{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T14:29:18Z","timestamp":1774448958496,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"13","license":[{"start":{"date-parts":[[2023,6,24]],"date-time":"2023-06-24T00:00:00Z","timestamp":1687564800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["62201418"],"award-info":[{"award-number":["62201418"]}]},{"name":"National Natural Science Foundation of China","award":["62192714"],"award-info":[{"award-number":["62192714"]}]},{"name":"National Natural Science Foundation of China","award":["61701379"],"award-info":[{"award-number":["61701379"]}]},{"name":"National Natural Science Foundation of China","award":["JKW202107"],"award-info":[{"award-number":["JKW202107"]}]},{"name":"National Natural Science Foundation of China","award":["XJS220203"],"award-info":[{"award-number":["XJS220203"]}]},{"name":"National Radar Signal Processing Laboratory","award":["62201418"],"award-info":[{"award-number":["62201418"]}]},{"name":"National Radar Signal Processing Laboratory","award":["62192714"],"award-info":[{"award-number":["62192714"]}]},{"name":"National Radar Signal Processing Laboratory","award":["61701379"],"award-info":[{"award-number":["61701379"]}]},{"name":"National Radar Signal Processing Laboratory","award":["JKW202107"],"award-info":[{"award-number":["JKW202107"]}]},{"name":"National Radar Signal Processing Laboratory","award":["XJS220203"],"award-info":[{"award-number":["XJS220203"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["62201418"],"award-info":[{"award-number":["62201418"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["62192714"],"award-info":[{"award-number":["62192714"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["61701379"],"award-info":[{"award-number":["61701379"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["JKW202107"],"award-info":[{"award-number":["JKW202107"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["XJS220203"],"award-info":[{"award-number":["XJS220203"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>The performance of traditional direction of arrival (DOA) estimation methods always deteriorates at a low signal-to-noise ratio (SNR) or without sufficient observations. This paper investigates the Bayesian DOA estimation problem aided by the prior knowledge from the target tracker. The Bayesian Cram\u00e9r\u2013Rao lower bounds (CRLB) and the expected CRLB are first derived to evaluate the theoretical performance of Bayesian DOA estimation. Based on the maximum a posterior (MAP) estimator in the Bayesian framework, two methods are proposed. One is a two-step grid search method for a single target DOA case. The other is a gradient-based iterative solution for multiple targets DOA case, which extends the traditional Newton method by incorporating the prior knowledge. We also propose a minimum mean square error (MMSE) estimator using a Monte Carlo method, which requires trading off accuracy against computational complexity. By comparing with the maximum likelihood (ML) estimators and the MUSIC algorithm, the proposed three Bayesian estimators improve the DOA estimation performance in low SNR or with limited snapshots. Moreover, the performance is not affected by the correlation between sources.<\/jats:p>","DOI":"10.3390\/rs15133255","type":"journal-article","created":{"date-parts":[[2023,6,26]],"date-time":"2023-06-26T03:14:56Z","timestamp":1687749296000},"page":"3255","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Bayesian Direction of Arrival Estimation with Prior Knowledge from Target Tracker"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7624-5448","authenticated-orcid":false,"given":"Tianyi","family":"Jia","sequence":"first","affiliation":[{"name":"National Key Laboratory of Radar Signal Processing, Xidian University, Xi\u2019an 710071, China"}]},{"given":"Hongwei","family":"Liu","sequence":"additional","affiliation":[{"name":"National Key Laboratory of Radar Signal Processing, Xidian University, Xi\u2019an 710071, China"}]},{"given":"Penghui","family":"Wang","sequence":"additional","affiliation":[{"name":"National Key Laboratory of Radar Signal Processing, Xidian University, Xi\u2019an 710071, China"}]},{"given":"Chang","family":"Gao","sequence":"additional","affiliation":[{"name":"National Key Laboratory of Radar Signal Processing, Xidian University, Xi\u2019an 710071, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.sigpro.2017.09.011","article-title":"Target localization based on structured total least squares with hybrid TDOA-AOA measurements","volume":"143","author":"Jia","year":"2018","journal-title":"Signal Process."