{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T10:04:09Z","timestamp":1780567449004,"version":"3.54.1"},"reference-count":25,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T00:00:00Z","timestamp":1747094400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62431009"],"award-info":[{"award-number":["62431009"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2242022k60006"],"award-info":[{"award-number":["2242022k60006"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["SL20230104"],"award-info":[{"award-number":["SL20230104"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National 2011 Collaborative Innovation Center of Wireless Communication Technologies","award":["62431009"],"award-info":[{"award-number":["62431009"]}]},{"name":"National 2011 Collaborative Innovation Center of Wireless Communication Technologies","award":["2242022k60006"],"award-info":[{"award-number":["2242022k60006"]}]},{"name":"National 2011 Collaborative Innovation Center of Wireless Communication Technologies","award":["SL20230104"],"award-info":[{"award-number":["SL20230104"]}]},{"name":"Research Project Fund of Songjiang Laboratory","award":["62431009"],"award-info":[{"award-number":["62431009"]}]},{"name":"Research Project Fund of Songjiang Laboratory","award":["2242022k60006"],"award-info":[{"award-number":["2242022k60006"]}]},{"name":"Research Project Fund of Songjiang Laboratory","award":["SL20230104"],"award-info":[{"award-number":["SL20230104"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Covert wireless communication is designed to securely transmit hidden information between two devices. Its primary objective is to conceal the existence of transmitted data, rendering communication signals difficult for unauthorized parties to detect, intercept, or decipher during transmission. In this paper, we propose a Noise-like Multi-Carrier Random Phase Communication System (NRPCS) to enhance covert wireless communication by significantly complicating the detection and interception of transmitted signals. The proposed system utilizes bipolar modulation and Cyclic Code Shift Keying (CCSK) modulation, complemented by a random sequence generation mechanism, to increase the randomness and complexity of the transmitted signals. A mathematical model of the NRPCS waveform is formulated, and detailed analyses of the system\u2019s time-domain basis functions, correlation properties, and power spectral characteristics are conducted to substantiate its noise-like behavior. Simulation results indicate that, compared to traditional fixed-frequency transmission methods, NRPCS substantially improves both the Low Probability of Detection (LPD) and the Low Probability of Interception (LPI). Further research results demonstrate that unauthorized eavesdroppers are unable to effectively demodulate signals without knowledge of the employed modulation scheme, thus significantly enhancing the overall security of communication.<\/jats:p>","DOI":"10.3390\/e27050520","type":"journal-article","created":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T03:59:26Z","timestamp":1747108766000},"page":"520","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Design of Covert Communication Waveform Based on Phase Randomization"],"prefix":"10.3390","volume":"27","author":[{"given":"Wenjie","family":"Zhou","sequence":"first","affiliation":[{"name":"School of Electronic and Information Engineering, Harbin Institute of Technology, Harbin 150006, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8236-7073","authenticated-orcid":false,"given":"Zhenyong","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Electronic and Information Engineering, Harbin Institute of Technology, Harbin 150006, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jun","family":"Shi","sequence":"additional","affiliation":[{"name":"School of Electronic and Information Engineering, Harbin Institute of Technology, Harbin 150006, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4209-5469","authenticated-orcid":false,"given":"Qing","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Electronic and Information Engineering, Harbin Institute of Technology, Harbin 150006, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4400","DOI":"10.1109\/TWC.2009.080057","article-title":"Reduction of peak-to-average power ratio in transform domain communication systems","volume":"8","author":"Martin","year":"2009","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1921","DOI":"10.1109\/JSAC.2013.130923","article-title":"Limits of reliable communication with low probability of detection on AWGN channels","volume":"31","author":"Bash","year":"2013","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1256","DOI":"10.1049\/el.2009.2280","article-title":"Stable non-gaussian noise parameter modulation in digital communication","volume":"45","author":"Ek","year":"2009","journal-title":"Electron. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1282","DOI":"10.1049\/iet-com.2016.1333","article-title":"Covert digital communication systems based on joint normal distribution","volume":"11","author":"Xu","year":"2017","journal-title":"IET Commun."