{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T04:59:03Z","timestamp":1776229143333,"version":"3.50.1"},"reference-count":47,"publisher":"MDPI AG","issue":"23","license":[{"start":{"date-parts":[[2020,12,4]],"date-time":"2020-12-04T00:00:00Z","timestamp":1607040000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Key R&amp;D Program of China","award":["2016YFC1400101"],"award-info":[{"award-number":["2016YFC1400101"]}]},{"name":"Provincial Funding Projects of National Key R&amp;D Program of China","award":["GX18C019"],"award-info":[{"award-number":["GX18C019"]}]},{"DOI":"10.13039\/501100012335","name":"National Defense Basic Scientific Research Program of China","doi-asserted-by":"publisher","award":["JCKY2019604B001"],"award-info":[{"award-number":["JCKY2019604B001"]}],"id":[{"id":"10.13039\/501100012335","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Array design is the primary consideration for array signal processing, and sparse array design is an important and challenging task. In underwater acoustic environments, the vector hydrophone array contains more information than the scalar hydrophone array, but there are few articles focused on the design of the vector hydrophone array. The difference between the vector hydrophone array and the scalar hydrophone array is that each vector hydrophone has three or four channels. When designing a sparse vector hydrophone array, these channels need to be optimized at the same time to ensure the sparsity of the array elements\u2019 number. To solve this problem, this paper introduced the compressed sensing (CS) theory into the vector hydrophone array design, constructed the vector hydrophone array design problem into a globally solvable optimization problem, proposed a CS-based algorithm with the L1 norm suitable for vector hydrophone array, and realized the simultaneous optimization of multiple channels from the same vector hydrophone. At the same time, the off-grid algorithm was added to obtain higher design accuracy. Two design examples verify the effectiveness of the proposed method. The theoretical analysis and simulation results show that compared with the conventional compressed sensing algorithm with the same aperture, the algorithm proposed in this paper used fewer vector hydrophone elements to obtain better fitting of the desired beam pattern.<\/jats:p>","DOI":"10.3390\/s20236949","type":"journal-article","created":{"date-parts":[[2020,12,7]],"date-time":"2020-12-07T21:37:42Z","timestamp":1607377062000},"page":"6949","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Vector Hydrophone Array Design Based on Off-Grid Compressed Sensing"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5478-7038","authenticated-orcid":false,"given":"Zhibo","family":"Shi","sequence":"first","affiliation":[{"name":"Acoustic Science and Technology Laboratory, Harbin Engineering University, Harbin 150001, China"},{"name":"Key Laboratory of Marine Information Acquisition and Security (Harbin Engineering University), Ministry of Industry and Information Technology, Harbin 150001, China"},{"name":"College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China"},{"name":"Qingdao Haina Underwater Information Technology Co., Ltd., Qingdao 266500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guolong","family":"Liang","sequence":"additional","affiliation":[{"name":"Acoustic Science and Technology Laboratory, Harbin Engineering University, Harbin 150001, China"},{"name":"Key Laboratory of Marine Information Acquisition and Security (Harbin Engineering University), Ministry of Industry and Information Technology, Harbin 150001, China"},{"name":"College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China"},{"name":"Qingdao Haina Underwater Information Technology Co., Ltd., Qingdao 266500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Longhao","family":"Qiu","sequence":"additional","affiliation":[{"name":"Acoustic Science and Technology Laboratory, Harbin Engineering University, Harbin 150001, China"},{"name":"Key Laboratory of Marine Information Acquisition and Security (Harbin Engineering University), Ministry of Industry and Information Technology, Harbin 150001, China"},{"name":"College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China"},{"name":"Qingdao Haina Underwater Information Technology Co., Ltd., Qingdao 266500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guangming","family":"Wan","sequence":"additional","affiliation":[{"name":"Acoustic