{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T15:56:20Z","timestamp":1776182180824,"version":"3.50.1"},"reference-count":31,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2018,11,26]],"date-time":"2018-11-26T00:00:00Z","timestamp":1543190400000},"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":["61401513"],"award-info":[{"award-number":["61401513"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This study discusses the localization problem based on time delay and Doppler shift for a far-field scenario. The conventional location methods employ two steps that first extract intermediate parameters from the received signals and then determine the source position from the measured parameters. As opposed to the traditional two-step methods, the direct position determination (DPD) methods accomplish the localization in a single step without computing intermediate parameters. However, the DPD cost function often remains non-convex, thereby it will cost a high amount of computational resources to find the estimated position through traversal search. Weiss proposed a DPD estimator to mitigate the computational complexity via eigenvalue decomposition. Unfortunately, when the computational resources are rather limited, Weiss\u2019s method fails to satisfy the timeliness. To solve this problem, this paper develops a DPD estimator using expectation maximization (EM) algorithm based on time delay and Doppler shift. The proposed method starts from choosing the transmitter-receiver range vector as the hidden variable. Then, the cost function is separated and simplified via the hidden variable, accomplishing the transformation from the high dimensional nonlinear search problem into a few one dimensional search subproblems. Finally, the expressions of EM repetition are obtained through Laplace approximation. In addition, we derive the Cram\u00e9r\u2013Rao bound to evaluate the best localization performance in this paper. Simulation results confirm that, on the basis of guaranteeing high accuracy, the proposed algorithm makes a good compromise in localization performance and computational complexity.<\/jats:p>","DOI":"10.3390\/s18124139","type":"journal-article","created":{"date-parts":[[2018,11,27]],"date-time":"2018-11-27T03:31:33Z","timestamp":1543289493000},"page":"4139","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Fast ML-Based Single-Step Localization Method Using EM Algorithm Based on Time Delay and Doppler Shift for a Far-Field Scenario"],"prefix":"10.3390","volume":"18","author":[{"given":"Tianzhu","family":"Qin","sequence":"first","affiliation":[{"name":"National Digital Switching System Engineering and Technology Research Center, Zhengzhou 450002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lin","family":"Li","sequence":"additional","affiliation":[{"name":"National Digital Switching System Engineering and Technology Research Center, Zhengzhou 450002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bin","family":"Ba","sequence":"additional","affiliation":[{"name":"National Digital Switching System Engineering and Technology Research Center, Zhengzhou 450002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daming","family":"Wang","sequence":"additional","affiliation":[{"name":"National Digital Switching System Engineering and Technology Research Center, Zhengzhou 450002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,11,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/j.sigpro.2015.09.019","article-title":"A rank-reduction based 2-D DOA estimation algorithm for three parallel uniform linear arrays","volume":"120","author":"Zhang","year":"2016","journal-title":"Signal Process."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3130","DOI":"10.1109\/TWC.2013.060413.112013","article-title":"Two-dimensional ESPRIT-like shift-invariant TOA estimation algorithm using multi-band chirp signals robust to carrier frequency Offset","volume":"12","author":"Oh","year":"2013","journal-title":"IEEE Trans. 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