## 2 System Model  
Traditionally, intensity modulation/direct detection based on an on-off keying (OOK) scheme is widely accepted in OWC owing to its easy implementation and lower cost [20]. Considering the low BER performance and spectrum efficiency of OOK, PPM has been considered to be used in OWC communications. Compared with OOK, the energy utilization is greatly increased. In addition, modulated QPSK has the characteristics of high spectrum utilization and strong anti-interference [21]. Therefore, combining the PPM and QPSK can improve the data transmission rate and the system reliability [22; 23].  
Figure 1 shows the schematic of the 4 PPM and QPSK hybrid modulated OWC system with FTN technology. The user data after Gray encoder are firstly mapped into 4 PPM and QPSK, respectively. Thereafter, the QPSK signal is loaded into the time slot of the 4 PPM signal to form the 4 PPM-QPSK hybrid modulated signal. Afterwards, the formatted 4 PPM-QPSK signal is sent to the FTN shaping filter for FTN signal forming. Subsequently, after digital-to-analogue conversion (DAC), the data are launched into the atmospheric channel. At the receiver end, the optical signal transmitted over the atmospheric channel is firstly detected by a photodiode (PD) and then sent for analog-to-digital converting (ADC), matched filtering, and sampling. Thereafter, the signal is sent to the DL module for data recovery.  
The \(S_{4 PPM-QPSK}\) signal formed by FTN shaping filter can be expressed as  
\[S_{4 PPM-QPSK-FTN}=\sqrt{E\tau/2}\sum_{\rho}S_{4 PPM-QPSK}\tau(t-\rho\tau T) \tag{1}\]  
where \(E\) is the pulse power; \(r(t)\) is the pulse shape of FTN; \(\tau\) is the time acceleration factor (\(0<\tau<1\)), which is the parameter to characterize the Nyquist compression ratio; \(\rho\) is the information carried by the \(\rho\) - th symbol on the \(S_{4 PPM-QPSK}\); and \(T\) is the symbol period.  
When the signal passes through the atmospheric channel, the received optical signal can be expressed as  
\[S_{u}(t)=h\cdot S(t)+Z_{n}(t) \tag{2}\]  
Figure 1: Schematic of atmospheric optical communication system based on the 4 PPM-QPSK–FTN modulation mode.  
where \(Z_{n}(t)\) is the channel additive noise, \(S(t)\) is the transmitted optical signal, and \(h\) is the channel fading coefficient and it follows Gamma-Gamma distribution. Therefore, the probability density function of \(h\) can be expressed as  
\[\mathrm{H}(h)=\frac{2(\alpha\beta)^{\frac{\alpha+\beta}{2}}}{\Gamma(\alpha) \Gamma(\beta)}\cdot h^{\frac{\alpha+\beta}{2}-1}\cdot Q_{\alpha-\beta}(2\sqrt {\alpha\beta h}),h>0 \tag{3}\]  
where \(Q_{\alpha-\beta}(\cdot)\) is the second class modified Bessel function of order \(\alpha-\beta\); \(\Gamma(\cdot)\) is the Gamma function; and \(\alpha\) and \(\beta\) are the large and small scale scattering coefficients, respectively.  
After electro-optic conversion, the electrical signal obtained can be expressed as  
\[S_{PD}=h\eta\sqrt{EG_{\mathrm{oc}}\tau/2}\int_{-\infty}^{\infty}S_{u}(t)r(t- \rho\tau T)dt+Z_{n}^{{}^{\prime}}(t) \tag{4}\]  
where \(\eta\) denotes the electro-optic conversion ratio, \(G_{\mathrm{OC}}\) denotes the average transmitted power, and \(Z_{n}^{{}^{\prime}}(t)\) denotes the overall interference carried in the signal.  
Sampling is carried out after ADC and matched filter, and its output can be written as  
\[\overline{S_{PD}}=h\eta\sqrt{EG_{\mathrm{oc}}\tau/2}\int_{-\infty}^{\infty}[ \sum_{n}a_{n}r(t-\rho\tau T)]r^{*}(t-\tau T)dt+\int_{-\infty}^{\infty}Z_{n}^{ {}^{\prime}}(t)r^{*}(t-\rho\tau T)dt \tag{5}\]  
where \(a_{n}\) denotes the data sequence and \(r^{*}(t)\) denotes the conjugate expression of \(r(t)\). Then, the \(\overline{S_{PD}}\) is sent to DL module for training and testing. Eventually, the mapping relationship corresponding to the origin signal is determined.