## 2 System and Channel Model
### Generalized Pointing Errors  
A generalized and realistic statistical model, which accurately describes the pointing errors effect, considering the influence of beam width, the detector's size, the different jitters for the elevation and the horizontal displacement, as well as the effect of non-zero boresight errors, is the Beckmann's distribution [9], which is approximated as [19]:  
\[f_{R_{\{i,j\}}}\left(R_{\{i,j\}}\right)=\frac{R_{\{i,j\}}}{\sigma_{\text{mod}, \{i,j\}}^{2}}\exp\Biggl{(}-\frac{R_{\{i,j\}}^{2}}{2\sigma_{\text{mod},\{i,j\}} ^{2}}\Biggr{)},\ \ R_{\{i,j\}}\geq 0 \tag{6}\]  
With \(R_{\{i,j\}}\) being the radial displacement in the \(j\)th hop in the \(i\)th path that is obtained as \(R_{\{i,j\}}=\left|\begin{subarray}{c}\rightarrow\\ R_{\{i,j\}}\end{subarray}\right|=\sqrt{R_{x,\{i,j\}}^{2}+R_{y,\{i,j\}}^{2}}\), while \(\overset{\rightarrow}{R}_{\{i,j\}}=\left[R_{x,\{i,j\}},R_{y,\{i,j\}}\right]^{T}\) stands for the radial displacement vector, with \(R_{x,\{i,j\}}\) and \(R_{y,\{i,j\}}\) representing the corresponding offsets located along the horizontal and elevation axes at the detector plane, which are expressed as non-zero mean Gaussian distributed random variables, i.e., \(R_{x,\{i,j\}}\sim N\Bigl{(}\mu_{x,\{i,j\}},\sigma_{x,\{i,j\}}^{2}\Bigr{)}\) and \(R_{y,\{i,j\}}\sim N\Bigl{(}\mu_{y,\{i,j\}},\sigma_{y,\{i,j\}}^{2}\Bigr{)}\), where the parameters \(\mu_{x,\{i,j\}}\), \(\mu_{y,\{i,j\}}\), represent their corresponding mean values, and \(\sigma_{x,\{i,j\}}\), \(\sigma_{y,\{i,j\}}\), the corresponding jitters for horizontal and elevation displacements, respectively. The joint standard deviation of \(\sigma_{x,\{i,j\}}\), \(\sigma_{y,\{i,j\}}\),in the \(j\)th hop in the \(i\)th path, can be obtained as [19],  
\[\sigma_{\text{mod},\{i,j\}}^{2}=\left(\frac{3\mu_{x,\{i,j\}}^{2}\sigma_{x,\{i, j\}}^{4}+3\mu_{y,\{i,j\}}^{2}\sigma_{y,\{i,j\}}^{4}+\sigma_{x,\{i,j\}}^{6}+ \sigma_{y,\{i,j\}}^{6}}{2}\right)^{1/3} \tag{7}\]  
Similarly, the PDF for \(I_{p,\{i,j\}}\) can be approximated as [19],  
\[f_{I_{p,\{i,j\}}}\left(I_{p,\{i,j\}}\right)=\frac{\psi_{\{i,j\}}^{2}\frac{- \psi_{\{i,j\}}^{2}}{2}}{A_{0,\{i,j\}}^{\psi_{\{i,j\}}^{2}}}I_{p,\{i,j\}}^{\psi_ {\{i,j\}}^{2}-1},\ \ 0\leq I_{p,\{i,j\}}\leq g_{\{i,j\}}A_{0,\{i,j\}}\] (8 fading, while, \(w_{z,\alpha_{i},\{i,j\}}\) denotes the equivalent beam radius in the \(j\)th hop in the \(i\)th path, which is given as \(w_{z,\alpha_{i},\{i,j\}}=\left[\sqrt{\pi}\text{erf}(v_{\{i,j\}})w_{z,\{i,j\}}^{2 }/2v_{\{i,j\}}\right)\exp(-v_{\{i,j\}}^{2})\right]^{1/2}\), where \(v_{\{i,j\}}=\sqrt{\pi}r_{a_{i}\{i,j\}}/\sqrt{2}w_{z,\{i,j\}}\), with \(r_{a_{i}\{i,j\}}\) being the radius of the circular detection aperture, and \(\text{erf}(.)\), the error function. Additionally \(A_{0,\{i,j\}}\) describes the fraction of the collected power at \(r_{a_{i}\{i,j\}}=0\), with \(A_{0,\{i,j\}}=\text{erf}^{2}(v_{\{i,j\}})\)[7; 19]. It is also worth mentioning that by setting in the \(j\)th hop in the \(i\)th path, the boresight displacement is equal to zero, i.e., \(s_{\{i,j\}}=\sqrt{n_{x,\{i,j\}}^{2}+\mu_{y,\{i,j\}}^{2}}=0\), the [PERSON]'s distribution reduces to the well-known [PERSON]'s distribution in [7], which accurately describes the corresponding zero boresight pointing errors effect [8]. Indeed, for \(s_{\{i,j\}}=0\) it holds that \(\mu_{x,\{i,j\}}=\mu_{y,\{i,j\}}=0\) and \(\sigma_{x,\{i,j\}}=\sigma_{y,\{i,j\}}\) and thus, (6) reduces to Equation (10) in [7] and (8) reduces to Equation (11) in [7].