## 4 Results and Discussion
Similar conclusions to the previous case can be drawn regarding the influence of parameter \(\Gamma\) on the BER value (Figure 5b). Namely, parameter \(\Gamma\) also has a strong influence on the BER values in the middle SNR range, and almost no effect for strong signal levels, i.e., when the system reaches the BER floor. Also, when the phase noise is stronger and has higher standard deviation, i.e., when the non-ideal extraction dominates, performance is less sensitive to parameter \(\Gamma\) changes, in comparison to the case where the reference phase recovery is more precise. In general, fading parameters \(K\) and \(\Gamma\) influence performance in a way that is not directly visible from the model presented. For example, if a system has fixed thermal SNR and phase noise levels, then its performance predominantly depends on these fading parameters. We explore this situation in Figure 6a. We analyze the case of BPSK by fixing the BER at \(10^{-5}\), and we calculate the combinations of fading parameters that are required to attain the requested BER performance. These constant BER curves show that an increase in the \(K\) parameter can be compensated by an increase in \(\Gamma\) along the presented curves. Below the curves and to the right lays the area of improved performance that corresponds to the LOS scenario (larger \(K\)) and/or single dominant specular component (smaller \(\Gamma\) values). Phase noise shifts the curves towards this direction, indicating that more favorable fading conditions are required when significant phase noise is present. Above the curves and to the left is the region that leads to poorer BER performance, and it corresponds to NLOS and/or balanced specular components. Figure 6b shows the dependence of BER on \(\rho_{b}\) for the BPSK modulation format. Firstly, it should be noted that, as in the case of QPSK (Figure 4a), for large values of \(\rho_{b}\), an error floor appears. Secondly, when in our numerical and simulation model, \(\sigma_{\varphi}=0^{\circ}\) is set, which means that the phase estimation is ideal; the same dependence is obtained  
Figure 6: Influence of imperfect reference signal phase recovery on BER performance. (**a**) Different fading conditions: \(\Gamma\) vs. \(K\), constant BER curves (BPSK) for fixed SNR, and phase error as parameter. (**b**) Different phase noise levels. Black line corresponds to [9]. as in ([9], Figure 6). Therefore, apart from the fact that the results obtained by analytical approach and numerical integration were verified by independent Monte Carlo simulations, this figure shows that our results are reduced to the results from [9], where the assumption about the ideal estimation of the carrier phase was set. Figure 7 illustrates the influence of the phase noise on mutual information. When transmitting an MPSK signal, the maximum mutual information value that can be achieved is \(\log_{2}M\) (bits/symbol). In order to reach this value of mutual information, the channel should be good enough, i.e., the signal power should be sufficiently large in relation to the noise power. Figure 7a shows the mutual information for the QPSK and 8 PSK formats. Firstly, it should be noted that the phase noise affects the mutual information in such a way that its maximum value cannot be reached in the presence of the phase noise regardless of the increase in signal power. For example, if \(\sigma_{\varphi}=20^{\circ}\), the maximum mutual information values for 8 PSK and QPSK are 1.8 bits/symbol and 1.75 bits/symbol, respectively, and not 3 bits/symbol and 2 bits/symbol, like it is the case when reference signal phase recovery is perfect. The relative decrease in mutual information is larger for 8 PSK than for QPSK. This saturation value of mutual information can be increased only by decreasing the phase noise standard deviation, i.e., by proper design of PLL, and not by increasing the signal power.