Science and Technology Laboratory, Harbin Engineering University, Harbin 150001, China"},{"name":"Key Laboratory of Marine Information Acquisition and Security (Harbin Engineering University), Ministry of Industry and Information Technology, Harbin 150001, China"},{"name":"College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China"},{"name":"Qingdao Haina Underwater Information Technology Co., Ltd., Qingdao 266500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lei","family":"Zhao","sequence":"additional","affiliation":[{"name":"Acoustic Science and Technology Laboratory, Harbin Engineering University, Harbin 150001, China"},{"name":"Key Laboratory of Marine Information Acquisition and Security (Harbin Engineering University), Ministry of Industry and Information Technology, Harbin 150001, China"},{"name":"College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China"},{"name":"Qingdao Haina Underwater Information Technology Co., Ltd., Qingdao 266500, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"87","DOI":"10.2528\/PIERL09090606","article-title":"Synthesis of thinned planar circular array antennas using modified particle swarm optimization","volume":"12","author":"Pathak","year":"2009","journal-title":"Prog. Electromagn. Res. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"822","DOI":"10.1049\/iet-map.2011.0484","article-title":"Synthesis of thinned planar concentric circular antenna arrays using biogeography-based optimisation","volume":"6","author":"Singh","year":"2012","journal-title":"IET Microw. Antennas Propag."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"63","DOI":"10.2528\/PIERB11020204","article-title":"Synthesis of thinned planar concentric circular antenna arrays\u2014A differential evolutionary approach","volume":"29","author":"Ghosh","year":"2011","journal-title":"Prog. Electromagn. Res. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2172","DOI":"10.1049\/iet-map.2009.0630","article-title":"Design of planar thinned arrays using a Boolean differential evolution algorithm","volume":"4","author":"Zhang","year":"2010","journal-title":"IET Microw. Antennas Propag."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6215","DOI":"10.1109\/TSP.2019.2952052","article-title":"Hybrid Sparse Array Beamforming Design for General Rank Signal Models","volume":"67","author":"Hamza","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"550","DOI":"10.1109\/TASLP.2017.2786544","article-title":"Microphone Subset Selection for MVDR Beamformer Based Noise Reduction","volume":"26","author":"Zhang","year":"2018","journal-title":"IEEE\/ACM Trans. Audio Speech Lang. Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2797","DOI":"10.1121\/1.3644914","article-title":"Optimum array design to maximize Fisher information for bearing estimation","volume":"130","author":"Tuladhar","year":"2011","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2926","DOI":"10.1121\/1.3488665","article-title":"Designing nonuniform linear arrays to maximize mutual information for bearing estimation","volume":"128","author":"Zhu","year":"2010","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"800","DOI":"10.1109\/TASLP.2014.2304635","article-title":"Design of Superdirective Planar Arrays with Sparse Aperiodic Layouts for Processing Broadband Signals via 3-D Beamforming","volume":"22","author":"Crocco","year":"2014","journal-title":"IEEE\/ACM Trans. Audio Speech Lang. Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2433","DOI":"10.1109\/TASL.2012.2203808","article-title":"Stochastic and Analytic Optimization of Sparse Aperiodic Arrays and Broadband Beamformers with Robust Superdirective Patterns","volume":"20","author":"Crocco","year":"2012","journal-title":"IEEE Trans. Audio Speech Lang. Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"31056","DOI":"10.3390\/s151229849","article-title":"Compressive Sensing Based Design of Sparse Tripole Arrays","volume":"15","author":"Hawes","year":"2015","journal-title":"Sensors"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2411","DOI":"10.1109\/TSP.2017.2655479","article-title":"Location and Orientation Optimization for Spatially Stretched Tripole Arrays Based on Compressive Sensing","volume":"65","author":"Hawes","year":"2017","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1109\/LAWP.2012.2186626","article-title":"Maximally Sparse Arrays Via Sequential Convex Optimizations","volume":"11","author":"Prisco","year":"2012","journal-title":"Antennas Wirel. Propag. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1109\/TAP.2007.913176","article-title":"Optimized element spacing for low sidelobe concentric ring arrays","volume":"56","author":"Haupt","year":"2008","journal-title":"IEEE Trans. Antennas Propagat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"147247","DOI":"10.1155\/2015\/147247","article-title":"Low Sidelobe Sparse Concentric Ring Arrays Optimization Using Modified GA","volume":"2015","author":"Chen","year":"2015","journal-title":"Int. J. Antennas Propag."},{"key":"ref_16","first-page":"21","article-title":"An optimum method of sparse plane arrays with circular boundary","volume":"28","author":"Tang","year":"2013","journal-title":"Chin. J. Radio"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Chatterjee, A., Mahanti, G.K., and Mahapatra, P.R.S. (2010, January 17\u201319). Optimum ring spacing and interelement distance for sidelobe reduction of a uniform concentric ring array antenna using differential evolution algorithm. Proceedings of the 2010 IEEE International Conference on Communication Systems, Singapore.","DOI":"10.1109\/ICCS.2010.5686077"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2528\/PIERL11112506","article-title":"Concentric circular antenna array synthesis using comprehensive learning particle swarm optimizer","volume":"29","author":"Elsaidy","year":"2012","journal-title":"Prog. Electromagn. Res. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1109\/TAP.2011.2167931","article-title":"A generalized hybrid approach for the synthesis of uniform amplitude pencil beam ring-arrays","volume":"60","author":"Bucci","year":"2012","journal-title":"IEEE Trans. Antennas Propagat."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1109\/TAP.2015.2504377","article-title":"A Hybrid Method for the Optimal Synthesis of 3-D Patterns of Sparse Concentric Ring Arrays","volume":"64","author":"Zhao","year":"2016","journal-title":"IEEE Trans. Antennas Propagat."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2006.871582","article-title":"Compressed sensing","volume":"52","author":"Donoho","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_22","first-page":"1433","article-title":"Compressive sampling","volume":"Volume 3","year":"2006","journal-title":"Proceedings of the International Congress of Mathematicians (ICM) 2006"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.sigpro.2017.03.013","article-title":"Design and analysis of compressive antenna arrays for direction of arrival estimation","volume":"138","author":"Ibrahim","year":"2017","journal-title":"Signal Process."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1109\/TSP.2008.2007095","article-title":"Sensor Selection via Convex Optimization","volume":"57","author":"Joshi","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5595","DOI":"10.1109\/TSP.2016.2573767","article-title":"Sensor Placement by Maximal Projection on Minimum Eigenspace for Linear Inverse Problems","volume":"64","author":"Jiang","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2141","DOI":"10.1049\/el.2015.2455","article-title":"Pattern synthesis of sparse linear array by off-grid Bayesian compressive sampling","volume":"51","author":"Lin","year":"2015","journal-title":"Electron. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"21981","DOI":"10.3390\/s141121981","article-title":"Off-Grid Direction of Arrival Estimation Based on Joint Spatial Sparsity for Distributed Sparse Linear Arrays","volume":"14","author":"Liang","year":"2014","journal-title":"Sensors"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Fan, Y., Wang, J., Du, R., and Lv, G. (2018). Sparse Method for Direction of Arrival Estimation Using Denoised Fourth-Order Cumulants Vector. Sensors, 18.","DOI":"10.3390\/s18061815"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.sigpro.2017.07.004","article-title":"Two sparse-based methods for off-grid direction-of-arrival estimation","volume":"142","author":"Wu","year":"2018","journal-title":"Signal Process."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.sigpro.2013.11.022","article-title":"Off-grid DOA estimation using array covariance matrix and block-sparse Bayesian learning","volume":"98","author":"Yi","year":"2014","journal-title":"Signal Process."},{"key":"ref_31","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 Process."