# Beyond 5G Fronthaul Based on FSO Using Spread Spectrum Codes and Graphene Modulators
## 4 Network Performance Evaluation
The photon absorption rate due to a mark transmission in the desired user antenna code can be written as [116]  
\[\lambda_{s}=\frac{\eta\lambda}{hc}P, \tag{7}\]  
where \(\eta\) is the APD efficiency, \(\lambda\) is the optical wavelength, \(h\) is [PERSON]'s constant, and \(c\) is the speed of light in vacuum. The receiver integrates the APD output over the code symbol interval \(T=LT_{c}\). Hence, the integral APD output of the desired user antenna is  
\[Z_{1}=\frac{1}{T_{c}}\int_{0}^{T}APD(r(t)c_{1}(t)-n_{b}(t))dt. \tag{8}\]By considering the occurrence of data bits \"0\" and \"1\" to be independent variables and that their probability density functions (PDF) can be represented by Gaussian functions [49], then, the mean and variance of the error probabilities conditioned to decision variables of the desired user antenna when it transmits the bit \"0\" and when \(I_{1}\) interfering signals are active in the channel will be [117]  
\[\mu_{b0}=GT_{c}\bigg{[}I_{1}\lambda_{s}+\frac{(WN-I_{1})\lambda_{s}}{M_{e}}+L \bigg{(}\lambda_{b}-\frac{I_{b}}{e}\bigg{)}\bigg{]}+LT_{c}\frac{Is}{e}, \tag{9}\]  
and  
\[\sigma_{b0}^{2}=G^{2}F_{e}T_{c}\bigg{[}I_{1}\lambda_{s}+\frac{(WN-I_{1}) \lambda_{s}}{M_{e}}+L\bigg{(}\lambda_{b}+\frac{I_{b}}{e}\bigg{)}\bigg{]}+L \bigg{(}T_{c}\frac{I_{s}}{e}+\sigma_{th}^{2}\bigg{)}, \tag{10}\]  
where \(G\) is the APD gain, \(F_{e}=k_{eff}G+(2-1/G)(1-k_{eff})\) is the excess noise factor, \(k_{eff}\) is the APD effective ionization ratio, \(W\) is the OOC code weight, \(M_{e}\) is the modulation extinction ratio, \(\lambda_{b}\) is the background light photon arrival rate, \(I_{b}\) is the APD bulk leakage current, \(I_{s}\) is the APD surface leakage current, \(\sigma_{th}^{2}=2k_{B}T_{R}T_{c}/e^{2}R_{L}\) is the thermal noise variance, \(k_{B}\) is Boltzmann's constant, \(T_{R}\) is the receiver noise temperature, and \(R_{L}\) is the receiver load resistor. The value of the receiver parameters is shown in Table 2. Moreover, the mean and variance of the error probabilities conditioned to decision variables of the desired user antenna when it transmits the bit \"1\" will be  
\[\mu_{b1}=GT_{c}\bigg{[}(W+I_{1})\lambda_{s}+\frac{(WN-(W+I_{1}))\lambda_{s}}{M _{e}}+L\bigg{(}\lambda_{b}+\frac{I_{b}}{e}\bigg{)}\bigg{]}+LT_{c}\frac{I_{s}} {e}, \tag{11}\]  
and  
\[s_{n}=Pb_{n}(t)c_{n}(t),\qquad 0\leq t\leq T=FT_{c}, \tag{4}\]0  
Further, the PDF of \(I_{1}\) (total undesired interference contribution to for a single-bit transmission to the desired receiver's accumulated output at the correlation time) can be written as [116]  
\[s_{n}=Pb_{n}(t)c_{n}(t),\qquad 0\leq t\leq T=FT_{c}, \tag{4}\]1  
where \(p=W^{2}/2L\), \(q=1-p\), and \(\delta(x)\) is the Dirac delta function. Because the receiver chooses a \"1\" if \(Z_{1}>T_{h}\), and chooses \"0\" otherwise, the optimum receiver uses the value of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Laser wavelength & \(\lambda\) & 1550 nm \\ APD efficiency & \(\eta\) & 0.85 \\ APD gain & \(G\) & 100 \\ APD effective ionization ratio & \(k_{eff}\) & 0.02 \\ APD bulk leakage current & \(I_{b}\) & 0.1 nA \\ APD surface leakage current & \(I_{s}\) & 10 nA \\ Background light photon arrival rate & \(\lambda_{b}\) & \(10^{9}\) electrons/s \\ Modulation extinction ratio & \(M_{e}\) & depends on \% graphene \\ Bandwidth & \(B\) & depends on \% graphene \\ Data bit rate & \(R_{b}=\frac{\sqrt{2}}{2}B=\frac{1}{T}\) & depends on \% graphene \\ Receiver noise temperature & \(T_{r}\) & \(1100\) K \\ Receiver load resistor & \(R_{L}\) & 1030 \(\Omega\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: FSO network parameters. \(T_{h}\) which minimizes the overall bit error probability. Then, the average bit error probability is given by  
\[