{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T18:25:20Z","timestamp":1761675920287,"version":"build-2065373602"},"reference-count":60,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2012,3,8]],"date-time":"2012-03-08T00:00:00Z","timestamp":1331164800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Joint estimation of direction-of-arrival (DOA) and polarization with electromagnetic vector-sensors (EMVS) is considered in the framework of complex-valued non-orthogonal joint diagonalization (CNJD). Two new CNJD algorithms are presented, which propose to tackle the high dimensional optimization problem in CNJD via a sequence of simple sub-optimization problems, by using LU or LQ decompositions of the target matrices as well as the Jacobi-type scheme. Furthermore, based on the above CNJD algorithms we present a novel strategy to exploit the multi-dimensional structure present in the second-order statistics of EMVS outputs for simultaneous DOA and polarization estimation. Simulations are provided to compare the proposed strategy with existing tensorial or joint diagonalization based methods.<\/jats:p>","DOI":"10.3390\/s120303394","type":"journal-article","created":{"date-parts":[[2012,3,8]],"date-time":"2012-03-08T11:27:41Z","timestamp":1331206061000},"page":"3394-3417","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Simultaneous Source Localization and Polarization Estimation via Non-Orthogonal Joint Diagonalization with Vector-Sensors"],"prefix":"10.3390","volume":"12","author":[{"given":"Xiao-Feng","family":"Gong","sequence":"first","affiliation":[{"name":"School of Information and Communication Engineering, Dalian University of Technology, No. 2 Linggong Road, Dalian 116024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ke","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Information and Communication Engineering, Dalian University of Technology, No. 2 Linggong Road, Dalian 116024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0145-7136","authenticated-orcid":false,"given":"Qiu-Hua","family":"Lin","sequence":"additional","affiliation":[{"name":"School of Information and Communication Engineering, Dalian University of Technology, No. 2 Linggong Road, Dalian 116024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhi-Wen","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Information and Electronics, Beijing Institute of Technology, 5 South Zhongguancun Street, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"You-Gen","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Information and Electronics, Beijing Institute of Technology, 5 South Zhongguancun Street, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,3,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"944","DOI":"10.1109\/TAP.1981.1142690","article-title":"The tripole antenna: An adaptive array with full polarization flexibility","volume":"29","author":"Compton","year":"1981","journal-title":"IEEE Trans. Antennas Propag"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1109\/78.275610","article-title":"Vector-sensor array processing for electromagnetic source localization","volume":"42","author":"Nehorai","year":"1994","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1111","DOI":"10.1109\/8.55625","article-title":"Maximum likelihood localization of diversely polarized sources by simulated annealing","volume":"38","author":"Ziskind","year":"1990","journal-title":"IEEE Trans. Antennas Propag"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1109\/TSP.2010.2084085","article-title":"Vector cross-product direction-finding with an electromagnetic vector-sensor of six orthogonally oriented but spatially noncollocating dipoles\/loops","volume":"59","author":"Wong","year":"2011","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Lo Monte, L., Elnour, B., Erricolo, D., and Nehorai, A. (2007, January 4\u20138). Design and Realization of a Distributed Vector Sensor for Polarization Diversity Applications. Pisa, Italy.","DOI":"10.1109\/WDDC.2007.4339443"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1109\/8.237623","article-title":"Maximum likelihood signal estimation for polarization sensitive arrays","volume":"41","author":"Weiss","year":"1993","journal-title":"IEEE Trans. Antennas Propag"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3114","DOI":"10.1109\/78.330371","article-title":"Efficient parameter estimation of partially polarized electromagnetic waves","volume":"42","author":"Li","year":"1994","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1109\/TAP.1983.1143038","article-title":"Direction