{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:55:12Z","timestamp":1760057712544,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,17]],"date-time":"2025-02-17T00:00:00Z","timestamp":1739750400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental Research Funds for the Central Universities","award":["2023YFB2906204","CUC230D038"],"award-info":[{"award-number":["2023YFB2906204","CUC230D038"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper investigates the joint optimization of active and passive beamforming in simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS)-assisted non-orthogonal multiple access (NOMA) systems, with the aim of maximizing system throughput and improving overall performance. To achieve this goal, we propose an iterative and efficient algorithmic framework. For active beamforming optimization, the fractional programming (FP) method is employed to reformulate the non-convex optimization problem into a convex problem, making it more tractable. Additionally, Nesterov\u2019s extrapolation technique is introduced to enhance the convergence rate and reduce computational overhead. For the phase optimization of the STAR-RIS, a binary phase design method is proposed, which reformulates the binary phase optimization problem as a segmentation problem on the unit circle. This approach enables a closed form solution that can be derived in linear time. Simulation results demonstrate that the proposed algorithmic framework outperforms existing benchmark algorithms in terms of both system throughput and computational efficiency, verifying its effectiveness and practicality in STAR-RIS-assisted NOMA systems.<\/jats:p>","DOI":"10.3390\/e27020210","type":"journal-article","created":{"date-parts":[[2025,2,17]],"date-time":"2025-02-17T04:40:25Z","timestamp":1739767225000},"page":"210","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Beamforming Design for STAR-RIS-Assisted NOMA with Binary and Coupled Phase-Shifts"],"prefix":"10.3390","volume":"27","author":[{"given":"Yongfei","family":"Liu","sequence":"first","affiliation":[{"name":"Engineering Research Center of Digital Audio and Video Ministry of Education, Communication University of China, Beijing 100024, China"}]},{"given":"Yuhuan","family":"Wang","sequence":"additional","affiliation":[{"name":"Engineering Research Center of Digital Audio and Video Ministry of Education, Communication University of China, Beijing 100024, China"},{"name":"State Key Laboratory of Media Convergence and Communication, Communication University of China, Beijing 100024, China"}]},{"given":"Weizhang","family":"Xu","sequence":"additional","affiliation":[{"name":"Engineering Research Center of Digital Audio and Video Ministry of Education, Communication University of China, Beijing 100024, China"},{"name":"State Key Laboratory of Media Convergence and Communication, Communication University of China, Beijing 100024, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1109\/MCOM.001.1900107","article-title":"Towards Smart and Reconfigurable Environment: Intelligent Reflecting Surface Aided Wireless Network","volume":"58","author":"Wu","year":"2019","journal-title":"IEEE Commun. Mag."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"116753","DOI":"10.1109\/ACCESS.2019.2935192","article-title":"Wireless Communications Through Reconfigurable Intelligent Surfaces","volume":"7","author":"Basar","year":"2019","journal-title":"IEEE Access"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1109\/MWC.001.2100191","article-title":"STAR: Simultaneous Transmission and Reflection for 360\u00b0 Coverage by Intelligent Surfaces","volume":"28","author":"Liu","year":"2021","journal-title":"IEEE Wirel. Commun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1109\/LCOMM.2020.3025978","article-title":"Sum Rate Maximization for IRS-Assisted Uplink NOMA","volume":"25","author":"Zeng","year":"2021","journal-title":"IEEE Commun. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6884","DOI":"10.1109\/TWC.2020.3006915","article-title":"Exploiting Intelligent Reflecting Surfaces in NOMA Networks: Joint Beamforming Optimization","volume":"19","author":"Mu","year":"2020","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"7653","DOI":"10.1109\/TWC.2022.3160151","article-title":"Resource Allocation in STAR-RIS-Aided Networks: OMA and NOMA","volume":"21","author":"Wu","year":"2022","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3404","DOI":"10.1109\/LCOMM.2023.3324488","article-title":"Phase Shift Configuration Strategies for Unbalanced T&R Users in STAR-RIS-Aided NOMA","volume":"27","author":"Wen","year":"2023","journal-title":"IEEE Commun. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2556","DOI":"10.1109\/JSAC.2022.3192053","article-title":"Hybrid Reinforcement Learning for STAR-RISs: A Coupled Phase-Shift Model Based Beamformer","volume":"40","author":"Zhong","year":"2022","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_9","first-page":"3537","article-title":"A Method for Maximizing Sum Rate in Downlink NOMA Systems Assisted by Dual STAR-RISs","volume":"46","author":"Tian","year":"2024","journal-title":"J. Electron. Inf. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"9031","DOI":"10.1109\/TVT.2023.3248789","article-title":"Joint Phase Shift and Beamforming Design in a Multi-User MISO STAR-RIS Assisted Downlink NOMA Network","volume":"72","author":"Wang","year":"2023","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1362","DOI":"10.1109\/LCOMM.2023.3261442","article-title":"Average Rate Maximization for Mobile STAR-RIS-Aided NOMA System","volume":"27","author":"Wang","year":"2023","journal-title":"IEEE Commun. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"11023","DOI":"10.1109\/TVT.2023.3261324","article-title":"Joint Location and Beamforming Optimization for STAR-RIS Aided NOMA-UAV Networks","volume":"72","author":"Su","year":"2023","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1097","DOI":"10.1109\/JSTSP.2022.3175030","article-title":"STAR-RISs: A Correlated T&R Phase-Shift Model and Practical Phase-Shift Configuration Strategies","volume":"16","author":"Xu","year":"2022","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2616","DOI":"10.1109\/TSP.2018.2812733","article-title":"Fractional Programming for Communication Systems\u2014Part I: Power Control and Beamforming","volume":"66","author":"Shen","year":"2018","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Liu, Y., Si, L., Wang, Y., Zhang, B., and Xu, W. (2023). Efficient Precoding and Power Allocation Techniques for Maximizing Spectral Efficiency in Beamspace MIMO-NOMA Systems. Sensors, 23.","DOI":"10.3390\/s23187996"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Zhang, Z., Zhao, Z., and Shen, K. (2023, January 4). Enhancing the Efficiency of WMMSE and FP for Beamforming by Minorization-Maximization. Proceedings of the ICASSP 2023\u20132023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Rhodes Island, Greece.","DOI":"10.1109\/ICASSP49357.2023.10096377"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"967","DOI":"10.1109\/JSTSP.2022.3176479","article-title":"Configuring Intelligent Reflecting Surface With Performance Guarantees: Optimal Beamforming","volume":"16","author":"Zhang","year":"2022","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4108","DOI":"10.1109\/TSP.2020.3001906","article-title":"Penalty Dual Decomposition Method for Nonsmooth Nonconvex Optimization\u2014Part I: Algorithms and Convergence Analysis","volume":"68","author":"Shi","year":"2020","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1109\/LWC.2022.3219020","article-title":"Coupled Phase-Shift STAR-RISs: A General Optimization Framework","volume":"12","author":"Wang","year":"2023","journal-title":"IEEE Wirel. Commun. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Guo, H., Liang, Y.-C., Chen, J., and Larsson, E.G. (2019, January 9\u201313). Weighted Sum-Rate Optimization for Intelligent Reflecting Surface Enhanced Wireless Networks. Proceedings of the IEEE Global Communications Conference (GLOBECOM), Waikoloa, HI, USA.","DOI":"10.1109\/GLOBECOM38437.2019.9013288"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/2\/210\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:36:02Z","timestamp":1760027762000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/2\/210"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,17]]},"references-count":20,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["e27020210"],"URL":"https:\/\/doi.org\/10.3390\/e27020210","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,2,17]]}}}