TY - JOUR

T1 - On hilbert spectral representation

T2 - A true time-frequency representation for nonlinear and nonstationary data

AU - Huang, Norden E.

AU - Chen, Xianyao

AU - Lo, Men Tzung

AU - Wu, Zhaohua

N1 - Funding Information:
N. E. Huang has been supported by a grant from the Federal Highway Administration, DTFH61-08-00028, and the grants NSC 98-2627-B-008-004 (Biology) and NSC 98-2611-M-008-004 (Geophysical) from the National Science Council, Taiwan, and finally a grant from NCU 965941 that have made the conclusion of this study possible. He is also supported by a KT Lee endowed Chair at NCU. Z.W. was also sponsored by the NSF of USA grant ATM-0917743.

PY - 2011/4

Y1 - 2011/4

N2 - 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.

AB - 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.

KW - Fourier spectrum

KW - Hilbert spectrum

KW - Nyquist frequency

KW - instantaneous frequency

KW - marginal Hilbert spectrum

KW - sampling rate

KW - spectral resolution

KW - uncertainty principle

UR - http://www.scopus.com/inward/record.url?scp=80052637482&partnerID=8YFLogxK

U2 - 10.1142/S1793536911000659

DO - 10.1142/S1793536911000659

M3 - 期刊論文

AN - SCOPUS:80052637482

SN - 1793-5369

VL - 3

SP - 63

EP - 93

JO - Advances in Adaptive Data Analysis

JF - Advances in Adaptive Data Analysis

IS - 1-2

ER -