{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T22:27:54Z","timestamp":1769552874321,"version":"3.49.0"},"reference-count":44,"publisher":"MDPI AG","issue":"22","license":[{"start":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T00:00:00Z","timestamp":1667952000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["62071257"],"award-info":[{"award-number":["62071257"]}]},{"name":"National Natural Science Foundation of China","award":["62161037"],"award-info":[{"award-number":["62161037"]}]},{"name":"National Natural Science Foundation of China","award":["NJYT-20-A11"],"award-info":[{"award-number":["NJYT-20-A11"]}]},{"name":"National Natural Science Foundation of China","award":["2021JQ07"],"award-info":[{"award-number":["2021JQ07"]}]},{"name":"Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region","award":["62071257"],"award-info":[{"award-number":["62071257"]}]},{"name":"Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region","award":["62161037"],"award-info":[{"award-number":["62161037"]}]},{"name":"Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region","award":["NJYT-20-A11"],"award-info":[{"award-number":["NJYT-20-A11"]}]},{"name":"Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region","award":["2021JQ07"],"award-info":[{"award-number":["2021JQ07"]}]},{"name":"Natural Science Foundation of Inner Mongolia Autonomous Region","award":["62071257"],"award-info":[{"award-number":["62071257"]}]},{"name":"Natural Science Foundation of Inner Mongolia Autonomous Region","award":["62161037"],"award-info":[{"award-number":["62161037"]}]},{"name":"Natural Science Foundation of Inner Mongolia Autonomous Region","award":["NJYT-20-A11"],"award-info":[{"award-number":["NJYT-20-A11"]}]},{"name":"Natural Science Foundation of Inner Mongolia Autonomous Region","award":["2021JQ07"],"award-info":[{"award-number":["2021JQ07"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Obtaining accurate angle parameters using direction-of-arrival (DOA) estimation algorithms is crucial for acquiring channel state information (CSI) in massive multiple-input multiple-output (MIMO) systems. However, the performance of the existing algorithms deteriorates severely due to mutual coupling between antenna elements in practical engineering. Therefore, for solving the array mutual coupling, the array output signal vector is modeled by mutual coupling coefficients and the DOA estimation problem is transformed into block sparse signal reconstruction and parameter optimization in this paper. Then, a novel sparse Bayesian learning (SBL)-based algorithm is proposed, in which the expectation-maximum (EM) algorithm is used to estimate the unknown parameters iteratively, and the convergence speed of the algorithm is enhanced by utilizing the approximate approximation. Moreover, considering the off-grid error caused by discretization processes, the grid refinement is carried out using the polynomial roots to realize the dynamic update of the grid points, so as to improve the DOA estimation accuracy. Simulation results show that compared with the existing algorithms, the proposed algorithm is more robust to mutual coupling and off-grid error and can obtain better estimation performance.<\/jats:p>","DOI":"10.3390\/s22228634","type":"journal-article","created":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T02:11:15Z","timestamp":1668046275000},"page":"8634","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["DOA Estimation for Massive MIMO Systems with Unknown Mutual Coupling Based on Block Sparse Bayesian Learning"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3873-4598","authenticated-orcid":false,"given":"Yang","family":"Liu","sequence":"first","affiliation":[{"name":"College of Electronic Information Engineering, Inner Mongolia University, Hohhot 010021, China"}]},{"given":"Na","family":"Dong","sequence":"additional","affiliation":[{"name":"College of Electronic Information Engineering, Inner Mongolia University, Hohhot 010021, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9249-1702","authenticated-orcid":false,"given":"Xiaohui","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Electronic Information Engineering, Inner Mongolia University, Hohhot 010021, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6015-7427","authenticated-orcid":false,"given":"Xin","family":"Zhao","sequence":"additional","affiliation":[{"name":"College