# Performance of SIMO FSO Links over Mixture Composite Irradiance Channels
## 2 Channel Model
### Composite Irradiance Model  
By following the methodology described in [5], and after making the necessary modifications in the PE model, the PDF of the composite model with turbulence effect along with NZB-PE, for \(I_{m}>0\) can be derived as:  
\[f_{I_{m}}(I_{m})=\frac{\psi_{m}^{2}I_{m}^{\psi_{m}^{2}-1}}{(A_{0,m}g_{m})\psi_{ m}^{2}}\sum_{i=1}^{N}a_{i}\zeta_{i}^{\psi_{m}^{2}-b_{i}}\Gamma\left(b_{i}-\psi_{m}^{ 2},\frac{\zeta_{i}I_{m}}{A_{0,m}g_{m}}\right), \tag{11}\]  
where \(\Gamma(\cdot,\cdot)\) refers to the upper incomplete Gamma function and \(\gamma(\cdot,\cdot)\) is the lower incomplete Gamma function defined in [6] (equation (8.350.2)) and [6] (equation (8.350.1)), respectively. Furthermore, the expression of the corresponding expected value for irradiance can be estimated through the general integral for the \(k\)-th moment estimation, \(E\left[I_{m}^{k}\right]=\int_{0}^{\infty}I^{k}f_{I_{m}}(I_{m})dI_{m}\), by assuming \(k=1\), as follows [5]:  
\[E[I_{m}]=\frac{\psi_{m}^{2}A_{0.m}g_{m}}{1+\psi_{m}^{2}}\sum_{i=1}^{N}a_{i} \zeta_{i}^{-(1+b_{i})}\Gamma(1+b_{i}) \tag{12}\]  
i.e., \(\sigma_{n,m}^{2}=\sigma_{n}^{2}/M=N_{0}/(2M)\). Therefore, the ABER of the SIMO FSO system, with OOK modulation scheme is obtained as [4]:  
\[P_{e,a\
u,M}^{\rm OOK}=\int_{I}f_{I}(I)Q\Bigg{(}\frac{1}{\sqrt{2 MN_{0}}}\sqrt{ \sum_{m=1}^{M}\left(\eta_{m}I_{m}\right)^{2}}\Bigg{)}dI, \tag{19}\]  
where \(\rm I=[I_{1},I_{2},\ldots,I_{M}]\) is the vector of the normalized irradiance at the receiver(s). By substituting the approximation \(Q(x)\approx(e^{-x^{2}/2}+3e^{-2x^{2}/3})/12\) for the Q-function proposed in [12] into the expression (19), an effective estimation yields as [4]:  
\[ \begin{array}{c}P_{e,a\
u,M}^{\rm OOK}\approx\frac{1}{\Omega}\prod\limits_{ m=1}^{M}\int_{0}^{\infty}\exp\left(-\frac{\eta_{m}^{2}I_{m}^{2}}{4 MN_{0}} \right)f_{I_{m}}(I_{m})dI_{m}\\ \qquad\qquad+\frac{1}{4}\prod\limits_{m=1}^{M}\int_{0}^{\infty}\exp\left(- \frac{\eta_{m}^{2}I_{m}^{2}}{3 MN_{0}}\right)f_{I_{m}}(I_{m})dI_{m}.\end{array}  \tag{20}\]  
Next, by substituting (11) into (20), we get:  
\[\begin{array}{c}P_{e,a\
u,M}^{\rm OOK}\approx\frac{1}{\Omega}\prod\limits_{ m=1}^{M}\frac{\psi_{m}^{2}}{\left(A_{0,m}g_{m}\right)^{\psi_{m}^{2}}}\sum \limits_{i=1}^{N}a_{i}\psi_{i}^{\psi_{i}^{2}-b_{i}}\  
and by using the same approximation for the Q-function as [4], the expression (24) can be written as:  
\[ \begin{array}{l}P_{\epsilon\mu\
u,M}^{\text{L-PPM}}\approx\frac{L}{24}\prod\limits _{m=1}^{M}\int_{0}^{\infty}\exp\left(-\frac{L\log_{2}L_{m}^{2}l_{m}^{2}}{8M_{ \text{N}0}}\right)f_{l_{m}}(I_{m})dl_{m}\\ \qquad+\frac{L}{8}\prod\limits_{m=1}^{M}\int_{0}^{\infty}\exp\left(-\frac{L\log _{2}L_{m}^{2}l_{m}^{2}}{6M_{\text{N}0}}\right)f_{l_{m}}(I_{m})dl_{m}.\end{array}  \tag{25}\]  
Next, by substituting (11) into (25) we obtain:  
\[\begin{array}{l}P_{\epsilon\mu\
u,M}^{\text{L-PPM}}\approx\frac{1}{12}\prod \limits_{m=1}^{M}\frac{\psi_{m}^{2}}{(A_{0,m}\beta_{m})^{\psi_{m}^{2}}}\sum \limits_{i=1}^{N}a_{i}\zeta  
\(M\) different branches of wavelength and spatial diversity schemes. Furthermore, we make the same assumptions for the PE parameters, i.e., \(\psi_{1}=\psi_{m}=\psi\), \(A_{0,1}=A_{0,m}=A_{0}\), \(g_{1}=g_{m}=g\), and finally for the average electrical SNR, \(\mu_{1}=\mu_{m}=\mu\). The parameter values used here are highlighted in Table 1 below. Additionally, for the Malaga distribution approximation, we consider that \(\varphi_{A}-\varphi_{B}=\pi/2\), \(\rho=0.9\), \(b_{0}=0.25\), and \(\gamma^{\prime}=0.05\). Hence, \(\Omega^{\prime}=0.95\), given the fact that the transmitted power is normalized as \(\Omega+2b_{0}=1\)[10]. Figure 1 demonstrates the OP versus the normalized outage threshold \(\mu/\gamma_{th,m}\) for the cases, where the MG model approximates the GG or Malaga distributions, while Figure 2 shows the OP of NE cases, accompanied by the corresponding Monte Carlo simulations with \(1\times 10^{9}\) samples.