{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:10:53Z","timestamp":1760130653784,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,8,27]],"date-time":"2023-08-27T00:00:00Z","timestamp":1693094400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Research Start-up Foundation of Recruiting Talents of Nanjing University of Posts and Telecommunications","award":["NY223019","62301277"],"award-info":[{"award-number":["NY223019","62301277"]}]},{"name":"National Natural Science Foundation of China","award":["NY223019","62301277"],"award-info":[{"award-number":["NY223019","62301277"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The performance analysis of polarization M-ary differential chaos shift keying (P-MDCSK) has been expressed using a tight upper bound with the Q-function. However, evaluating the Q-function directly is not a closed expression and there has been less work on closed expression for the upper bound. In order to solve the problem, this paper presents approximate closed-form expressions on the error probability of P-MDCSK. This expression is derived by employing a polynomial approximation of the Q-function. These closed-form expressions are verified through simulations conducted under both additive white Gaussian noise (AWGN) and multipath Rayleigh fading channels. The simulation results reveal that there exists only a negligible gap between the simulations and the derived closed-form expressions. For example, it is observed that the theoretical approximate closed-form expressions exhibit a marginal deviation of approximately 0.4 dB from the simulations when the bit error rate (BER) reaches 10\u22124. Although the proposed method can only give approximate closed-form expressions of the upper bound, it provides an effective method for other communication schemes where the exact BER closed-form formula cannot be obtained.<\/jats:p>","DOI":"10.3390\/e25091267","type":"journal-article","created":{"date-parts":[[2023,8,28]],"date-time":"2023-08-28T01:56:57Z","timestamp":1693187817000},"page":"1267","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Closed-Form Expressions of Upper Bound for Polarization-MDCSK System"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6252-9271","authenticated-orcid":false,"given":"Meiyuan","family":"Miao","sequence":"first","affiliation":[{"name":"School of Communication and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210023, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6698-129X","authenticated-orcid":false,"given":"Lin","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Information and Communication Engineering, Xiamen University, Xiamen 361005, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7172-1807","authenticated-orcid":false,"given":"Weikai","family":"Xu","sequence":"additional","affiliation":[{"name":"Department of Information and Communication Engineering, Xiamen University, Xiamen 361005, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5280","DOI":"10.1109\/TCOMM.2017.2743166","article-title":"Capacity analysis of index modulations over spatial, polarization, and frequency dimensions","volume":"65","author":"Henarejos","year":"2017","journal-title":"IEEE Trans. Commun."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"10139","DOI":"10.1109\/TVT.2017.2755401","article-title":"Polarization shift keying (PolarSK): System scheme and performance analysis","volume":"66","author":"Zhang","year":"2017","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"908","DOI":"10.1109\/LWC.2022.3149618","article-title":"Binary polarization shift keying with reconfigurable intelligent surfaces","volume":"11","author":"Ibrahim","year":"2022","journal-title":"IEEE Wirel. Commun. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3803","DOI":"10.1109\/TCOMM.2015.2461221","article-title":"Dual polarized modulation and reception for next generation mobile satellite communications","volume":"63","author":"Henarejos","year":"2015","journal-title":"IEEE Trans. Commun."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5305","DOI":"10.1109\/TCOMM.2018.2850814","article-title":"3D polarized modulation: System analysis and performances","volume":"66","author":"Henarejos","year":"2018","journal-title":"IEEE Trans. Commun."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1109\/COMST.2016.2606639","article-title":"Advances on exploiting polarization in wireless communications: Channels, technologies, and applications","volume":"19","author":"Guo","year":"2017","journal-title":"IEEE Commun. Surv. Tutor."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"130588","DOI":"10.1109\/ACCESS.2020.3006833","article-title":"A secure constellation design for polarized modulation in wireless communications","volume":"8","author":"Sun","year":"2020","journal-title":"IEEE Access"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"6813","DOI":"10.1109\/TVT.2020.2987524","article-title":"A polarization MDCSK modulation without PDL over multipath Rayleigh fading channels","volume":"66","author":"Miao","year":"2020","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3760","DOI":"10.1109\/TCOMM.2022.3169214","article-title":"A survey on DCSK-based communication systems and their application to UWB scenarios","volume":"70","author":"Tao","year":"2022","journal-title":"IEEE Trans. Commun."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2029","DOI":"10.1109\/LCOMM.2021.3061675","article-title":"Differential permutation index DCSK modulation for chaotic communication system","volume":"25","author":"Liu","year":"2021","journal-title":"IEEE Commun. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1726","DOI":"10.1109\/TCOMM.2023.3234995","article-title":"Design and performance analysis of RIS-aided DCSK-WPC system with energy buffer","volume":"71","author":"Xie","year":"2023","journal-title":"IEEE Trans. Commun."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1109\/LCOMM.2016.2614682","article-title":"Design of an adaptive multiresolution M-ary DCSK system","volume":"21","author":"Cai","year":"2017","journal-title":"IEEE Commun. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5197","DOI":"10.1109\/TWC.2015.2434820","article-title":"Design and performance analysis of a new multiresolution M-ary differential chaos shift keying communication system","volume":"14","author":"Wang","year":"2015","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1109\/TCOMM.2006.887503","article-title":"Approximate minimum BER power allocation for MIMO spatial multiplexing systems","volume":"55","author":"Wang","year":"2007","journal-title":"IEEE Trans. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2245","DOI":"10.1109\/JSAC.2023.3280984","article-title":"Predictive Precoder Design for OTFS-Enabled URLLC: A Deep Learning Approach","volume":"41","author":"Liu","year":"2023","journal-title":"IEEE J. Sel. Area Commun."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1192","DOI":"10.1109\/LCOMM.2020.2981060","article-title":"Closed-Form BER Expressions of M-Ary DCSK Systems Over Multipath Rayleigh Fading Channels","volume":"24","author":"Cai","year":"2020","journal-title":"IEEE Commun. Lett."},{"key":"ref_17","first-page":"431","article-title":"Design of simultaneous wireless information and power transfer scheme for short reference DCSK communication systems","volume":"65","author":"Kaddoum","year":"2017","journal-title":"IEEE Trans. Commun."},{"key":"ref_18","first-page":"1859","article-title":"A framework for the lower bound on the BER of DCSK systems over multi-path Nakagami-m fading channels","volume":"67","author":"Dawa","year":"2020","journal-title":"IEEE Trans. Circuits Syst. II"},{"key":"ref_19","first-page":"321","article-title":"A Generalized lower bound on the bit error rate of DCSK system over multi-path Rayleigh fading channels","volume":"65","author":"Dawa","year":"2018","journal-title":"IEEE Trans. Circuits Syst. II"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2473","DOI":"10.1109\/LCOMM.2020.3012837","article-title":"SEP bounds for MPSK with low SNR","volume":"24","author":"Rugini","year":"2020","journal-title":"IEEE Commun. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2939","DOI":"10.1109\/TCOMM.2011.072011.100049","article-title":"Chernoff-type bounds for the Gaussian error fuction","volume":"59","author":"Chang","year":"2011","journal-title":"IEEE Trans. Commun."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/9\/1267\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:40:13Z","timestamp":1760128813000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/9\/1267"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,27]]},"references-count":21,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["e25091267"],"URL":"https:\/\/doi.org\/10.3390\/e25091267","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,8,27]]}}}