# Error Probability of a Coherent \(M\)-ary PSK FSO System Influenced by Phase Noise
## 3 Error Rate Analysis
### Representation of PDF of Signal Phase in Fourier Series Form  
In order to evaluate the SEP performance of the coherent FSO system, the PDF of the composite received signal phase \(\psi\) is represented in the Fourier series form as it was suggested in [32]. The conditional PDF of the composite signal phase in a Fourier series form due to additive noise can be presented as [32]  
\[f(\psi|\gamma)=\frac{1}{2\pi}+\sum_{n=1}^{\infty}a_{n}(\gamma)\cos(n\psi),\ \ |\psi|<\pi, \tag{9}\]where \(a_{n}(\gamma)\) is the Fourier coefficient for the AWGN channel defined as [32]  
\[a_{n}(\gamma)=\frac{\Gamma\big{(}\frac{n}{2}\!+\!1\big{)}}{n!\pi}\gamma^{\frac{n} {2}}\exp(-\gamma)_{1}F_{1}\Big{(}\frac{n}{2}+1;n+1;\gamma\Big{)}, \tag{10}\]  
where \({}_{1}F_{1}(\cdot;\cdot;\cdot)\) denotes the confluent hypergeometric function ([33], (9.21)). The average PDF of the resulting received signal phase can be determined as  
\[f_{\psi}(\psi)=\int\limits_{0}^{\infty}f(\psi|\gamma)f_{\gamma}(\gamma)\mathrm{ d}\gamma, \tag{11}\]  
where \(f_{\gamma}(\gamma)\) is the PDF of the instantaneous SNR given by (5). After substituting (5), (9) and (10) into (11), the PDF of phase \(\psi\) is given as  
\[ \begin{split} f_{\psi}(\psi)&=\frac{1}{2\pi}+\frac {\xi^{2}A}{2\pi}\sum\limits_{n=1}^{\infty}\sum\limits_{k=1}^{\beta}\frac{a_{k} \Gamma\big{(}\frac{n}{2}+1\big{)}}{n!}\cos(n\psi)\\ &\times\int\limits_{0}^{\infty}\gamma^{\frac{n}{2}-1}\exp(-\gamma )_{1}F_{1}\Big{(}\frac{n}{2}+1;n+1;\gamma\Big{)}G_{1,3}^{\,3,0}\Big{(}\frac{ \alpha\beta\kappa(g+\Omega^{\prime})\gamma}{(g\beta+\Omega^{\prime})\bar{ \gamma}}\Big{|}\begin{smallmatrix}\xi^{2}+1\\ \xi^{2},\,\alpha,k\end{smallmatrix}\Big{)}\,\mathrm{d}\gamma.\end{split}  \tag{12}\]  
The product of exponential and confluent hypergeometric functions in (12) is represented in terms of Meijer's \(G\)-function based on ([34], (07.20.26.0015.01)) as  
\[\exp(-\gamma)_{1}F_{1}\Big{(}\frac{n}{2}\!+\!1;n\!+\!1;\gamma\Big{)}\!=\!\frac {\Gamma(n\!+\!1)}{\Gamma\big{(}\frac{n}{2}\big{)}}G_{1,2}^{\,1,1}\Big{(}\gamma \left| \begin{smallmatrix}1-\frac{n}{2}\\ 0,-\,n\end{smallmatrix} \Big{)}. \tag{13}\]  
Furthermore, by inserting (13) into (12), and after applying ([34], (06.05.16.0002.01) and (06.05.03. 0001.01)), the PDF of phase \(\psi\) is re-written as  
\[ \begin{split} f_{\psi}(\psi)&=\frac{1}{2\pi}+\frac {\xi^{2}A}{4\pi}\sum\limits_{n=1}^{\infty}\sum\limits_{k=1}^{\beta}a_{k}n\cos (n\psi)\\ &\times\int\limits_{0}^{\infty}\gamma^{\frac{n}{2}-1}G_{1,2}^{\,1,1}\Big{(}\gamma\left|\begin{smallmatrix}1-\frac{n}{2}\\ 0,-\,n\end{smallmatrix}\Big{)}G_{1,3}^{\,3,0}\Big{(}\frac{\alpha\beta\kappa( g+\Omega^{\prime})\gamma}{(g\beta+\Omega^{\prime})\bar{\gamma}}\Big{|} \begin{smallmatrix}\xi^{2}+1\\ \xi^{2},\,\alpha,k\end{smallmatrix}\Big{)}\mathrm{d}\gamma.\end{split}  \tag{14}\]  
The integral in (14) is solved by utilizing ([34], (07.34.21.0013.01)) as  
\[\int\limits_{0}^{\infty}\gamma^{\frac{n}{2}-1}G_{1,2}^{\,1,1}\Big{(}\gamma \left| \begin{smallmatrix}1-\frac{n}{2}\\ 0,-\,n\end{smallmatrix} \Big{)}G_{1,3}^{\,3,0}\Big{(}\frac{\alpha\beta\kappa(g+ \Omega^{\prime})\gamma}{(g\beta+\Omega^{\prime})\bar{\gamma}}\Big{|} \begin{smallmatrix}\xi^{2}+1\\ \xi^{2},\,\alpha,k\end{smallmatrix} \Big{)}\mathrm{d}\gamma=G_{3,4}^{\,4,1} \Big{(}\frac{\alpha\beta\kappa(g+\Omega^{\prime})}{(g\beta+\Omega^{\prime}) \bar{\gamma}}\Big{|} \begin{smallmatrix}1-\frac{n}{2},1+\frac{n}{2},\xi^{2}+1 \\ \xi^{2},\alpha,k,0\end{smallmatrix} \Big{)}. \tag{15}\]  
The final PDF of phase \(\psi\) is derived as  
\[f_{\psi}(\psi)=\frac{1}{2\pi}+\frac{\xi^{2}A}{4\pi}\sum\limits_{n=1}^{\infty} \sum\limits_{k=1}^{\beta}a_{k}nG_{3,4}^{\,4,1}\Big{(}\frac{\alpha\beta\kappa(g+ \Omega^{\prime})}{(g\beta+\Omega^{\prime})\bar{\gamma}}\Big{|} \begin{smallmatrix} 1-\frac{n}{2},1+\frac{n}{2},\xi^{2}+1\\ \xi^{2},\alpha,k,0\end{smallmatrix} \Big{)}\cos(n\psi).