{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T14:06:10Z","timestamp":1773065170170,"version":"3.50.1"},"reference-count":49,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,2,13]],"date-time":"2023-02-13T00:00:00Z","timestamp":1676246400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100010418","name":"Institute for Information &amp; Communications Technology Promotion (IITP) grant funded by the Korea government (MSIT)","doi-asserted-by":"publisher","award":["2017-0-00217"],"award-info":[{"award-number":["2017-0-00217"]}],"id":[{"id":"10.13039\/501100010418","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100010418","name":"Institute for Information &amp; Communications Technology Promotion (IITP) grant funded by the Korea government (MSIT)","doi-asserted-by":"publisher","award":["2020R1A6A1A03038540"],"award-info":[{"award-number":["2020R1A6A1A03038540"]}],"id":[{"id":"10.13039\/501100010418","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Basic Science Research Program through the National Research Foundation of Korea (NRF)","award":["2017-0-00217"],"award-info":[{"award-number":["2017-0-00217"]}]},{"name":"Basic Science Research Program through the National Research Foundation of Korea (NRF)","award":["2020R1A6A1A03038540"],"award-info":[{"award-number":["2020R1A6A1A03038540"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper proposes a novel hybrid beamforming and relay selection scheme for spectral efficiency maximization in a non-regenerative multi-relay multi-input multi-output (MIMO) system. The analog beamforming component in the radio-frequency (RF) domain must follow an element-wise constant modulus constraint, which makes the underlying design problem mathematically intractable and therefore, it is quite challenging to obtain the global optimal solution. To address this problem, phase-only precoding\/combining matrices are derived by maximizing the end-to-end received signal-to-noise ratio (SNR) under transmit power constraint at the source and each relay node. This task is achieved by decomposing the original complicated optimization problem into two independent components. The first component designs the RF precoder\/combiner at source and relay nodes by maximizing the received SNR at relay nodes. While the second component attempts to derive the analog precoder\/combiner at relay nodes and destination by maximizing the received SNR at the destination. Digital baseband processing matrices are obtained by deriving the closed-form expression, which minimizes interference among different sub-channels. Finally, the relay selection is made by maximizing the overall SNR from the source to the destination. Computer simulations reveal that the performance of the proposed algorithm is close to its fully digital counterpart and approximately 6% higher than the specified relay-assisted hybrid beamforming techniques. Moreover, the proposed method achieves more than 15% higher performance in a sparse scattering environment when compared with the given relay selection techniques.<\/jats:p>","DOI":"10.3390\/s23042079","type":"journal-article","created":{"date-parts":[[2023,2,13]],"date-time":"2023-02-13T02:14:11Z","timestamp":1676254451000},"page":"2079","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Hybrid Beamforming and Relay Selection for End-to-End SNR Maximization in Single-User Multi-Relay MIMO Systems"],"prefix":"10.3390","volume":"23","author":[{"given":"Hafiz Muhammad Tahir","family":"Mustafa","sequence":"first","affiliation":[{"name":"Department of Information and Communication Engineering, Sejong University, Seoul 05006, Republic of Korea"},{"name":"Department of Convergence Engineering for Intelligent Drone, Sejong University, Seoul 05006, Republic of Korea"}]},{"given":"Jung-In","family":"Baik","sequence":"additional","affiliation":[{"name":"Department of Information and Communication Engineering, Sejong University, Seoul 05006, Republic of Korea"},{"name":"Department of Convergence Engineering for Intelligent Drone, Sejong University, Seoul 05006, Republic of Korea"}]},{"given":"Young-Hwan","family":"You","sequence":"additional","affiliation":[{"name":"Department of Convergence Engineering for Intelligent Drone, Sejong University, Seoul 05006, Republic of Korea"},{"name":"Department of Computer Engineering, Sejong University, Seoul 05006, Republic of Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3274-4982","authenticated-orcid":false,"given":"Hyoung-Kyu","family":"Song","sequence":"additional","affiliation":[{"name":"Department of Information and Communication Engineering, Sejong University, Seoul 05006, Republic of Korea"},{"name":"Department of Convergence Engineering for