## 2 The CDMA RoFSO System with Spatial Diversity  
### Forward Link  
A thorough analysis of the CDMA point to point (P2P) RoFSO link is presented in Ref. [19]. The proposed spatially diverse system is composed of \(M\)-individual P2P links, where a set of \(M\) laser emitters transmit towards an array of \(M\) receivers. So, at the CBS \(K\) CDMA user's signals proceed for optical transmission through the wireless spatially diverse optical link. Each laser transmitter is linked with a specific receiver as it can be seen in Figure 1. In the case of the forward link transmission, the detected carrier to noise plus interference power ratio for each \(k\)-th user (\(\textit{CNIR}_{k}\)) in a specific \(m\)-th P2P link is given as, [19]:  
Figure 1: The system model for the CDMA RoFSO system with spatial diversity for both directions of the forward and the reverse link. (SPL: splitter, MRC: maximum ratio combiner).  
\[CNIR_{k,f,m}(I_{m})\approx\frac{\left(m_{k,m}\rho_{m}L_{tot,m}P_{0}I_{m}\right)^{2 }}{2\Big{(}\langle N_{0,m}B\rangle_{AV}+\Big{\langle}n_{NLD,f,k,m}\Big{\rangle} _{AV}\Big{\rangle}}, \tag{1}\]  
where the subscript symbol \(f\) denotes the forward link, \(m_{k,m}\) is the optical modulation index (OMI) for each user, \(\rho_{m}\) is the optoelectronic conversion ratio of the photo-detector (PD), \(L_{tot,m}\) corresponds to the total losses of the P2P wireless optical link due to path loss, geometrical loss etc., \(P_{0}\) is the average transmitted optical power and \(I_{m}\) is the total normalized instantaneous irradiance at the receiver's input. The random variable of \(I_{m}\) is a product of two random factors which correspond to the atmospheric turbulence \(I_{t,m}\) and the PE \(I_{p,m}\) i.e., \(I_{m}=I_{t,m}I_{p,m}\). The power spectral density \(N_{0,m}\) of the optical link noise for the \(m\)th receiver is given as, [4; 19]:  
\[N_{0,m}=N_{th,m}+N_{shot,m}+N_{RIN,m}=\frac{4K_{B}T}{R_{L}}+2 qI_{ph,m}+I_{ph,m} {}^{2}(RIN)\Bigg{(}1+\sum_{k=1}^{K}m_{k,m}^{2}\Big{\langle}s_{k}^{2}(t)\Big{ }\Big{\rangle}\Bigg{)}, \tag{2}\]  
where \(K_{B}\) is the Boltzmann's constant, \(T\) is the absolute temperature, \(R_{L}\) is the receiver circuit load resistor, \(q\) is the electron charge, \(RIN\) is the relative intensity noise factor of the laser diode (LD) and \(I_{ph,m}\) is the dc value of the photo-induced current at the \(m\)th receiver and is given as \(I_{ph,m}=\rho_{m}L_{tot,m}P_{0}I_{m}=\rho_{m}P_{R,m}I_{m}\) with \(P_{R,m}\) being the average received optical power at the \(m\)th receiver. The parameter \(B\) is the bandpass filter bandwidth of the receiver centered at the RF carrier frequency \(f_{c}\). We assume that the receivers have very narrow field of view (FOV) and employ optical bandpass filters (OBPF) in the form of coatings on the receivers' optics. The OBPF have \(\Delta\lambda<1\) nm in order to reduce significantly the background radiation \(I_{B}\) from the sun and the sky. In case this is not happened, the background noise contributes to the shot noise term i.e., \(N_{shot,m}=2q\Big{(}I_{ph,m}+I_{B,m}\Big{)}\), [2] (Equation (2.44)). Finally, the total nonlinear distortion for the forward link, i.e., \(n_{NLD,f,m}\), is a sum of the 3 rd order inter-modulation distortion (IMD3) and the clipping distortion \(n_{NLD,f,k,m}=\sigma_{IMD,f,k,m}^{2}+\sigma_{cl,k,m}^{2}\) which are given as, [19]:  
\[\sigma_{IMD,f,k,m}^{2}=\frac{9\alpha_{3}^{2}m_{k,m}^{6}(K-1)I_{ph,m}^{2}}{128} \tag{3}\]  
and  
\[\sigma_{cl,k,m}^{2}=\frac{I_{ph,m}^{2}m_{k,m}^{6}K^{3}}{27.2}\exp\Bigg{(}- \frac{1}{2m_{k,m}^{2}K}\Bigg{)} \tag{4}\]