finding with an array of antennas having diverse polarizations","volume":"31","author":"Ferrara","year":"1983","journal-title":"IEEE Trans. Antennas Propag"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1235","DOI":"10.1109\/8.884492","article-title":"Self-initiating MUSIC-based direction finding and polarization estimation in spatio-polarizational beamspace","volume":"48","author":"Wong","year":"2000","journal-title":"IEEE Trans. Antennas Propag"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1109\/LAWP.2004.831083","article-title":"Root-MUSIC-based direction-finding and polarization estimation using diversely polarized possibly collocated antennas","volume":"3","author":"Wong","year":"2004","journal-title":"IEEE Antennas Wirel. Propag. Lett"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1687","DOI":"10.1109\/8.273313","article-title":"Subspace fitting with diversely polarized antenna arrays","volume":"41","author":"Swindlehurst","year":"1993","journal-title":"IEEE Trans. Antennas Propag"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1109\/8.489306","article-title":"Efficient direction and polarization estimation with a COLD array","volume":"44","author":"Li","year":"1996","journal-title":"IEEE Trans. Antennas Propag"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1109\/8.233120","article-title":"Direction and polarization estimation using arrays with small loops and short dipoles","volume":"41","author":"Li","year":"1993","journal-title":"IEEE Trans. Antennas Propag"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1467","DOI":"10.1109\/8.633852","article-title":"Uni-vector-sensor ESPRIT for multisource azimuth, elevation, and polarization estimation","volume":"45","author":"Wong","year":"1997","journal-title":"IEEE Trans. Antennas Propag"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1109\/8.855485","article-title":"Closed-form direction-finding with arbitrarily spaced electromagnetic vector-sensors at unknown locations","volume":"48","author":"Wong","year":"2000","journal-title":"IEEE Trans. Antennas Propag"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2195","DOI":"10.1109\/78.852000","article-title":"ESPRIT-based 2D direction finding with a sparse array of electromagnetic vector-sensors","volume":"48","author":"Zoltowski","year":"2000","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2205","DOI":"10.1109\/78.852001","article-title":"Closed-form eigenstructure-based direction finding using arbitrary but identical subarrays on a sparse uniform Cartesian array grid","volume":"48","author":"Zoltowski","year":"2000","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1109\/7.913668","article-title":"Blind beamforming\/geolocation for wideband-FFHs with unknown hop-sequences","volume":"37","author":"Wong","year":"2001","journal-title":"IEEE Trans. Aerosp. Electron. Syst"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2863","DOI":"10.1109\/78.790667","article-title":"Cross-product algorithms for source tracking using an EM vector sensor","volume":"47","author":"Nehorai","year":"1999","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Xu, Y.G., and Liu, Z.W. (2006, January 27\u201330). Adaptive Quasi-Cross-Product Algorithm for Uni-Tripole Tracking of Moving Source. Guilin, China.","DOI":"10.1109\/ICCT.2006.341732"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3096","DOI":"10.1109\/TSP.2004.836456","article-title":"Source localization using vector sensor array in a multipath environment","volume":"52","author":"Rahamim","year":"2004","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1327","DOI":"10.1109\/TSP.2005.863124","article-title":"Spatial polarimetric time-frequency distributions for direction-of-arrival estimations","volume":"54","author":"Zhang","year":"2006","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1049\/iet-rsn:20050108","article-title":"Polarimetric angular smoothing algorithm for an electromagnetic vector-sensor array","volume":"1","author":"Xu","year":"2007","journal-title":"IET Radar Sonar Navig"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1049\/iet-rsn.2008.0118","article-title":"Computationally efficient two-dimensional direction-of-arrival estimation of electromagnetic sources using the propagator method","volume":"3","author":"He","year":"2007","journal-title":"IET Radar Sonar Navig"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2628","DOI":"10.1109\/78.229894","article-title":"Analysis