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Xu, X., and Jing, T. (2019, January 16\u201319). Design of strong signal masking covert communication transmission scheme based on OFDM System. Proceedings of the IEEE 19th International Conference on Communication Technology (ICCT), Xi\u2019an, China.","DOI":"10.1109\/ICCT46805.2019.8947096"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1251","DOI":"10.1007\/s11432-010-0073-1","article-title":"Research on the application of 4-weighted fractional Fourier transform in communication system","volume":"53","author":"Mei","year":"2010","journal-title":"Sci. China Inf. Sci."},{"key":"ref_7","unstructured":"Ni, L., Da, X.Y., and Hu, H. (,  2018). Research on chaotic encrypted satellite covert communication based on MPWFRFT. Proceedings of the 14th Annual Conference of Satellite Communications, Beijing, China."},{"key":"ref_8","first-page":"98","article-title":"Research satellite chaotic encryption communication based on DL-MPWFRFT","volume":"38","author":"Wang","year":"2019","journal-title":"Meas. Control Technol."},{"key":"ref_9","first-page":"825","article-title":"Analysis and implementation of constellation precoding system based on multiple parameters weighted-type fractional Fourier transform","volume":"40","author":"Liang","year":"2018","journal-title":"J. Electron. Inf. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1109\/JSTSP.2007.914879","article-title":"SNR walls for signal detection","volume":"2","author":"Tandra","year":"2008","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_11","first-page":"107566107573","article-title":"A time-varying chaotic multitone communication method based on OFDM for low detection probability of wardens","volume":"9","author":"Guo","year":"2021","journal-title":"IEEE Access"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1974","DOI":"10.1109\/TWC.2019.2900915","article-title":"Multi antenna covert communications in random wireless networks","volume":"18","author":"Zheng","year":"2019","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1109\/LWC.2018.2864210","article-title":"PAPR reduction of TDCS based on chirp noise sequence","volume":"8","author":"Shi","year":"2019","journal-title":"IEEE Wirel. Commun. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"7026","DOI":"10.1109\/TWC.2021.3080382","article-title":"Wireless covert communications aided by distributed cooperative jamming over slow fading channels","volume":"20","author":"Zheng","year":"2021","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1109\/LCOMM.2018.2889062","article-title":"Secure UAV communication with cooperative jamming and trajectory control","volume":"23","author":"Zhong","year":"2019","journal-title":"IEEE Commun. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1049\/iet-com.2010.0032","article-title":"Transform domain communication systems from a multidimensional perspective, impacts on bit error rate and spectrum efficiency","volume":"5","author":"Fumat","year":"2011","journal-title":"IET Commun."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1109\/LCOMM.2014.2359200","article-title":"Secret Communication Using Parallel Combinatory Spreading WFRFT","volume":"19","author":"Fang","year":"2015","journal-title":"IEEE Commun. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"941","DOI":"10.1109\/LCOMM.2016.2647716","article-title":"On covert communication with noise uncertainty","volume":"21","author":"He","year":"2017","journal-title":"IEEE Commun. Lett."},{"key":"ref_19","first-page":"27855","article-title":"Covert communication in wireless networks: A survey and new directions","volume":"9","author":"Zhang","year":"2021","journal-title":"IEEE Access"},{"key":"ref_20","first-page":"1536","article-title":"Covert communication under wireless jamming: A deep learning-based solution","volume":"69","author":"Zhang","year":"2021","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_21","first-page":"606","article-title":"Rendezvous in Cognitive Radio Networks: A Survey","volume":"25","author":"Liu","year":"2014","journal-title":"J. Softw."},{"key":"ref_22","first-page":"3436","article-title":"Analysis and design of covert communication schemes in mobile networks","volume":"21","author":"Chaudhary","year":"2022","journal-title":"IEEE Trans. Mob. Comput."},{"key":"ref_23","first-page":"113065","article-title":"Covert communication with optimal modulation in low-power sensor networks","volume":"334","author":"Xiao","year":"2022","journal-title":"IEEE Trans. Sens. Actuators A Phys."},{"key":"ref_24","first-page":"3042","article-title":"Covert communication with optimal channel selection and secrecy in multi-user environments","volume":"71","author":"Wang","year":"2023","journal-title":"IEEE Trans. Commun."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Miao, G., Zander, J., Sung, K.W., and Slimane, B. (2016). Fundamentals of Mobile Data Networks, Cambridge University Press.","DOI":"10.1017\/CBO9781316534298"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/5\/520\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:31:37Z","timestamp":1760031097000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/5\/520"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,13]]},"references-count":25,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["e27050520"],"URL":"https:\/\/doi.org\/10.3390\/e27050520","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,13]]}}}