## 2 System and Channel Model
### Non-Zero Boresight Pointing Error Model  
In practice, pointing errors are determined by boresight and jitter components. The former denotes the fixed displacement between the beam and detector center, while the latter is the random beam offset [15]. The Beckmann model is a general and realistic statistical distribution model that accurately describes the pointing mismatch by taking into account the beam width, detector size, and different jitter for elevation and horizontal displacements, as well as the effect of the non-zero boresight error; it is expressed by [16]: \begin{table}
\begin{tabular}{c c} \hline \hline
**Distribution Model** & **Generation** \\ \hline Lognormal & \(\rho=0\), Var \(\left[\,\left|\,\mathrm{U}_{\mathrm{L}}\,\right|\,\right]=0\), \(c\rightarrow\infty\) \\ Gamma & \(\rho=0\), \(c=0\) \\ K & \(\Omega=0\) and \(\rho=0\) or \(b=1\) \\ Gamma-Gamma & \(\rho=1\), \(c=0\), \(\Omega=1\) \\ Exponential & \(\Omega=0\), \(\rho=0\), \(a\rightarrow\infty\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter set for the generation of widely used distribution models from the unifying \(M\)(alaga) distribution model, where \(U_{L}\) represents the line-of-sight (LOS) component, Var\(\left[\cdot\right]\) denotes the variance operator, and \(\left|\cdot\right|\) stands for the absolute value [11]. \[f_{R}(R)=\frac{R}{2\pi\,\sigma_{x}\sigma_{y}}\int\limits_{0}^{2\pi}\exp\left(- \frac{(R\cos\theta-\mu_{x})^{2}}{2\sigma_{x}^{2}}-\frac{(R\sin\theta-\mu_{y})^{2 }}{2\sigma_{y}^{2}}\right)d\theta, \tag{6}\]  
where \(\theta\) is the divergence angle describing the increase of the beam radius with distance \(z\) from the transmitter, which can be approximated by \(w_{z}=\theta\,z\) for larger values of \(z\). Additionally, \(R=\sqrt{R_{x}^{2}+R_{y}^{2}}\) is the radial displacement, with \(R_{x}\) and \(R_{y}\) representing the deviation along the horizontal and elevation axes of the detector plane, respectively. These random variables are considered as non-zero mean Gaussian random variables, i.e., \(R_{x}\sim N(\mu_{x},\sigma_{x}^{2})\) and \(R_{y}\sim N(\mu_{y},\sigma_{y}^{2})\), where the parameters \(\mu_{x},\,\mu_{y}\) denote the mean values, and \(\sigma_{x},\,\sigma_{y}\) symbolize the standard deviation for the horizontal and elevation displacements, respectively [16]. The Beckmann model can be suitably approximated by a modified Rayleigh distribution as [16; 17]:  
\[f_{R}(R)=\frac{R}{\sigma_{\text{mod}}^{2}}\exp\left(-\frac{R^{2}}{2\sigma_{ \text{mod}}^{2}}\right),\ R\geq 0, \tag{7}\]  
where:  
\[\sigma_{\text{mod}}^{2}=\left(\frac{3\mu_{x}^{2}\sigma_{x}^{4}+3\mu_{y}^{2} \sigma_{y}^{4}+\sigma_{x}^{6}+\sigma_{y}^{6}}{2}\right)^{1/3}. \tag{8}\]  
Furthermore, the PDF for the irradiance depending on pointing errors can be derived as:  
\[f_{I_{p}}(I_{p})=\frac{q^{2}}{(A_{0}\,g)^{q^{2}}}\,{l_{p}^{q^{2}-1}},\,0\leq I _{p}\leq A_{0}\,g, \tag{9}\]  
with \(g\) defined by:  
\[g=\exp\left(\frac{1}{q^{2}}-\frac{1}{2q_{x}^{2}}-\frac{1}{2q_{y}^{2}}-\frac{ \mu_{x}^{2}}{2\sigma_{x}^{2}q_{x}^{2}}-\frac{\mu_{y}^{2}}{2\sigma_{y}^{2}q_{y }^{2}}\right) \tag{10}\]  
and \(q_{x}=w_{z,eq}/2\sigma_{x},\ q_{y}=w_{z,eq}/2\sigma_{y}\), and \(q=w_{z,eq}/2\sigma_{\text{mod}}\), where \(w_{z,eq}\) represents the equivalent beam radius at the receiver expressed by \(w_{z,eq}=w_{z}[\sqrt{\pi}\text{erf}(v)/(2v\,e^{-v^{2}})]^{1/2}\). Moreover, \(v=\sqrt{\pi}r_{a}/(\sqrt{2}w_{z})\),with \(r_{a}\) being the radius of the circular detection aperture, and \(\text{erf}(\cdot)\) stands for the error function. In addition, \(A_{0}=\left[\text{erf}(v)\right]^{2}\) is the fraction of the collected power at the center of the receiver [18]. Note that the Beckmann distribution incorporates some classical fading models such as the Rician, Hoyt (Nakagami-q), and Rayleigh distributions as particular cases [19].