{n}\) from the transmitter, whereas the beam width could be approximated as \(w_{z,n}\approx\zeta_{n}z_{n}\) for relatively long propagation distances. Additionally, \(R_{n}=\sqrt{R_{x,n}^{2}+R_{y,n}^{2}}\) where \(R_{x,n}^{2}\) and \(R_{y,n}^{2}\) denote the offsets along the horizontal and elevation axes at the detector plane, respectively. In this context, the random variables \(R_{x,n}\) and \(R_{y,n}\) are considered as nonzero mean Gaussian random variables, i.e., \(R_{x,n}\sim N\left(\mu_{x,n},\sigma_{x,n}^{2}\right)\) and \(R_{y,n}\sim N\left(\mu_{y,n},\sigma_{y,n}^{2}\right)\) with \(\mu_{x,n}\) and \(\mu_{y,n}\) representing the mean values, whereas \(\sigma_{x,n}\) and \(\sigma_{y,n}\) are the standard deviations for horizontal and elevation displacements, respectively. According to the analysis which has been performed in [69], the Beckmann distribution above can be accurately simplified through a modified Rayleigh distribution as  
\[f_{R_{n}}(R_{n})=\frac{R_{n}}{\sigma_{mod,n}^{2}}exp\left(-\frac{R_{n}^{2}}{2 \sigma_{mod,n}^{2}}\right),R_{n}\geq 0, \tag{12}\]  
with \(\sigma_{mod,n}\) being the joint standard deviation of \(\sigma_{x,n}\) and \(\sigma_{y,n}\) for the \(n\)-th hop, expressed as follows [69],  
\[\sigma_{mod,n}^{2}=\left(\frac{3\mu_{x,n}^{2}\sigma_{x,n}^{4}+3\mu_{y,n}^{2} \sigma_{y,n}^{4}+\sigma_{x,n}^{6}+\sigma_{y,n}^{6}}{2}\right)^{\frac{1}{3}} \tag{13}\]  
Therefore, according to [69], the PDF of the random variable \(h_{p,n}\) can be written as  
\[f_{h_{p,n}}\left(h_{p,n}\right)=\frac{\psi_{n}^{2}}{\left(A_{0,n}g_{n}\right) ^{\psi_{n}^{2}}}h_{p,n}^{\psi_{n}^{2}-1},0\leq h_{p,n}\leq g_{n}A_{0,n}, \tag{14}\]  
where \(\psi_{n}=w_{z,eq,n}/2\sigma_{mod,n}\) refers to the total amount of the pointing mismatch at the \(n\)-th receiver aperture. Indeed, larger values of \(\psi_{n}\) denote weaker generalized pointing errors, and conversely, smaller values of \(\psi_{n}\) correspond to a stronger total amount of pointing mismatch [22]. Similarly, \(\psi_{x,n}=w_{z,eq,n}/2\sigma_{x,n}\) and \(\psi_{y,n}=w_{z,eq,n}/2\sigma_{y,n}\) indicate the specific generalized misalignment strengths along the horizontal and elevation axis, respectively. It is worth noting here that \(w_{z,eq,n}=\left[\sqrt{\pi\text{erf}}(v_{n})w_{z,n}^{2}/2v_{n}exp\left(-v_{n} ^{2}\right)\right]^{1/2}\), with \(A_{0,n}=\text{erf}^{2}(v_{n})\) representing the fraction of the collected power at \(r_{n}=0\), with \(\text{erf}(. )\) standing for the error function [70], Equation (7.1.1), and \(r_{n}\) being the radius of the \(n\)-th receiver aperture, while \(v_{n}=\sqrt{\pi}r_{n}/\sqrt{2}w_{n,z}\), and \(w_{n,z}\) standing for the corresponding beam waist on the receiver plane at propagating distance \(z_{n}\)[19; 24]. Additionally, the parameter \(g_{n}=exp\left(\frac{1}{\psi_{n}^{2}}-\frac{1}{2\psi_{n}^{2}}-\frac{1}{2\psi_{ n}^{2}}-\frac{\mu_{x,eq}^{2}}{2\sigma_{x,n}^{2}\psi_{n,z}^{2}}-\frac{\mu_{x,eq}^{2}}{ 2\sigma_{x,n}^{2}\psi_{n,z}^{2}}-\frac{\mu_{x,eq}^{2}}{2\sigma_{x,n}^{2}\psi_{ n,z}^{2}}\right)\) determines the presence of ZB or NZB pointing errors.