{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T20:36:45Z","timestamp":1771706205994,"version":"3.50.1"},"reference-count":56,"publisher":"Springer Science and Business Media LLC","issue":"26","license":[{"start":{"date-parts":[[2024,12,19]],"date-time":"2024-12-19T00:00:00Z","timestamp":1734566400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,12,19]],"date-time":"2024-12-19T00:00:00Z","timestamp":1734566400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"National Key Research and Development Program of China","award":["2021QY1303"],"award-info":[{"award-number":["2021QY1303"]}]},{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"crossref","award":["62301473"],"award-info":[{"award-number":["62301473"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Jiangsu Province Industry-University-Research Co-operation Project","award":["BY20231068"],"award-info":[{"award-number":["BY20231068"]}]},{"DOI":"10.13039\/501100013280","name":"Major Basic Research Project of the Natural Science Foundation of the Jiangsu Higher Education Institutions","doi-asserted-by":"publisher","award":["2023JSJG399"],"award-info":[{"award-number":["2023JSJG399"]}],"id":[{"id":"10.13039\/501100013280","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Comput & Applic"],"published-print":{"date-parts":[[2025,9]]},"DOI":"10.1007\/s00521-024-10526-2","type":"journal-article","created":{"date-parts":[[2024,12,19]],"date-time":"2024-12-19T05:05:53Z","timestamp":1734584753000},"page":"22161-22178","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["RSMA-assisted SHAPTINs: secrecy performance under imperfect hardware and channel estimation errors"],"prefix":"10.1007","volume":"37","author":[{"given":"Zhou","family":"Feng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7535-2057","authenticated-orcid":false,"given":"Guo","family":"Kefeng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cheng","family":"Jian","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Khowaja","family":"Sunder Ali","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dev","family":"Kapal","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gadekallu Thippa","family":"Reddy","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hamadi","family":"Hussam Al","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,12,19]]},"reference":[{"issue":"1","key":"10526_CR1","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1109\/COMST.2020.3028247","volume":"23","author":"O Kodheli","year":"2021","unstructured":"Kodheli O et al (2021) Satellite communications in the new space era: a survey and future challenges. IEEE Commun Surv Tut 23(1):70\u2013109","journal-title":"IEEE Commun Surv Tut"},{"issue":"11","key":"10526_CR2","first-page":"15133","volume":"73","author":"Y Yang","year":"2023","unstructured":"Yang Y et al (2023) Semantic sensing performance analysis: assessing keyword coverage in text data. IEEE Trans Vehi Technol 73(11):15133\u201315137","journal-title":"IEEE Trans Vehi Technol"},{"issue":"21","key":"10526_CR3","doi-asserted-by":"publisher","first-page":"18890","DOI":"10.1109\/JIOT.2023.3281942","volume":"10","author":"Z Han","year":"2023","unstructured":"Han Z et al (2023) Smart optimization solution for channel access attack defense under UAV-aided heterogeneous network. IEEE Internet Things J 10(21):18890\u201318897","journal-title":"IEEE Internet Things J"},{"issue":"1","key":"10526_CR4","first-page":"12","volume":"7","author":"M Wu","year":"2024","unstructured":"Wu M et al (2024) Optimization design in RIS-assisted integrated satellite-UAV served 6G IoT: a deep reinforcement learning