{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T03:38:47Z","timestamp":1771472327414,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,2,7]],"date-time":"2018-02-07T00:00:00Z","timestamp":1517961600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Applied Basic Research Program of Wuhan Municipality","award":["2015011701011639"],"award-info":[{"award-number":["2015011701011639"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["41074023"],"award-info":[{"award-number":["41074023"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["41304006"],"award-info":[{"award-number":["41304006"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Basic Research Program of China (973 Program)","award":["2013CB733205"],"award-info":[{"award-number":["2013CB733205"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>High-performance oscillators, atomic clocks for instance, are important in modern industries, finance and scientific research. In this paper, the authors study the estimation and prediction of long-term stability based on convex optimization techniques and compressive sensing. To take frequency drift into account, its influence on Allan and modified Allan variances is formulated. Meanwhile, expressions for the expectation and variance of discrete-time Hadamard variance are derived. Methods that reduce the computational complexity of these expressions are also introduced. Tests against GPS precise clock data show that the method can correctly predict one-week frequency stability from 14-day measured data.<\/jats:p>","DOI":"10.3390\/s18020502","type":"journal-article","created":{"date-parts":[[2018,2,7]],"date-time":"2018-02-07T12:20:29Z","timestamp":1518006029000},"page":"502","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Predicting Long-Term Stability of Precise Oscillators under Influence of Frequency Drift"],"prefix":"10.3390","volume":"18","author":[{"given":"Weiwei","family":"Cheng","sequence":"first","affiliation":[{"name":"GNSS (Global Navigation Satellite System) Research Center, Wuhan University, 129 Luoyu Road, 430079 Wuhan, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guigen","family":"Nie","sequence":"additional","affiliation":[{"name":"GNSS (Global Navigation Satellite System) Research Center, Wuhan University, 129 Luoyu Road, 430079 Wuhan, China"},{"name":"Collaborative Innovation Center for Geospatial Information Technology, Wuhan University, 129 Luoyu Road, 430079 Wuhan, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,2,7]]},"reference":[{"key":"ref_1","unstructured":"Shorter, G., and Miller, R. (2014). High-Frequency Trading: Background, Concerns, and Regulatory Developments, Congressional Research Service."},{"key":"ref_2","unstructured":"McCarthy, D., and Seidelmann, P. (2009). Time, WILEY-VCH Verlag GmbH & Co. KGaA."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1109\/PROC.1982.12282","article-title":"1\/f noise","volume":"70","author":"Keshner","year":"1982","journal-title":"Proc. IEEE"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1253","DOI":"10.1109\/TUFFC.2005.1509784","article-title":"Enhancements to GPS Operations and Clock Evaluations Using a \u201cTotal\u201d Hadamard Deviation","volume":"52","author":"Howe","year":"2005","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1088\/0026-1394\/43\/4\/S20","article-title":"Th\u00eaoH: A Hybrid, High-Confidence Statistic that Improves on the Allan Deviation","volume":"43","author":"Howe","year":"2006","journal-title":"Metrologia"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1109\/TUFFC.2015.2499325","article-title":"The Parabolic Variance (PVAR): A Wavelet Variance Based on the Least-Square Fit","volume":"63","author":"Vernotte","year":"2016","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_7","unstructured":"Cheng, W., Nie, G., Wang, P., Zhang, C., and Gao, Y. Predicting Long-Term Frequency Stability: Stochastic Oscillator Noise Analysis. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, in press."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Manolakis, D., and Ingle, V. (2011). Applied Digital Signal Processing, Cambridge University Press.","DOI":"10.1017\/CBO9780511835261"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1088\/0026-1394\/52\/2\/222","article-title":"Metrology and 1\/f noise: Linear regressions and confidence intervals in flicker noise context","volume":"52","author":"Vernotte","year":"2015","journal-title":"Metrologia"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1063\/1.3681448","article-title":"The statistical modeling of atomic clocks and the design of time sacles","volume":"83","author":"Levine","year":"2012","journal-title":"Rev. Sci. Instrum."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1088\/0026-1394\/49\/1\/008","article-title":"A new prediction algorithm for the generation of International Atomic Time","volume":"49","author":"Panfilo","year":"2012","journal-title":"Metrologia"},{"key":"ref_12","unstructured":"Institute of Electrical and Electronics Engineers (2009). IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology\u2013Random Instabilites, IEEE."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Riley, W. (2008). Handbook of Frequency Stability Analysis, NIST Special Publication.","DOI":"10.6028\/NIST.SP.1065"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Kobayashi, H., Mark, B.L., and Turin, W. (2012). Probability, Random Processes, and Statistical Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511977770"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"802","DOI":"10.1109\/5.381848","article-title":"Discrete simulation of colored noise and stochastic processes and l\/f\u2212\u03b1 power law generation","volume":"83","author":"Kasdin","year":"1995","journal-title":"Proc. IEEE"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1109\/19.286357","article-title":"Characterizing frequency stability: A continuous power-law model with discrete sampling","volume":"43","author":"Walter","year":"1994","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1109\/PROC.1966.4630","article-title":"A statistical model of flicker noise","volume":"54","author":"Barnes","year":"1966","journal-title":"Proc. IEEE"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1109\/19.278579","article-title":"Oscillator Noise Analysis: Multivariance Measurement","volume":"42","author":"Vernotte","year":"1993","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1088\/0026-1394\/50\/6\/586","article-title":"An adaptive oscillator noise analysis: using factor analysis","volume":"50","author":"Cheng","year":"2013","journal-title":"Metrologia"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1207","DOI":"10.1002\/cpa.20124","article-title":"Stable Signal Recovery from Incomplete and Inaccurate Measurements","volume":"59","author":"Candes","year":"2006","journal-title":"Commun. Pure Appl. 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