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"094801","DOI":"10.1121\/10.0006389","article-title":"Direction-of-arrival estimation for coherent signals through covariance-based grid free compressive sensing","volume":"1","author":"Zhang","year":"2021","journal-title":"JASA Express Lett."},{"key":"ref_3","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_4","doi-asserted-by":"crossref","first-page":"103028","DOI":"10.1016\/j.dsp.2021.103028","article-title":"Maximum-likelihood direction of arrival estimation under intermittent jamming","volume":"113","author":"Akdemir","year":"2021","journal-title":"Digit. Signal Process."},{"key":"ref_5","unstructured":"Ottersten, B., Viberg, M., Stoica, P., and Nehorai, A. (1993). Radar Array Processing, Springer."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Pal, P., and Vaidyanathan, P.P. (2011, January 4\u20137). Coprime sampling and the MUSIC algorithm. Proceedings of the 2011 Digital Signal Processing and Signal Processing Education Meeting (DSP\/SPE), Sedona, AZ, USA.","DOI":"10.1109\/DSP-SPE.2011.5739227"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Jia, T., Wang, H., Shen, X., and Liu, X. (2016, January 10\u201313). Direction of arrival estimation with co-prime arrays via compressed sensing methods. Proceedings of the OCEANS 2016\u2013Shanghai, Shanghai, China.","DOI":"10.1109\/OCEANSAP.2016.7485484"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1438","DOI":"10.1109\/LSP.2015.2409153","article-title":"Remarks on the spatial smoothing step in coarray MUSIC","volume":"22","author":"Liu","year":"2015","journal-title":"IEEE Signal Process. Lett."},{"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":"509","DOI":"10.1016\/B978-0-12-811887-0.00011-0","article-title":"Sparse methods for direction-of-arrival estimation","volume":"Volume 7","author":"Yang","year":"2018","journal-title":"Academic Press Library in Signal Processing: Array, Radar and Communications Engineering"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5591","DOI":"10.1109\/TSP.2017.2739105","article-title":"Sparsity-based two-dimensional DOA estimation for coprime array: From sum-difference coarray viewpoint","volume":"65","author":"Shi","year":"2017","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.sigpro.2016.03.024","article-title":"An efficient off-grid DOA estimation approach for nested array signal processing by using sparse Bayesian learning strategies","volume":"128","author":"Yang","year":"2016","journal-title":"Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1007\/s11235-020-00676-8","article-title":"A study of compressed sensing single-snapshot DOA estimation based on the RIPless theory","volume":"74","author":"Jia","year":"2020","journal-title":"Telecommun. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1257","DOI":"10.1109\/TSP.2015.2496294","article-title":"Off-the-grid line spectrum denoising and estimation with multiple measurement vectors","volume":"64","author":"Li","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_15","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_16","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":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1744","DOI":"10.1109\/LSP.2021.3104503","article-title":"Sparse Bayesian learning using generalized double Pareto prior for DOA estimation","volume":"28","author":"Wang","year":"2021","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_18","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_19","doi-asserted-by":"crossref","first-page":"2004","DOI":"10.1109\/JSEN.2015.2508059","article-title":"Direction of arrival estimation for off-grid signals based on sparse Bayesian learning","volume":"16","author":"Wu","year":"2016","journal-title":"IEEE Sens. J."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1109\/TSP.2017.2773420","article-title":"Sparse Bayesian learning approach for outlier-resistant direction-of-arrival estimation","volume":"66","author":"Dai","year":"2018","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1109\/TSP.2018.2881663","article-title":"Off-grid DOA estimation using sparse Bayesian learning in MIMO radar with unknown mutual coupling","volume":"67","author":"Chen","year":"2018","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4997","DOI":"10.1109\/TSP.2014.2343940","article-title":"Joint sparse recovery method for compressed sensing with structured dictionary mismatches","volume":"62","author":"Tan","year":"2014","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1109\/LSP.2016.2636319","article-title":"Root sparse Bayesian learning for off-grid DOA