approach. IEEE Internet J Mag 7(1):12\u201318","journal-title":"IEEE Internet J Mag"},{"key":"10526_CR5","unstructured":"Liu R et al (2024) RIS-empowered satellite-aerial-terrestrial networks with PDNOMA. IEEE Commun Surv Tut 1\u201332 (early access)"},{"issue":"2","key":"10526_CR6","doi-asserted-by":"publisher","first-page":"294","DOI":"10.1109\/MNET.003.2200519","volume":"37","author":"C Wang","year":"2023","unstructured":"Wang C et al (2023) Metaverse-inspired cybertwin-based space-air-ground integrated networks. IEEE Net 37(2):294\u2013300","journal-title":"IEEE Net"},{"issue":"2","key":"10526_CR7","doi-asserted-by":"publisher","first-page":"102","DOI":"10.1109\/MNET.006.2200281","volume":"37","author":"H Dahrouj","year":"2023","unstructured":"Dahrouj H et al (2023) Machine leanning-based user scheduling in integrated satellite-HAPS-ground networks. IEEE Net 37(2):102\u2013109","journal-title":"IEEE Net"},{"issue":"9","key":"10526_CR8","doi-asserted-by":"publisher","first-page":"6298","DOI":"10.1109\/TWC.2023.3241341","volume":"22","author":"J Xu","year":"2023","unstructured":"Xu J et al (2023) Space-air-ground-sea integrated networks: modeling and coverage analysis. IEEE Trans Wirel Commun 22(9):6298\u20136313","journal-title":"IEEE Trans Wirel Commun"},{"issue":"4","key":"10526_CR9","doi-asserted-by":"publisher","first-page":"5953","DOI":"10.1109\/TVT.2023.3330329","volume":"73","author":"K Guo","year":"2024","unstructured":"Guo K et al (2024) Outage performance of RIS-assisted cognitive non-terrestrial network with NOMA. IEEE Trans Vehi Technol 73(4):5953\u20135958","journal-title":"IEEE Trans Vehi Technol"},{"key":"10526_CR10","unstructured":"ETSI TS 102 585 V1.1.2 (2008) Digital video broadcasting (DVB); system specifications for satellite services to handheld devices (SH) below 3 GHz. Apr 2008"},{"issue":"2","key":"10526_CR11","doi-asserted-by":"publisher","first-page":"1307","DOI":"10.1109\/TIV.2022.3215011","volume":"8","author":"K Guo","year":"2023","unstructured":"Guo K et al (2023) Integrated satellite multiple two-way relay networks: secrecy performance under multiple eves and vehicles with non-ideal hardware. IEEE Trans Intel Vehi 8(2):1307\u20131318","journal-title":"IEEE Trans Intel Vehi"},{"issue":"8","key":"10526_CR12","doi-asserted-by":"publisher","first-page":"6211","DOI":"10.1109\/TWC.2022.3147823","volume":"21","author":"L Qu","year":"2022","unstructured":"Qu L et al (2022) UAV-assisted RF\/FSO relay system for space-air-ground integrated network: a performance analysis. IEEE Trans Wirel Commun 21(8):6211\u20136225","journal-title":"IEEE Trans Wirel Commun"},{"issue":"1","key":"10526_CR13","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1109\/LCOMM.2013.111313.131963","volume":"18","author":"MR Bhatnagar","year":"2014","unstructured":"Bhatnagar MR et al (2014) On the closed-form performance analysis of maximal ratio combining in shadowed-rician fading LMS channels. IEEE Commun Lett 18(1):54\u201357","journal-title":"IEEE Commun Lett"},{"issue":"7","key":"10526_CR14","doi-asserted-by":"publisher","first-page":"1157","DOI":"10.1109\/LCOMM.2015.2428256","volume":"19","author":"K An","year":"2015","unstructured":"An K et al (2015) Outage analysis of multi-atenna cognitive hybrid satellite terrestrial relay networks with beamforming. IEEE Commun Lett 19(7):1157\u20131160","journal-title":"IEEE Commun Lett"},{"issue":"11","key":"10526_CR15","doi-asserted-by":"publisher","first-page":"1947","DOI":"10.1109\/LCOMM.2014.2361517","volume":"18","author":"K An","year":"2014","unstructured":"An K et al (2014) Symbol error analysis of hybrid satellite-terrestrial