## 4 Aggregated Channel Attenuation Model
### Aggregated Channel Attenuation Model  
The PDF of the jointly attenuated channel gain \(h\) could be obtained by evaluating the product of the PDFs of the three independent attenuation factors that affect the channel gain. \[f_{h}(h)=\int f_{h|h_{a}}(h|h_{a})f_{h_{a}}(h_{a})dh_{a} \tag{13}\]  
where the function \(f_{h|h_{a}}(h|h_{a})\) represents the probability density function that depends on the atmospheric turbulence \(h_{a}\), and it is defined as follows:  
\[f_{h|h_{a}}(h|h_{a})=\frac{1}{h_{a}h_{l}}f_{h_{p}}\left(\frac{h}{h_{a}h_{l}} \right)=\frac{\gamma^{2}}{A_{0}\gamma^{2}h_{a}h_{l}}\left(\frac{h}{h_{a}h_{l} }\right)^{\gamma^{2}-1},\;0\leq h\leq A_{0}h_{a}h_{l} \tag{14}\]  
By utilizing Equations (9), (13) and (14), and by applying the relevant integral formula of the Meijer G function found on the Mathematical Functions website [21], it was possible to calculate a closed-form expression for the probability density function (PDF) of the channel state \(h\) in the M distribution model as follows:  
\[f_{h}(h)=\frac{A\gamma^{2}}{2h}\sum\limits_{k=1}^{\beta}a_{k}\left(\frac{ \alpha\beta}{\xi\beta+\Omega^{\prime}}\right)^{-\frac{\alpha+k}{2}}G_{1}^{3} \!\!\!\!\!\!\!  
By the approximate computation of the Gaussian Q-function \(Q(x)\approx\frac{1}{12}\exp\left(-\frac{y^{2}}{2}\right)+\frac{1}{4}\exp\left(- \frac{2x^{2}}{3}\right)\)[23], Equation (18) could be further written as:  
\[P\left(\mathbf{S}_{i}\rightarrow\mathbf{S}_{j}\left|\mathbf{H}\right.\right)= \frac{1}{12}\exp\left(-\frac{\rho}{4}\sum_{n=1}^{Nr}|\alpha_{n}|^{2}\right)+ \frac{1}{4}\exp\left(-\frac{\rho}{3}\sum_{n=1}^{Nr}|\alpha_{n}|^{2}\right) \tag{19}\]  
Thus, the PEP could be expressed as follows [17]:  
\[P\left(\mathbf{S}_{i}\rightarrow\mathbf{S}_{j}\right)=\mathbf{E}\left(P\left( \mathbf{S}_{i}\rightarrow\mathbf{S}_{j}\left|\mathbf{H}\right.\right)\right) \tag{20}\]  
where \(\mathbf{E}()\) represents the expectation operator. Combined with Equations (18) and (20), PEP could be expressed as follows:  
\[P\left(\mathbf{S}_{i}\rightarrow\mathbf{S}_{j}\right)=\mathbf{E}\left(\frac{ 1}{12}\exp\left(-\frac{\rho}{4}\sum_{n=1}^{Nr}|\alpha_{n}|^{2}\right)+\frac{1} {4}\exp\left(-\frac{\rho}{3}\sum_{n=1}^{Nr}|\alpha_{n}|^{2}\right)\right) \tag{21}\]  
Let \(Y=|\alpha_{n}|^{2}\), PEP can be further expressed as follows:  
\[P\left(\mathbf{S}_{i}\rightarrow\mathbf{S}_{j}\right)=\frac{1}{12}\prod_{n=1 }^{Nr}\int\limits_{Y}\exp\left(-\frac{\rho}{4}Y\right)f_{Y}(Y)dY+\frac{1}{4} \prod_{n=1}^{Nr}\int\limits_{Y}\exp\left(-\frac{\rho}{3}Y\right)f_{Y}(Y)dY \tag{22}\]  
With the help of the Moment Generating function (MGF), the above equation could be written as follows:  
\[P\left(\mathbf{S}_{i}\rightarrow\mathbf{S}_{j}\right)=\frac{1}{12}\prod_{m=1 }^{Nr}M_{|\alpha_{n}|^{2}}\left(\frac{\rho}{4}\right)+\frac{1}{4}\prod_{m=1}^ {Nr}M_{|\alpha_{n}|^{2}}\left(\frac{\rho}{3}\right) \tag{23}\]  
By putting the joint channel attenuation Equations (15) and (23) into Equation (16) and using the relevant integral formula of Meijer G function in [21], the upper bound expression of ABER could be calculated as follows:  
\[\begin{split} P_{b}\leq&\frac{1}{TR_{p}^{{}^{\prime} 2}T_{P}^{{}^{\prime}}}\sum_{i=1}^{2^{T_{P}^{\prime}}}\sum_{j=1,j\
eq i}^{2^{T _{P}^{\prime}}}d_{H}\left(\mathbf{S}_{i},\mathbf{S}_{j}\right)\left(\frac{1}{ 12}\prod_{m=1}^{Nr}M_{|  
conventional OSM ML decoder uses this definition to compute the actual operation total count, and the complexity of its receiver can be expressed as [24]:  
\[C_{OSM}=8N_{r}\times 2^{R_{p}^{OSM}} \tag{25}\]  
In the same way, the complexity of the OGSM receiver is calculated. However, since multiple lasers in OGSM are activated to emit the same constellation symbol, the complexity of its receiver is increased compared to OSM.