{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T13:11:41Z","timestamp":1781529101385,"version":"3.54.1"},"reference-count":54,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,4]],"date-time":"2026-02-04T00:00:00Z","timestamp":1770163200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62501101"],"award-info":[{"award-number":["62501101"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100012906","name":"Shandong Provincial Education Department","doi-asserted-by":"publisher","award":["2024KJN033v"],"award-info":[{"award-number":["2024KJN033v"]}],"id":[{"id":"10.13039\/100012906","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Coprime arrays enable enhanced degrees of freedom through the construction of virtual array equivalent signals. However, the presence of large \u201choles\u201d leads to discontinuous co-arrays, which severely hampers direction-of-arrival (DOA) estimation techniques that rely on uniform array structures. This paper explores the practical application of co-array domain signal processing for underwater acoustic coprime arrays. We propose a novel array configuration based on coprime minimum disordered pairs, enabling the formation of continuously connected co-arrays without interpolating. To address the challenge of limited snapshots in underwater environments, DOA estimation can be achieved by utilizing traditional multipath matching pursuit (MMP) algorithms under the proposed continuous co-array implementation scheme. In practical applications, physical array element failures are inevitable, and faulty elements can create holes in the originally continuous co-array. While interpolation techniques can mitigate small gaps, their performance deteriorates significantly in the presence of large holes or uneven data distribution. To overcome these limitations, we introduce a sparse signal recovery (SSR) method using a fragment array data validation technique for sparse DOA estimation with an underwater acoustic coprime array. Based on the designed continuous array expansion scheme, the resulting continuous co-array is used to map the positions of element failures, revealing the gaps in the co-array. A validation model is established for partially continuous sub-arrays within the discontinuous co-array, enabling signal direction estimation based on the fragmented array validation. Both simulation and sea trial results confirm that the proposed approach maximizes the utilization of co-array elements without relying on interpolation or prediction, offering a robust solution for scenarios involving sensor failures.<\/jats:p>","DOI":"10.3390\/a19020125","type":"journal-article","created":{"date-parts":[[2026,2,4]],"date-time":"2026-02-04T12:51:56Z","timestamp":1770209516000},"page":"125","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["DOA Estimation for Underwater Coprime Arrays with Sensor Failure Based on Segmented Array Validation and Multipath Matching Pursuit"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2547-1736","authenticated-orcid":false,"given":"Xiao","family":"Chen","sequence":"first","affiliation":[{"name":"School of Information Science and Technology, Dalian Maritime University, Dalian 116026, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ying","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Electronic Engineering, Ocean University of China, Qingdao 266005, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,4]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"562","DOI":"10.1109\/PROC.1977.10517","article-title":"Applications of numbered undirected graphs","volume":"65","author":"Bloom","year":"1977","journal-title":"Proc. IEEE"},{"key":"ref_3","first-page":"972","article-title":"Enhanced MISC-based sparse array with high uDOFs and low mutual coupling","volume":"71","author":"Sheng","year":"2023","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"EL306","DOI":"10.1121\/1.5126924","article-title":"Array gain of coprime arrays","volume":"146","author":"Yang","year":"2019","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"964","DOI":"10.1049\/iet-rsn.2016.0517","article-title":"Passive beamforming with coprime arrays","volume":"11","author":"Iodice","year":"2017","journal-title":"IET Radar Sonar Navig."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1121\/1.1909060","article-title":"Theory of Time-Averaged-Product Arrays","volume":"29","author":"Berman","year":"1957","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_7","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":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Tan, Z., Nehorai, A., and Eldar, Y.C. (2014). Continuous sparse recovery for direction of arrival estimation with co-prime arrays. Proceedings of the 2014 IEEE 8th Sensor Array and Multichannel Signal Processing Workshop (SAM), A Coruna, Spain, 22\u201325 June 2014, IEEE.","DOI":"10.1109\/SAM.2014.6882425"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Liu, Y., and Buck, J.R. (2015). Detecting Gaussian signals in the presence of interferers using the coprime sensor arrays with the min processor. Proceedings of the 2015 49th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA, 8\u201311 November 2015, IEEE.","DOI":"10.1109\/ACSSC.2015.7421150"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Liu, Y., and Buck, J.R. (2016). Spatial spectral estimation using a coprime sensor array with the min processor. Proceedings of the 2016 IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM), Rio de Janeiro, Brazil, 10\u201313 July 2016, IEEE.","DOI":"10.1109\/SAM.2016.7569626"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1109\/TSP.2017.2762284","article-title":"Gaussian source detection and spatial spectral estimation using a coprime sensor array with the min processor","volume":"66","author":"Liu","year":"2017","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Adhikari, K., Buck, J.R., and Wage, K.E. (2013). Beamforming with extended co-prime sensor arrays. Proceedings of the 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, Vancouver, BC, Canada, 26\u201331 May 2013, IEEE.","DOI":"10.1109\/ICASSP.2013.6638447"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1186\/1687-6180-2014-148","article-title":"Extending coprime sensor arrays to achieve the peak side lobe height of a full uniform linear array","volume":"2014","author":"Adhikari","year":"2014","journal-title":"EURASIP J. Adv. Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1049\/iet-spr.2018.5517","article-title":"Novel method for digital beamforming in co-prime sensor arrays using product and min processors","volume":"13","year":"2019","journal-title":"IET Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zhou, C., Gu, Y., Song, W.Z., Xie, Y., and Shi, Z. (2016). Robust adaptive beamforming based on DOA support using decomposed coprime subarrays. Proceedings of the 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Shanghai, China, 20\u201325 March 2016, IEEE.","DOI":"10.1109\/ICASSP.2016.7472225"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1099","DOI":"10.1109\/TVT.2017.2704610","article-title":"A robust and efficient algorithm for coprime array adaptive beamforming","volume":"67","author":"Zhou","year":"2017","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Zhou, C., Shi, Z., Gu, Y., and Goodman, N.A. (2015). DOA estimation by covariance matrix sparse reconstruction of coprime array. Proceedings of the 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), South Brisbane, QLD, Australia, 19\u201324 April 2015, IEEE.","DOI":"10.1109\/ICASSP.2015.7178395"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Zhou, C., Shi, Z., Gu, Y., and Zhang, Y.D. (2018). Coarray interpolation-based coprime array DOA estimation via covariance matrix reconstruction. Proceedings of the 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Calgary, AB, Canada, 15\u201320 April 2018, IEEE.","DOI":"10.1109\/ICASSP.2018.8461721"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3272","DOI":"10.1109\/TSP.2019.2914879","article-title":"Sparse sampling for inverse problems with tensors","volume":"67","author":"Coutino","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Liu, C.L., and Vaidyanathan, P. (2015). Tensor MUSIC in multidimensional sparse arrays. Proceedings of the 2015 49th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA, 8\u201311 November 2015, IEEE.","DOI":"10.1109\/ACSSC.2015.7421458"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1109\/LCOMM.2020.3029322","article-title":"Improved coprime linear array configuration for moving platform in DOA estimation","volume":"25","author":"Yang","year":"2020","journal-title":"IEEE Commun. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"108061","DOI":"10.1016\/j.sigpro.2021.108061","article-title":"Efficient direction-of-arrival estimation via annihilating-based denoising with coprime array","volume":"184","author":"Pan","year":"2021","journal-title":"Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Pal, P., and Vaidyanathan, P.P. (2011). Coprime sampling and the MUSIC algorithm. Proceedings of the 2011 Digital Signal Processing and Signal Processing Education Meeting (DSP\/SPE), Sedona, AZ, USA, 4\u201311 January 2011, IEEE.","DOI":"10.1109\/DSP-SPE.2011.5739227"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1109\/LSP.2020.2982769","article-title":"Direction-of-arrival estimation of coherent signals via coprime array interpolation","volume":"27","author":"Zheng","year":"2020","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Liu, C.L., Vaidyanathan, P., and Pal, P. (2016). Coprime coarray interpolation for DOA estimation via nuclear norm minimization. Proceedings of the 2016 IEEE International Symposium on Circuits and Systems (ISCAS), Montreal, QC, Canada, 22\u201325 May 2016, IEEE.","DOI":"10.1109\/ISCAS.2016.7539135"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"8223","DOI":"10.1109\/TVT.2017.2695226","article-title":"A Toeplitz covariance matrix reconstruction approach for direction-of-arrival estimation","volume":"66","author":"Wu","year":"2017","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"5956","DOI":"10.1109\/TSP.2018.2872012","article-title":"Direction-of-arrival estimation for coprime array via virtual array interpolation","volume":"66","author":"Zhou","year":"2018","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1109\/LSP.2020.3044019","article-title":"Hole-free coprime array for DOA estimation: Augmented uniform co-array","volume":"28","author":"Ma","year":"2020","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2621","DOI":"10.1109\/TVT.2023.3317864","article-title":"Multi-layer filled coprime arrays for DOA estimation with extended hole-free coarray","volume":"73","author":"Lai","year":"2023","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1825","DOI":"10.1109\/LSP.2025.3563118","article-title":"Super Fragmented Coprime Arrays for DOA Estimation","volume":"32","author":"Shaikh","year":"2025","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"109273","DOI":"10.1016\/j.sigpro.2023.109273","article-title":"Fragmented coprime arrays with optimal inter subarray spacing for DOA estimation: Increased DOF and reduced mutual coupling","volume":"215","author":"Lai","year":"2024","journal-title":"Signal Process."