{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T05:01:36Z","timestamp":1777266096276,"version":"3.51.4"},"reference-count":37,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,9]],"date-time":"2024-07-09T00:00:00Z","timestamp":1720483200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Major Innovation Project of CIOMP, CAS","award":["E20961X6CZ00"],"award-info":[{"award-number":["E20961X6CZ00"]}]},{"name":"Major Innovation Project of CIOMP, CAS","award":["SKLLIM2105"],"award-info":[{"award-number":["SKLLIM2105"]}]},{"name":"Major Innovation Project of CIOMP, CAS","award":["61904178"],"award-info":[{"award-number":["61904178"]}]},{"name":"Fund Project of the State Key Laboratory of Laser-Matter Interaction","award":["E20961X6CZ00"],"award-info":[{"award-number":["E20961X6CZ00"]}]},{"name":"Fund Project of the State Key Laboratory of Laser-Matter Interaction","award":["SKLLIM2105"],"award-info":[{"award-number":["SKLLIM2105"]}]},{"name":"Fund Project of the State Key Laboratory of Laser-Matter Interaction","award":["61904178"],"award-info":[{"award-number":["61904178"]}]},{"name":"National Natural Science Foundation","award":["E20961X6CZ00"],"award-info":[{"award-number":["E20961X6CZ00"]}]},{"name":"National Natural Science Foundation","award":["SKLLIM2105"],"award-info":[{"award-number":["SKLLIM2105"]}]},{"name":"National Natural Science Foundation","award":["61904178"],"award-info":[{"award-number":["61904178"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>To solve the separation of multi-source signals and detect their features from a single channel, a signal separation method using multi-constraint non-negative matrix factorization (NMF) is proposed. In view of the existing NMF algorithm not performing well in the underdetermined blind source separation, the \u03b2-divergence constraints and determinant constraints are introduced in the NMF algorithm, which can enhance local feature information and reduce redundant components by constraining the objective function. In addition, the Sine-bell window function is selected as the processing method for short-time Fourier transform (STFT), and it can preserve the overall feature distribution of the original signal. The original vibration signal is first transformed into time\u2013frequency domain with the STFT, which describes the local characteristic of the signal from the time\u2013frequency distribution. Then, the multi-constraint NMF is applied to reduce the dimensionality of the data and separate feature components in the low dimensional space. Meanwhile, the parameter WK is constructed to filter the reconstructed signal that recombined with the feature component in the time domain. Ultimately, the separated signals will be subjected to envelope spectrum analysis to detect fault features. The simulated and experimental results indicate the effectiveness of the proposed approach, which can realize the separation of multi-source signals and their fault diagnosis of bearings. In addition, it is also confirmed that the proposed method, juxtaposed with the NMF algorithm of the traditional objective function, is more applicable for compound fault diagnosis of the rotating machinery.<\/jats:p>","DOI":"10.3390\/e26070583","type":"journal-article","created":{"date-parts":[[2024,7,9]],"date-time":"2024-07-09T08:38:04Z","timestamp":1720514284000},"page":"583","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Separation and Extraction of Compound-Fault Signal Based on Multi-Constraint Non-Negative Matrix Factorization"],"prefix":"10.3390","volume":"26","author":[{"given":"Mengyang","family":"Wang","sequence":"first","affiliation":[{"name":"Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenbao","family":"Zhang","sequence":"additional","affiliation":[{"name":"Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mingzhen","family":"Shao","sequence":"additional","affiliation":[{"name":"Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guang","family":"Wang","sequence":"additional","affiliation":[{"name":"Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.measurement.2018.04.063","article-title":"Non-stationary vibration feature extraction method based on sparse decomposition and order tracking for gearbox fault diagnosis","volume":"124","author":"Li","year":"2018","journal-title":"Measurement"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"066118","DOI":"10.1088\/1361-6501\/ad31f7","article-title":"An improved tracking method of bearing characteristic frequencies in the time-frequency representation of vibration signal","volume":"35","author":"Chen","year":"2024","journal-title":"Meas. 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