{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T02:47:21Z","timestamp":1769914041848,"version":"3.49.0"},"reference-count":28,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2015,10,16]],"date-time":"2015-10-16T00:00:00Z","timestamp":1444953600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Sparse Bayesian learning (SBL) has given renewed interest to the problem of direction-of-arrival (DOA) estimation. It is generally assumed that the measurement matrix in SBL is precisely known. Unfortunately, this assumption may be invalid in practice due to the imperfect manifold caused by unknown or misspecified mutual coupling. This paper describes a modified SBL method for joint estimation of DOAs and mutual coupling coefficients with uniform linear arrays (ULAs). Unlike the existing method that only uses stationary priors, our new approach utilizes a hierarchical form of the Student t prior to enforce the sparsity of the unknown signal more heavily. We also provide a distinct Bayesian inference for the expectation-maximization (EM) algorithm, which can update the mutual coupling coefficients more efficiently. Another difference is that our method uses an additional singular value decomposition (SVD) to reduce the computational complexity of the signal reconstruction process and the sensitivity to the measurement noise.<\/jats:p>","DOI":"10.3390\/s151026267","type":"journal-article","created":{"date-parts":[[2015,10,16]],"date-time":"2015-10-16T14:46:27Z","timestamp":1445006787000},"page":"26267-26280","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Sparse Bayesian Learning for DOA Estimation with Mutual Coupling"],"prefix":"10.3390","volume":"15","author":[{"given":"Jisheng","family":"Dai","sequence":"first","affiliation":[{"name":"School of Electrical and Information Engineering, Jiangsu University, 301 Xuefu Road, Zhenjiang 212013, China"},{"name":"National Mobile Communications Research Laboratory, Southeast University, 2 Sipailou Road, Nanjing 210096, China"}]},{"given":"Nan","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Electronic and Information Engineering, Soochow University, 178 East Ganjiang Road, Suzhou 215006, China"}]},{"given":"Weichao","family":"Xu","sequence":"additional","affiliation":[{"name":"Department of Automatic Control, Guangdong University of Technology, 100 Huanxi Road, Guangzhou 510006, China"}]},{"given":"Chunqi","family":"Chang","sequence":"additional","affiliation":[{"name":"School of Electronic and Information Engineering, Soochow University, 178 East Ganjiang Road, Suzhou 215006, China"}]}],"member":"1968","published-online":{"date-parts":[[2015,10,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1109\/79.526899","article-title":"Two decades of array signal processing research","volume":"13","author":"Krim","year":"1996","journal-title":"IEEE Signal Proc. Mag."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"277","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_3","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1109\/TSP.2006.885732","article-title":"A novel online mutual coupling compensation algorithm for uniform and linear arrays","volume":"55","author":"Sellone","year":"2007","journal-title":"IEEE Trans. Signal Proc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1109\/TAP.2007.915461","article-title":"On the resiliency of MUSIC direction finding against antenna sensor coupling","volume":"56","author":"Ye","year":"2008","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1109\/TAES.2009.4805279","article-title":"DOA estimation for uniform linear array with mutual coupling","volume":"45","author":"Ye","year":"2009","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1049\/iet-spr.2008.0100","article-title":"Spatial smoothing for DOA estimation of coherent signals in the presence of unknown mutual coupling","volume":"5","author":"Dai","year":"2011","journal-title":"IET Signal Proc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2056","DOI":"10.1016\/j.sigpro.2012.01.017","article-title":"Real-valued DOA estimation for uniform linear array with unknown mutual coupling","volume":"92","author":"Dai","year":"2012","journal-title":"Signal Proc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"20064","DOI":"10.3390\/s141120064","article-title":"An efficient algorithm for direction finding against unknown mutual coupling","volume":"14","author":"Wang","year":"2014","journal-title":"Sensors"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1109\/TAES.2012.6129676","article-title":"DOA estimation and tracking of ULAs with mutual coupling","volume":"48","author":"Liao","year":"2012","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1109\/LAWP.2012.2196017","article-title":"Adaptive beamforming for uniform linear arrays with unknown mutual coupling","volume":"11","author":"Liao","year":"2012","journal-title":"IEEE Antennas Wirel. Propag. