## 5 Conclusions  
In this paper, we analyzed the transmission of PSK signals over a TWDP channel in the presence of imperfect reference signal phase estimation. Non-perfect estimation of the incoming signal phase directly causes the appearance of the BER floor. The specific value of this BER floor is reducible only by decreasing the phase noise level, i.e., by improving the received signal phase estimation, while the BER floor is insensitive to a further increase in signal power, as well as changing the values of parameters \(K\) and \(\Gamma\). The obtained results showed that parameters \(K\) and \(\Gamma\) strongly influence BER over the range of moderate SNR values. In addition, the results showed that for typical channel conditions, the BPSK modulation format can tolerate phase noise until its standard deviation reaches 16 degrees, while in the same environment, the QPSK/8 PSK format is resistant to this phenomenon only up to the standard deviation of eight/four degrees. The results illustrated that the maximum value of mutual information cannot be reached in the presence of the phase noise regardless of the increase in signal power. The 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. In an illustrative scenario, we showed that the maximum value of mutual information for SPSK, QPSK and BPSK begins to decrease at \(\sigma_{\varphi}=5^{\circ}\), \(15^{\circ}\) and \(25^{\circ}\), respectively. The higher the order of modulation, the stronger the mutual information shows sensitivity to phase noise. These values of standard deviation are the starting condition for the correct design of the phase estimator, i.e., the design of the extraction circuit should be undertaken in such a way that the predetermined phase noise levels in terms of its standard deviation are not exceeded. This standard deviation value is determined based on the tolerable BER (mutual information) value and under given channel conditions. In this paper, we focused on MPSK because it is a simple example that is very close to practical applications. We emphasized that the solution for \(M=4\) directly corresponds to the case of the 4 QAM modulation format proposed for modern 5G systems. Our solution for Fourier coefficients in composite signal phase expansion will be useful for all propagation environments that can be described by the TWDP model. In further work, our attention  
Figure 8: Power penalty versus transmitted information decrease threshold for 8 PSK. will be focused on finding a general solution for MQAM formats as well. We will improve the model of the receiver part for reference signal recovery, and will consider recovery circuits based on signal squaring and maximum a posteriori estimation. Conceptualization, G.T.D. and J.M. ; methodology, G.T.D., D.N.M. and B.V. (Bata Vasic); software, G.T.D., D.N.M. and B.V. ([PERSON]); formal analysis, G.T.D., J.M. and D.N.M. ; investigation, G.T.D. and D.N.M. ; resources, G.T.D., J.M. and D.N.M. ; data curation, G.T.D. ; writing--original draft preparation G.T.D., D.N.M., B.V. ([PERSON]) and J.M. ; writing--review and editing, B.V. ([PERSON]); visualization, G.T.D. and D.N.M. ; supervision, B.V. ([PERSON]); project administration, G.T.D. ; funding acquisition, G.T.D. All authors have read and agreed to the published version of the manuscript. This work is partially supported by The Polish National Agency for Academic Exchange (NAWA) under grant No. PPN/ULM/2020/1/00256/DEC/1, as well as by Ministry of Science and Technological Development of the Republic of Serbia and Science Fund of the Republic of Serbia (Grant Agreement No: 7750824).