{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T17:48:26Z","timestamp":1767116906132,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T00:00:00Z","timestamp":1752105600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China (NSFC)","award":["U2241201"],"award-info":[{"award-number":["U2241201"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Harmonics are a common phenomenon widely present in power systems. The presence of harmonics not only increases the energy consumption of equipment but also poses hidden risks to the safety and stealth performance of large ships. Thus, there is an urgent need for a detection method for the harmonic characteristics of time series. We propose a novel harmonic feature estimation method, termed Harmonic Aggregation Entropy (HaAgEn), which effectively discriminates against background noise. The method is based on bispectrum analysis; utilizing the distribution characteristics of harmonic signals in the bispectrum matrix, a new Diagonal Bi-directional Integral Bispectrum (DBIB) method is employed to effectively extract harmonic features within the bispectrum matrix. This approach addresses the issues associated with traditional time\u2013frequency analysis methods, such as the large computational burden and lack of specificity in feature extraction. The integration results, Ix and Iy, of DBIB on different frequency axes are calculated using cross-entropy to derive HaAgEn. It is verified that HaAgEn is significantly more sensitive to harmonic components in the signal compared to other types of entropy, thereby better addressing harmonic detection issues and reducing feature redundancy. The detection accuracy of harmonic components in the shaft-rate electromagnetic field signal, as evidenced by sea trial data, reaches 96.8%, which is significantly higher than that of other detection methods. This provides a novel technical approach for addressing the issue of harmonic detection in industrial applications.<\/jats:p>","DOI":"10.3390\/e27070738","type":"journal-article","created":{"date-parts":[[2025,7,11]],"date-time":"2025-07-11T13:44:19Z","timestamp":1752241459000},"page":"738","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Harmonic Aggregation Entropy: A Highly Discriminative Harmonic Feature Estimator for Time Series"],"prefix":"10.3390","volume":"27","author":[{"given":"Ye","family":"Wang","sequence":"first","affiliation":[{"name":"Naval Submarine Academy, Qingdao 266199, China"}]},{"given":"Zhentao","family":"Yu","sequence":"additional","affiliation":[{"name":"Naval Submarine Academy, Qingdao 266199, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9056-0498","authenticated-orcid":false,"given":"Cheng","family":"Chi","sequence":"additional","affiliation":[{"name":"Naval Submarine Academy, Qingdao 266199, China"}]},{"given":"Bozhong","family":"Lei","sequence":"additional","affiliation":[{"name":"Naval Submarine Academy, Qingdao 266199, China"}]},{"given":"Jianxin","family":"Pei","sequence":"additional","affiliation":[{"name":"College of Marine Geoscience, Ocean University of China, Qingdao 266100, China"}]},{"given":"Dan","family":"Wang","sequence":"additional","affiliation":[{"name":"Naval Submarine Academy, Qingdao 266199, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2522","DOI":"10.1109\/TASLP.2024.3393727","article-title":"Harmonic Detection from Noisy Speech with Auditory Frame Gain for Intelligibility Enhancement","volume":"32","author":"Queiroz","year":"2024","journal-title":"IEEE\/ACM Trans. Audio Speech Lang. Process."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Birsan, M. (2011). Measurement of the extremely low frequency (ELF) magnetic field emission from a ship. Meas. Sci. Technol., 22.","DOI":"10.1088\/0957-0233\/22\/8\/085709"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1016\/j.oceaneng.2018.06.024","article-title":"Influence of anode location and quantity for the reduction of underwater electric fields under cathodic protection","volume":"163","author":"Kim","year":"2018","journal-title":"Ocean Eng."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Wang, D., Yang, L., and Ni, L. (2024). Performance and harmonic detection algorithm of phase locked Loop for parallel APF. Energy Inform., 7.","DOI":"10.1186\/s42162-024-00325-3"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Xu, M., Sang, Z., Li, X., You, Y., and Dai, D. (2022). An Observer-Based Harmonic Extraction Method with Front SOGI. Machines, 10.","DOI":"10.3390\/machines10020095"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1109\/TPWRD.2007.911141","article-title":"Extraction of Harmonics Using Composite Observers","volume":"23","author":"Selvajyothi","year":"2008","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1235","DOI":"10.1109\/TPWRD.2004.834311","article-title":"A New Method for Power Signal Harmonic Analysis","volume":"20","author":"Yang","year":"2005","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_8","unstructured":"He, W., Zhang, J., Yao, W., and Tang, L. (2017). FFT-based Amplitude Estimation of Power Distribution Systems Signal Distorted by Harmonics and Noise. IEEE Trans. Ind. Inform., 10."