## 2 System Description
NF and G are the noise figure and OA gain G respectively, \(\mathrm{L}=\mathrm{B_{o}m_{t}t_{s}}\) encompass the system modes for the spatial and temporal method [2; 8; 9; 35; 36; 37; 38; 39; 40; 41; 42], \(\mathrm{B_{o}}\) is the bandwidth of the optical noise for the demux channel and \(\mathrm{m_{t}}\) is the number of ASEN states. \(N_{o\_XT}\) is the PSD-ASEN at the PD and the signal-to-crosstalk ratio \(\mathrm{C_{XT}}=\mathrm{P_{tr}}/\mathrm{P_{XT}}\). The total MGF for Gaussian zero-mean, including the thermal noise variance (TNV) is calculated as [5; 9; 35; 36; 37; 38; 39; 40; 41; 42]:  
\[\mathrm{M_{Y_{sig}}(\Delta t)}(\mathrm{s})=\mathrm{M_{Y_{sig}}(\Delta t)}( \mathrm{s})\ \mathrm{exp}\left(\frac{\mathrm{s}^{2}\sigma_{\mathrm{th-DPPM-PAPM}}^{2}}{2}\right) \tag{2}\]  
where \(\sigma_{\mathrm{th-DPPM}}^{2}\) is the DPPM-PAPM TNV. The means and variances are given as [5; 9]:  
\[\mathrm{\mu_{Y_{sig}}(\Delta t)}(s)=\frac{LR^{\prime}qN_{o}}{t_{s}}+R^{\prime} Gq\left(\mathrm{sig}\ \mathrm{P_{tr}}\frac{\mathrm{P_{XT}}\Delta t}{t_{s}}\right) \tag{3}\]  
\[\sigma_{\mathrm{X_{sig}}(\Delta t)}^{2}=\sigma_{\mathrm{th-DPPM-PAPM}}^{2}+ \left(\frac{LR^{\prime}q^{2}N_{o}(1+R^{\prime}N_{o})}{t_{s}^{2}}\right)+R^{ \prime}Gq^{2}\left[\left(1+2R^{\prime}N_{o}\right)\frac{\mathrm{sigP_{tr}}}{t_ {s}}\right]+R^{\prime}_{i}Gq^{2}\left[\left(1+2R^{\prime}_{i}N_{o_{XT}}\right) \frac{\mathrm{P_{XT}}\Delta t}{t_{s}^{2}}\right] \tag{4}\]  
Figure 1: System model for 8 channels PON-DWDM-HFFSO for OOK-NRZ/M-ary DPPM-M-PAPM under moderate turbulence (MT), strong turbulence (ST), and weak turbulence (WT): (**a**) proposed framework for interchannel crosstalk (ICC) evaluation and (**b**) schematic of the receiver system. We reproduced it from [9]. The symbol error probability (SER) is received in the exist of ICC \(\mathrm{P_{ws(I_{i-\mathrm{r_{i}}})}}\):  
\[\mathrm{P_{ws(I_{i-\mathrm{r_{i}}})}}\geq\prod_{ \begin{array}{c}\mathrm{j=1}\\ \mathrm{j\
eq\mathrm{sigslot}}\end{array} }^{\mathrm{n}}\mathrm{P}\big{(}\mathrm{ X_{1}(\Delta t)>X_{\mathrm{j}}}\big{)} \tag{5}\]  
where \(\mathrm{X_{j}}\) denotes the non-signal slot \(\mathrm{X_{o}}\big{(}\Delta t_{\mathrm{j}}\big{)}\) and \(\Delta t_{\mathrm{j}}\) is the \(\mathrm{j}\) (empty) slot overlap with the ICC. Using the GA, the expression \(\mathrm{P}\big{\{}\mathrm{X_{o}}\big{(}\Delta t_{\mathrm{j}}\big{)}>X_{1}( \Delta t)\big{\}}\) is explained from [1; 2; 5; 9; 10; 29]:  
\[\mathrm{P}\big{\{}\mathrm{X_{o}}\big{(}\Delta t_{\mathrm{j}}\big{)}>X_{1}( \Delta t)\big{\}}=0.5\mathrm{erfc}\left(\frac{\mu_{X_{1}(\Delta t)}-\mu_{X_{0}( \Delta t_{\mathrm{j}})}}{\sqrt{2\Big{(}\sigma_{X_{1}(\Delta t)}^{2}+\sigma_{X_ {0}(\Delta t_{\mathrm{j}})}^{2}\Big{)}}}\right) \tag{6}\]  
In the proposed CB, we have fixed threshold \(\mathrm{T_{th}}\) and the general form for RV (X) variable and a is \(\mathrm{P}\big{(}\mathrm{X>T_{th}}\big{)}\leq\mathrm{E}\left\{\exp\left[\mathrm{ s}\big{(}\mathrm{X-T_{th}}\big{)}\right]\right\}\), \(\mathrm{s}>0\):  
\[\mathrm{P}\big{\{}\mathrm{X_{o}}\big{(}\Delta t_{\mathrm{j}}\big{)}>X_{1}( \Delta t)\big{\}}\leq\mathrm{M_{X_{1}(\Delta t)}(-\mathrm{s})\ M_{X_{0}( \Delta t)}(\mathrm{s})\ (\mathrm{s}>0) \tag{7}\]  
For the MCB [2; 5] of two RVs for \(\mathrm{X_{o}}\big{(}\Delta t_{\mathrm{j}}\big{)}\) and \(\mathrm{X_{1}(\Delta t)},\mathrm{P}(\mathrm{X>T_{th}})\leq\mathrm{M_{X}( \mathrm{s})e^{-\mathrm{sT_{th}}}/\mathrm{s}\sigma_{th}\sqrt{\pi}}\)[5; 9; 10].