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Qin, Y., Liu, Y., Liu, J., and Yu, Z. (2018). Underdetermined Wideband DOA Estimation for Off-Grid Sources with Coprime Array Using Sparse Bayesian Learning. Sensors, 18.","DOI":"10.3390\/s18010253"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1109\/LSP.2013.2289740","article-title":"Sparse Direction of Arrival Estimation Using Co-Prime Arrays with Off-Grid Targets","volume":"21","author":"Tan","year":"2013","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1710","DOI":"10.1109\/LSP.2018.2872400","article-title":"Off-Grid Direction-of-Arrival Estimation Using Coprime Array Interpolation","volume":"25","author":"Zhou","year":"2018","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Kou, J., Li, M., and Jiang, C. (2019). A robust DOA estimator based on compressive sensing for coprime array in the presence of miscalibrated sensors. Sensors, 19.","DOI":"10.3390\/s19163538"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1719","DOI":"10.1049\/iet-com.2016.1048","article-title":"Compressive sensing-based coprime array direction-of-arrival estimation","volume":"11","author":"Zhou","year":"2017","journal-title":"IET Commun."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"4650","DOI":"10.1007\/s00034-020-01391-0","article-title":"Off-Grid DOA Estimation Based on Alternating Iterative Weighted Least Squares for Acoustic Vector Hydrophone Array","volume":"39","author":"Wang","year":"2020","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Liang, Y., Meng, Z., Chen, Y., Zhang, Y., Wang, M., and Zhou, X. (2020). A data fusion orientation algorithm based on the weighted histogram statistics for vector hydrophone vertical array. Sensors, 20.","DOI":"10.3390\/s20195619"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Zhu, C., Seri, S.G., Mohebbi-Kalkhoran, H., and Ratilal, P. (2020). Long-range automatic detection, acoustic signature characterization and bearing-time estimation of multiple ships with coherent hydrophone array. Remote Sens., 12.","DOI":"10.3390\/rs12223731"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1564","DOI":"10.1109\/JOE.2019.2944444","article-title":"Performance of a Passive Acoustic Linear Array in a Tidal Channel","volume":"45","author":"Auvinen","year":"2020","journal-title":"IEEE J. Ocean. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Liang, Y., Meng, Z., Chen, Y., Zhang, Y., Zhou, X., and Wang, M. (2020). Research on passive ranging technology of moving ship based on vertical hydrophone array. Appl. Sci., 10.","DOI":"10.3390\/app10217396"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Cao, R., Yang, K., Yang, Q., Chen, P., Sun, Q., and Xue, R. (2019). Localization of two sound sources based on compressed matched field processing with a short hydrophone array in the deep ocean. Sensors, 19.","DOI":"10.3390\/s19173810"},{"key":"ref_43","unstructured":"Sun, S., Qin, S., Hao, Y., Zhang, G., and Zhao, C. (2020). Underwater Acoustic Localization of the Black Box Based on Generalized Second-Order Time Difference of Arrival (GSTDOA). IEEE Trans. Geosci. Remote Sens., 1\u201311."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1268","DOI":"10.1109\/JOE.2019.2950954","article-title":"Underwater Acoustical Localization of the Black Box Utilizing Single Autonomous Underwater Vehicle Based on the Second-Order Time Difference of Arrival","volume":"45","author":"Sun","year":"2020","journal-title":"IEEE J. Ocean. Eng."},{"key":"ref_45","first-page":"943","article-title":"Broadband Beamforming with Minimum Sidelobe and Constant Beamwidth Based on Convex Optimization","volume":"41","author":"Zhan","year":"2013","journal-title":"Chin. J. Electron."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"606","DOI":"10.1109\/JSTSP.2007.910971","article-title":"An interior-point method for large-scale \u21131-regularized least squares","volume":"1","author":"Kim","year":"2007","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_47","first-page":"439","article-title":"A new formulation for the design of Chebyshev arrays","volume":"42","year":"2002","journal-title":"IEEE Trans. Antennas Propag."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/23\/6949\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:41:49Z","timestamp":1760179309000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/23\/6949"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,4]]},"references-count":47,"journal-issue":{"issue":"23","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["s20236949"],"URL":"https:\/\/doi.org\/10.3390\/s20236949","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,4]]}}}