## 2 Analytical Modeling  
The mixed FSO and fiber link system is introduced in this paper, purposing to minimize the nonlinear issues and FSO related impairments such as RF alignment issues, FSO pointing errors, and co-channel interference. This section includes the analytical calculations for the proposed FSO and optical systems. The channel model of FSO is defined [14; 15] as  
\[F_{ch}=\psi_{a}\psi_{p}\psi_{l}. \tag{1}\]  
where \(F_{ch}\) presents the FSO channel, \(\psi_{a}\) is the atmospheric turbulence loss, \(\psi_{l}\) is the geometric loss, and \(\psi_{p}\) is the FSO pointing issues. Three components are considered for optical signal transmission [16]: (1) line of sight component, (2) line of sight coupled with the scattered component, and (3) independent scattered component. The power distribution function (pdf) for free space is expressed [17; 18; 19; 20] as  
\[f_{F_{ch}}(F_{ch})=B\Sigma_{n=1}^{\beta_{m}}a_{n}F_{ch}^{\frac{n \alpha+n}{2}-1}N_{\alpha-n}(2\sqrt{\frac{\alpha_{m}\beta_{m}F_{ch}}{\gamma \beta_{m}+P^{\prime}}}) \tag{2}\]  
and \(B\) means  
\[B=\frac{2\alpha_{m}^{\alpha_{m}/2}}{\gamma^{1+\alpha_{m}/2}\varphi (\alpha_{m})}(\frac{\gamma\beta_{m}}{\beta_{m}+P^{\prime}})^{\beta_{m}+\alpha _{m}/2} \tag{3}\]where \(\alpha_{m}\) is the large scale scattering process, \(\beta_{m}\) is the fading parameter, \(\gamma\) is the independent scattering component, and \(\varphi(. )\) represents gamma function. The \(p^{\prime}\) is defined as  
\[p^{\prime}=p+2\tau b_{0}, \tag{4}\]  
where the \(p\) is the power for the first line of sight component, and \(2\tau b_{0}\) is the coupled line of sight and scattered component. The parameter \(a_{n}\) is further explained [21] as  
\[a_{n}= \begin{bmatrix}\beta_{m}-1\\ n-1\end{bmatrix} \frac{(\gamma\beta_{m}+p^{\prime})^{1-\frac{n}{2}}}{(n-1)! }( \frac{p^{\prime}}{U})^{n-1}(\frac{\alpha_{m}}{\beta_{m}})^{\frac{n}{2}} \tag{5}\]  
where \(\beta_{m}\) is the fading element, and \(\alpha_{m}\) is the effective number of large scale scattering process. The FSO system performance is conditioned by the transceiver and structural ways; this leads to FSO pointing impairments, and it is calculated in terms of PFD [22; 23] as  
\[f_{\psi_{p}}=\frac{u^{2}}{A_{0}^{u^{2}}}(\psi_{p})^{u^{2}-1},0\leq\psi_{p}\leq A _{0} \tag{6}\]  
where \(A_{0}\) is integrated optical power function, and \(u\) is related to jitter deviation and equal to \(\frac{\alpha_{m}}{2\pi^{2}}\). The width of the data carrying laser beam is denoted by \(\omega_{z}\). The statistical analysis of FSO pointing errors, turbulence fading, and co-channel interference is expressed [24; 25] as  
\[f_{F_{ch}}(F_{ch})=\int f_{F_{ch}/\psi_{a}}(F_{ch}/\psi_{a}).f_{\psi_{a}}(\psi_{ a})d\psi_{a} \tag{7}\]  
In Equation (7), the \(f_{F_{ch}/\psi_{a}}(F_{ch}/\psi_{a})\) declares the conditional probability. By substituting Equation (1) to Equation (6) into Equation (7), the CDF of the N channel is defined as  
\[f_{F_{ch}}(F_{ch})=\frac{u^{2}B}{2}\Sigma_{n=1}^{\beta_{m}}(a_{n}[\frac{1}{A_{ m}}]^{\frac{n+n}{2}})G_{2A}^{3,1}(\frac{F_{ch}}{AB_{0}I_{l}}) \tag{8}\]  
where \(G_{2A}^{3,1}\) is the Meijer's G function. On the transmitter side, multi-pulse position modulation (MPPM) based intensity modulation direct detection system is used for the FSO system. The electrical filter is installed on the receiver side to remove unwanted signals from the original signals. The output of the filtered signal is calculated as  
\[y(t)=R\frac{N}{n}P_{R}\Sigma_{n=0}^{N-1}C_{n}+k(t) \tag{9}\]  
The average received optical power, \(R\), is the photodetector responsitivity, and \(k(t)\) is the additive white Gaussian (AWG); \(C_{n}\) is the signal time slot. The transmitter and receiver telescope gains are expressed [18; 26; 27] as  
\[G_{t}=G_{r}=(\pi d/\lambda_{k})^{2} \tag{10}\]  
where \(G_{t}\) is the transmitter gain, \(G_{r}\) is the receiver gain, and \(d\) is the diameter.