{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:13:03Z","timestamp":1760238783236,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"18","license":[{"start":{"date-parts":[[2020,9,7]],"date-time":"2020-09-07T00:00:00Z","timestamp":1599436800000},"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":["61671012"],"award-info":[{"award-number":["61671012"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"the National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61671012"],"award-info":[{"award-number":["61671012"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The closed-form robust Chinese Remainder Theorem (CRT) is a powerful approach to achieve single-frequency estimation from noisy undersampled waveforms. However, the difficulty of CRT-based methods\u2019 extension into the multi-tone case lies in the fact it is complicated to explore the mapping relationship between an individual tone and its corresponding remainders. This work deals with this intractable issue by means of decomposing the desired multi-tone estimator into several single-tone estimators. Firstly, high-accuracy harmonic remainders are calculated by applying all-phase Discrete Fourier Transform (apDFT) and spectrum correction operations on the undersampled waveforms. Secondly, the aforementioned mapping relationship is built up by a novel frequency classifier which fully captures the amplitude and phase features of remainders. Finally, the frequencies are estimated one by one through directly applying the closed-form robust CRT into these remainder groups. Due to all the components (including closed-form CRT, the apDFT, the spectrum corrector and the remainder classifier) only involving slight computation complexity, the proposed scheme is of high efficiency and consumes low hardware cost. Moreover, numeral results also show that the proposed method possesses high accuracy.<\/jats:p>","DOI":"10.3390\/s20185066","type":"journal-article","created":{"date-parts":[[2020,9,7]],"date-time":"2020-09-07T09:18:16Z","timestamp":1599470296000},"page":"5066","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Multi-Tone Frequency Estimation Based on the All-Phase Discrete Fourier Transform and Chinese Remainder Theorem"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1217-3809","authenticated-orcid":false,"given":"Xiangdong","family":"Huang","sequence":"first","affiliation":[{"name":"School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lu","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.sigpro.2015.05.005","article-title":"Frequency estimation by two- or three-point interpolated Fourier algorithms based on cosine windows","volume":"117","author":"Belega","year":"2015","journal-title":"Signal Process."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.sigpro.2015.03.009","article-title":"Fine resolution frequency estimation from three DFT samples: Case of windowed data","volume":"114","author":"Candan","year":"2015","journal-title":"Signal Process."},{"key":"ref_3","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. 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