# Error Probability of a Coherent \(M\)-ary PSK FSO System Influenced by Phase Noise
## 4 Numerical Results
The value of this floor is affected only by the standard deviation of the phase noise. The higher it is, the lower the value of the floor. It should be noted that even very small changes in the standard deviation can significantly change the value of the floor. For example, if the standard deviation is increased from 2.5 degrees to 3 degrees, the floor increases by more than one order of magnitude. It is very important to notice the SNR area when the floor is reached, and it is even more important to estimate this floor. The red lines show the floor values, which are estimated based on the derived formula (40). The same figure also shows the dependence of BER on SNR when there is no phase noise, i.e., when the phase error compensation in the receiver is perfect. This dependence is shown in blue. It should be noted that this dependence is essentially different compared with a case when there is a phase noise in the receiver. Namely, in this case, BER decreases monotonically for all SNR values, i.e., there is no occurrence of floor. Based on this, it can be concluded that the influence of the phase noise is of essential importance in coherent FSO systems and that this phenomenon should be given special attention when designing the receiver. Figure 4 aims to show the influence of the pointing errors on the SEP for different degrees of hardware imperfections. The influence of the phase noise is actually defined by the value of the standard deviation (\(\sigma_{\varphi}\)) of the phase noise. The higher this value is, the stronger is the influence of the phase noise. Similarly, the impact of misalignment is defined by the value of the standard deviation (\(\sigma_{s}\)) of pointing errors. The higher its value is, the more pronounced is the misalignment. When \(\sigma_{\varphi}=8^{0}\), then the SEP is insensitive to the increase in the \(\sigma_{s}\) up to 0.125 m. In the case when \(\sigma_{\varphi}=12^{0}\), the SEP is insensitive to pointing errors as long as the \(\sigma_{s}\) is less than 0.2 m. Therefore, the influence of pointing errors is stronger in conditions when the phase noise is weaker. The average SEP dependence on the phase noise standard deviation for different values of modulation order is shown in Figure 5. Numerical results are obtained for different atmospheric turbulence conditions based on the values of the parameters given  
Figure 3: Average BER vs. average SNR per bit. in ([9], Table 1). The results show that the BPSK format is insensitive to the phase noise as long as the \(\sigma_{\varphi}\) is less than 16 degrees, while the QPSK becomes sensitive already when the \(\sigma_{\varphi}\) is 8 degrees, while in the case of applying the 8 PSK format, the SEP begins to increase significantly already for a \(\sigma_{\varphi}\) value of 2.5 degrees. In addition, the effect of phase noise on SEP values is stronger when atmospheric turbulence is weaker (\(C_{n}^{2}\) is lower). In other words, when \(C_{n}^{2}\) is lower, the SEP begins increasing at a lower value of \(\sigma_{\varphi}\). The numerical results illustrate that developed mathematical framework allows us to connect the required error rate performance of the system with the modulation order and parameters of the receiver under given channel conditions. Our work should not be understood only as performance analysis, but we also contribute to the design of such systems. For given conditions over the FSO channel, from the condition that a certain SEP  
Figure 4: Average SEP vs. standard deviation of the pointing errors. Figure 5: Average SEP vs. standard deviation of the phase noise. value should be reached, it is possible to estimate the maximum value of the standard deviation of the phase noise for which this will be achieved. This value of the standard deviation of the phase noise plays a very important role in receiver design.