{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T22:45:56Z","timestamp":1772232356549,"version":"3.50.1"},"reference-count":30,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2012,9,12]],"date-time":"2012-09-12T00:00:00Z","timestamp":1347408000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>To overcome the performance degradation in the presence of steering vector mismatches, strict restrictions on the number of available snapshots, and numerous interferences, a novel beamforming approach based on nonlinear least-square support vector regression machine (LS-SVR) is derived in this paper. In this approach, the conventional linearly constrained minimum variance cost function used by minimum variance distortionless response (MVDR) beamformer is replaced by a squared-loss function to increase robustness in complex scenarios and provide additional control over the sidelobe level. Gaussian kernels are also used to obtain better generalization capacity. This novel approach has two highlights, one is a recursive regression procedure to estimate the weight vectors on real-time, the other is a sparse model with novelty criterion to reduce the final size of the beamformer. The analysis and simulation tests show that the proposed approach offers better noise suppression capability and achieve near optimal signal-to-interference-and-noise ratio (SINR) with a low computational burden, as compared to other recently proposed robust beamforming techniques.<\/jats:p>","DOI":"10.3390\/s120912424","type":"journal-article","created":{"date-parts":[[2012,9,12]],"date-time":"2012-09-12T12:11:41Z","timestamp":1347451901000},"page":"12424-12436","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["A Nonlinear Adaptive Beamforming Algorithm Based on Least Squares Support Vector Regression"],"prefix":"10.3390","volume":"12","author":[{"given":"Lutao","family":"Wang","sequence":"first","affiliation":[{"name":"School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gang","family":"Jin","sequence":"additional","affiliation":[{"name":"China Aerodynamics Research & Development Center, Mianyang 621000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhengzhou","family":"Li","sequence":"additional","affiliation":[{"name":"School of Communication, Chongqing University, Chongqing 400044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongbin","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,9,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Van Trees, H.L. (2002). Optimum Array Processing, John Wiley & Sons, Inc.","DOI":"10.1002\/0471221104"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1109\/LSP.2003.817852","article-title":"Adaptive beamforming with sidelobe control: A second-order cone programming approach","volume":"10","author":"Liu","year":"2003","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2407","DOI":"10.1109\/TSP.2004.831998","article-title":"Doubly constrained robust Capon beamformer","volume":"52","author":"Li","year":"2004","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Li, J., and Stoica, P. (2006). Robust Adaptive Beamforming, John Wiley & Sons, Inc.","DOI":"10.1002\/0471733482"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1573","DOI":"10.1016\/j.sigpro.2009.10.027","article-title":"Robust capon beamformer under norm constraint","volume":"90","author":"Liu","year":"2010","journal-title":"Signal Process"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1702","DOI":"10.1109\/TSP.2003.812831","article-title":"On robust capon beamforming and diagonal loading","volume":"51","author":"Li","year":"2003","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1684","DOI":"10.1109\/TSP.2005.845436","article-title":"Robust minimum variance beamforming","volume":"53","author":"Lorenz","year":"2005","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_8","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"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"733","DOI":"10.1109\/LSP.2008.2001115","article-title":"Robust adaptive beamforming using sequential quadratic programming: An iterative solution to the mismatch problem","volume":"15","author":"Hassanien","year":"2008","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4483","DOI":"10.1109\/TSP.2011.2157500","article-title":"Adaptive uncertainty based iterative robust capon beamformer using steering vector mismatch estimation","volume":"59","author":"Lie","year":"2011","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_11","unstructured":"Landau, L., de, Lamare, R.C., and Haardt, M. (June, January 28\u2013). Robust Adaptive Beamforming Algorithms Using Low-Complexity Mismatch Estimation. Jachranka, Poland."