of a signal estimation algorithm for diversely polarized arrays","volume":"41","author":"Weiss","year":"1993","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1109\/TAES.2009.4805266","article-title":"CRB: Sinusoid-sources\u2019s estimation using collocated dipoles\/loops","volume":"45","author":"Wong","year":"2009","journal-title":"IEEE Trans. Aerosp. Electron. Syst"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1036","DOI":"10.1109\/78.492566","article-title":"Uniqueness study of measurements obtainable with arrays of electromagnetic vector sensors","volume":"44","author":"Tan","year":"1996","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3099","DOI":"10.1109\/78.553483","article-title":"Linear independence of steering vectors of an electromagnetic vector sensor","volume":"44","author":"Tan","year":"1996","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1109\/78.482014","article-title":"Identifiability in array processing models with vector-sensor applications","volume":"44","author":"Hochwald","year":"1996","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1291","DOI":"10.1109\/TAES.2008.4667710","article-title":"Virtual-manifold ambiguity in HOS-based direction-finding with electromagnetic vector-sensors","volume":"44","author":"Xu","year":"2008","journal-title":"IEEE Trans. Aerosp. Electron. Syst"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"5337","DOI":"10.1109\/TSP.2007.899367","article-title":"Higher order direction finding from arrays with diversely polarized antennas: The PD-2q-MUSIC algorithms","volume":"55","author":"Chevalier","year":"2007","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2377","DOI":"10.1109\/78.852018","article-title":"Parallel factor analysis in sensor array processing","volume":"48","author":"Sidiropoulos","year":"2000","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"831","DOI":"10.1016\/j.sigpro.2008.10.034","article-title":"Direction-of-arrival estimation via twofold mode-projection","volume":"89","author":"Gong","year":"2009","journal-title":"Signal Process"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1049\/iet-spr.2009.0221","article-title":"Regularised parallel factor analysis for the estimation of direction-of-arrival and polarisation with a single electromagnetic vector-sensor","volume":"5","author":"Gong","year":"2011","journal-title":"IET Signal Process"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Gong, X.F. (2010, January 10\u201312). Source Localization via Trilinear Decomposition of Cross Covariance Tensor with Vector-Sensor Arrays. Yantai, China.","DOI":"10.1109\/FSKD.2010.5569727"},{"key":"ref_36","unstructured":"Guo, X., Miron, S., and Brie, D. (April, January 31). Identifiability of the PARAFAC Model for Polarized Source Mixture on a Vector Sensor Array. Las Vegas, NV, USA."},{"key":"ref_37","unstructured":"Miron, S., Guo, X., and Brie, D. (2008, January 25\u201329). DOA Estimation for Polarized Sources on a Vector-Sensor Array by PARAFAC Decomposition of the Fourth-Order Covariance Tensor. Lausanne, Switzerland."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1218","DOI":"10.1109\/TSP.2006.870630","article-title":"Quaternion-MUSIC for vector-sensor array processing","volume":"54","author":"Miron","year":"2006","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"4523","DOI":"10.1109\/TSP.2007.896067","article-title":"MUSIC algorithm for vector-sensors array using biquaternions","volume":"55","author":"Miron","year":"2007","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Gong, X.F., Liu, Z.W., and Xu, Y.G. (2008). Quad-quaternion MUSIC for DOA estimation using electromagnetic vector-sensors. EURASIP J. Adv. Signal Process.","DOI":"10.1155\/2008\/213293"},{"key":"ref_41","unstructured":"Gong, X.F., Xu, Y.G., and Liu, Z.W. (2008, January 26\u201329). Quaternion ESPRIT for Direction Finding with Polarization Sensitive Array. Beijing, China."