{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T03:44:19Z","timestamp":1761709459842,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,2,6]],"date-time":"2018-02-06T00:00:00Z","timestamp":1517875200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper presents an analytic algorithm for estimating three-dimensional (3-D) localization of a single source with uniform circular array (UCA) interferometers. Fourier transforms are exploited to expand the phase distribution of a single source and the localization problem is reformulated as an equivalent spectrum manipulation problem. The 3-D parameters are decoupled to different spectrums in the Fourier domain. Algebraic relations are established between the 3-D localization parameters and the Fourier spectrums. Fourier sampling theorem ensures that the minimum element number for 3-D localization of a single source with a UCA is five. Accuracy analysis provides mathematical insights into the 3-D localization algorithm that larger number of elements gives higher estimation accuracy. In addition, the phase-based high-order difference invariance (HODI) property of a UCA is found and exploited to realize phase range compression. Following phase range compression, ambiguity resolution is addressed by the HODI of a UCA. A major advantage of the algorithm is that the ambiguity resolution and 3-D localization estimation are both analytic and are processed simultaneously, hence computationally efficient. Numerical simulations and experimental results are provided to verify the effectiveness of the proposed 3-D localization algorithm.<\/jats:p>","DOI":"10.3390\/s18020484","type":"journal-article","created":{"date-parts":[[2018,2,6]],"date-time":"2018-02-06T15:18:05Z","timestamp":1517930285000},"page":"484","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Analytic and Unambiguous Phase-Based Algorithm for 3-D Localization of a Single Source with Uniform Circular Array"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7582-4248","authenticated-orcid":false,"given":"Le","family":"Zuo","sequence":"first","affiliation":[{"name":"Department of Microwave Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China"},{"name":"School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jin","family":"Pan","sequence":"additional","affiliation":[{"name":"Department of Microwave Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Boyuan","family":"Ma","sequence":"additional","affiliation":[{"name":"Department of Microwave Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,2,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1096","DOI":"10.1109\/LAWP.2014.2327992","article-title":"Closed-Form Algorithm for 3-D single-source localization with uniform circular array","volume":"13","author":"Jung","year":"2014","journal-title":"IEEE Antennas Wirel. 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