{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T10:25:48Z","timestamp":1760955948222,"version":"build-2065373602"},"reference-count":49,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T00:00:00Z","timestamp":1558483200000},"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":["61374027"],"award-info":[{"award-number":["61374027"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In this paper, an     l 1    -penalized maximum likelihood (ML) approach is developed for estimating the directions of arrival (DOAs) of source signals from the complex elliptically symmetric (CES) array outputs. This approach employs the     l 1    -norm penalty to exploit the sparsity of the gridded directions, and the CES distribution setting has a merit of robustness to the uncertainty of the distribution of array output. To solve the constructed non-convex penalized ML optimization for spatially either uniform or non-uniform sensor noise, two majorization-minimization (MM) algorithms based on different majorizing functions are developed. The computational complexities of the above two algorithms are analyzed. A modified Bayesian information criterion (BIC) is provided for selecting an appropriate penalty parameter. The effectiveness and superiority of the proposed methods in producing high DOA estimation accuracy are shown in numerical experiments.<\/jats:p>","DOI":"10.3390\/s19102356","type":"journal-article","created":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T03:22:03Z","timestamp":1558581723000},"page":"2356","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Direction of Arrival Estimation in Elliptical Models via Sparse Penalized Likelihood Approach"],"prefix":"10.3390","volume":"19","author":[{"given":"Chen","family":"Chen","sequence":"first","affiliation":[{"name":"College of Mathematics, Sichuan University, Chengdu 610064, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6203-3583","authenticated-orcid":false,"given":"Jie","family":"Zhou","sequence":"additional","affiliation":[{"name":"College of Mathematics, Sichuan University, Chengdu 610064, China"}]},{"given":"Mengjiao","family":"Tang","sequence":"additional","affiliation":[{"name":"Center for Information Engineering Science Research, School of Electronics and Information Engineering, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,22]]},"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-estimation 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":"1132","DOI":"10.1109\/29.57542","article-title":"Maximum likelihood methods for direction-of-arrival estimation","volume":"38","author":"Stoica","year":"1990","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/0165-1684(93)90048-F","article-title":"The root-MUSIC algorithm for direction finding with interpolated arrays","volume":"30","author":"Friedlander","year":"1993","journal-title":"Signal Process."},{"key":"ref_5","unstructured":"Chellappa, R., and Theodoridis, S. (2018). Sparse methods for direction-of-arrival estimation. Academic Press Library in Signal Processing, Volume 7: Array, Radar and Communications Engineering, Academic Press. Chapter 11."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1109\/78.558475","article-title":"Sparse signal reconstruction from limited data using FOCUSS: A re-weighted minimum norm algorithm","volume":"45","author":"Gorodnitsky","year":"1997","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_7","first-page":"211","article-title":"Sparse Bayesian learning and the relevance vector machine","volume":"1","author":"Tipping","year":"2001","journal-title":"J. Mach. Learn. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.1109\/TSP.2004.831016","article-title":"Sparse Bayesian learning for basis selection","volume":"52","author":"Wipf","year":"2004","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_9","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_10","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1109\/TIP.2009.2032894","article-title":"Bayesian compressive sensing using Laplace priors","volume":"19","author":"Babacan","year":"2010","journal-title":"IEEE Trans. Image Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4646","DOI":"10.1109\/TSP.2010.2050477","article-title":"Direction-of-arrival estimation using a mixed l2,0 norm approximation","volume":"58","author":"Hyder","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1109\/LSP.2012.2183592","article-title":"DOA estimation based on sparse signal recovery utilizing weighted l1-norm penalty","volume":"19","author":"Xu","year":"2012","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_13","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_14","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1109\/TSP.2010.2086452","article-title":"New method of sparse parameter estimation in separable models and its use for spectral analysis of irregularly sampled data","volume":"59","author":"Stoica","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1580","DOI":"10.1016\/j.sigpro.2011.11.010","article-title":"SPICE and LIKES: Two hyperparameter-free methods for sparse-parameter estimation","volume":"92","author":"Stoica","year":"2012","journal-title":"Signal Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.dsp.2014.06.010","article-title":"Weighted SPICE: A unifying approach for hyperparameter-free sparse estimation","volume":"33","author":"Stoica","year":"2014","journal-title":"Digit. Signal Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4489","DOI":"10.1109\/TSP.2011.2158425","article-title":"Direction-of-arrival estimation using a sparse representation of array covariance vectors","volume":"59","author":"Yin","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2767","DOI":"10.1109\/TSP.2013.2256903","article-title":"Sparse spatial spectral estimation: A covariance fitting algorithm, performance and regularization","volume":"61","author":"Zheng","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1710","DOI":"10.1109\/TAES.2013.6558014","article-title":"Array signal processing via sparsity-inducing representation of the array covariance matrix","volume":"49","author":"Liu","year":"2013","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.dsp.2014.02.013","article-title":"Covariance sparsity-aware DOA estimation for nonuniform noise","volume":"28","author":"He","year":"2014","journal-title":"Digit. Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Chen, T., Wu, H., and Zhao, Z. (2016). The real-valued sparse direction of arrival (DOA) estimation based on the Khatri-Rao product. Sensors, 16.","DOI":"10.3390\/s16050693"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.sigpro.2017.09.029","article-title":"Sparsity-aware DOA estimation of quasi-stationary signals using nested arrays","volume":"144","author":"Wang","year":"2018","journal-title":"Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3609","DOI":"10.1109\/TSP.2011.2140106","article-title":"Sparse variational Bayesian SAGE algorithm with application to the estimation of multipath wireless channels","volume":"59","author":"Shutin","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3809","DOI":"10.1109\/TSP.2012.2193392","article-title":"Compressed sensing of complex sinusoids: An approach based on dictionary refinement","volume":"60","author":"Hu","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4658","DOI":"10.1109\/TSP.2012.2201152","article-title":"Robustly stable signal recovery in compressed sensing with structured matrix perturbation","volume":"60","author":"Yang","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1109\/TSP.2012.2222378","article-title":"Off-grid direction of arrival estimation using sparse Bayesian inference","volume":"61","author":"Yang","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Fan, Y., Wang, J., Du, R., and Lv, G. (2018). Sparse method for direction of arrival estimation using denoised fourth-order cumulants vector. Sensors, 18.","DOI":"10.3390\/s18061815"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.sigpro.2017.07.004","article-title":"Two sparse-based methods for off-grid direction-of-arrival estimation","volume":"142","author":"Wu","year":"2018","journal-title":"Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"5987","DOI":"10.1109\/TSP.2013.2273443","article-title":"Atomic norm denoising with applications to line spectral estimation","volume":"61","author":"Bhaskar","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"4959","DOI":"10.1109\/TSP.2014.2339792","article-title":"A discretization-free sparse and parametric approach for linear array signal processing","volume":"62","author":"Yang","year":"2014","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"995","DOI":"10.1109\/TSP.2015.2493987","article-title":"Enhancing sparsity and resolution via reweighted atomic norm minimization","volume":"64","author":"Yang","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Wang, X., Wang, W., Li, X., Liu, Q., and Liu, J. (2016). Sparsity-aware DOA estimation scheme for noncircular source in MIMO Radar. Sensors, 16.","DOI":"10.3390\/s16040539"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1066","DOI":"10.1007\/s00034-016-0339-y","article-title":"A sparse recovery method for DOA estimation based on the sample covariance vectors","volume":"36","author":"Jing","year":"2017","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"962","DOI":"10.1109\/TSP.2011.2175222","article-title":"Sparse estimation of spectral lines: Grid selection problems and their solutions","volume":"60","author":"Stoica","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"5597","DOI":"10.1109\/TSP.2012.2212433","article-title":"Complex elliptically symmetric distributions: Survey, new results and applications","volume":"60","author":"Ollila","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3576","DOI":"10.1109\/TSP.2016.2546222","article-title":"Robust estimation of structured covariance matrix for heavy-tailed elliptical distributions","volume":"64","author":"Sun","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_37","unstructured":"Pourahmadi, M. (2013). High-Dimensional Covariance Estimation, John Wiley & Sons Inc."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1109\/TAES.2012.6129621","article-title":"Coherent radar target detection in heavy-tailed compound-Gaussian clutter","volume":"48","author":"Sangston","year":"2012","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1198\/0003130042836","article-title":"A tutorial on MM algorithms","volume":"58","author":"Hunter","year":"2004","journal-title":"Am. Stat."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"794","DOI":"10.1109\/TSP.2016.2601299","article-title":"Majorization-Minimization algorithms in signal processing, communications, and machine learning","volume":"65","author":"Sun","year":"2017","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1080\/02664768800000029","article-title":"Maximum likelihood estimation for the wrapped Cauchy distribution","volume":"15","author":"Kent","year":"1988","journal-title":"J. Appl. Stat."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/BF00939948","article-title":"On the convergence of the coordinate descent method for convex differentiable minimization","volume":"72","author":"Luo","year":"1992","journal-title":"J. Optim. Theory Appl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1214\/aos\/1176344136","article-title":"Estimating the dimension of a model","volume":"6","author":"Schwarz","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"288","DOI":"10.1111\/j.1467-9574.2011.00515.x","article-title":"Estimating the evidence\u2014A review","volume":"66","author":"Friel","year":"2012","journal-title":"Stat. Neerl."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"4422","DOI":"10.1109\/TSP.2015.2440215","article-title":"An adaptive population importance sampler: Learning from uncertainty","volume":"63","author":"Martino","year":"2015","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1007\/s11222-016-9642-5","article-title":"Layered adaptive importance sampling","volume":"27","author":"Martino","year":"2017","journal-title":"Stat. Comput."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1348","DOI":"10.1198\/016214501753382273","article-title":"Variable selection via nonconcave penalized likilihood and its oracle properties","volume":"96","author":"Fan","year":"2001","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1418","DOI":"10.1198\/016214506000000735","article-title":"The adaptive Lasso and its oracle properties","volume":"101","author":"Zou","year":"2006","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1214\/08-AOAS215","article-title":"Network exploration via the adaptive LASSO and SCAD penalties","volume":"3","author":"Fan","year":"2009","journal-title":"Ann. Appl. Stat."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/10\/2356\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:54:14Z","timestamp":1760187254000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/10\/2356"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,22]]},"references-count":49,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["s19102356"],"URL":"https:\/\/doi.org\/10.3390\/s19102356","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2019,5,22]]}}}