{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T09:03:42Z","timestamp":1790154222694,"version":"4.0.1"},"reference-count":43,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T00:00:00Z","timestamp":1790121600000},"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>Mutual coupling among array receiving antennas in passive detection systems can introduce amplitude and phase inconsistencies between antenna elements. The resulting amplitude and phase errors across receiving channels not only affect the spectral peak values in subsequent MUSIC-based direction finding, but also cause shifts in the locations of the spectral peaks, thereby increasing detection errors. To reduce geometry-level sensitivity to this effect, this paper presents an analytically constructed L-shaped sparse array for two-dimensional direction-of-arrival (2D DOA) estimation. The array comprises two orthogonal three-subarray branches that share the sensor at the origin. The electrical angle of one branch is first estimated by a one-dimensional (1D) spatial spectrum; the corresponding electrical angle of the other branch is then obtained from a source-conditioned 1D spectrum, yielding paired 2D directional parameters. With 23 physical sensors, the proposed geometry provides 83 central contiguous difference-coarray lags and a one-arm aperture of 43d, where d is one-half wavelength. Under the normalized parametric coupling model adopted in this work, the reported numerical results show lower DOA-estimation RMSE than the compared L-shaped arrays over the investigated ranges. These results establish an analytical geometry and signal-model comparison rather than antenna-specific electromagnetic validation.<\/jats:p>","DOI":"10.3390\/s26196015","type":"journal-article","created":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T08:14:39Z","timestamp":1790151279000},"page":"6015","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An L-Shaped Sparse Array with Reduced Sensitivity to Parametric Mutual Coupling for 2D DOA Estimation"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-0265-1390","authenticated-orcid":false,"given":"Xiaoming","family":"Wu","sequence":"first","affiliation":[{"name":"Qianyuan Laboratory, Hangzhou 310013, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"methodology","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-1317-2183","authenticated-orcid":false,"given":"Wencan","family":"Cheng","sequence":"additional","affiliation":[{"name":"Hangzhou Research Institute of Xidian University, Hangzhou 311231, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"writing-original-draft","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-8145-9966","authenticated-orcid":false,"given":"Fengtong","family":"Mei","sequence":"additional","affiliation":[{"name":"Hangzhou Research Institute of Xidian University, Hangzhou 311231, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"validation","vocabulary":"credit"},{"role":"data-curation","vocabulary":"credit"},{"role":"supervision","vocabulary":"credit"},{"role":"resources","vocabulary":"credit"},{"role":"software","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-2328-7645","authenticated-orcid":false,"given":"Yazhou","family":"Yang","sequence":"additional","affiliation":[{"name":"Hangzhou Research Institute of Xidian University, Hangzhou 311231, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"software","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"},{"role":"data-curation","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7714-7620","authenticated-orcid":false,"given":"Yifeng","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Electronics and Communication Engineering, Sun Yat-sen University, Shenzhen 518107, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"project-administration","vocabulary":"credit"},{"role":"funding-acquisition","vocabulary":"credit"},{"role":"supervision","vocabulary":"credit"}]}],"member":"1968","published-online":{"date-parts":[[2026,9,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2136","DOI":"10.1109\/TAP.2010.2046838","article-title":"Joint elevation and azimuth direction finding using L-shaped array","volume":"58","author":"Liang","year":"2010","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3197","DOI":"10.1109\/TSP.2011.2144591","article-title":"Computationally efficient subspace-based method for two-dimensional direction estimation with L-shaped array","volume":"59","author":"Wang","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1077","DOI":"10.1109\/LAWP.2015.2493099","article-title":"Two-dimensional DOA estimation using cross-correlation matrix with L-shaped array","volume":"15","author":"Tayem","year":"2016","journal-title":"IEEE Antennas Wirel. Propag. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1109\/TAP.1968.1139138","article-title":"Minimum-redundancy linear arrays","volume":"16","author":"Moffet","year":"1968","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4167","DOI":"10.1109\/TSP.2010.2049264","article-title":"Nested arrays: A novel approach to array processing with enhanced degrees of freedom","volume":"58","author":"Pal","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1109\/TSP.2010.2089682","article-title":"Sparse sensing with co-prime samplers and arrays","volume":"59","author":"Vaidyanathan","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3997","DOI":"10.1109\/TSP.2016.2558159","article-title":"Super nested arrays: Linear sparse arrays with reduced mutual coupling-Part I: Fundamentals","volume":"64","author":"Liu","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4203","DOI":"10.1109\/TSP.2016.2558167","article-title":"Super nested arrays: Linear sparse arrays with reduced mutual coupling-Part II: High-order extensions","volume":"64","author":"Liu","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5549","DOI":"10.1109\/TSP.2017.2736493","article-title":"Augmented nested arrays with enhanced DOF and reduced mutual coupling","volume":"65","author":"Liu","year":"2017","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1109\/JSEN.2023.3335257","article-title":"Enhanced dilated nested arrays with reduced mutual coupling for DOA estimation","volume":"24","author":"Zhou","year":"2024","journal-title":"IEEE Sens. J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1109\/LSP.2023.3340599","article-title":"Super augmented nested arrays: A new sparse array for improved DOA estimation accuracy","volume":"31","author":"Wang","year":"2024","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"10380","DOI":"10.1109\/TVT.2023.3260196","article-title":"Generalized nested array configuration family for direction-of-arrival estimation","volume":"72","author":"Zhao","year":"2023","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Zhang, Y., Hu, G., Zhou, H., Bai, J., Zhan, C., and Guo, S. (2023). Direction of arrival estimation of generalized nested array via difference-sum co-array. Sensors, 23.","DOI":"10.3390\/s23020906"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1007\/s11045-019-00657-4","article-title":"L-shaped coprime array structures for DOA estimation","volume":"31","author":"Elbir","year":"2020","journal-title":"Multidimens. Syst. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"140030","DOI":"10.1109\/ACCESS.2020.3012685","article-title":"L-shaped sparse array structure for 2-D DOA estimation","volume":"8","author":"Wu","year":"2020","journal-title":"IEEE Access"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1934","DOI":"10.1109\/LCOMM.2020.2998529","article-title":"2-D DOA estimation for L-shaped sparse array via joint use of spatial and temporal information","volume":"24","author":"Peng","year":"2020","journal-title":"IEEE Commun. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Sun, Y., Tian, Q., Yao, D., and Zhang, Q. (2025). Low-complexity 2D DOA estimation via L-shaped array for underwater hexapod robot. J. Mar. Sci. Eng., 13.","DOI":"10.3390\/jmse13020229"},{"key":"ref_18","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_19","doi-asserted-by":"crossref","first-page":"3150","DOI":"10.1109\/TAP.2008.929446","article-title":"2-D DOA estimation in the presence of mutual coupling","volume":"56","author":"Ye","year":"2008","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_20","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_21","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_22","doi-asserted-by":"crossref","unstructured":"Famoriji, O.J., and Shongwe, T. (2021). Direction-of-arrival estimation of electromagnetic wave impinging on spherical antenna array in the presence of mutual coupling using a multiple signal classification method. Electronics, 10.","DOI":"10.3390\/electronics10212651"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"154453","DOI":"10.1016\/j.aeue.2022.154453","article-title":"Toeplitz rectification based DOA estimation under the coexistence of mutual coupling and nonuniform noise","volume":"158","author":"Wen","year":"2023","journal-title":"AEU\u2014Int. J. Electron. Commun."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"109940","DOI":"10.1016\/j.sigpro.2025.109940","article-title":"Direction of arrival estimation for sparse arrays with gain-phase errors in nonuniform noise environment","volume":"233","author":"Zhang","year":"2025","journal-title":"Signal Process."