cooperative networks with cochannel interference. IEEE Commun Lett 18(11):1947\u20131950","journal-title":"IEEE Commun Lett"},{"issue":"3","key":"10526_CR16","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1109\/TWC.2003.811182","volume":"2","author":"A Abdi","year":"2003","unstructured":"Abdi A et al (2003) A new simple model for land mobile satellite channels: first-and second-order statistics. IEEE Trans Wirel Commun 2(3):519\u2013528","journal-title":"IEEE Trans Wirel Commun"},{"issue":"12","key":"10526_CR17","doi-asserted-by":"publisher","first-page":"5052","DOI":"10.1109\/TCOMM.2015.2495278","volume":"63","author":"L Yang","year":"2015","unstructured":"Yang L et al (2015) Performance analysis of amplify-and-forward hybrid satellite terrestrial networks with cochannel interference. IEEE Trans Commun 63(12):5052\u20135061","journal-title":"IEEE Trans Commun"},{"issue":"2","key":"10526_CR18","doi-asserted-by":"publisher","first-page":"268","DOI":"10.1109\/LCOMM.2015.2502599","volume":"20","author":"PK Upadhyay","year":"2016","unstructured":"Upadhyay PK, Sharma PK (2016) Max-max user-relay selection scheme in multiuser and multirelay hybrid satellite-terrestrial relay systems. IEEE Commun Lett 20(2):268\u2013271","journal-title":"IEEE Commun Lett"},{"issue":"10","key":"10526_CR19","doi-asserted-by":"publisher","first-page":"6586","DOI":"10.1109\/TWC.2017.2725950","volume":"16","author":"PK Sharma","year":"2017","unstructured":"Sharma PK, Upadhyay PK, Costa DB, Bithas PS, Kanatas AG (2017) Performance analysis of overlay spectrum sharing in hybrid satellite-terrestrial systems with secondary network selection. IEEE Trans Wirel Commun 16(10):6586\u20136601","journal-title":"IEEE Trans Wirel Commun"},{"issue":"9","key":"10526_CR20","doi-asserted-by":"publisher","first-page":"4031","DOI":"10.1109\/TVT.2014.2361832","volume":"64","author":"NI Miridakis","year":"2015","unstructured":"Miridakis NI et al (2015) Dual-hop communication over a satellite relay and shadowed rician channels. IEEE Trans Veh Technol 64(9):4031\u20134040","journal-title":"IEEE Trans Veh Technol"},{"issue":"10","key":"10526_CR21","doi-asserted-by":"publisher","first-page":"1722","DOI":"10.1109\/LCOMM.2015.2466535","volume":"19","author":"K An","year":"2015","unstructured":"An K et al (2015) On the performance of multiuser hybrid satellite-terrestrial relay networks with opportunistic scheduling. IEEE Commun Lett 19(10):1722\u20131725","journal-title":"IEEE Commun Lett"},{"issue":"11","key":"10526_CR22","doi-asserted-by":"publisher","first-page":"4512","DOI":"10.1109\/TCOMM.2013.100913.130282","volume":"61","author":"E Bjornson","year":"2013","unstructured":"Bjornson E, Matthaiou M, Debbah M (2013) A new look at dual-hop relaying: performance limits with hardware impairments. IEEE Trans Commun 61(11):4512\u20134525","journal-title":"IEEE Trans Commun"},{"issue":"2","key":"10526_CR23","doi-asserted-by":"publisher","first-page":"1490","DOI":"10.1109\/TCOMM.2018.2878848","volume":"67","author":"K Guo","year":"2019","unstructured":"Guo K et al (2019) On the performance of LMS communication with hardware impairments and interference. IEEE Trans Commun 67(2):1490\u20131505","journal-title":"IEEE Trans Commun"},{"issue":"3","key":"10526_CR24","doi-asserted-by":"publisher","first-page":"2297","DOI":"10.1109\/JSYST.2019.2901800","volume":"13","author":"K Guo","year":"2019","unstructured":"Guo K et al (2019) On the performance of the uplink satellite multi-terrestrial relay networks with hardware impairments and interference. IEEE Syst J 13(3):2297\u20132308","journal-title":"IEEE Syst