{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T23:23:35Z","timestamp":1778628215172,"version":"3.51.4"},"reference-count":35,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,3,27]],"date-time":"2020-03-27T00:00:00Z","timestamp":1585267200000},"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":["61501471"],"award-info":[{"award-number":["61501471"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>When jammers move rapidly or an antenna platform travels at high speed, interference signals may move out of the null width in the array beampattern. Consequently, the interference suppression performance can be significantly degraded. To solve this problem, both the null broadening technique and robust adaptive beamforming are considered in this paper. A novel null broadening beamforming method based on reconstruction of the interference-plus-noise covariance (INC) matrix is proposed, in order to broaden the null width and offset the motion of the interfering signals. In the moving case, a single interference signal can have multiple directions of arrival, which is equivalent to the existence of multiple interference sources. In the reconstruction of the INC matrix, several virtual interference sources are set up around each of the actual jammers, such that the nulls can be broadened. Based on the reconstructed INC and signal-plus-noise covariance (SNC) matrices, the steering vector of the desired signal can be obtained by solving a new convex optimization problem. Simulation results show that the proposed beamformer can effectively broaden the null width and deepen the null depth, and its performance in interference cancellation is robust against fast-moving jammers or array platform motion. Furthermore, the null depth can be controlled by adjusting the power parameters in the reconstruction process and, if the direction of interference motion is known, the virtual interference sources can be set to achieve better performance.<\/jats:p>","DOI":"10.3390\/s20071865","type":"journal-article","created":{"date-parts":[[2020,4,1]],"date-time":"2020-04-01T03:44:13Z","timestamp":1585712653000},"page":"1865","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Robust Null Broadening Beamforming Based on Covariance Matrix Reconstruction via Virtual Interference Sources"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0632-5805","authenticated-orcid":false,"given":"Jian","family":"Yang","sequence":"first","affiliation":[{"name":"School of Electronic Engineering, Xidian University, Xi\u2019an 710071, China"},{"name":"School of Engineering, Rocket Force University of Engineering, Xi\u2019an 710025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Engineering, Rocket Force University of Engineering, Xi\u2019an 710025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinxin","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Engineering, Rocket Force University of Engineering, Xi\u2019an 710025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guisheng","family":"Liao","sequence":"additional","affiliation":[{"name":"School of Electronic Engineering, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,27]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1109\/JSEN.2016.2637059","article-title":"Source estimation using coprime array: A sparse reconstruction perspective","volume":"17","author":"Shi","year":"2017","journal-title":"IEEE Sens. J."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Singh, R.-K., Basu, A., and Koul, S.-K. (2016, January 5\u20139). Efficient Null Broadening and Steering Using Slot Antenna Array for Radar Applications. Proceedings of the 2016 Asia-Pacific Microwave Conference (APMC), New Delhi, India.","DOI":"10.1109\/APMC.2016.7931260"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"8495","DOI":"10.1109\/TVT.2018.2849646","article-title":"Covariance Matrix Reconstruction With Interference Steering Vector and Power Estimation for Robust Adaptive Beamforming","volume":"67","author":"Zheng","year":"2018","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3881","DOI":"10.1109\/TSP.2012.2194289","article-title":"Robust adaptive beamforming based on interference covariance matrix reconstruction and steering vector estimation","volume":"60","author":"Gu","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1016\/j.sigpro.2013.10.009","article-title":"Robust adaptive beamforming based on interference covariance matrix sparse reconstruction","volume":"96","author":"Gu","year":"2014","journal-title":"Signal Process."},{"key":"ref_7","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":"2018","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1109\/LSP.2015.2504954","article-title":"Interference-plus-noise covariance matrix reconstruction via spatial power spectrum sampling for robust adaptive beamforming","volume":"23","author":"Zhang","year":"2016","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1636","DOI":"10.1109\/LCOMM.2015.2455503","article-title":"Robust adaptive beamforming based on steering vector estimation and covariance matrix reconstruction","volume":"19","author":"Shen","year":"2015","journal-title":"IEEE Commun. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ma, Y.-X., Lu, D., Wang, W.-Y., Wang, L., and Wu, R.-B. (2014). A high-dynamic null-widen GPS anti-jamming algorithm based on statistical model of the changing interference DOA. China Satellite Navigation Conference (CSNC) 2014 Proceedings: Volume I, Springer.","DOI":"10.1007\/978-3-642-54737-9_60"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1049\/el:19950537","article-title":"Covariance matrix augmentation to produce adaptive array pattern troughs","volume":"31","author":"Mailloux","year":"1995","journal-title":"Electron. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1109\/78.752596","article-title":"Theory and application of covariance matrix tapers for robust adaptive beamforming","volume":"47","author":"Guerci","year":"1999","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1796","DOI":"10.1109\/78.845937","article-title":"Comment on theory and application of covariance matrix tapers for robust adaptive beam forming","volume":"48","author":"Zatman","year":"2000","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Rubsamen, M., Gerlach, C., and Gershman, A.