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Daniel, K., K\u00fctt, L., Iqbal, M.N., Shabbir, N., Raja, H.A., and Sardar, M.U. (2024). A Review of Harmonic Detection, Suppression, Aggregation, and Estimation Techniques. Appl. Sci., 14.","DOI":"10.3390\/app142310966"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Wang, X., Wang, B., Wang, W., Yu, M., Wang, Z., and Chang, Y. (2015). Harmonic signal extraction from chaotic interference based on synchrosqueezed wavelet transform. Acta Phys. Sin., 64.","DOI":"10.7498\/aps.64.100201"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Liu, L., Zhi, Z., Yang, Y., Shirmohammadi, S., and Liu, D. (2023). Harmonic reducer fault detection with acoustic emission. IEEE Trans. Instrum. Meas., 72.","DOI":"10.1109\/TIM.2023.3291747"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Kumaraswamy, B. (2023). An Improved Sub-Harmonic to Harmonic Ratio Method for Pitch Estimation and Shadja Detection. Concurr. Comput. Pract. Exp., 35.","DOI":"10.1002\/cpe.7604"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1109\/JESTPE.2014.2299240","article-title":"Current Control for Dual Three-Phase Permanent Magnet Synchronous Motors Accounting for Current Unbalance and Harmonics","volume":"2","author":"Hu","year":"2014","journal-title":"IEEE J. Emerg. Sel. Top. Power Electron."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Zhu, Z., Xu, M., Li, P., and Wang, Q.S. (2022). Rapid Harmonic Detection Scheme Based on Expanded Input Observer. Electronics, 11.","DOI":"10.3390\/electronics11182860"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zhang, J., Wang, K., Yang, D., Yuan, Y., and Yang, S. (2025). Nonlinear Vibro-Acoustic Modulation for Microcrack Detection of Steel Strands Based on S-Transform Bispectrum. Appl. Acoust., 227.","DOI":"10.1016\/j.apacoust.2024.110237"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2038","DOI":"10.1109\/59.544682","article-title":"Discrete Wavelet Analysis of Power System Transients","volume":"11","author":"Wilkinson","year":"1996","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Lu, Y., Li, B., Teng, G., Zhang, Z., and Xu, X. (2025). A harmonic current detection algorithm for aviation active power filter based on generalized delayed signal superposition. Sci. Rep., 15.","DOI":"10.1038\/s41598-025-94829-x"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1109\/59.317683","article-title":"Artificial neural networks for real-time estimation of basic waveforms of voltages and currents","volume":"9","author":"Cichocki","year":"1994","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Sun, H., Wang, L., Qi, L., Yan, J., and Jiang, M. (2024). Composite Harmonic Source Detection with Multi-Label Approach Using Advanced Fusion Method. Electronics, 13.","DOI":"10.3390\/electronics13071275"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1109\/TNN.2010.2045129","article-title":"A New One-Layer Neural Network for Linear and Quadratic Programming","volume":"21","author":"Gao","year":"2010","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TASLP.2022.3164199","article-title":"Overlapped Speech-Music Detection Using Harmonic-Percussive Features and Multi-Task Learning","volume":"31","author":"Bhattacharjee","year":"2023","journal-title":"IEEE\/ACM Trans. Audio Speech Lang. Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.cmpb.2010.07.011","article-title":"Prediction of paroxysmal atrial fibrillation based on non-linear analysis and spectrum and bispectrum features of the heart rate variability signal","volume":"105","author":"Mohebbi","year":"2012","journal-title":"Comput. Methods Programs Biomed."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.apacoust.2017.02.007","article-title":"The deterministic bispectrum of coupled harmonic random signals and its application to rotor faults diagnosis considering noise immunity","volume":"122","author":"Saidi","year":"2017","journal-title":"Appl. Acoust."