{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T02:04:36Z","timestamp":1771466676262,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,3,17]],"date-time":"2023-03-17T00:00:00Z","timestamp":1679011200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["42130202"],"award-info":[{"award-number":["42130202"]}]},{"name":"National Natural Science Foundation of China","award":["41931073"],"award-info":[{"award-number":["41931073"]}]},{"name":"National Natural Science Foundation of China","award":["183311KYSB20200003"],"award-info":[{"award-number":["183311KYSB20200003"]}]},{"name":"National Natural Science Foundation of China","award":["XDA15350201"],"award-info":[{"award-number":["XDA15350201"]}]},{"name":"International Partnership Program of Chinese Academy of Sciences","award":["42130202"],"award-info":[{"award-number":["42130202"]}]},{"name":"International Partnership Program of Chinese Academy of Sciences","award":["41931073"],"award-info":[{"award-number":["41931073"]}]},{"name":"International Partnership Program of Chinese Academy of Sciences","award":["183311KYSB20200003"],"award-info":[{"award-number":["183311KYSB20200003"]}]},{"name":"International Partnership Program of Chinese Academy of Sciences","award":["XDA15350201"],"award-info":[{"award-number":["XDA15350201"]}]},{"name":"Strategic Pioneer Program on Space Science, the Chinese Academy of Sciences","award":["42130202"],"award-info":[{"award-number":["42130202"]}]},{"name":"Strategic Pioneer Program on Space Science, the Chinese Academy of Sciences","award":["41931073"],"award-info":[{"award-number":["41931073"]}]},{"name":"Strategic Pioneer Program on Space Science, the Chinese Academy of Sciences","award":["183311KYSB20200003"],"award-info":[{"award-number":["183311KYSB20200003"]}]},{"name":"Strategic Pioneer Program on Space Science, the Chinese Academy of Sciences","award":["XDA15350201"],"award-info":[{"award-number":["XDA15350201"]}]},{"name":"Pandeng Program of National Space Science Center, Chinese Academy of Sciences","award":["42130202"],"award-info":[{"award-number":["42130202"]}]},{"name":"Pandeng Program of National Space Science Center, Chinese Academy of Sciences","award":["41931073"],"award-info":[{"award-number":["41931073"]}]},{"name":"Pandeng Program of National Space Science Center, Chinese Academy of Sciences","award":["183311KYSB20200003"],"award-info":[{"award-number":["183311KYSB20200003"]}]},{"name":"Pandeng Program of National Space Science Center, Chinese Academy of Sciences","award":["XDA15350201"],"award-info":[{"award-number":["XDA15350201"]}]},{"name":"Ground-Based Space Environment Monitoring Network (Meridian Project II)","award":["42130202"],"award-info":[{"award-number":["42130202"]}]},{"name":"Ground-Based Space Environment Monitoring Network (Meridian Project II)","award":["41931073"],"award-info":[{"award-number":["41931073"]}]},{"name":"Ground-Based Space Environment Monitoring Network (Meridian Project II)","award":["183311KYSB20200003"],"award-info":[{"award-number":["183311KYSB20200003"]}]},{"name":"Ground-Based Space Environment Monitoring Network (Meridian Project II)","award":["XDA15350201"],"award-info":[{"award-number":["XDA15350201"]}]},{"name":"Specialized Research Fund for State Key Laboratories","award":["42130202"],"award-info":[{"award-number":["42130202"]}]},{"name":"Specialized Research Fund for State Key Laboratories","award":["41931073"],"award-info":[{"award-number":["41931073"]}]},{"name":"Specialized Research Fund for State Key Laboratories","award":["183311KYSB20200003"],"award-info":[{"award-number":["183311KYSB20200003"]}]},{"name":"Specialized Research Fund for State Key Laboratories","award":["XDA15350201"],"award-info":[{"award-number":["XDA15350201"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Adaptive array processing technology for a phased array radar is usually based on the assumption of a stationary environment; however, in real-world scenarios, nonstationary interference and noise deteriorate the performance of the traditional gradient descent algorithm, in which the learning rate of the tap weights is fixed, leading to errors in the beam pattern and a reduced output signal-to-noise