J"},{"key":"10526_CR25","doi-asserted-by":"publisher","first-page":"35103","DOI":"10.1109\/ACCESS.2019.2905129","volume":"7","author":"H Wu","year":"2019","unstructured":"Wu H et al (2019) Impact of hardware impairments on outage performance of hybrid satellite-terrestrial relay systems. IEEE Access 7:35103\u201335112","journal-title":"IEEE Access"},{"key":"10526_CR26","doi-asserted-by":"crossref","unstructured":"Singh V et al (2020) Impact of hardware impairments on cognitive satellite-terrestrial relaying with a direct link. In: IEEE CNCC 2017, Las Vegas, NV, USA, Mar 2020, pp 1\u20135","DOI":"10.1109\/CCNC46108.2020.9045578"},{"issue":"1","key":"10526_CR27","doi-asserted-by":"publisher","first-page":"192","DOI":"10.1109\/JSYST.2020.2967836","volume":"15","author":"V Singh","year":"2021","unstructured":"Singh V et al (2021) Performance analysis of hardware-impaired overlay cognitive satellite-terrestrial networks with adaptive relaying protocol. IEEE Syst J 15(1):192\u2013203","journal-title":"IEEE Syst J"},{"issue":"2","key":"10526_CR28","doi-asserted-by":"publisher","first-page":"2311","DOI":"10.1109\/JSYST.2021.3090799","volume":"16","author":"PK Sharma","year":"2022","unstructured":"Sharma PK et al (2022) Outage performance of multi-UAV relaying-based imperfect hardware hybrid satellite-terrestrial networks. IEEE Syst J 16(2):2311\u20132314","journal-title":"IEEE Syst J"},{"key":"10526_CR29","doi-asserted-by":"crossref","unstructured":"Zhao J et al (2022) Performance analysis of full-duplex NOMA assisted satellite terrestrial systems with hardware impairments and imperfect CSI. In: IEEE ICICN 2022, Zhangye, China, Aug 2022, pp 1\u20136","DOI":"10.1109\/ICICN56848.2022.10006588"},{"issue":"2","key":"10526_CR30","doi-asserted-by":"publisher","first-page":"8009","DOI":"10.1109\/TWC.2016.2610965","volume":"15","author":"M Yang","year":"2016","unstructured":"Yang M, Guo D, Huang Y, Duong TQ, Zhang B (2016) Secure multiuser scheduling in downlink dual-hop regenerative relay networks over nakagami-m fading channels. IEEE Trans Wirel Commun 15(2):8009\u20138024","journal-title":"IEEE Trans Wirel Commun"},{"key":"10526_CR31","doi-asserted-by":"publisher","first-page":"128896","DOI":"10.1109\/ACCESS.2019.2939436","volume":"7","author":"X Tang","year":"2019","unstructured":"Tang X et al (2019) On the performance of two-way multiple relay non-orthogonal multiple access based networks with hardware impairments. IEEE Access 7:128896\u2013128909","journal-title":"IEEE Access"},{"issue":"7","key":"10526_CR32","doi-asserted-by":"publisher","first-page":"7011","DOI":"10.1109\/TVT.2019.2916992","volume":"68","author":"W Zeng","year":"2019","unstructured":"Zeng W et al (2019) Two-way hybrid terrestrial-satellite relaying systems: performance analysis and relay selection. IEEE Trans Vehi Technol 68(7):7011\u20137023","journal-title":"IEEE Trans Vehi Technol"},{"issue":"3","key":"10526_CR33","doi-asserted-by":"publisher","first-page":"3799","DOI":"10.1007\/s11277-017-4207-2","volume":"96","author":"Y Zhao","year":"2017","unstructured":"Zhao Y, Xie L, Chen H, Wang K (2017) Ergodic channel capacity analysis of downlink in the hybrid satellite-terrestrial cooperative system. Wirel Personal Commun 96(3):3799\u20133815","journal-title":"Wirel Personal Commun"},{"issue":"18","key":"10526_CR34","doi-asserted-by":"publisher","first-page":"17334","DOI":"10.1109\/JIOT.2022.3157673","volume":"9","author":"K Guo","year":"2022","unstructured":"Guo K, Dong C, An K (2022) NOMA-based cognitive satellite terrestrial relay network: secrecy performance under channel estimation errors and hardware impairments. IEEE