-B. (April, January 31). Low-rank covariance matrix tapering for robust adaptive beamforming. Proceedings of the 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, Las Vegas, NV, USA.","DOI":"10.1109\/ICASSP.2008.4518114"},{"key":"ref_15","unstructured":"Zhou, Z., Zhang, Z., and Lai, S.-M. (2018, January 18\u201320). An Analytical Approach to Null Broadening. Proceedings of the 2018 IEEE International Conference on Computer and Communication Engineering Technology (CCET), Beijing, China."},{"key":"ref_16","first-page":"11","article-title":"New directions in communications","volume":"13","author":"Turner","year":"1995","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Qian, J., He, Z., Zhang, Y., and Wu, B. (2016, January 6\u201310). Null Broadening Adaptive Beamforming Based on Semidefinite Programming. Proceedings of the 2016 IEEE 13th International Conference on Signal Processing (ICSP), Chengdu, China.","DOI":"10.1109\/ICSP.2016.7878097"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13634-016-0440-1","article-title":"Null Broadening Adaptive Beamforming Based on covariance matrix reconstruction and similarity constraint","volume":"2017","author":"Qian","year":"2017","journal-title":"EURASIP J. Adv. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1109\/JOE.2003.814055","article-title":"Null broadening with snapshot-deficient covariance matrices in passive sonar","volume":"28","author":"Song","year":"2003","journal-title":"IEEE J. Ocean. Eng."},{"key":"ref_20","first-page":"2882","article-title":"A new algorithm for null broadening beamforming","volume":"36","author":"Li","year":"2014","journal-title":"J. Electron. Inf. Technol."},{"key":"ref_21","first-page":"565","article-title":"Robust Sidelobes Cancellation Algorithm with Null Broadening","volume":"38","author":"Liu","year":"2016","journal-title":"J. Electron. Inf. Technol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1049\/el.2018.1228","article-title":"Adaptive null broadening method in wideband beamforming for rapidly moving interference suppression","volume":"54","author":"Yang","year":"2018","journal-title":"Electron. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Li, W., and Yang, B. (2017, January 26\u201330). An Improved Null Broadening Beamforming Method based on Covariance Matrix Reconstruction. Proceedings of the 2017 International Applied Computational Electromagnetics Society Symposium\u2014Italy (ACES), Firenze, Italy.","DOI":"10.23919\/ROPACES.2017.7916028"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ors, B., and Suleesathira, R. (2019, January 27\u201329). First and Second Order Iterative Null Broadening Beamforming. Proceedings of the 2019 3rd International Conference on Imaging, Signal Processing and Communication (ICISPC), Singapore.","DOI":"10.1109\/ICISPC.2019.8935850"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2249","DOI":"10.1109\/LAWP.2019.2925419","article-title":"Data-Based Model for Wide Nulling Problem in Adaptive Digial Beamforming Antenna Array","volume":"18","author":"Xiao","year":"2019","journal-title":"IEEE Antennas Wirel. Propag. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"11213","DOI":"10.1109\/JSEN.2019.2936681","article-title":"Robust Adaptive Beamforming for Sidelobe Canceller with Null Widening","volume":"19","author":"Liu","year":"2019","journal-title":"IEEE Sens. J."},{"key":"ref_27","first-page":"163","article-title":"A null broadening beamforming against array calibration errors","volume":"39","author":"Zhao","year":"2018","journal-title":"J. Harbin Eng. Univ."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"5380","DOI":"10.1109\/TAP.2019.2916607","article-title":"Robust Wideband Adaptive Beamforming with Null Broadening and Constant Beamwidth","volume":"67","author":"Yang","year":"2019","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"486","DOI":"10.1049\/iet-spr.2018.5264","article-title":"Robust adaptive beamforming for fast moving interference based on the covariance matrix reconstruction","volume":"13","author":"Mohammadzadeh","year":"2019","journal-title":"IET Signal Proc."},{"key":"ref_30","first-page":"94","article-title":"A null widening anti-jamming algorithm for GNSS receivers","volume":"51","author":"Wang","year":"2019","journal-title":"J. Harbin Ins. Technol."},{"key":"ref_31","first-page":"1","article-title":"High Dynamic GNSS Anti-jamming Algorithms Based on Nulling Widening and Deepening","volume":"38","author":"Cong","year":"2020","journal-title":"J. Jilin Univ."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1408","DOI":"10.1109\/PROC.1969.7278","article-title":"High-resolution frequency-wavenumber spectrum analysis","volume":"57","author":"Capon","year":"1969","journal-title":"Proc. IEEE"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Gass, S.I., and Harris, C.M. (2001). Karush-Kuhn-Tucker (KKT) conditions. Encyclopedia of Operations Research and Management Science, Springer.","DOI":"10.1007\/1-4020-0611-X_495"},{"key":"ref_34","unstructured":"Grant, M., Boyd, S., and Ye, S. (2019, May 16). CVX: Matlab Software for Disciplined Convex Programming, Version 2.1. Mar. Available online: http:\/\/cvxr.com\/cvx."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1109\/TSP.2002.806865","article-title":"Robust adaptive beamforming using worst-case performance optimization: A solution to the signal mismatch problem","volume":"51","author":"Vorobyov","year":"2003","journal-title":"IEEE Trans. Signal Process."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/7\/1865\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:12:24Z","timestamp":1760173944000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/7\/1865"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,27]]},"references-count":35,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["s20071865"],"URL":"https:\/\/doi.org\/10.3390\/s20071865","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3,27]]}}}