# Error Probability of a Coherent \(M\)-ary PSK FSO System Influenced by Phase Noise
## 4 Numerical Results  
Based on the convergence analysis presented in Section 3.3 ((31) and (32)) for a given value of the upper limit of the truncation error, we determine the number of terms \(N\) (\(|E_{N}|\leq q_{N+1}\) ) in the sum for the symbol error probability given by (22) (or (20) in the case of perfect compensation of phase error). Furthermore, the value of SEP is estimated by using (22), i.e., (20), in combination with (18), i.e., (8). The basic parameters influencingthe error rate system performance are: turbulence strength parameters (\(C_{n}^{2}\), \(\alpha\), \(\beta\), \(\rho\)) and standard deviation (\(\sigma_{s}\)) of pointing error (related to misalignment between transmitter laser and receiver photodiode), signal-to-noise ratio per bit (\(\mu_{b}\)), wavelength of light (\(\lambda\)) and modulation order (\(M\)), as well as the standard deviation of phase noise (\(\sigma_{\varphi}\)). For obtaining numerical results, we use values of parameters from experimental measurements [9]. The Rytov variance determines the intensity of atmospheric turbulence, with the wavelength \(\lambda=785\) nm and the FSO link distance \(L=1\) km. The pointing errors effect is described by the jitter standard deviation, \(\sigma_{s}\). The radius of a circular detector aperture is \(a=5\) cm, the beam radius at the waist is \(a_{0}=5\) cm, and the radius of curvature takes a value \(F_{0}=-10\). Normalized average optical power of the FSO hop is \(\Omega+2b_{0}=1\). All numerical results are confirmed by independent Monte Carlo simulations. Figure 2 represents the average SEP dependence on the average SNR per bit assuming different values of phase noise standard deviation. SEP decreases with increasing \(\mu_{b}\), but only in the range of medium values of \(\mu_{b}\). In the region of large values of \(\mu_{b}\), SEP tends to a constant value called the SEP floor. This SEP floor cannot be reduced by neither increasing the signal power, nor by improving the channel conditions, but depends only on the phase noise standard deviation. Furthermore, we observe the SEP dependence on a different amount of scattering power coupled to the LOS. A higher value of \(\rho\) means that the power of scattered component by off-axis eddies is smaller. Since the presented results are obtained for the same value of the Rytov variance [9], the scattering component by off-axis eddies will have a crucial impact on the system performance. The results show that a larger value of the parameter \(\rho\) improves the system performance. It is expected because, when the value of parameter \(\rho\) tends to one, the whole amount of scattering components power belongs to that scattering component coupled to an LOS component, and there is not a component scattered by offline eddies. In the case of \(\rho=1\), the SEP performance is the best possible, since the power of \(C_{GS}\) equals zero (the total received optical power is equal to the sum of the power of an LOS component and power of a scattered component coupled to an LOS component, while the component \(C_{GS}\) scattered by the off-axis does not exist). Additionally, it can be observed that the amount of the scattering components has a stronger impact on the average SEP when the FSO receiver is affected by weaker phase noise, i.e., for lower values of \(\sigma_{\varphi}\). When hardware imperfections are more pronounced, the effect of the FSO channel conditions on the SEP is minor. Figure 2: Average SEP vs. average SNR per bit. Figure 3 shows the dependence of the bit error probability on the signal-to-noise ratio per bit when detecting 16 PSK signals. It should be noted that BER decreases with increasing SNR in the area of low and medium SNR values. However, for high SNR values, i.e., when the signal power is higher, the BER dependencies enter saturation. When BER curve reaches saturation, i.e., when the floor is reached, it no longer makes sense to increase the signal power because the performance is not improved.