{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,11]],"date-time":"2026-08-11T04:31:02Z","timestamp":1786422662307,"version":"build-2736575974"},"reference-count":30,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2026,8,8]],"date-time":"2026-08-08T00:00:00Z","timestamp":1786147200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"crossref","award":["62271192"],"award-info":[{"award-number":["62271192"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Direction-of-arrival (DOA) estimation with coprime arrays can synthesize an enlarged virtual aperture from a limited number of physical sensors. However, coherent incident sources lead to rank deficiency of the source covariance matrix, while unknown nonuniform sensor noise mainly contaminates the zero-lag component of the difference-coarray covariance. These two effects jointly degrade conventional Coarray Root-MUSIC, Coarray ESPRIT, and interpolation-based virtual-array methods. To address this problem, this paper proposes a Toeplitz\u2013Hankel structured covariance reconstruction method for coherent-source DOA estimation with coprime arrays under unknown nonuniform noise. The method first performs redundancy-aware difference-coarray lag averaging. The zero-lag component is then suppressed during missing-lag interpolation to reduce the bias caused by sensor-dependent noise powers. A Toeplitz positive semidefinite projection is used to enforce covariance validity, and a relaxed Hankel truncated-singular-value-decomposition refinement is introduced to enhance the low-rank spectral structure of the reconstructed virtual covariance sequence. Finally, multi-scale forward\u2013backward spatial smoothing MUSIC is applied for coherent-source DOA estimation. Simulation results with a coprime array of M=4 and N=5 show that the proposed method provides more accurate and stable DOA estimates than Coarray Root-MUSIC, Coarray ESPRIT, and RV-TSI. Compared with CVX-based THSCR, the proposed method avoids semidefinite programming and nuclear-norm optimization and reduces the average runtime from approximately 11.2 s per trial to approximately 0.11 s per trial under the tested setting.<\/jats:p>","DOI":"10.3390\/s26165041","type":"journal-article","created":{"date-parts":[[2026,8,10]],"date-time":"2026-08-10T09:24:53Z","timestamp":1786353893000},"page":"5041","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Toeplitz\u2013Hankel Structured Covariance Reconstruction for DOA Estimation of Coherent Sources with Coprime Arrays Under Nonuniform Noise"],"prefix":"10.3390","volume":"26","author":[{"given":"Heng","family":"Zhao","sequence":"first","affiliation":[{"name":"Ocean College, Jiangsu University of Science and Technology, Zhenjiang 212100, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-6150-2155","authenticated-orcid":false,"given":"Ying","family":"Hu","sequence":"additional","affiliation":[{"name":"Ocean College, Jiangsu University of Science and Technology, Zhenjiang 212100, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zijing","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4488-8692","authenticated-orcid":false,"given":"Fei","family":"Zhang","sequence":"additional","affiliation":[{"name":"Ocean College, Jiangsu University of Science and Technology, Zhenjiang 212100, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,8]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"984","DOI":"10.1109\/29.32276","article-title":"ESPRIT\u2014Estimation of signal parameters via rotational invariance techniques","volume":"37","author":"Roy","year":"1989","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1939","DOI":"10.1109\/29.45540","article-title":"Performance analysis of root-MUSIC","volume":"37","author":"Rao","year":"1989","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1109\/79.526899","article-title":"Two decades of array signal processing research: The parametric approach","volume":"13","author":"Krim","year":"1996","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_5","unstructured":"Stoica, P., and Moses, R. (2005). Spectral Analysis of Signals, Pearson Prentice Hall."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Van Trees, H.L. (2002). Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, Wiley.","DOI":"10.1002\/0471221104"},{"key":"ref_7","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_8","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_9","doi-asserted-by":"crossref","unstructured":"Pal, P., and Vaidyanathan, P.P. (2011, January 4\u20137). Coprime sampling and the MUSIC algorithm. Proceedings of the IEEE Digital Signal Processing Workshop and IEEE Signal Processing Education Workshop, Sedona, AZ, USA.","DOI":"10.1109\/DSP-SPE.2011.5739227"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5565","DOI":"10.1109\/TSP.2014.2354316","article-title":"Direction of arrival estimation using co-prime arrays: A super resolution viewpoint","volume":"62","author":"Tan","year":"2014","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_11","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_12","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1109\/29.17496","article-title":"Forward\/backward spatial smoothing techniques for coherent signal identification","volume":"37","author":"Pillai","year":"1989","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hu, Y., Liu, Y., and Wang, X. (2017). DOA estimation of coherent signals on coprime arrays exploiting fourth-order cumulants. Sensors, 17.","DOI":"10.3390\/s17040682"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1109\/TASLP.2023.3334482","article-title":"Coherent signal DOA estimation with coprime array: Exploiting signal subspace reconstructing strategy","volume":"32","author":"Ma","year":"2024","journal-title":"IEEE\/ACM Trans. Audio Speech Lang. Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3471","DOI":"10.1109\/TAP.2025.3650631","article-title":"Measurement and calibration methods for full-polarization characteristics of radar targets at subterahertz frequencies","volume":"74","author":"Xu","year":"2026","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2458","DOI":"10.1109\/78.709534","article-title":"Positive-definite Toeplitz completion in DOA estimation for nonuniform linear antenna arrays. I. Fully augmentable arrays","volume":"46","author":"Abramovich","year":"1998","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1502","DOI":"10.1109\/78.765119","article-title":"Positive-definite Toeplitz completion in DOA estimation for nonuniform linear antenna arrays. II. Partially augmentable arrays","volume":"47","author":"Abramovich","year":"1999","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_18","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 Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1109\/TSP.2010.2090525","article-title":"SPICE: A sparse covariance-based estimation method for array processing","volume":"59","author":"Stoica","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2265","DOI":"10.1109\/LCOMM.2021.3075227","article-title":"Rank minimization-based Toeplitz reconstruction for DOA estimation using coprime array","volume":"25","author":"Liu","year":"2021","journal-title":"IEEE Commun. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Meng, Z., and Zhou, W. (2019). Direction-of-arrival estimation in coprime array using the ESPRIT-based method. Sensors, 19.","DOI":"10.3390\/s19030707"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Park, Y., and Gerstoft, P. (2021, January 6\u201311). Alternating projections gridless covariance-based estimation for DOA. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Toronto, ON, Canada.","DOI":"10.1109\/ICASSP39728.2021.9414972"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"574","DOI":"10.1109\/LSP.2026.3652124","article-title":"Covariance tensor decomposition for NLOS direction finding in RIS-aided bistatic MIMO radar","volume":"33","author":"Xie","year":"2026","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6003404","DOI":"10.1109\/LSENS.2025.3546092","article-title":"Localization of RFID tags through real-time angle-of-arrival estimation","volume":"9","author":"Florio","year":"2025","journal-title":"IEEE Sens. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"B\u00f6ller, S., B\u00f6dder, A., Riegler, J., and Grenter, T. (2024, January 18\u201320). SHF RFID based localization: Combining time-of-flight and angle-of-arrival. Proceedings of the 2024 IEEE International Conference on RFID Technology and Applications (RFID-TA), Daytona Beach, FL, USA.","DOI":"10.1109\/RFID-TA64374.2024.10965186"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"8001913","DOI":"10.1109\/TIM.2026.3662891","article-title":"Low-complexity angle-of-arrival estimation using 1-bit conversion","volume":"75","author":"Florio","year":"2026","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1111","DOI":"10.1109\/LSP.2026.3667810","article-title":"Real-valued DOA estimation of coprime array via Toeplitz construction-based spline interpolation","volume":"33","author":"Pan","year":"2026","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Zhao, H., Hu, Y., Zhang, Z., and Zhang, F. (2026). DOA estimation with coprime arrays using Toeplitz and Hankel-based structured covariance reconstruction. Electronics, 15.","DOI":"10.3390\/electronics15102118"},{"key":"ref_29","unstructured":"Texas Instruments (2020). AWR1243 Single-Chip 77- and 79-GHz FMCW Transceiver, Texas Instruments."},{"key":"ref_30","unstructured":"Texas Instruments (2020). AWR1243BOOST Evaluation Module, Texas Instruments."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/16\/5041\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,11]],"date-time":"2026-08-11T04:13:25Z","timestamp":1786421605000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/16\/5041"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,8]]},"references-count":30,"journal-issue":{"issue":"16","published-online":{"date-parts":[[2026,8]]}},"alternative-id":["s26165041"],"URL":"https:\/\/doi.org\/10.3390\/s26165041","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,8]]}}}