{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,21]],"date-time":"2026-03-21T17:50:00Z","timestamp":1774115400975,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,15]],"date-time":"2023-09-15T00:00:00Z","timestamp":1694736000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"The Polish National Agency for Academic Exchange (NAWA)","award":["PPN\/ULM\/2020\/1\/00256\/DEC\/1"],"award-info":[{"award-number":["PPN\/ULM\/2020\/1\/00256\/DEC\/1"]}]},{"name":"The Polish National Agency for Academic Exchange (NAWA)","award":["7750824"],"award-info":[{"award-number":["7750824"]}]},{"name":"The Polish National Agency for Academic Exchange (NAWA)","award":["CIF-1855879"],"award-info":[{"award-number":["CIF-1855879"]}]},{"name":"The Polish National Agency for Academic Exchange (NAWA)","award":["CCF 2106189"],"award-info":[{"award-number":["CCF 2106189"]}]},{"name":"The Polish National Agency for Academic Exchange (NAWA)","award":["CCSS-2027844"],"award-info":[{"award-number":["CCSS-2027844"]}]},{"name":"The Polish National Agency for Academic Exchange (NAWA)","award":["CCSS-2052751"],"award-info":[{"award-number":["CCSS-2052751"]}]},{"name":"The Polish National Agency for Academic Exchange (NAWA)","award":["CCF-2100013"],"award-info":[{"award-number":["CCF-2100013"]}]},{"name":"Ministry of Science and Technological Development of the Republic of Serbia and Science Fund of the Republic of Serbia","award":["PPN\/ULM\/2020\/1\/00256\/DEC\/1"],"award-info":[{"award-number":["PPN\/ULM\/2020\/1\/00256\/DEC\/1"]}]},{"name":"Ministry of Science and Technological Development of the Republic of Serbia and Science Fund of the Republic of Serbia","award":["7750824"],"award-info":[{"award-number":["7750824"]}]},{"name":"Ministry of Science and Technological Development of the Republic of Serbia and Science Fund of the Republic of Serbia","award":["CIF-1855879"],"award-info":[{"award-number":["CIF-1855879"]}]},{"name":"Ministry of Science and Technological Development of the Republic of Serbia and Science Fund of the Republic of Serbia","award":["CCF 2106189"],"award-info":[{"award-number":["CCF 2106189"]}]},{"name":"Ministry of Science and Technological Development of the Republic of Serbia and Science Fund of the Republic of Serbia","award":["CCSS-2027844"],"award-info":[{"award-number":["CCSS-2027844"]}]},{"name":"Ministry of Science and Technological Development of the Republic of Serbia and Science Fund of the Republic of Serbia","award":["CCSS-2052751"],"award-info":[{"award-number":["CCSS-2052751"]}]},{"name":"Ministry of Science and Technological Development of the Republic of Serbia and Science Fund of the Republic of Serbia","award":["CCF-2100013"],"award-info":[{"award-number":["CCF-2100013"]}]},{"name":"NSF","award":["PPN\/ULM\/2020\/1\/00256\/DEC\/1"],"award-info":[{"award-number":["PPN\/ULM\/2020\/1\/00256\/DEC\/1"]}]},{"name":"NSF","award":["7750824"],"award-info":[{"award-number":["7750824"]}]},{"name":"NSF","award":["CIF-1855879"],"award-info":[{"award-number":["CIF-1855879"]}]},{"name":"NSF","award":["CCF 2106189"],"award-info":[{"award-number":["CCF 2106189"]}]},{"name":"NSF","award":["CCSS-2027844"],"award-info":[{"award-number":["CCSS-2027844"]}]},{"name":"NSF","award":["CCSS-2052751"],"award-info":[{"award-number":["CCSS-2052751"]}]},{"name":"NSF","award":["CCF-2100013"],"award-info":[{"award-number":["CCF-2100013"]}]},{"name":"Jet Propulsion Laboratory, California Institute of Technology","award":["PPN\/ULM\/2020\/1\/00256\/DEC\/1"],"award-info":[{"award-number":["PPN\/ULM\/2020\/1\/00256\/DEC\/1"]}]},{"name":"Jet Propulsion Laboratory, California Institute of Technology","award":["7750824"],"award-info":[{"award-number":["7750824"]}]},{"name":"Jet Propulsion Laboratory, California Institute of Technology","award":["CIF-1855879"],"award-info":[{"award-number":["CIF-1855879"]}]},{"name":"Jet Propulsion Laboratory, California Institute of Technology","award":["CCF 2106189"],"award-info":[{"award-number":["CCF 2106189"]}]},{"name":"Jet Propulsion Laboratory, California Institute of Technology","award":["CCSS-2027844"],"award-info":[{"award-number":["CCSS-2027844"]}]},{"name":"Jet Propulsion Laboratory, California Institute of Technology","award":["CCSS-2052751"],"award-info":[{"award-number":["CCSS-2052751"]}]},{"name":"Jet Propulsion Laboratory, California Institute of Technology","award":["CCF-2100013"],"award-info":[{"award-number":["CCF-2100013"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We examine the effects of imperfect phase estimation of a reference signal on the bit error rate and mutual information over a communication channel influenced by fading and thermal noise. The Two-Wave Diffuse-Power (TWDP) model is utilized for statistical characterization of propagation environment where there are two dominant line-of-sight components together with diffuse ones. We derive novel analytical expression of the Fourier series for probability density function arising from the composite received signal phase. Further, the expression for the bit error rate is presented and numerically evaluated. We develop efficient analytical, numerical and simulation methods for estimating the value of the error floor and identifying the range of acceptable signal-to-noise ratio (SNR) values in cases when the floor is present during the detection of multilevel phase-shift keying (PSK) signals. In addition, we use Monte Carlo simulations in order to evaluate the mutual information for modulation orders two, four and eight, and identify its dependence on receiver hardware imperfections under the given channel conditions. Our results expose direct correspondence between bit error rate and mutual information value on one side, and the parameters of TWDP channel, SNR and phase noise standard deviation on the other side. The results illustrate that the error floor values are strongly influenced by the phase noise when signals propagate over a TWDP channel. In addition, the phase noise considerably affects the mutual information.<\/jats:p>","DOI":"10.3390\/e25091341","type":"journal-article","created":{"date-parts":[[2023,9,17]],"date-time":"2023-09-17T06:12:21Z","timestamp":1694931141000},"page":"1341","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the Effect of Imperfect Reference Signal Phase Recovery on Performance of PSK System Influenced by TWDP Fading"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6431-4186","authenticated-orcid":false,"given":"Goran T.","family":"Djordjevic","sequence":"first","affiliation":[{"name":"Faculty of Electronic Engineering, University of Nis, 18104 Nis, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6472-2027","authenticated-orcid":false,"given":"Dejan N.","family":"Milic","sequence":"additional","affiliation":[{"name":"Faculty of Electronic Engineering, University of Nis, 18104 Nis, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8667-1507","authenticated-orcid":false,"given":"Bata","family":"Vasic","sequence":"additional","affiliation":[{"name":"Faculty of Electronic Engineering, University of Nis, 18104 Nis, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9048-1837","authenticated-orcid":false,"given":"Jaros\u0142aw","family":"Makal","sequence":"additional","affiliation":[{"name":"Faculty of Electrical Engineering, Bia\u0142ystok University of Technology, 15-351 Bia\u0142ystok, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2365-4106","authenticated-orcid":false,"given":"Bane","family":"Vasic","sequence":"additional","affiliation":[{"name":"Department of Electrical & Computer Engineering, University of Arizona, Tucson, AZ 85721-0104, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"S\u00e1nchez, J.D.V., Urquiza-Aguiar, L., and Paredes Paredes, M.C. (2021). Fading Channel Models for mm-Wave Communications. Electronics, 10.","DOI":"10.3390\/electronics10070798"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Simon, M.K., and Alouini, M.S. (2004). Digital Communication over Fading Channels, John Wiley & Sons, Ltd.. [2nd ed.].","DOI":"10.1002\/0471715220"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1002\/ett.1145","article-title":"Average SEP for M-ary CPSK with noisy phase reference in Nakagami fading and Gaussian noise","volume":"18","year":"2007","journal-title":"Eur. Trans. Telecommun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"965","DOI":"10.1016\/j.jfranklin.2017.10.029","article-title":"Truncation error analysis in computing of SEP and SEP floor for partially coherent receiver of MPSK signals over composite fading channels","volume":"355","year":"2018","journal-title":"J. Frankl. Inst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1109\/TWC.2006.1611079","article-title":"Performance analysis of generalized-faded coherent PSK channels with equal-gain combining and carrier phase error","volume":"5","author":"Smadi","year":"2006","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_6","unstructured":"Lindsey, W.C., and Simon, M.K. (1991). Telecommunication Systems Engineering, Courier Corporation."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1109\/TCOMM.2002.1010620","article-title":"New analytical models and probability density functions for fading in wireless communications","volume":"50","author":"Durgin","year":"2002","journal-title":"IEEE Trans. Commun."