Intelligent Drone, Sejong University, Seoul 05006, Republic of Korea"}]},{"given":"Zunira","family":"Abbasi","sequence":"additional","affiliation":[{"name":"Department of Information and Communication Engineering, Sejong University, Seoul 05006, Republic of Korea"},{"name":"Department of Convergence Engineering for Intelligent Drone, Sejong University, Seoul 05006, Republic of Korea"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"99878","DOI":"10.1109\/ACCESS.2019.2930208","article-title":"Cell-free massive MIMO: A new next-generation paradigm","volume":"7","author":"Zhang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"5491","DOI":"10.1109\/TVT.2016.2625805","article-title":"Multiuser steered multiset space-time shift keying for millimeter-wave communications","volume":"66","author":"Hemadeh","year":"2016","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"870","DOI":"10.1109\/COMST.2017.2783541","article-title":"Millimeter-wave communications: Physical channel models, design considerations, antenna constructions, and link-budget","volume":"20","author":"Hemadeh","year":"2017","journal-title":"IEEE Commun. Surv. Tutor."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"7208","DOI":"10.1109\/TWC.2018.2865786","article-title":"On the energy efficiency of MIMO hybrid beamforming for millimeter-wave systems with nonlinear power amplifiers","volume":"17","author":"Moghadam","year":"2018","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1186\/s13638-019-1370-z","article-title":"Performance analysis and optimization for non-uniformly deployed mmWave cellular network","volume":"2019","author":"Zhang","year":"2019","journal-title":"EURASIP J. Wirel. Commun. Netw."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1109\/JSTSP.2016.2523924","article-title":"An overview of signal processing techniques for millimeter wave MIMO systems","volume":"10","author":"Heath","year":"2016","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"49611","DOI":"10.1109\/ACCESS.2022.3172436","article-title":"Performance and user association optimization for uav relay-assisted mm-wave massive mimo systems","volume":"10","author":"Belaoura","year":"2022","journal-title":"IEEE Access"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1880","DOI":"10.1109\/JSAC.2020.3000808","article-title":"Hybrid transceiver optimization for multi-hop communications","volume":"38","author":"Xing","year":"2020","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1663","DOI":"10.1109\/JSAC.2021.3071836","article-title":"Multi-hop RIS-empowered terahertz communications: A DRL-based hybrid beamforming design","volume":"39","author":"Huang","year":"2021","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5075","DOI":"10.1109\/TWC.2021.3136788","article-title":"Multi-hop relaying distribution strategies for terahertz-band communication networks: A cross-layer analysis","volume":"21","author":"Xia","year":"2021","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1016\/j.comcom.2021.09.025","article-title":"Hybrid beamforming NOMA for mmWave half-duplex UAV relay-assisted B5G\/6G IoT networks","volume":"180","author":"Du","year":"2021","journal-title":"Comput. Commun."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Jing, Y., and Jafarkhani, H. (2008, January 19\u201323). Network beamforming with channel means and covariances at relays. Proceedings of the 2008 IEEE International Conference on Communications, Beijing, China.","DOI":"10.1109\/ICC.2008.703"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Jing, Y., and Jafarkhani, H. (2007, January 15\u201320). Network beamforming using relays with perfect channel information. Proceedings of the 2007 IEEE International Conference on Acoustics, Speech and Signal Processing-ICASSP\u201907, Honolulu, HI, USA.","DOI":"10.1109\/ICASSP.2007.366575"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2086","DOI":"10.1109\/JSAC.2020.3000838","article-title":"Two-timescale hybrid analog-digital beamforming for mmWave full-duplex MIMO multiple-relay aided systems","volume":"38","author":"Cai","year":"2020","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Song, N., Yang, T., and Sun, H. (2019, January 15\u201318). Efficient hybrid beamforming for relay assisted millimeter-wave multi-user massive MIMO. Proceedings of the 2019 IEEE Wireless Communications and Networking Conference (WCNC), Marrakesh, Morocco.","DOI":"10.1109\/WCNC.2019.8886016"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"13913","DOI":"10.1109\/JIOT.2022.3142932","article-title":"Joint hybrid and passive RIS-assisted beamforming for mmWave MIMO systems relying on dynamically configured subarrays","volume":"9","author":"Feng","year":"2022","journal-title":"IEEE Internet Things J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"157212","DOI":"10.1109\/ACCESS.2019.2949786","article-title":"Optimal hybrid beamforming design for millimeter-wave massive multi-user MIMO relay systems","volume":"7","author":"Zhang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3060","DOI":"10.1109\/COMST.2018.2843719","article-title":"A survey on hybrid beamforming techniques in 5G: Architecture and system model perspectives","volume":"20","author":"Ahmed","year":"2018","journal-title":"IEEE Commun. Surv. Tutor."