},{"key":"ref_42","first-page":"2268","article-title":"Geometric algebra of Euclidean 3-space for electromagnetic vector-sensor array processing, part I: Modeling","volume":"47","author":"Jiang","year":"2011","journal-title":"IEEE Trans. Antennas Propag"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2268","DOI":"10.1109\/TAES.2011.5937301","article-title":"Coherent source localization: Bicomplex polarimetric smoothing with electromagnetic vector-sensors","volume":"47","author":"Gong","year":"2011","journal-title":"IEEE Trans. Aerosp. Electron. Syst"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1016\/j.sigpro.2010.08.015","article-title":"Direction finding via biquaternion matrix diagonalization with vector-sensors","volume":"91","author":"Gong","year":"2011","journal-title":"Signal Process"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1137\/S0895479893259546","article-title":"Jacobi angles for simultaneous diagonalization","volume":"17","author":"Cardoso","year":"1996","journal-title":"SIAM J. Matrix Anal. Appl"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1545","DOI":"10.1109\/TSP.2002.1011195","article-title":"Non-orthogonal joint diagonalization in the least-squares sense with application in blind source separation","volume":"50","author":"Yeredor","year":"2002","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1136","DOI":"10.1137\/S089547980035689X","article-title":"Joint approximate diagonalization of positive definite Hermitian matrices","volume":"22","author":"Pham","year":"2001","journal-title":"SIAM J. Matrix Anal. Appl"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1803","DOI":"10.1109\/TSP.2006.889983","article-title":"Nonorthogonal joint diagonaliza-tion free of degenerate solution","volume":"55","author":"Li","year":"2007","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"878","DOI":"10.1109\/TSP.2008.2009271","article-title":"Fast approximate joint diagonalization incorporating weight matrices","volume":"57","author":"Tichavsky","year":"2009","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Guo, X.J., Zhu, S.H., Miron, S., and Brie, D. (2010, January 14\u201319). Approximate Joint Diagonalization by Nonorthogonal Nonparametric Jacobi Transformations. Dallas, TX, USA.","DOI":"10.1109\/ICASSP.2010.5495846"},{"key":"ref_51","unstructured":"Joho, M., and Rahbar, K. (2002, January 4\u20136). Joint Diagonalization of Correlation Matrices by Using Newton Methods with Application to Blind Signal Separation. Champaign, IL, USA."},{"key":"ref_52","unstructured":"Joho, M., and Rahbar, K. (2002, January 4\u20136). Joint Diagonalization of Correlation Matrices by Using Gradient Methods with Application to Blind Signal Separation. Champaign, IL, USA."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Afsari, B. (2006, January 5\u20138). Simple LU and QR Based Non-Orthogonal Matrix Joint Diagonalization. Charleston, SC, USA.","DOI":"10.1007\/11679363_1"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"2222","DOI":"10.1109\/TSP.2009.2016997","article-title":"Nonorthogonal joint diagonalization by combining givens and hyperbolic rotations","volume":"57","author":"Souloumiac","year":"2009","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Luciani, X., and Albera, L. (2010, January 27\u201330). Joint Eigenvalue Decomposition Using Polar Matrix Factorization. St. Malo, France.","DOI":"10.1007\/978-3-642-15995-4_69"},{"key":"ref_56","unstructured":"Hao, S., and Huper, K. (2009, January 19\u201324). Block Jacobi-Type Methods for Non-Orthogonal Joint Diagonalisation. Taipei, China."},{"key":"ref_57","unstructured":"Bin, S., Florian, R., and Martin, H. (2010, January 14\u201319). Using a New Structured Joint Congruence (STJOCO) Transformation of Hermitian Matrices for Precoding in Multi-User MIMO Systems. Dallas, TX, USA."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.3758\/BRM.41.4.1073","article-title":"The LMPCA program: A graphical user interface for fitting the linked-mode PARAFAC-PCA model to coupled real-valued data","volume":"41","author":"Wilderjans","year":"2009","journal-title":"Behav. Res. Methods"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"642","DOI":"10.1137\/040608830","article-title":"A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization","volume":"28","year":"2006","journal-title":"SIAM J. Matrix Anal. Appl"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"810","DOI":"10.1109\/78.824675","article-title":"Blind PARAFAC receivers for DS-CDMA systems","volume":"48","author":"Sidiropoulos","year":"2000","journal-title":"IEEE Trans. Signal Process"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/12\/3\/3394\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:49:18Z","timestamp":1760219358000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/12\/3\/3394"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,3,8]]},"references-count":60,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2012,3]]}},"alternative-id":["s120303394"],"URL":"https:\/\/doi.org\/10.3390\/s120303394","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2012,3,8]]}}}