Internet Things J 9(18):17334\u201317347","journal-title":"IEEE Internet Things J"},{"issue":"20","key":"10526_CR35","doi-asserted-by":"publisher","first-page":"1059","DOI":"10.1049\/el.2020.1814","volume":"56","author":"F Zhou","year":"2020","unstructured":"Zhou F, Wang R, Guo K (2020) Secrecy outage analysis of satellite communication networks with hardware impairments and channel estimation errors. IET Electron Lett 56(20):1059\u20131062","journal-title":"IET Electron Lett"},{"issue":"5","key":"10526_CR36","first-page":"7004","volume":"59","author":"TN Nguyen","year":"2023","unstructured":"Nguyen TN et al (2023) Security-reliability tradeoffs for satellite-terrestrial relay networks with a friendly jammer and imperfect CSI. IEEE Trans Aero Elec Syst 59(5):7004\u20137019","journal-title":"IEEE Trans Aero Elec Syst"},{"issue":"7","key":"10526_CR37","doi-asserted-by":"publisher","first-page":"4386","DOI":"10.1109\/TCOMM.2022.3179389","volume":"70","author":"X Wang","year":"2022","unstructured":"Wang X et al (2022) Joint bayesian channel estimation and data detection for OFTS systems in LEO satellite communications. IEEE Trans Commun 70(7):4386\u20134399","journal-title":"IEEE Trans Commun"},{"key":"10526_CR38","doi-asserted-by":"publisher","first-page":"126293","DOI":"10.1109\/ACCESS.2019.2939165","volume":"7","author":"X Yan","year":"2019","unstructured":"Yan X et al (2019) Effect of imperfect channel estimation on the performance of cognitive satellite terrestrial networks. IEEE Access 7:126293\u2013126304","journal-title":"IEEE Access"},{"issue":"11","key":"10526_CR39","doi-asserted-by":"publisher","first-page":"13002","DOI":"10.1109\/JIOT.2023.3337124","volume":"11","author":"K Guo","year":"2024","unstructured":"Guo K et al (2024) Power allocation and performance evaluation for NOMA-aided integrated satellite-HAP-terrestrial networks under practical limitations. IEEE Internet Things J 11(11):13002\u201313017","journal-title":"IEEE Internet Things J"},{"issue":"7","key":"10526_CR40","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1109\/MCOM.004.2100956","volume":"60","author":"AS de Sena","year":"2022","unstructured":"de Sena AS et al (2022) Rate-splitting multiple access and its interpaly with intelligent reflecting surfaces. IEEE Commun Mag 60(7):52\u201357","journal-title":"IEEE Commun Mag"},{"issue":"1","key":"10526_CR41","doi-asserted-by":"publisher","first-page":"184","DOI":"10.23919\/JCC.2023.01.015","volume":"20","author":"T Li","year":"2023","unstructured":"Li T et al (2023) Max-min fair RIS-aided rate-splitting multiple access for multigroup multicast communcations. China Commun 20(1):184\u2013198","journal-title":"China Commun"},{"issue":"2","key":"10526_CR42","first-page":"102","volume":"6","author":"B Zhuo","year":"2023","unstructured":"Zhuo B et al (2023) RIS-IoE for date-dirven networks: new mentalities, trends and preliminary solutions. IEEE Internet J Mag 6(2):102\u2013107","journal-title":"IEEE Internet J Mag"},{"issue":"6","key":"10526_CR43","doi-asserted-by":"publisher","first-page":"5393","DOI":"10.1109\/TWC.2023.3326245","volume":"23","author":"Z Li","year":"2024","unstructured":"Li Z et al (2024) Dynamic multiple access based on RSMA and spectrum sharing for integrated satellite-terrestrial networks. IEEE Trans Wirel Commun 23(6):5393\u20135408","journal-title":"IEEE Trans Wirel Commun"},{"key":"10526_CR44","doi-asserted-by":"crossref","unstructured":"Can M et al (2023) Rate-splitting multiple access-based uplink satellite networks with receive antenna selection. In: IEEE BlacksSeaCom 2023, Istanbul, Turkiye, pp 