of Electronic Information Engineering, Inner Mongolia University, Hohhot 010021, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6105-0664","authenticated-orcid":false,"given":"Yinghui","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Electronic Information Engineering, Inner Mongolia University, Hohhot 010021, China"}]},{"given":"Tianshuang","family":"Qiu","sequence":"additional","affiliation":[{"name":"Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1109\/TSP.2021.3129337","article-title":"Improving Sum-Rate of Cell-Free Massive MIMO With Expanded Compute-and-Forward","volume":"70","author":"Zhang","year":"2022","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1109\/TVT.2021.3125499","article-title":"DNN-Aided Codebook Based Beamforming for FDD Millimeter-Wave Massive MIMO Systems Under Multipath","volume":"71","author":"Xu","year":"2022","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Dicandia, F.A., Fonseca, N.J.G., Bacco, M., Mugnaini, S., and Genovesi, S. (2022). Space-Air-Ground Integrated 6G Wireless Communication Networks: A Review of Antenna Technologies and Application Scenarios. Sensors, 22.","DOI":"10.3390\/s22093136"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2188","DOI":"10.1109\/TCOMM.2020.2969351","article-title":"Efficient Angle-Domain Processing for FDD-Based Cell-Free Massive MIMO Systems","volume":"68","author":"Abdallah","year":"2020","journal-title":"IEEE Trans. Commun."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ma, J., Zhang, J., Yang, Z., and Qiu, T. (2022). Off-Grid DOA Estimation Using Sparse Bayesian Learning for MIMO Radar under Impulsive Noise. Sensors, 22.","DOI":"10.3390\/s22166268"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Gong, P., and Chen, X. (2022). Computationally Efficient Direction-of-Arrival Estimation Algorithms for a Cubic Coprime Array. Sensors, 22.","DOI":"10.3390\/s22010136"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1656","DOI":"10.1109\/TVT.2021.3132673","article-title":"Direction-of-Arrival Estimation of Coherent Signals for Uniform Linear Antenna Arrays With Mutual Coupling in Unknown Nonuniform Noise","volume":"71","author":"Fang","year":"2022","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Xu, W., Chen, B., Li, Y., Hu, Y., Li, J., and Zeng, Z. (2022). Dir-MUSIC Algorithm for DOA Estimation of Partial Discharge Based on Signal Strength Represented by Antenna Gain Array Manifold. Sensors, 22.","DOI":"10.3390\/s22145406"},{"key":"ref_9","first-page":"1","article-title":"DOA Estimation Method Based on EMD and MUSIC for Mutual Interference in FMCW Automotive Radars","volume":"19","author":"Liu","year":"2022","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Jiang, X., and Qian, S. (2021, January 20\u201322). DOA estimation of coherent signals based on modified music algorithm. Proceedings of the 2021 IEEE 3rd International Conference on Civil Aviation Safety and Information Technology (ICCASIT), Changsha, China.","DOI":"10.1109\/ICCASIT53235.2021.9633755"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Zhang, Q., Liu, Y., Long, X., Song, K., He, X., Ren, X., and Qiu, T. (2021, January 11\u201314). A Cyclostationarity Based Esprit Algorithm for DOA Estimation of Uniform Circular Array. Proceedings of the 2021 IEEE Statistical Signal Processing Workshop (SSP), Rio de Janeiro, Brazil.","DOI":"10.1109\/SSP49050.2021.9513864"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Daponte, P., De Vito, L., Picariello, F., Rapuano, S., and Tudosa, I. (2018, January 20\u201322). Compressed Sensing Technologies and Challenges for Aerospace and Defense RF Source Localization. Proceedings of the 2018 5th IEEE International Workshop on Metrology for AeroSpace (MetroAeroSpace), Rome, Italy.","DOI":"10.1109\/MetroAeroSpace.2018.8453560"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5185","DOI":"10.1109\/JSEN.2020.3030043","article-title":"Adaptive Beamforming Design of Planar Arrays Based on Bayesian Compressive Sensing","volume":"21","author":"Lin","year":"2021","journal-title":"IEEE Sens. J."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/j.dsp.2018.08.004","article-title":"Sparse Bayesian learning for off-grid DOA estimation with nested arrays","volume":"82","author":"Chen","year":"2018","journal-title":"Digit. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Pandey, R., Nannuru, S., and Siripuram, A. (2021, January 6\u201311). Sparse Bayesian Learning for Acoustic Source Localization. Proceedings of the ICASSP 2021\u20132021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Toronto, ON, Canada.","DOI":"10.1109\/ICASSP39728.2021.9413960"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4977","DOI":"10.1109\/TSP.2021.3106741","article-title":"Real-Valued Sparse Bayesian Learning for DOA Estimation With Arbitrary Linear Arrays","volume":"69","author":"Dai","year":"2021","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1109\/LCOMM.2021.3131727","article-title":"Adaptive Antenna Diagnosis Based on Clustering Block Sparse Bayesian Learning","volume":"26","author":"Liu","year":"2022","journal-title":"IEEE Commun. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"70071","DOI":"10.1109\/ACCESS.2021.3077759","article-title":"SBL-Based 2-D DOA Estimation for L-Shaped Array With Unknown Mutual Coupling","volume":"9","author":"Xiong","year":"2021","journal-title":"IEEE Access"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Ufiteyezu, E., and Yun, L. (2021, January 23\u201326). Dynamic Power Allocation for Cooperative Multi- Antenna Networks in Presence of Mutual Coupling. Proceedings of the 2021 IEEE 6th International Conference on Computer and Communication Systems (ICCCS), Chengdu, China.","DOI":"10.1109\/ICCCS52626.2021.9449251"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1109\/TSP.2018.2881663","article-title":"Off-Grid DOA Estimation Using Sparse Bayesian Learning in MIMO Radar With Unknown Mutual Coupling","volume":"67","author":"Chen","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"126859","DOI":"10.1109\/ACCESS.2020.3008162","article-title":"A Metasurface Superstrate for Mutual Coupling Reduction of Large Antenna Arrays","volume":"8","author":"Tang","year":"2020","journal-title":"IEEE Access"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Bhattacharjee, S., and Ghosh, C.K. (2020, January 30). Reduction of Mutual Coupling between two adjacent Microstrip antennas using I-shaped Resonators. Proceedings of the 2020 IEEE International Conference on Advent Trends in Multidisciplinary Research and Innovation (ICATMRI), Buldhana, India.","DOI":"10.1109\/ICATMRI51801.2020.9398327"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"213206","DOI":"10.1109\/ACCESS.2020.3041726","article-title":"Fast Reconstruction and Iterative Updating of Spatial Covariance Matrix for DOA Estimation in Hybrid Massive MIMO","volume":"8","author":"Fu","year":"2020","journal-title":"IEEE Access"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"145060","DOI":"10.1109\/ACCESS.2021.3122810","article-title":"Adaptive Reconstruction for Spatial Covariance Matrix in Hybrid Massive MIMO Systems","volume":"9","author":"Yan","year":"2021","journal-title":"IEEE Access"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3628","DOI":"10.1109\/JSYST.2020.3012775","article-title":"Two-Dimensional Localization: Low-Rank Matrix Completion With Random Sampling in Massive MIMO System","volume":"15","author":"Liu","year":"2021","journal-title":"IEEE Syst. J."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1109\/LCOMM.2019.2960341","article-title":"A Higher-Order Propagator Method for 2D-DOA Estimation in Massive MIMO Systems","volume":"24","author":"Ahmed","year":"2020","journal-title":"IEEE Commun. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1109\/LWC.2019.2942595","article-title":"Low-Complexity Deep-Learning-Based DOA Estimation for Hybrid Massive MIMO Systems With Uniform Circular Arrays","volume":"9","author":"Hu","year":"2020","journal-title":"IEEE Wireless Commun. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3152","DOI":"10.1109\/TWC.2020.3047866","article-title":"DOA and Polarization Estimation for Non-Circular Signals in 3-D Millimeter Wave Polarized Massive MIMO Systems","volume":"20","author":"Wan","year":"2021","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_29","first-page":"1","article-title":"Multi-BS Spatial Spectrum Fusion for 2-D DOA Estimation and Localization Using UCA in Massive MIMO System","volume":"70","author":"He","year":"2021","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"869","DOI":"10.1109\/TWC.2021.3100073","article-title":"Millidegree-Level Direction-of-Arrival Estimation and Tracking for Terahertz Ultra-Massive MIMO Systems","volume":"21","author":"Chen","year":"2022","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1143","DOI":"10.1109\/TWC.2021.3102483","article-title":"2-D DOA Estimation of Incoherently Distributed Sources Considering Gain-Phase Perturbations in Massive MIMO Systems","volume":"21","author":"Tian","year":"2022","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"912","DOI":"10.1109\/LCOMM.2022.3148260","article-title":"Real-Valued DOA Estimation Utilizing Enhanced Covariance Matrix With Unknown Mutual Coupling","volume":"26","author":"Tian","year":"2022","journal-title":"IEEE Commun. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1109\/8.76322","article-title":"Direction finding in the presence of mutual coupling","volume":"39","author":"Friedlander","year":"1991","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1109\/TSP.2019.2956677","article-title":"A Block Sparsity Based Estimator for mmWave Massive MIMO Channels With Beam Squint","volume":"68","author":"Wang","year":"2020","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2009","DOI":"10.1109\/TSP.2013.2241055","article-title":"Extension of SBL Algorithms for the Recovery of Block Sparse Signals With Intra-Block Correlation","volume":"61","author":"Zhang","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_36","first-page":"1","article-title":"Sparse Reconstruction Using Block Sparse Bayesian Learning With Fast Marginalized Likelihood Maximization for Near-Infrared Spectroscopy","volume":"71","author":"Pan","year":"2022","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_37","first-page":"211","article-title":"Sparse Bayesian learning and the relevance vector machine","volume":"1","author":"Tipping","year":"2001","journal-title":"J. Mach. Learn. Res."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Pan, Y., Tai, N., Cheng, S., and Yuan, N. (2015, January 19\u201322). Joint estimation of DOA and mutual coupling via block sparse Bayesian learning. Proceedings of the 2015 IEEE International Conference on Signal Processing, Communications and Computing (ICSPCC), Ningbo, China.","DOI":"10.1109\/ICSPCC.2015.7338843"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Wang, H., Wang, X., Huang, M., Lan, X., and Wan, L. (2022, January 25\u201327). Off-grid DOA Estimation for Temporally Correlated Source via Robust Block-SBL in Mutual Coupling. Proceedings of the 2022 Photonics and Electromagnetics Research Symposium (PIERS), Hangzhou, China.","DOI":"10.1109\/PIERS55526.2022.9793226"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1109\/LSP.2016.2636319","article-title":"Root Sparse Bayesian Learning for Off-Grid DOA Estimation","volume":"24","author":"Dai","year":"2017","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1109\/TAP.1986.1143830","article-title":"Multiple emitter location and signal parameter estimation","volume":"34","author":"Schmidt","year":"1986","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Dou, H., Liang, X., and Zhang, W. (2019, January 28\u201330). Research on DOA Estimation Method Based on RM-FOCUSS Improved Algorithms. Proceedings of the 2019 IEEE 2nd International Conference on Information Communication and Signal Processing (ICICSP), Weihai, China.","DOI":"10.1109\/ICICSP48821.2019.8958608"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"3786","DOI":"10.1109\/TSP.2013.2262682","article-title":"A Unified Framework and Sparse Bayesian Perspective for Direction-of-Arrival Estimation in the Presence of Array Imperfections","volume":"61","author":"Liu","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2622","DOI":"10.1109\/LCOMM.2017.2747547","article-title":"Effective Block Sparse Representation Algorithm for DOA Estimation With Unknown Mutual Coupling","volume":"21","author":"Wang","year":"2017","journal-title":"IEEE Commun. Lett."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/22\/8634\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:13:06Z","timestamp":1760145186000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/22\/8634"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,9]]},"references-count":44,"journal-issue":{"issue":"22","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["s22228634"],"URL":"https:\/\/doi.org\/10.3390\/s22228634","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,9]]}}}