## 4 Results and Discussion  
On the basis of definitions introduced in Section 2 and expressions presented in Section 3, we present some numerical and simulation results in this Section and offer some discussions of the results. Figures 4a,b and 5a,b present numerical results evaluated on the basis of Formula (37) together with Monte Carlo simulation results that verify the correctness of the mathematical derivations. Figure 6a,b present results evaluated by applying numerical integration based on (38). Although Monte Carlo simulation results are not presented in these figures due to clarity, we claim that they overlap with those obtained by numerical integration. The numerical results related to the mutual information presented in Figures 7a,b and 8 are evaluated by means of Monte Carlo simulations based on explanations from Section 3.2. A typical influence of average signal-to-noise ratio per bit \(\rho_{b}\) on system BER is illustrated in Figure 4a. Increasing \(\rho_{b}\) has an effect of decreasing the BER, which is an improvement in system performance. However, the BER decrease is not uniform in the entire range of \(\rho_{b}\) shown, but is pronounced for low and moderate values of \(\rho_{b}\). In the range of higher \(\rho_{b}\) values, we observe that the BER values do not decrease significantly following the increase in \(\rho_{b}\). On the contrary, the BER tends towards a constant value, which is called the error floor. The appearance of this floor is directly caused by the phase noise resulting from non-ideal recovery of the reference carrier phase. A particular value of the BER floor is dependant on standard deviation of the phase noise. Increase in the phase noise level directly corresponds to higher BER floors and poorer system performance. The BER floor value cannot be reduced by increasing the signal power, and it can only be influenced by the correct design of the subsystem responsible for estimating the incoming signal reference phase. Figure 4b shows the influence of phase noise level on the BER values during the demodulation of BPSK, QPSK and 8 PSK signals. Firstly, it is obvious that BER increases with increasing standard deviation of phase noise. Secondly, the 8 PSK modulation format starts to be sensitive to phase noise starting from four degrees of standard deviation. Regarding the QPSK format, the BER value remains unaffected until the standard deviation reaches about 8 degrees, while the BPSK format is insensitive to phase noise below 16 degrees. In other words, the BPSK format is more resistant to the influence of the phase noise. This result is also logical because the areas of decision-making in the BPSK format are wider than those in the QPSK and 8 PSK formats, so that the phase noise has less possibility to shift the point from one to another area of decision-making. Figure 4: (**a**) BER performance of QPSK receiver for different values of standard deviation of phase noise; (**b**) BER performance of BPSK and QPSK modulation formats in the presence of phase noise. Figure 5: BER performance for different values of fading parameters. (**a**) BER dependence on parameter \(K\); (**b**) BER dependence on parameter \(\Gamma\). The influence of the average power ratio of specular-to-diffuse components (i.e., parameter \(K\)) is shown in Figure 5a. In the range of large values of \(\rho_{b}\), error floor appears. In that range, the phase noise has the dominant effect on system performance and the value of error probability does not depend on the value of parameter \(K\), but only on the value of standard deviation of phase noise. However, in the range of low and moderate values of \(\rho_{b}\), the effect of the propagation environment conditions is stronger in comparison to effects of phase noise. Also, when the value of the standard deviation is lower, i.e., under conditions when PLL better follows changes in stochastic received signal phase, the effect of parameter \(K\) is stronger.