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2243","DOI":"10.1016\/j.sigpro.2005.10.005","article-title":"A robust minimum variance beamformer with new constraint on uncertainty of steering vector","volume":"86","author":"Yu","year":"2006","journal-title":"Signal Process"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2198","DOI":"10.1109\/TAP.2009.2021971","article-title":"A robust adaptive beamforming framework with beampattern shaping constraints","volume":"57","author":"Nai","year":"2009","journal-title":"IEEE Trans. Antennas Propagat"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2615","DOI":"10.1109\/TSP.2009.2017004","article-title":"Robust adaptive beamformers based on worst-case optimization and constraints on magnitude response","volume":"57","author":"Yu","year":"2009","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"642","DOI":"10.1109\/TAP.2007.891550","article-title":"Kernel antenna array processing","volume":"55","author":"Christodoulou","year":"2007","journal-title":"IEEE Trans. Antennas Propagat"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1109\/8.135474","article-title":"A neural network approach to MVDR beamforming problem","volume":"40","author":"Chang","year":"1992","journal-title":"IEEE Trans. Antennas Propagat"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Vapnik, V.N. (1995). The Nature of Statistical Learning Theory, Springer-Verlag Inc.","DOI":"10.1007\/978-1-4757-2440-0"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"574","DOI":"10.1109\/TSP.2006.885720","article-title":"Roubust array beamforming with sidelobe control using support vector machines","volume":"55","author":"Gaudes","year":"2007","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1109\/LAWP.2005.860196","article-title":"Beamforming using support vector machines","volume":"4","author":"Ramon","year":"2005","journal-title":"IEEE Antenn. Wirel. Propag. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/S0925-2312(01)00644-0","article-title":"Weighted least squares support vector machines: Robustness and sparse approximation","volume":"48","author":"Suykens","year":"2002","journal-title":"Neurocomputing"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1109\/TNN.2004.841785","article-title":"An improved conjugate gradient scheme to the solution of least squares svm","volume":"16","author":"Chu","year":"2005","journal-title":"IEEE Trans. Neural Networks"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3264","DOI":"10.1016\/j.patcog.2008.10.023","article-title":"Model selection for LSSVM application to handwriting recognition","volume":"42","author":"Adankon","year":"2009","journal-title":"Pattern Recognit"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/j.neucom.2004.04.013","article-title":"Subset based least squares subspace regression in RKHS","volume":"63","author":"Hoegaerts","year":"2005","journal-title":"Neurocomputting"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1109\/TNN.2006.889500","article-title":"Fast sparse approximation for least square support vector machine","volume":"18","author":"Jiao","year":"2007","journal-title":"IEEE Trans. Neural Networks"},{"key":"ref_25","unstructured":"Vaerenbergh, S.V., Via, J., and Santamaria, I. (2006, January 14\u201319). A Sliding-Window Kernel RLS Algorithm and Its Application to Nonlinear Channel Indentification. Toulouse, France."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2781","DOI":"10.1109\/TSP.2008.917376","article-title":"On line classification using kernels and projection based adaptive algorithm","volume":"56","author":"Slavakis","year":"2008","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1162\/neco.1991.3.2.213","article-title":"A resource allocating network for function interpolation","volume":"3","author":"Platt","year":"1991","journal-title":"Neural Comput."},{"key":"ref_28","unstructured":"Liu, W.F., Jose, C.P., and Simon, H. (2010). Kernel Adaptive Filtering, John Wiley & Sons, Inc."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2453","DOI":"10.1109\/78.482097","article-title":"Generalized eigenspace-based beamformers","volume":"43","author":"Yu","year":"1995","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.sigpro.2007.07.003","article-title":"Automatic robust adaptive beamforming via ridge regression","volume":"88","author":"Selen","year":"2008","journal-title":"Signal Process"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/12\/9\/12424\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:52:19Z","timestamp":1760219539000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/12\/9\/12424"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,9,12]]},"references-count":30,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2012,9]]}},"alternative-id":["s120912424"],"URL":"https:\/\/doi.org\/10.3390\/s120912424","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,9,12]]}}}