{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T16:39:05Z","timestamp":1781714345503,"version":"3.54.5"},"reference-count":27,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,5,16]],"date-time":"2023-05-16T00:00:00Z","timestamp":1684195200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005374","name":"Open Research Project of Jiangsu Provincial Key Laboratory of Photonic and Electronic Materials Sciences and Technology","doi-asserted-by":"publisher","award":["NJUZDS 2022-008"],"award-info":[{"award-number":["NJUZDS 2022-008"]}],"id":[{"id":"10.13039\/501100005374","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The existing expectation maximization (EM) and space-alternating generalized EM (SAGE) algorithms are only applied to direction of arrival (DOA) estimation in known noise. In this paper, the two algorithms are designed for DOA estimation in unknown uniform noise. Both the deterministic and random signal models are considered. In addition, a new modified EM (MEM) algorithm applicable to the noise assumption is also proposed. Next, these EM-type algorithms are improved to ensure the stability when the powers of sources are not equal. After being improved, simulation results illustrate that the EM algorithm has similar convergence with the MEM algorithm, the SAGE algorithm outperforms the EM and MEM algorithms for the deterministic signal model, and the SAGE algorithm cannot always outperform the EM and MEM algorithms for the random signal model. Furthermore, simulation results show that processing the same snapshots from the random signal model, the SAGE algorithm for the deterministic signal model can require the fewest computations.<\/jats:p>","DOI":"10.3390\/s23104811","type":"journal-article","created":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T01:58:06Z","timestamp":1684288686000},"page":"4811","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["EM and SAGE Algorithms for DOA Estimation in the Presence of Unknown Uniform Noise"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4052-1498","authenticated-orcid":false,"given":"Ming-Yan","family":"Gong","sequence":"first","affiliation":[{"name":"School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5077-2576","authenticated-orcid":false,"given":"Bin","family":"Lyu","sequence":"additional","affiliation":[{"name":"Key Laboratory of Ministry of Education in Broadband Wireless Communication and Sensor Network Technology, Nanjing University of Posts and Telecommunications, Nanjing 210003, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,16]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"1195","DOI":"10.1109\/5.622504","article-title":"Application of antenna arrays to mobile communications. II. Beam-forming and direction-of-arrival considerations","volume":"85","author":"Godara","year":"1997","journal-title":"Proc. IEEE"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","article-title":"Maximum likelihood from incomplete data via the EM algorithm","volume":"39","author":"Dempster","year":"1977","journal-title":"J. R. Stat. Soc. Ser. B (Methodol.)"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1111\/1467-9868.00082","article-title":"The EM algorithm\u2013an old folk-song sung to a fast new tune","volume":"59","author":"Meng","year":"1997","journal-title":"J. R. Stat. Soc. Ser. B (Methodol.)"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1109\/29.1552","article-title":"Parameter estimation of superimposed signals using the EM algorithm","volume":"36","author":"Feder","year":"1988","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1560","DOI":"10.1109\/29.60075","article-title":"Maximum-likelihood narrow-band direction finding and the EM algorithm","volume":"38","author":"Miller","year":"1990","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2664","DOI":"10.1109\/78.324732","article-title":"Space-alternating generalized expectation-maximization algorithm","volume":"42","author":"Fessler","year":"1994","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2940","DOI":"10.1109\/78.969503","article-title":"Comparative convergence analysis of EM and SAGE algorithms in DOA estimation","volume":"49","author":"Chung","year":"2001","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"9634","DOI":"10.1109\/TVT.2021.3106794","article-title":"Alternating maximization and the EM algorithm in maximum-likelihood direction finding","volume":"70","author":"Gong","year":"2021","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1553","DOI":"10.1109\/29.7543","article-title":"Maximum likelihood localization of multiple sources by alternating projection","volume":"36","author":"Ziskind","year":"1988","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"720","DOI":"10.1109\/29.17564","article-title":"MUSIC, maximum likelihood, and Cramer-Rao bound","volume":"37","author":"Stoica","year":"1989","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1783","DOI":"10.1109\/29.60109","article-title":"Performance study of conditional and unconditional direction-of-arrival estimation","volume":"38","author":"Stoica","year":"1990","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3008","DOI":"10.1109\/TSP.2016.2537265","article-title":"Iterative methods for subspace and DOA estimation in nonuniform noise","volume":"64","author":"Liao","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"8982","DOI":"10.1109\/JSEN.2016.2621057","article-title":"New approaches to direction-of-arrival estimation with sensor arrays in unknown nonuniform noise","volume":"16","author":"Liao","year":"2016","journal-title":"IEEE Sens. J."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"848","DOI":"10.1109\/LSP.2019.2909587","article-title":"Non-iterative subspace-based DOA estimation in the presence of nonuniform noise","volume":"26","author":"Esfandiari","year":"2019","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Esfandiari, M., and Vorobyov, S.A. (2022, January 22\u201327). A novel angular estimation method in the presence of nonuniform noise. Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Singapore.","DOI":"10.1109\/ICASSP43922.2022.9747777"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"108100","DOI":"10.1016\/j.sigpro.2021.108100","article-title":"Iterative methods for DOA estimation of correlated sources in spatially colored noise fields","volume":"185","author":"Yang","year":"2021","journal-title":"Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"108879","DOI":"10.1016\/j.sigpro.2022.108879","article-title":"Efficient computation of ML DOA estimates under unknown nonuniform sensor noise powers","volume":"205","author":"Selva","year":"2023","journal-title":"Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1310","DOI":"10.1109\/78.928686","article-title":"Maximum-likelihood direction-of-arrival estimation in the presence of unknown nonuniform noise","volume":"49","author":"Pesavento","year":"2001","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3038","DOI":"10.1109\/TSP.2008.917364","article-title":"Stochastic maximum-likelihood DOA estimation in the presence of unknown nonuniform noise","volume":"56","author":"Chen","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1093\/biomet\/80.2.267","article-title":"Maximum likelihood estimation via the ECM algorithm: A general framework","volume":"80","author":"Meng","year":"1993","journal-title":"Biometrika"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1753","DOI":"10.1016\/S0165-1684(02)00337-7","article-title":"DOA estimation using fast EM and SAGE algorithms","volume":"82","author":"Chung","year":"2002","journal-title":"Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"688","DOI":"10.1109\/TAC.1971.1099833","article-title":"A tutorial introduction to estimation and filtering","volume":"16","author":"Rhodes","year":"1971","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_24","unstructured":"Jaffer, A.G. (1988, January 11\u201314). Maximum likelihood direction finding of stochastic sources: A separable solution. Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (ICASSP), New York, NY, USA."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1007\/BF01213963","article-title":"On the concentrated stochastic likelihood function in array signal processing","volume":"14","author":"Stoica","year":"1995","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_26","first-page":"95","article-title":"On the convergence properties of the EM algorithm","volume":"11","year":"1983","journal-title":"Ann. Stat."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Boyd, S., and Vandenberghe, L. (2004). Convex Optimization, Cambridge University Press. [1st ed.].","DOI":"10.1017\/CBO9780511804441"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/10\/4811\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:36:22Z","timestamp":1760124982000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/10\/4811"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,16]]},"references-count":27,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["s23104811"],"URL":"https:\/\/doi.org\/10.3390\/s23104811","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,16]]}}}