1\u20136","DOI":"10.1109\/BlackSeaCom58138.2023.10299739"},{"issue":"5","key":"10526_CR45","first-page":"6508","volume":"59","author":"M Can","year":"2023","unstructured":"Can M et al (2023) Outage probability analysis of rate splitting multiple access-based hybrid satellite-terrestrial relay networks with relay selection. IEEE Trans Aero Elec Syst 59(5):6508\u20136517","journal-title":"IEEE Trans Aero Elec Syst"},{"issue":"3","key":"10526_CR46","doi-asserted-by":"publisher","first-page":"657","DOI":"10.1109\/JSTSP.2019.2899731","volume":"13","author":"Z Lin","year":"2019","unstructured":"Lin Z, Lin M, Wang J-B, de Cola T, Wang J (2019) Joint beamforming and power allocation for satellite-terrestrial integrated networks with non-orthogonal multiple access. IEEE J Sel Topic S Proc 13(3):657\u2013670","journal-title":"IEEE J Sel Topic S Proc"},{"issue":"3","key":"10526_CR47","doi-asserted-by":"publisher","first-page":"6345","DOI":"10.1109\/TCOMM.2021.3088898","volume":"69","author":"Z Lin","year":"2021","unstructured":"Lin Z et al (2021) Secrecy-energy efficient hybrid beamforming for satellite-terrestrial integrated networks. IEEE Trans Commun 69(3):6345\u20136360","journal-title":"IEEE Trans Commun"},{"issue":"1","key":"10526_CR48","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1109\/JIOT.2019.2943900","volume":"7","author":"B Li","year":"2020","unstructured":"Li B, Fei Z, Zhou C, Zhang Y (2020) Physical layer security in space information networks: a survey. IEEE Internet Things J 7(1):33\u201352","journal-title":"IEEE Internet Things J"},{"issue":"5","key":"10526_CR49","doi-asserted-by":"publisher","first-page":"5129","DOI":"10.1109\/TVT.2020.2979496","volume":"69","author":"K Guo","year":"2020","unstructured":"Guo K, An K, Zhang B et al (2020) Physical layer security for multiuser satellite communication systems with threshold-based scheduling scheme. IEEE Trans Veh Technol 69(5):5129\u20135141","journal-title":"IEEE Trans Veh Technol"},{"issue":"4","key":"10526_CR50","doi-asserted-by":"publisher","first-page":"3661","DOI":"10.1109\/TVT.2021.3068062","volume":"70","author":"K Guo","year":"2021","unstructured":"Guo K et al (2021) On the secrecy performance of NOMA-based integrated satellite multiple-terrestrial relay networks with hardware impairments. IEEE Trans Vehi Technol 70(4):3661\u20133676","journal-title":"IEEE Trans Vehi Technol"},{"issue":"2","key":"10526_CR51","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1109\/LWC.2020.3026700","volume":"10","author":"Z Lin","year":"2021","unstructured":"Lin Z et al (2021) Secure and energy efficient transmission for RSMA-based cognitive satellite-terrestrial networks. IEEE Wirel Commun Lett 10(2):251\u2013255","journal-title":"IEEE Wirel Commun Lett"},{"issue":"11","key":"10526_CR52","doi-asserted-by":"publisher","first-page":"11756","DOI":"10.1109\/TVT.2022.3192042","volume":"71","author":"Y Li","year":"2022","unstructured":"Li Y et al (2022) Rate-splitting multiple access-enabled security analysis in cognitive satellite terrestrial networks. IEEE Trans Veh Technol 71(11):11756\u201311771","journal-title":"IEEE Trans Veh Technol"},{"issue":"11","key":"10526_CR53","doi-asserted-by":"publisher","first-page":"6286","DOI":"10.1109\/TWC.2014.2329682","volume":"13","author":"J Arnau","year":"2014","unstructured":"Arnau J, Christopoulos D, Chatzinotas S, Mosquera C, Ottersten B (2014) Performance of the multibeam satellite return link with correlated rain attenuation. IEEE Trans Wirel Commun 13(11):6286\u20136299","journal-title":"IEEE Trans Wirel Commun"},{"key":"10526_CR54","volume-title":"Table of integrals, series and products","author":"IS