ratio (SNR). In this paper, we use the incremental delta-bar-delta (IDBD) algorithm, which has been widely used for system identification problems in nonstationary environments, to control the time-varying learning rates of the tap weights. The designed iteration formula for the learning rate ensures that the tap weights adaptively track the Wiener solution. The results of numerical simulations show that in a nonstationary environment, the traditional gradient descent algorithm with a fixed learning rate has a distorted beam pattern and reduced output SNR; however, the IDBD-based beamforming algorithm, in which a secondary control mechanism is used to adaptively update the learning rates, showed a similar beam pattern and output SNR to a traditional beamformer in a Gaussian white noise background; that is, the main beam and null satisfied the pointing constraints, and the optimal output SNR was obtained. Although the proposed algorithm contains a matrix inversion operation, which has considerable computational complexity, this operation could be replaced by the Levinson\u2013Durbin iteration due to the Toeplitz characteristic of the matrix; therefore, the computational complexity could be decreased to O(n), so additional computing resources are not required. Moreover, according to some intuitive interpretations, the reliability and stability of the algorithm are guaranteed.<\/jats:p>","DOI":"10.3390\/s23063211","type":"journal-article","created":{"date-parts":[[2023,3,17]],"date-time":"2023-03-17T05:06:01Z","timestamp":1679029561000},"page":"3211","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["IDBD-Based Beamforming Algorithm for Improving the Performance of Phased Array Radar in Nonstationary Environments"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2558-3666","authenticated-orcid":false,"given":"Shihan","family":"Wang","sequence":"first","affiliation":[{"name":"National Space Science Center, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"School of Earth and Planetary Sciences, University of Chinese Academy of Sciences, Beijing 100049, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9790-2358","authenticated-orcid":false,"given":"Tao","family":"Chen","sequence":"additional","affiliation":[{"name":"National Space Science Center, Chinese Academy of Sciences, Beijing 100190, China"}]},{"given":"Hongjian","family":"Wang","sequence":"additional","affiliation":[{"name":"National Space Science Center, Chinese Academy of Sciences, Beijing 100190, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1002\/j.1538-7305.1943.tb01306.x","article-title":"A mathematical theory of linear arrays","volume":"22","author":"Schelkunoff","year":"1943","journal-title":"Bell Syst. Tech. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1109\/TAP.1964.1138222","article-title":"Antenna synthesis by weighted Fourier coefficients","volume":"12","author":"Jaeckle","year":"1964","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_3","unstructured":"Liao, G.S., Tao, H.H., and Zeng, C. (2017). Digital Beamforming Technique of Phased Array Radar, National Defence Industry Press."},{"key":"ref_4","unstructured":"Mailloux, R.J. (2008). Phased Array Antenna Handbook, Publishing House of Electronics Industry. [2nd ed.]."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4517","DOI":"10.1109\/TAP.2019.2905723","article-title":"Fast Beamforming with Fault-Tolerance in Massive Phased-Arrays using Intelligent Learning Control","volume":"67","author":"Hamici","year":"2019","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Iskander, M.F., Kim, W., Bell, J., Celik, N., Yun, Z., and Youn, H.S. (2010, January 9\u201311). Antenna arrays technologies for advanced wireless systems. Proceedings of the IEEE International Conference on Microwaves, Tel Aviv, Israel.","DOI":"10.1109\/COMCAS.2009.5385955"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1109\/LGRS.2016.2542104","article-title":"Improved Phase-Encoding Calibration for Active Phased-Array Antennas of SAR","volume":"13","author":"Gong","year":"2017","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1436","DOI":"10.1109\/LGRS.2015.2406815","article-title":"On the Pulse Extension