# Error Probability of a Coherent \(M\)-ary PSK FSO System Influenced by Phase Noise
## 2 System and Channel Model  
After optical signal transmission over an FSO channel influenced by atmospheric turbulence and pointing errors, coherent detection is performed by mixing received optical signals with the LO beam. Next, the PIN photodetector (PD) is used to converting an optical signal into an electrical one, which is further recovered by the PSK demodulator in the electrical domain. The block-scheme of the system considered here is presented in Figure 1. Based on the quite similar analysis presented in [28; 29], the electrical signal at the MPSK receiver input can be presented as  
Figure 1: Model of the system. \[i(t)=2R\sqrt{(P_{\rm s}P_{\rm LO})}\cos(\omega_{\rm IF}t+\phi_{m}+\phi)+n(t), \tag{1}\]  
where \(R\) denotes the PD responsivity, optical signal power is equal to \(P_{\rm s}=A_{r}I\), where \(A_{r}\) is detector area, and \(I\) represents the optical signal irradiance. The LO signal power is denoted by \(P_{\rm LO}\), and \(\omega_{\rm IF}=\omega_{\rm OS}-\omega_{\rm LO}\) is the difference between the optical and LO signal frequencies, respectively. The encoded signal phase information is denoted by \(\phi_{m}\), and \(\phi\) is the random signal phase generated due to transmission over an FSO channel. The system is affected by zero-mean additive white Gaussian noise (AWGN) with variance \(\sigma_{n}^{2}=qRP_{\rm LO}T/2\), where \(q\) is an electronic charge, and \(T\) is symbol interval [28]. The instantaneous SNR per symbol is defined as [28]  
\[\gamma=\frac{4R^{2}P_{\rm s}P_{\rm LO}}{2\sigma_{n}^{2}}=\frac{RP_{\rm s}T}{q} =\frac{RA_{r}T}{q}I=\frac{\eta_{e}A_{r}T}{h\
u}I=C_{\rm c}I. \tag{2}\]  
The responsivity is equal to \(R=\eta_{e}q/(h\
u)\), where \(\eta_{e}\) is the detector quantum efficiency, \(h\) is [PERSON]'s constant, and \(\
u\) is the frequency of the received optical signal. The multiplicative constant for coherent FSO system is defined as \(C_{c}=\eta_{e}A_{r}T/(h\
u)\). The optical signal transmission via free space is affected by \(\mathcal{M}\)-distributed atmospheric turbulence and pointing errors. Since it takes into account the effect of multiple scattered components, the \(\mathcal{M}\) distribution represents a more general model compared to mostly adopted Gamma-Gamma distribution [9; 10]. Besides the component \(C_{L}\) occurring due to LOS contribution, there is the component \(C_{CS}\) scattered by the eddies on the propagation axis, and it is coupled to the component \(C_{L}\). The \(\mathcal{M}\) distribution also joins the component \(C_{GS}\) scattered to the receiver by the off-axis eddies. This component is statistically independent from both components \(C_{L}\) and \(C_{CS}\)[10]. The PDF of irradiance \(I\) accounted for both \(\mathcal{M}\)-distributed atmospheric turbulence and pointing errors is [10; (21)]  
\[f_{I}(I)=\frac{\xi^{2}A}{2}I^{-1}\sum_{k=1}^{\beta}a_{k}G_{1,\lambda}^{3,0} \left(\frac{\alpha\beta I}{(g\beta+\Omega^{\prime})A_{0}}\left| \begin{smallmatrix} \xi^{2+1}\\ \xi^{2},a,k\end{smallmatrix} \right.\right), \tag{3}\]  
where \(G_{p,q}^{m,n}(\cdot)\) is [PERSON]'s \(G\)-function ([33], (9.301)), while constants \(A\) and \(a_{k}\) are defined as ([10], (8))  
\[ \begin{split}& A\triangleq\frac{2\alpha^{\frac{\alpha}{2}}}{g^{1+ \frac{\alpha}{2}}\Gamma(\alpha)}\left(\frac{g\beta}{g\beta+\Omega^{\prime}} \right)^{\beta+\frac{\alpha}{2}},\\ & a_{k}\triangleq\binom{\beta-1}{k-1}\frac{(g\beta+\Omega^{ \prime})^{1-\frac{1}{2}}}{(k-1)!}\left(\frac{\Omega^{\prime}}{g}\right)^{k-1 }\left(\frac{\alpha}{\beta}\right)^{\frac{k}{2}}.\end{split}  \tag{4}\]  
The natural number \(\beta\) defines the amount of fading parameter, while a positive parameter \(\alpha\) is related to the effective number of large-scale cells of the scattering process.