## 2 System Description  
The length of slots \(\mathrm{t_{s}}=\mathrm{MT_{b}}/\mathrm{n}\) is for the DPPM frames, \(n=2^{\mathrm{M}}\) where \(\mathrm{T_{b}}\)\(=\)\(1/\mathrm{R_{b}}\) is for the bit cycle, \(\mathrm{R_{b}}\) for the data rate and M is for the coding level (CL) [9]. Figure 1a shows the HFFSO-PON systems where optical signals (OSs) suffer from the ASE noise (ASEN), beam-spreading, beam-absorption, attenuation, ICC, and splitting losses at the optical band-pass filter (OBPF)/demux. The proposed system is modeled and simulated using MATLAB software (2013). The results are based on Monte-Carlo simulations. We use eight channels WDM-DWDM starting from 1550 nm over single-mode fiber and the channel spacing in the C-band of 100 GHz of the ITU (International Telecommunication Union). The results demonstrate (2.5 Gbps data rate \(\times\)8 channels) 20 Gbit/s-4000 m transmission with favorable performance. We investigate eight-wavelength channels that have been transmitted over turbulent HFFSO links using the WDM technique. The feeder fiber path length of 20 km is proposed for the system model [8; 15; 17]. By using the optical amplifier (OA) gain G on the remote node, we improve the device model after receiver collection lens (RCL) as seen in Figure 1a. We propose that the OA system is used to reduce the inter-channel crosstalk for the proposed system. Figure 1b shows the proposed architecture for all scenarios [9]. As seen in Figure 1, M-ary DPPM is transmitted over the OA and OBPF (b). The receiver converts the OS into an electrical signal. The module of the receivers consists of a photodetector (PD), an electric amplifier, a filter, and a compared circuitry for the decision circuit and AO. We propose the hybrid pulse modulation/M-ary DPPM-M-PAPM for the DWDM-FSO and hybrid optical fiber over the atmospheric turbulence channel. Information is transmitted as a series of DPPM-M-PAPM pulses. The modulated signal from M-ary DPPM-M-PAPM is connected to OA and OBPF to provide M-ary DPPM-M-PAPM output as shown in Figure 1b [9]. For our calculations, we assume the loss of the signal multiplexer (mux)/demultiplexer (demux) (\(\leq\)3.5 dB) [8; 15]. Figure 1: _Cont._  
**3. M-ary Digital Pulse Position Modulation and M-pulse Amplitude and Position Modulation (M-ary DPPM-M-PAPM) Scheme**  
In this section, the random variable (RV) of the MGF describes the current \(\mathrm{Y_{sig}}(\Delta t)\) where \(\mathrm{sig}\in\{0,1\}\) which depends on the pulses or is not transmitted for the signal pulses, \(\Delta t\) is the duration of the crosstalk pulse. It is written as [5; 9; 26; 27; 28]:  
\[\mathrm{M_{Y_{sig}}(\Delta t)}(\mathrm{s})=\left\{\frac{R^{\prime}G\left( \mathrm{e^{\mathrm{sq/t_{b}}}-1}\right)\int_{\mathrm{t_{b}}}\mathrm{sigP_{tr} }(t)dt}{1-R^{\prime}N_{o}\left(\mathrm{e^{\mathrm{sq/t}}}-1\right)}\right\}\ \frac{\ \ \ \mathrm{exp}\left\{\frac{R^{\prime}_{c}(G \mathrm{e^{\mathrm{sq/t_{b}}}}-1)\int_{\Delta t}\mathrm{P_{XT}}(t)dt}{1-R^{ \prime}N_{o}\left(\mathrm{e^{\mathrm{sq/t_{b}}}}-1\right)}\right\}}{\left[1-R^ {\prime}N_{o}\left(\mathrm{e^{\mathrm{sq/t_{b}}}}-1\right)\right]^{\mathrm{L}}} \tag{1}\]  
where \(\Delta t=\mathrm{t_{s}}\) is the time slots align with the OS slots otherwise \(\mathrm{t_{1}}\) or \(\mathrm{t_{2}}\), and \(\Delta t=0\) for no crosstalk in the slot. Furthermore, the DPPM-PAPM pulse and ICC pulse strength are \(\mathrm{P_{tr}}\), \(\mathrm{P_{XT}}\), respectively for the hybrid modulation techniques over the FSO link. \(R^{\prime}=\eta/\hbar v_{i}\eta\) is the PD quantum efficiency, \(v_{i}\), \(v\) and are the optical frequencies of ICC wavelengths and signal respectively and \(h\) is [PERSON]'s constant, \(\mathrm{q}\) is the electron charge, [5; 8; 9; 10; 29; 30; 31; 32; 33; 34], \(N_{o}=0.5(\mathrm{NF}\times\mathrm{G}-1)\,hv\) is the OA power spectral density (PSD) at the single polarization ASEN.