Gradshteyn","year":"2007","unstructured":"Gradshteyn IS, Ryzhik IM, Jeffrey A et al (2007) Table of integrals, series and products, 7th edn. Elsevier, Amsterdam","edition":"7"},{"key":"10526_CR55","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","author":"M Abramowitz","year":"1972","unstructured":"Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables, 9th edn. Dover, New York","edition":"9"},{"issue":"12","key":"10526_CR56","doi-asserted-by":"publisher","first-page":"9715","DOI":"10.1109\/TVT.2016.2526628","volume":"65","author":"P Patcharamaneepakorn","year":"2016","unstructured":"Patcharamaneepakorn P et al (2016) Spectral, energy, and economic efficiency of 5G multicell massive MIMO systems with generalized spatial modulation. IEEE Trans Veh Technol 65(12):9715\u20139731","journal-title":"IEEE Trans Veh Technol"}],"container-title":["Neural Computing and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-024-10526-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00521-024-10526-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-024-10526-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,3]],"date-time":"2025-09-03T11:04:48Z","timestamp":1756897488000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00521-024-10526-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,19]]},"references-count":56,"journal-issue":{"issue":"26","published-print":{"date-parts":[[2025,9]]}},"alternative-id":["10526"],"URL":"https:\/\/doi.org\/10.1007\/s00521-024-10526-2","relation":{},"ISSN":["0941-0643","1433-3058"],"issn-type":[{"value":"0941-0643","type":"print"},{"value":"1433-3058","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,19]]},"assertion":[{"value":"13 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 September 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 December 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"There are no conflicts of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"By considering (), the first task is to derive the expression of  and . In the following analysis, the detailed expressions for these two factors will be given, respectively.At the very beginning, the expression for the  will be obtained. As discussed in the system model, proposed HAP selection scheme is utilized in the  link, and thus, the OP for the first time slot is given aswherewhere  represents the OP for the -th  by decrypting the common signal and  denotes the OP for the -th  by deciphering the private signal.With the help of () and (),  can be derived asThen, through the similar method for obtaining , the detailed expression for  can be obtained asAfter these steps, by inserting () and () into (),  will be obtained. Next, by submitting () into (),  will be obtained.The second thing is to obtain the expression of , in what follows,  will be derived.From (),  has the following expression aswhere .From (A5),  has an expression asFrom (A6), we realize that the serious thing is to derive the CDF of . With the help of () and (),  can be rewritten aswhere , ,  and .Then, by using (), () and () can be derived.Next, by inserting () and () into (A6),  can be rewritten asIt is a pity that, doing the best efforts of ours, (A8) cannot be obtained through the regular way, thus, we use the Gaussian\u2013Chebyshev quadrature in [], and  has an expression asThe next step is to submit (A9) into (A5), and the expression for  will be derived.At last, by inserting  and  into (), after several calculating steps, the SOP will be obtained.The proof is over.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Appendix A: Proof of Theorem 1"}}]}}