# Error Probability of a Coherent \(M\)-ary PSK FSO System Influenced by Phase Noise
## 2 System and Channel Model
Furthermore, the parameter \(g\) represents the average power of the component \(C_{GS}\) scattered by off-axis eddies, defined as \(g=\mathrm{E}\!\left[\left|C_{GS}\right|^{2}\right]=2b_{0}(1-\rho)\), where \(2b_{0}=\mathrm{E}\!\left[\left|C_{CS}\right|^{2}+\left|C_{GS}\right|^{2}\right]\) is the average total power of the scatter components. The parameter \(\rho\)\((0\leq\rho\leq 1)\) represents the contribution of \(C_{GS}\) into the total power of scattered components. Next, \(\Omega^{\prime}=\Omega+2b_{0}\rho+2\sqrt{2b_{0}\rho\Omega}\cos(\phi_{A}- \phi_{B})\) is the average power from the coherent contributions, with the average power of the LOS defined as \(\Omega=\mathrm{E}\!\left[\left|C_{L}\right|^{2}\right]\). The deterministic phases of the LOS and the coupled to LOS scatter term are represented as \(\phi_{A}\) and \(\phi_{B}\), respectively [10]. Related to the pointing errors, the parameter \(\xi\) is defined as \(\xi=a_{e}/(2\sigma_{\rm s})\), where \(a_{e}\) is the equivalent beam radius at the receiver, and \(\sigma_{\rm s}\) is jitter standard deviation. Parameter \(a_{e}\) is related to the parameter \(a_{L}\) being the optical beam radius at the distance \(L\), as \(a_{e}^{2}=a_{L}^{2}\sqrt{\pi}\operatorname{erf}(v)/\left(2v\exp(-v^{2})\right)\), with \(A_{0}=\operatorname{erf}^{2}(v)\), \(v=\sqrt{\pi}\,a/\left(\sqrt{2}\,a_{d}\right)\), where \(a\) denotes the radius of circular detector [13; 14] and \(\operatorname{erf}(\cdot)\) is the error function ([33], (8.250.1)). The parameter \(a_{L}\) is related with the beam radius at the waist, \(a_{0}\), and the radius of curvature,\(F_{0}\), by \(a_{L}=a_{0}((\Theta_{0}+\Lambda_{0})(1+1.63\sigma_{R}^{12/5}\Lambda_{1}))^{1/2}\), where \(\Theta_{0}=1-L/F_{0}\), \(\Lambda_{0}=2L/(ka_{0}^{2})\), \(\Lambda_{1}=\Lambda_{0}/(\Theta_{0}^{2}+\Lambda_{0}^{2})\)[13; 14]. The Rytov variance is defined as \(\sigma_{R}^{2}=1.23C_{n}^{2}(2\pi/\lambda)^{7/6}d^{11/6}\), The Rytov variance determines intensity of atmospheric turbulence being defined as \(\sigma_{R}^{2}=1.23C_{n}^{2}(2\pi/\lambda)^{7/6}L^{11/6}\), where \(\lambda\) is the wavelength, and \(C_{n}^{2}\) is the refractive index structure parameter. Based on the instantaneous SNR definition in (2) and the PDF in (3), the PDF of \(\gamma\) is derived as [31]  
\[f_{\gamma}(\gamma)=\frac{\xi^{2}A}{2\gamma}\sum_{k=1}^{\beta}a_{k}G_{1,3}^{3, 0}\bigg{(}\frac{\alpha\beta\kappa(g+\Omega^{\prime})\gamma}{(g\beta+\Omega^ {\prime})\tilde{\gamma}}\bigg{|}\frac{\xi^{2}+1}{\xi^{2},\kappa,k}\bigg{)}, \tag{5}\]  
where \(\kappa=\xi^{2}/\big{(}\xi^{2}+1\big{)}\), and the average SNR per symbol for the coherent detection is \(\tilde{\gamma}=\mathrm{E}[\gamma]=\mathrm{E}[I]C_{c}\)[31]. The average SNR per bit, \(\mu_{b}\), is defined as \(\mu_{b}=\tilde{\gamma}/\log_{2}M\). At the receiver, coherent detection is performed by combining received optical signal with the LO beam. There is a difference between received signal random phase (\(\phi_{\mathrm{OS}}\)) and LO output signal random phase (\(\phi_{\mathrm{LO}}\)). This difference, denoted by \(\phi=\phi_{\mathrm{OS}}-\phi_{\mathrm{LO}}\), is also a random process. This random process is compensated by PLL in the receiver in the electrical domain. However, the compensation is not perfect, i.e., there is a difference between phase \(\phi\) and voltage-controlled local oscillator output signal phase \(\phi\). This difference, \(\varphi=\phi-\phi\), is a random process fluctuating over time, and it is known as a phase noise. The phase noise is described by a Tikhonov PDF given by [21]  
\[f_{\varphi}(\varphi)=\frac{\exp(b\cos(\varphi))}{2\pi I_{0}(b)},\quad|\varphi| \leq\pi, \tag{6}\]  
where \(I_{0}(\cdot)\) is the zero order modified Bessel function of the first kind, while the \(n\)-th order modified Bessel function of the first kind is defied by ([33], (8.431)), \(b=1/\sigma_{\varphi}^{2}\), and \(\sigma_{\varphi}^{2}\) is the variance of the phase noise.