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.acha.2020.10.005","article-title":"Defining the wavelet bispectrum","volume":"51","author":"Newman","year":"2021","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Cui, L., Xu, H., Ge, J., Cao, M., Xu, Y., Xu, W., and Sumarac, D. (2021). Use of Bispectrum Analysis to Inspect the Non-Linear Dynamic Characteristics of Beam-Type Structures Containing a Breathing Crack. Sensors, 21.","DOI":"10.3390\/s21041177"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1949","DOI":"10.1109\/TIM.2008.917192","article-title":"Application of Bispectrum Estimation for Time-Frequency Analysis of Ground Surveillance Doppler Radar Echo Signals","volume":"57","author":"Astola","year":"2008","journal-title":"IEEE Trans. Instrum. Meas. J."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Tassiopoulou, S., Koukiou, G., and Anastassopoulos, V. (2024). Revealing Coupled Periodicities in Sunspot Time Series Using Bispectrum\u2014An Inverse Problem. Appl. Sci., 14.","DOI":"10.3390\/app14031318"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1109\/TAES.2022.3184619","article-title":"Semi-Supervised Specific Emitter Identification Based on Bispectrum Feature Extraction CGAN in Multiple Communication Scenarios","volume":"57","author":"Tan","year":"2023","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2009","DOI":"10.1007\/s11760-022-02162-x","article-title":"Deep learning-based specific emitter identification using integral bispectrum and the slice of ambiguity function","volume":"16","author":"Wan","year":"2022","journal-title":"Signal Image Video Process."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Ying, W., Tong, J., Dong, Z., Pan, H., Liu, Q., and Zheng, J. (2022). Composite multivariate multi-scale permutation entropy and Laplacian score based fault diagnosis of rolling bearing. Entropy, 24.","DOI":"10.3390\/e24020160"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1007\/s11760-021-02053-7","article-title":"A novel method for fault diagnosis in rolling bearings based on bispectrum signals and combined feature extraction algorithms","volume":"16","author":"Hashempour","year":"2022","journal-title":"Signal Image Video Process."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Wang, Q., Jia, X., Luo, T., Yu, J., and Xia, S. (2023). Deep learning algorithm using bispectrum analysis energy feature maps based on ultrasound radio frequency signals to detect breast cancer. Front. Oncol., 13.","DOI":"10.3389\/fonc.2023.1272427"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Sharma, A., Patra, G.K., and Naidu, V.P.S. (2024). Bispectral analysis and information fusion technique for bearing fault classification. Meas. Sci. Technol., 35.","DOI":"10.1088\/1361-6501\/acffe4"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3189","DOI":"10.1177\/14759217221144724","article-title":"A local modulation signal bispectrum for multiple amplitude and frequency modulation demodulation in gearbox fault diagnosis","volume":"22","author":"Guo","year":"2023","journal-title":"Struct. Health Monit."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Xu, Y., Tang, X., Sun, X., Gu, F., and Ball, A.D. (2022). A Squeezed Modulation Signal Bispectrum Method for Motor Current Signals Based Gear Fault Diagnosis. IEEE Trans. Instrum. Meas., 71.","DOI":"10.1109\/TIM.2022.3201549"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"5419","DOI":"10.1109\/TII.2020.3022369","article-title":"Multiscale Diversity Entropy: A Novel Dynamical Measure for Fault Diagnosis of Rotating Machinery","volume":"17","author":"Wang","year":"2021","journal-title":"IEEE Trans. Ind. Inf."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Koukiou, G. (2024). Identifying System Non-Linearities by Fusing Signal Bispectral Signatures. Electronics, 13.","DOI":"10.3390\/electronics13071287"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Li, Y., Wang, S., Yang, Y., and Deng, Z. (2022). Multiscale symbolic fuzzy entropy: An entropy denoising method for weak feature extraction of rotating machinery. Mech. Syst. Signal Process., 162.","DOI":"10.1016\/j.ymssp.2021.108052"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Liu, X., Jiang, A., Xu, N., and Xue, J. (2016). Increment Entropy as a Measure of Complexity for Time Series. Entropy, 18.","DOI":"10.3390\/e18010022"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Espinosa, R., Bail\u00f3n, R., and Laguna, P. (2021). Two-Dimensional EspEn: A New Approach to Analyze Image Texture by Irregularity. Entropy, 23.","DOI":"10.3390\/e23101261"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/7\/738\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:07:56Z","timestamp":1760033276000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/7\/738"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,10]]},"references-count":40,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["e27070738"],"URL":"https:\/\/doi.org\/10.3390\/e27070738","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,7,10]]}}}