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3243","DOI":"10.1109\/TSP.2019.2911255","article-title":"A family of hybrid analog\u2013digital beamforming methods for massive MIMO systems","volume":"67","author":"Ioushua","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1445","DOI":"10.1109\/TWC.2017.2778258","article-title":"Low-complexity and robust hybrid beamforming design for multi-antenna communication systems","volume":"17","author":"Molu","year":"2017","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1109\/JSTSP.2016.2523903","article-title":"Alternating minimization algorithms for hybrid precoding in millimeter wave MIMO systems","volume":"10","author":"Yu","year":"2016","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1109\/LWC.2015.2409268","article-title":"An iterative hybrid transceiver design algorithm for millimeter wave MIMO systems","volume":"4","author":"Chen","year":"2015","journal-title":"IEEE Wirel. Commun. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1499","DOI":"10.1109\/TWC.2014.011714.130846","article-title":"Spatially sparse precoding in millimeter wave MIMO systems","volume":"13","author":"Rajagopal","year":"2014","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4980","DOI":"10.1109\/TCOMM.2017.2730878","article-title":"Codebook design for beam alignment in millimeter wave communication systems","volume":"65","author":"Zhang","year":"2017","journal-title":"IEEE Trans. Commun."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5289","DOI":"10.1109\/TSP.2017.2723353","article-title":"Codebook-based hybrid precoding for millimeter wave multiuser systems","volume":"65","author":"He","year":"2017","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1109\/JSTSP.2018.2824246","article-title":"Spectral efficiency optimization for millimeter wave multiuser MIMO systems","volume":"12","author":"Shi","year":"2018","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"9698","DOI":"10.1109\/TVT.2019.2932128","article-title":"Weighted spectral efficiency optimization for hybrid beamforming in multiuser massive MIMO-OFDM systems","volume":"68","author":"Du","year":"2019","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3693","DOI":"10.1109\/TCOMM.2019.2893632","article-title":"Hybrid beamforming for millimeter wave systems using the MMSE criterion","volume":"67","author":"Lin","year":"2019","journal-title":"IEEE Trans. Commun."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"95021","DOI":"10.1109\/ACCESS.2019.2928559","article-title":"Energy-efficient hybrid precoding with low complexity for mmWave massive MIMO systems","volume":"7","author":"Liu","year":"2019","journal-title":"IEEE Access"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"998","DOI":"10.1109\/JSAC.2016.2549418","article-title":"Energy-efficient hybrid analog and digital precoding for mmWave MIMO systems with large antenna arrays","volume":"34","author":"Gao","year":"2016","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"9895","DOI":"10.1109\/ACCESS.2017.2649539","article-title":"Energy-efficient transceiver design for hybrid sub-array architecture MIMO systems","volume":"4","author":"He","year":"2016","journal-title":"IEEE Access"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Gao, X., Dai, L., Sun, Y., Han, S., and Chih-Lin, I. (2017, January 21\u201325). Machine learning inspired energy-efficient hybrid precoding for mmWave massive MIMO systems. Proceedings of the IEEE International Conference on Communications (ICC), Paris, France.","DOI":"10.1109\/ICC.2017.7997065"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Wang, Z., Li, M., Li, H., and Liu, Q. (2018, January 20\u201324). Hybrid beamforming with one-bit quantized phase shifters in mmWave MIMO systems. Proceedings of the IEEE International Conference on Communications (ICC), Kansas City, MO, USA.","DOI":"10.1109\/ICC.2018.8422249"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Lee, J., and Lee, Y.H. (2014, January 10\u201314). AF relaying for millimeter wave communication systems with hybrid RF\/baseband MIMO processing. Proceedings of the IEEE International Conference on Communications (ICC), Sydney, NSW, Australia.","DOI":"10.1109\/ICC.2014.6884253"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Abbas, H., and Hamdi, K. (2017, January 19\u201322). Millimeter wave communications over relay networks. Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC), San Francisco, CA, USA.","DOI":"10.1109\/WCNC.2017.7925836"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Abbas, H., and Hamdi, K. (2017, January 19\u201322). Hybrid beamformers design for MIMO relay networks in millimeter wave. Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC), San Francisco, CA, USA.","DOI":"10.1109\/WCNC.2017.7925884"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1109\/JCN.2017.000078","article-title":"Hybrid digital and analogue beamforming design for millimeter wave relaying systems","volume":"19","author":"Sheu","year":"2017","journal-title":"J. Commun. Netw."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Tsinos, C.G., Chatzinotas, S., and Ottersten, B. (2018, January 4\u20136). Hybrid analog-digital transceiver designs for mmWave amplify-and-forward relaying systems. Proceedings of the 41st International Conference on Telecommunications and Signal Processing (TSP), Athens, Greece.","DOI":"10.1109\/TSP.2018.8441203"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3027","DOI":"10.1109\/TVT.2019.2893928","article-title":"Deep-learning-based millimeter-wave massive MIMO for hybrid precoding","volume":"68","author":"Huang","year":"2019","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/s11277-016-3459-6","article-title":"Adaptive relay selection and data transmission scheme for cooperative communication","volume":"91","author":"Ahn","year":"2016","journal-title":"Wirel. Pers. Commun."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2200","DOI":"10.1587\/transfun.E100.A.2200","article-title":"Relay selection scheme for improved performance in the wireless communication systems based on OFDM","volume":"100","author":"Kim","year":"2017","journal-title":"IEICE Trans. Fundam. Electron. Commun. Comput. Sci."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"66141","DOI":"10.1109\/ACCESS.2021.3073847","article-title":"Efficient hybrid beamforming design in mmWave massive MU-MIMO DF relay systems with the mixed-structure","volume":"9","author":"Han","year":"2021","journal-title":"IEEE Access"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"5775","DOI":"10.1109\/TVT.2021.3076691","article-title":"Constrained tensor decomposition-based hybrid beamforming for mmWave massive MIMO-OFDM communication systems","volume":"70","author":"Zilli","year":"2021","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"75592","DOI":"10.1109\/ACCESS.2020.2988485","article-title":"Multilinear singular value decomposition for millimeter wave channel parameter estimation","volume":"8","author":"Ruble","year":"2020","journal-title":"IEEE Access"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1109\/TCOMM.2015.2502954","article-title":"Hybrid block diagonalization for massive multiuser MIMO systems","volume":"64","author":"Ni","year":"2015","journal-title":"IEEE Trans. Commun."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"5346","DOI":"10.1109\/TCOMM.2019.2911276","article-title":"Relay selection in network-coded cooperative MIMO systems","volume":"67","author":"Heidarpour","year":"2019","journal-title":"IEEE Trans. Commun."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"102706","DOI":"10.1016\/j.adhoc.2021.102706","article-title":"A realistic relay selection scheme for cooperative MIMO networks","volume":"124","author":"Zytoune","year":"2022","journal-title":"Ad Hoc Netw."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Gopal, L., Rong, Y., and Zang, Z. (2011, January 2\u20135). Joint MMSE transceiver design in non-regenerative MIMO relay systems with covariance feedback. Proceedings of the 17th Asia Pacific Conference on Communications, Sabah, Malaysia.","DOI":"10.1109\/APCC.2011.6152821"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1109\/CC.2018.8388000","article-title":"Geometric mean decomposition based hybrid precoding for millimeter-wave massive MIMO","volume":"15","author":"Xie","year":"2018","journal-title":"China Commun."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/4\/2079\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:32:47Z","timestamp":1760121167000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/4\/2079"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,13]]},"references-count":49,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["s23042079"],"URL":"https:\/\/doi.org\/10.3390\/s23042079","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,13]]}}}