estimation","volume":"24","author":"Dai","year":"2017","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"7465","DOI":"10.1109\/TIT.2013.2277451","article-title":"Compressed sensing off the grid","volume":"59","author":"Tang","year":"2013","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3139","DOI":"10.1109\/TSP.2015.2420541","article-title":"On gridless sparse methods for line spectral estimation from complete and incomplete data","volume":"63","author":"Yang","year":"2015","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"5342","DOI":"10.1109\/TSP.2015.2452223","article-title":"Spectral super-resolution with prior knowledge","volume":"63","author":"Mishra","year":"2015","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1427","DOI":"10.1109\/JSTSP.2015.2465304","article-title":"Cognitive radar framework for target detection and tracking","volume":"9","author":"Bell","year":"2015","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1109\/MSP.2006.1593335","article-title":"Cognitive radar: A way of the future","volume":"23","author":"Haykin","year":"2006","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1109\/MAES.2019.2953762","article-title":"An overview of cognitive radar: Past, present, and future","volume":"34","author":"Gurbuz","year":"2019","journal-title":"IEEE Aerosp. Electron. Syst. Mag."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Guerci, J.R. (2010). Cognitive Radar: A Knowledge-Aided Fully Adaptive Approach, Artech House.","DOI":"10.1109\/RADAR.2010.5494403"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1109\/MSP.2018.2822847","article-title":"Cognitive radars: On the road to reality: Progress thus far and possibilities for the future","volume":"35","author":"Greco","year":"2018","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1007\/s11804-011-1098-6","article-title":"Research of new concept sonar-cognitive sonar","volume":"10","author":"Li","year":"2011","journal-title":"J. Marine Sci. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1264","DOI":"10.1109\/TAES.2006.314571","article-title":"Road-map assisted ground moving target tracking","volume":"42","author":"Ulmke","year":"2006","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1454","DOI":"10.1109\/TAES.2007.4407470","article-title":"Efficient particle filtering for road-constrained target tracking","volume":"43","author":"Cheng","year":"2007","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"4729","DOI":"10.1109\/TSP.2013.2273441","article-title":"Mean-squared-error prediction for Bayesian direction-of-arrival estimation","volume":"61","author":"Kantor","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_36","unstructured":"Trees, H.L.V. (2004). Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, John Wiley & Sons."},{"key":"ref_37","unstructured":"Stoica, P., and Moses, R.L. (2005). Spectral Analysis of Signals, Pearson Prentice Hall."},{"key":"ref_38","unstructured":"Kay, S.M. (1993). Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice Hall PTR."},{"key":"ref_39","unstructured":"Trees, H.L.V., and Bell, K.L. (2007). Bayesian Bounds for Parameter Estimation and Nonlinear Filtering\/Tracking, Wiley."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2146","DOI":"10.1109\/TIT.2006.872975","article-title":"Mean-squared error and threshold SNR prediction of maximum-likelihood signal parameter estimation with estimated colored noise covariances","volume":"52","author":"Richmond","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1109\/TAES.2010.5417172","article-title":"Source localization and sensing: A nonparametric iterative adaptive approach based on weighted least squares","volume":"46","author":"Yardibi","year":"2010","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1109\/TSP.2017.2764865","article-title":"Fast frequency estimation with prior information","volume":"66","author":"Mahata","year":"2018","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.sigpro.2017.07.028","article-title":"Fast convex optimization method for frequency estimation with prior knowledge in all dimensions","volume":"142","author":"Yang","year":"2018","journal-title":"Signal Process."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/13\/3255\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:59:50Z","timestamp":1760126390000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/13\/3255"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,24]]},"references-count":43,"journal-issue":{"issue":"13","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["rs15133255"],"URL":"https:\/\/doi.org\/10.3390\/rs15133255","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,24]]}}}