},{"key":"ref_8","first-page":"500","article-title":"Comprehensive analysis of the impact of TWDP fading on the achievable error rate performance of BPSK signaling","volume":"101","author":"Kim","year":"2017","journal-title":"IEICE Trans. Commun."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Rao, M., Lopez-Martinez, F.J., and Goldsmith, A. (2014, January 19\u201321). Statistics and system performance metrics for the two wave with diffuse power fading model. Proceedings of the 2014 48th Annual Conference on Information Sciences and Systems (CISS), Princeton, NJ, USA.","DOI":"10.1109\/CISS.2014.6814106"},{"key":"ref_10","first-page":"2548","article-title":"MGF approach to the analysis of generalized two-ray fading models","volume":"14","author":"Rao","year":"2015","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Maric, A., Kaljic, E., and Njemcevic, P. (2021). An alternative statistical characterization of TWDP fading model. Sensors, 21.","DOI":"10.3390\/s21227513"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Njemcevic, P., Kaljic, E., and Maric, A. (2022). Moment-Based Parameter Estimation for the \u0393-Parameterized TWDP Model. Sensors, 22.","DOI":"10.3390\/s22030774"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1186\/s13638-018-1336-6","article-title":"Better than Rician: Modelling millimetre wave channels as two-wave with diffuse power","volume":"2019","author":"Caban","year":"2019","journal-title":"EURASIP J. Wirel. Commun. Netw."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1049\/el:19780329","article-title":"Envelope probability density function of the sum of signal, noise and interference","volume":"14","author":"Kostic","year":"1978","journal-title":"Electron. Lett."},{"key":"ref_15","unstructured":"Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing), John Wiley & Sons, Ltd.. [2nd ed.]."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Ivani\u0161, P., and Draji\u0107, D. (2017). Information Theory and Coding-Solved Problems, Springer.","DOI":"10.1007\/978-3-319-49370-1"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Baig, I., Farooq, U., Hasan, N.U., Zghaibeh, M., and Jeoti, V. (2020). A multi-carrier waveform design for 5G and beyond communication systems. Mathematics, 8.","DOI":"10.3390\/math8091466"},{"key":"ref_18","unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (2000). Table of Integrals, Series, and Products, Academic Press. [6th ed.]."},{"key":"ref_19","unstructured":"Kostic, I. (1996, January 26\u201328). Cumulative distribution function of envelope of sum of signal, noise and interference. Proceedings of the Telecommunication Forum (TELFOR), Belgrade, Yugoslavia."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Marjanovic, Z., Milic, D.N., and Djordjevic, G.T. (2022). Estimation of Truncation Error in Statistical Description of Communication Signals over mm-Wave Channels. Axioms, 11.","DOI":"10.3390\/axioms11100569"},{"key":"ref_21","unstructured":"Wolfram Research, Inc (2023, September 04). The Mathematical Functions Site. 1998\u20132023. Available online: http:\/\/functions.wolfram.com."},{"key":"ref_22","unstructured":"Erldelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. (1955). Higher Transcendental Functions, McGraw-Hill."},{"key":"ref_23","unstructured":"Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V., Cohl, H.S., and McClain, M.A. (2023, September 04). NIST Digital Library of Mathematical Functions, Available online: http:\/\/dlmf.nist.gov\/."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1109\/TAES.1976.308305","article-title":"PSK performance with imperfect carrier phase recovery","volume":"AES-12","author":"Prabhu","year":"1976","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_25","unstructured":"Milovanovic, G., and Djordjevic, R. (2005). Mathematical Analysis I, Faculty of Electronic Engineering, University of Nis."},{"key":"ref_26","unstructured":"Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. (2007). Numerical Recipes: The Art of Scientific Computing, Cambridge University Press. [3rd ed.]."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Jeruchim, M.C., Balaban, P., and Shanmugan, K.S. (2000). Simulation of Communication Systems: Modeling, Methodology and Techniques, Springer Science & Business Media.","DOI":"10.1007\/b117713"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/9\/1341\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:52:00Z","timestamp":1760129520000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/9\/1341"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,15]]},"references-count":27,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["e25091341"],"URL":"https:\/\/doi.org\/10.3390\/e25091341","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,15]]}}}