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Florio, A., Avitabile, G., and Coviello, G. (2022). A linear technique for artifacts correction and compensation in phase interferometric angle of arrival estimation. Sensors, 22.","DOI":"10.3390\/s22041427"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2052","DOI":"10.1109\/TSP.2019.2901380","article-title":"Thinned coprime array for second-order difference co-array generation with reduced mutual coupling","volume":"67","author":"Raza","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1728","DOI":"10.1109\/TSP.2019.2897954","article-title":"MISC array: A new sparse array design achieving increased degrees of freedom and reduced mutual coupling effect","volume":"67","author":"Zheng","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"9934097","DOI":"10.1155\/2021\/9934097","article-title":"The design of a novel sparse array using two uniform linear arrays considering mutual coupling","volume":"2021","author":"Xu","year":"2021","journal-title":"J. Sens."},{"key":"ref_29","first-page":"8800833","article-title":"The limited aperture sparse array for DOA estimation of coherent signals: A mutual-coupling-optimized array and coherent DOA estimation algorithm","volume":"2022","author":"Zhang","year":"2022","journal-title":"Math. Probl. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1612","DOI":"10.1049\/rsn2.12453","article-title":"Hole-free sparse array for highly efficient direction of arrival estimation: Enhanced uniform difference co-array","volume":"17","author":"Zhang","year":"2023","journal-title":"IET Radar Sonar Navig."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2849","DOI":"10.1109\/TSP.2023.3300635","article-title":"Third-order nested array: An optimal geometry for third-order cumulants based array processing","volume":"71","author":"Sharma","year":"2023","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1545","DOI":"10.1109\/LSP.2021.3099074","article-title":"Coupled coarray tensor CPD for DOA estimation with coprime L-shaped array","volume":"28","author":"Zheng","year":"2021","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Du, B., Cui, W., Ba, B., Xu, H., and Gao, W. (2023). An L-shaped three-level and single common element sparse sensor array for 2-D DOA estimation. Sensors, 23.","DOI":"10.3390\/s23146625"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Zhang, L., Ren, S., Li, X., Ren, G., and Wang, X. (2018). Generalized L-shaped nested array concept based on the fourth-order difference co-array. Sensors, 18.","DOI":"10.3390\/s18082482"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3374","DOI":"10.1109\/TAP.2024.3369718","article-title":"A second-order L-shaped full digital receiving sparse array for direction-of-arrival estimation","volume":"72","author":"Xu","year":"2024","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"604","DOI":"10.1109\/TAES.2023.3326793","article-title":"Tensor-based 2-D DOA estimation for L-shaped nested array","volume":"60","author":"Xu","year":"2024","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1109\/TASSP.1985.1164557","article-title":"Detection of signals by information theoretic criteria","volume":"33","author":"Wax","year":"1985","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1109\/8.68174","article-title":"An L-shaped array for estimating 2-D directions of wave arrival","volume":"39","author":"Hua","year":"1991","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1007\/s11045-016-0402-7","article-title":"2D DOA estimation for noncircular sources using L-shaped sparse array","volume":"29","author":"Liu","year":"2018","journal-title":"Multidimens. Syst. Signal Process."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"806","DOI":"10.1109\/TASSP.1985.1164649","article-title":"On spatial smoothing for direction-of-arrival estimation of coherent signals","volume":"33","author":"Shan","year":"1985","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1438","DOI":"10.1109\/LSP.2015.2409153","article-title":"Remarks on the spatial smoothing step in coarray MUSIC","volume":"22","author":"Liu","year":"2015","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Zhang, X., Li, J., and Xu, L. (2011). Novel two-dimensional DOA estimation with L-shaped array. EURASIP J. Adv. Signal Process., 2011.","DOI":"10.1186\/1687-6180-2011-50"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Citroni, R., and Livreri, P. (2023). A novel shape of bowtie antenna arranged in a linear array for energy harvesting in MID-IR band. 2023 12th International Conference on Renewable Energy Research and Applications (ICRERA), Oshawa, ON, Canada, 29 August 2023\u20131 September 2023, IEEE.","DOI":"10.1109\/ICRERA59003.2023.10269377"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/19\/6015\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T08:20:14Z","timestamp":1790151614000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/19\/6015"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,23]]},"references-count":43,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2026,10]]}},"alternative-id":["s26196015"],"URL":"https:\/\/doi.org\/10.3390\/s26196015","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,23]]}}}