Loss in Digital Beamforming SAR","volume":"12","author":"Younis","year":"2015","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1819","DOI":"10.1109\/TAP.2004.831314","article-title":"Fast beamforming of electronically steerable parasitic array radiator antennas: Theory and experiment","volume":"52","author":"Sun","year":"2004","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1109\/TAP.1970.1139620","article-title":"Directivity of basic linear arrays","volume":"18","author":"Bach","year":"2003","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1109\/TAP.2014.2382637","article-title":"Waveform Optimization for Transmit Beamforming With MIMO Radar Antenna Arrays","volume":"63","author":"Guo","year":"2015","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_12","unstructured":"Haykin, S. (2002). Adaptive Filter Theory, Publishing House of Electronics Industry. [5th ed.]."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1109\/LSP.2018.2880086","article-title":"Time-Varying Autoregressive Model for Characterizing Nonstationary Processes","volume":"26","author":"Kuhn","year":"2019","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"762","DOI":"10.1109\/LSP.2016.2553042","article-title":"A Novel Data-Driven Learning Method for Radar Target Detection in Nonstationary Environments","volume":"23","author":"Akcakaya","year":"2016","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"80985","DOI":"10.1109\/ACCESS.2019.2923680","article-title":"Fast Nonstationary Noise Tracking Based on Log-Spectral Power MMSE Estimator and Temporal Recursive Averaging","volume":"7","author":"Zhang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Mahmood, A.R., Sutton, R.S., Degris, T., and Pilarski, P.M. (2012, January 25\u201330). Tuning-free step-size adaptation. Proceedings of the 2012 IEEE International Conference on Acoustics, Kyoto, Japan.","DOI":"10.1109\/ICASSP.2012.6288330"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/0893-6080(88)90003-2","article-title":"Increased Rates of Convergence through Learning Rate Adaptation","volume":"1","author":"Jacobs","year":"1988","journal-title":"Neural Netw."},{"key":"ref_18","unstructured":"Sutton, R.S. (1992). Adapting bias by gradient descent: An incremental version of delta-bar-delta. Proc. AAII-92, 171\u2013176."},{"key":"ref_19","first-page":"75","article-title":"Adaptive beamforming in colored noise environment","volume":"26","author":"Zhang","year":"1998","journal-title":"Acta Electron. Sin."},{"key":"ref_20","unstructured":"Tsakalides, P., and Nikias, C.L. (1996, January 9). Robust adaptive beamforming in alpha-stable noise environments. Proceedings of the 1996 IEEE International Conference on Acoustics, Speech, and Signal Processing Conference Proceedings, Atlanta, GA, USA."},{"key":"ref_21","unstructured":"Richmond, C.D. (1996, January 24\u201326). Adaptive array processing in non-Gaussian environments. Proceedings of the IEEE Signal Processing Workshop on Statistical Signal & Array Processing, Corfu, Greece."},{"key":"ref_22","unstructured":"Cox, H. (2003, January 3\u20136). Adaptive beamforming in non- stationary environments. Presented at the Conference on Signals, Systems & Computers, Pacific Grove, CA, USA."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"926","DOI":"10.1109\/PROC.1972.8817","article-title":"An Algorithm for Linearly Constrained Adaptive Array Processing","volume":"60","author":"Frost","year":"1972","journal-title":"Proc. IEEE"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Wang, S.H., Li, L., Chen, T., Ti, S., Cai, C.L., Li, W., and Luo, J. (2022). Statistical Characterization of the Magnetic Field in Space during Magnetic Storms. Atmosphere, 13.","DOI":"10.3390\/atmos13101578"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/6\/3211\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:57:29Z","timestamp":1760122649000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/6\/3211"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,17]]},"references-count":24,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["s23063211"],"URL":"https:\/\/doi.org\/10.3390\/s23063211","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,17]]}}}