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2509","DOI":"10.1109\/LSP.2025.3577927","article-title":"Lengthened Coprime Arrays with Hole-free Coarrays and Reduced Mutual Coupling","volume":"32","author":"Yan","year":"2025","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1109\/TIT.2005.862083","article-title":"Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information","volume":"52","author":"Romberg","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"574","DOI":"10.1109\/7.144583","article-title":"Direction finding using spatial smoothing with interpolated arrays","volume":"28","author":"Friedlander","year":"2002","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"165","DOI":"10.2528\/PIER12010802","article-title":"On adaptive beamforming for coherent interference suppression via virtual antenna array","volume":"125","author":"Li","year":"2012","journal-title":"Prog. Electromagn. Res."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"588","DOI":"10.13164\/re.2017.0588","article-title":"Doa estimation of uncorrelated, partly correlated and coherent signals using alternating oblique projection","volume":"26","author":"Hou","year":"2017","journal-title":"Radioengineering"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1843","DOI":"10.1109\/TSP.2002.800420","article-title":"Maximum-likelihood source localization and unknown sensor location estimation for wideband signals in the near-field","volume":"50","author":"Chen","year":"2002","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_38","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":"2016","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_39","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_40","doi-asserted-by":"crossref","first-page":"110570","DOI":"10.1016\/j.apacoust.2025.110570","article-title":"High gain beamforming method in low-frequency band using virtual array migration","volume":"232","author":"Hu","year":"2025","journal-title":"Appl. Acoust."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Chavali, V., Wage, K.E., and Buck, J.R. (2014). Coprime processing for the Elba Island sonar data set. Proceedings of the 2014 48th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA, 2\u20135 November 2014, IEEE.","DOI":"10.1109\/ACSSC.2014.7094791"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"3262","DOI":"10.1121\/1.5064458","article-title":"Multiplicative and min processing of experimental passive sonar data from thinned arrays","volume":"144","author":"Chavali","year":"2018","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_43","unstructured":"Chavali, V. (2017). Coprime and Nested Array Processing of the Elba Island Sonar Data Set. [Master\u2019s Thesis, University of Mumbai]."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"109329","DOI":"10.1016\/j.apacoust.2023.109329","article-title":"Improving the beamforming performance of a vector sensor line array with a coprime array configuration","volume":"207","author":"Chen","year":"2023","journal-title":"Appl. Acoust."},{"key":"ref_45","unstructured":"Chan, Y. (2012). Underwater Acoustic Data Processing, Springer Science & Business Media."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3426","DOI":"10.1121\/10.0011470","article-title":"Robust high-resolution direction-of-arrival estimation method using DenseBlock-based U-net","volume":"151","author":"Sun","year":"2022","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"107292","DOI":"10.1016\/j.apacoust.2020.107292","article-title":"Improving the resolution of underwater acoustic image measurement by deconvolution","volume":"165","author":"Sun","year":"2020","journal-title":"Appl. Acoust."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"107986","DOI":"10.1016\/j.apacoust.2021.107986","article-title":"High-resolution and low-sidelobe forward-look sonar imaging using deconvolution","volume":"178","author":"Liu","year":"2021","journal-title":"Appl. Acoust."},{"key":"ref_49","unstructured":"Zhang, H., Gao, Z., and Fu, H. (2013). A high resolution random linear sonar array based MUSIC method for underwater DOA estimation. Proceedings of the 32nd Chinese Control Conference, Xi\u2019an, China, 26\u201328 July 2013, IEEE."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Van Trees, H.L. (2002). Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, John Wiley & Sons.","DOI":"10.1002\/0471221104"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"3592","DOI":"10.1109\/TSP.2011.2135348","article-title":"Theory of sparse coprime sensing in multiple dimensions","volume":"59","author":"Vaidyanathan","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"4444","DOI":"10.1109\/TSP.2024.3461720","article-title":"Weight-constrained sparse arrays for direction of arrival estimation under high mutual coupling","volume":"72","author":"Kulkarni","year":"2024","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"2986","DOI":"10.1109\/TIT.2014.2310482","article-title":"Multipath matching pursuit","volume":"60","author":"Kwon","year":"2014","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Zhang, J., Chen, L., Boufounos, P.T., and Gu, Y. (2014). On the theoretical analysis of cross validation in compressive sensing. Proceedings of the 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Florence, Italy, 4\u20139 May 2014, IEEE.","DOI":"10.1109\/ICASSP.2014.6854225"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/19\/2\/125\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T05:11:41Z","timestamp":1770873101000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/19\/2\/125"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,4]]},"references-count":54,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,2]]}},"alternative-id":["a19020125"],"URL":"https:\/\/doi.org\/10.3390\/a19020125","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,4]]}}}