{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T08:27:54Z","timestamp":1774513674346,"version":"3.50.1"},"reference-count":21,"publisher":"World Scientific Pub Co Pte Lt","issue":"01n02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Adv. Adapt. Data Anal."],"published-print":{"date-parts":[[2011,4]]},"abstract":"<jats:p> As the original definition on Hilbert spectrum was given in terms of total energy and amplitude, there is a mismatch between the Hilbert spectrum and the traditional Fourier spectrum, which is defined in terms of energy density. Rigorous definitions of Hilbert energy and amplitude spectra are given in terms of energy and amplitude density in the time-frequency space. Unlike Fourier spectral analysis, where the resolution is fixed once the data length and sampling rate is given, the time-frequency resolution could be arbitrarily assigned in Hilbert spectral analysis (HSA). Furthermore, HSA could also provide zooming ability for detailed examination of the data in a specific frequency range with all the resolution power. These complications have made the conversion between Hilbert and Fourier spectral results difficult and the conversion formula is elusive until now. We have derived a simple relationship between them in this paper. The conversion factor turns out to be simply the sampling rate for the full resolution cases. In case of zooming, there is another additional multiplicative factor. The conversion factors have been tested in various cases including white noise, delta function, and signals from natural phenomena. With the introduction of this conversion, we can compare HSA and Fourier spectral analysis results quantitatively. <\/jats:p>","DOI":"10.1142\/s1793536911000659","type":"journal-article","created":{"date-parts":[[2011,9,8]],"date-time":"2011-09-08T09:53:56Z","timestamp":1315475636000},"page":"63-93","source":"Crossref","is-referenced-by-count":65,"title":["ON HILBERT SPECTRAL REPRESENTATION: A TRUE TIME-FREQUENCY REPRESENTATION FOR NONLINEAR AND NONSTATIONARY DATA"],"prefix":"10.1142","volume":"03","author":[{"given":"NORDEN E.","family":"HUANG","sequence":"first","affiliation":[{"name":"Research Center for Adaptive Data Analysis, National Central University, Zhongli, Taiwan 32001, Taiwan"}]},{"given":"XIANYAO","family":"CHEN","sequence":"additional","affiliation":[{"name":"The First Institute of Oceanography, State Ocean Administration, Qingdao, China"}]},{"given":"MEN-TZUNG","family":"LO","sequence":"additional","affiliation":[{"name":"Research Center for Adaptive Data Analysis, National Central University, Zhongli, Taiwan 32001, Taiwan"}]},{"given":"ZHAOHUA","family":"WU","sequence":"additional","affiliation":[{"name":"Department of Earth, Ocean and Atmospheric Science, &amp; Center for Ocean-Atmospheric Prediction Studies, Florida State University, Tallahassee, FL 32306, USA"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf2","unstructured":"P.\u00a0Flandrin, Time-Frequency\/Time-Scale Analysis (Academic Press, San Diego, 1995)\u00a0p. 386."},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1109\/LSP.2003.821662"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1142\/S0219691304000561"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1142\/9789812703347_0003"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1142\/S179353690900031X"},{"key":"rf8","volume":"3","author":"Hou T. 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