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.sigpro.2014.04.013","article-title":"A cumulant-based approach for direction finding in the presence of mutual coupling","volume":"104","author":"Liao","year":"2014","journal-title":"Signal Proc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1210","DOI":"10.1109\/LAWP.2012.2223651","article-title":"A sparse representation method for DOA estimation with unknown mutual coupling","volume":"11","author":"Dai","year":"2012","journal-title":"IEEE Antennas Wirel. Propag. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"760","DOI":"10.1109\/TAES.2013.6494379","article-title":"DOA estimation for sparse array via sparse signal reconstruction","volume":"49","author":"Hu","year":"2013","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3010","DOI":"10.1109\/TSP.2005.850882","article-title":"A sparse signal reconstruction perspective for source localization with sensor arrays","volume":"53","author":"Malioutov","year":"2005","journal-title":"IEEE Trans. Signal Proc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1109\/LAWP.2013.2252415","article-title":"Direction-of-arrival estimation via real-valued sparse representation","volume":"12","author":"Dai","year":"2013","journal-title":"IEEE Antennas Wirel. Propag. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1109\/LSP.2013.2272462","article-title":"An improved auto-calibration algorithm based on sparse Bayesian learning framework","volume":"20","author":"Zhao","year":"2013","journal-title":"IEEE Signal Proc. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1109\/TSP.2012.2222378","article-title":"Off-grid direction of arrival estimation using sparse bayesian inference","volume":"61","author":"Yang","year":"2013","journal-title":"IEEE Trans. Signal Proc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3828","DOI":"10.1109\/TAP.2013.2256093","article-title":"Directions-of-arrival estimation through Bayesian compressive sensing strategies","volume":"61","author":"Carlin","year":"2013","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_19","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_20","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.1109\/TSP.2004.831016","article-title":"Sparse bayesian learning for basis selection","volume":"52","author":"Wipf","year":"2004","journal-title":"IEEE Trans. Signal Proc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2346","DOI":"10.1109\/TSP.2007.914345","article-title":"Bayesian compressive sensing","volume":"56","author":"Ji","year":"2008","journal-title":"IEEE Trans. Signal Proc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1109\/JSTSP.2009.2039170","article-title":"General deviants: An analysis of perturbations in compressed sensing","volume":"4","author":"Herman","year":"2010","journal-title":"IEEE J. Sel. Top. Signal Proc."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4658","DOI":"10.1109\/TSP.2012.2201152","article-title":"Robustly stable signal recovery in compressed sensing with structured matrix perturbation","volume":"60","author":"Yang","year":"2012","journal-title":"IEEE Trans. Signal Proc."},{"key":"ref_24","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 Proc."},{"key":"ref_25","first-page":"1","article-title":"Perspectives on sparse bayesian learning","volume":"16","author":"Wipf","year":"2004","journal-title":"Adv. Neural Inf. Proc. Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1109\/TAP.1983.1143128","article-title":"Effect of mutual coupling in the performance of adaptive arrays","volume":"31","author":"Gupta","year":"1983","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1109\/LAWP.2004.841209","article-title":"A new definition of mutual impedance for application in dipole receiving antenna arrays","volume":"3","author":"Hui","year":"2004","journal-title":"IEEE Antennas Wirel. Propag. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0962492904000236","article-title":"Random matrix theory","volume":"14","author":"Edelman","year":"2005","journal-title":"Acta Numer."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/15\/10\/26267\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:50:15Z","timestamp":1760215815000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/15\/10\/26267"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,10,16]]},"references-count":28,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2015,10]]}},"alternative-id":["s151026267"],"URL":"https:\/\/doi.org\/10.3390\/s151026267","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,10,16]]}}}