On the quadrature approximation to the Hilbert transform of modulated signals

Nuttall, Albert H.; Bedrosian, E.

doi:10.1109/proc.1966.5138

Cited by 220 publications

(103 citation statements)

References 4 publications

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“…To estimate instantaneous frequency accurately via commonly used Hilbert transform, it requires the signal to be mono-component and satisfy Bedrosian and Nuttall theorem, i.e., the spectra of carrier signal and amplitude envelope do not overlap in frequency domain [37]. However, the IMFs obtained from VMD do not satisfy this condition automatically.…”

Section: Empirical Am-fm Decompositionmentioning

confidence: 99%

“…We propose a criterion to select the sensitive IMF for further demodulation analysis, based on the modulation characteristics of gear vibration signals and the wavelet packet like decomposition property of VMD. (3) Accurate instantaneous frequency estimation is necessary for effective frequency demodulation analysis, but conventional Hilbert transform (HT) based approach is subject to the constraint imposed by Bedrosian and Nuttall theorem [37]. We preprocess the IMFs with empirical AM-FM decomposition by exploiting its capability to separate the carrier signal from amplitude envelope of an AM-FM signal, to satisfy the non-overlap requirement of amplitude envelope and carrier signal spectra by Bedrosian and Nuttall theorem.…”

Section: Introductionmentioning

confidence: 99%

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Planetary Gearbox Fault diagnosis via Joint Amplitude and Frequency Demodulation Analysis Based on Variational Mode Decomposition

2017

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Abstract:Planetary gearbox vibration signals have strong modulation features due to the amplitude modulation and frequency modulation (AM-FM) effect of gear faults, as well as the amplitude modulation (AM) effect of time-varying vibration transfer paths, on gear meshing vibrations. This results in an involute sidebands structure in Fourier spectrum, possibly misleading fault diagnosis. The modulating frequency of both amplitude modulation (AM) and frequency modulation (FM) parts is closely related to the gear fault characteristic frequency. This inspires the idea of joint amplitude and frequency demodulation analysis, thus addressing the complex sidebands issue inherent in Fourier spectrum. Demodulation analysis requires mono-component signals for accurate estimation of instantaneous frequency, and proper selection of an AM-FM component sensitive to gear fault. To this end, we firstly decompose the complex signal into intrinsic mode functions (IMFs) via variational mode decomposition (VMD), by exploiting its capability in decomposing complex modulated signal into constituent AM-FM components. For effective application of VMD in complex planetary gearbox signal analysis, we propose a method to determine a key parameter in VMD, i.e., the number of IMFs to be separated. For accurate instantaneous frequency estimation, we decompose IMFs via empirical AM-FM decomposition, to remove the influence of AM on instantaneous frequency estimation. Then, we select the sensitive IMF that contains the main gear fault information for further demodulation analysis. In order to properly select the sensitive IMF, we propose a criterion based on the gear vibration characteristics and the VMD properties. Finally, we obtain the amplitude and frequency demodulated spectra by applying Fourier transform to the amplitude envelope and instantaneous frequency of the selected sensitive IMF. According to the characteristics exhibited in the demodulated spectra, we can detect planetary gearbox fault. The proposed method is illustrated via a numerical simulated planetary gearbox vibration signal, and is further validated using lab experimental vibration signals of a planetary gearbox. Faults on all the three types of gear (sun, planet and ring) are successfully identified.

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Section: Empirical Am-fm Decompositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Planetary Gearbox Fault diagnosis via Joint Amplitude and Frequency Demodulation Analysis Based on Variational Mode Decomposition

2017

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“…Before proceeding, we comment that most time series we could acquire in the real world are real, so we consider only real model but not the exponential complex model. We mention that obtaining the imaginary part of a given real signal to recover its exponential complex form is not a trivial problem [30,31,32].…”

Section: The Adaptive Harmonic Model and Instantaneous Nyquist Ratementioning

confidence: 99%

When Interpolation-Induced Reflection Artifact Meets Time–Frequency Analysis

Lin¹,

Flandrin

2016

IEEE Trans. Biomed. Eng.

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ABSTRACT. While extracting the temporal dynamical features based on the time-frequency analyses, like the reassignment and synchrosqueezing transform, attracts more and more interest in bio-medical data analysis, we should be careful about artifacts generated by interpolation schemes, in particular when the sampling rate is not significantly higher than the frequency of the oscillatory component we are interested in. In this study, we formulate the problem called the reflection effect and provide a theoretical justification of the statement. We also show examples in the anesthetic depth analysis with clear but undesirable artifacts. The results show that the artifact associated with the reflection effect exists not only theoretically but practically. Its influence is pronounced when we apply the time-frequency analyses to extract the time-varying dynamics hidden inside the signal. In conclusion, we have to carefully deal with the artifact associated with the reflection effect by choosing a proper interpolation scheme.

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“…In (9), we chose to conduct Hilbert transform on IMFN k (x) rather than on IMF k (x), and this to avoid the Bedrosian and Nuttall restrictions [9][10][11][12].…”

Section: Analytic Signal Constructionmentioning

confidence: 99%

“…It consists of three parts: empirical mode decomposition (EMD), intrinsic mode functions (IMF) normalization, and Hilbert spectral analysis (HSA). The NHHT was developed by Huang et al [4][5][6][7] to overcome the limitations set by the Bedrosian and Nuttall theorems when applying the regular Hilbert transform [8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Single Frame Fringe Pattern Analysis for Phase Recovery with Analytic Signal

et al. 2012

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We consider a new application of the normalized Hilbert-Huang transform to extract directly the phase from a single fringe pattern. We present a technique to provide, with good accuracy, the phase distribution from a single interferogram without unwrapping step and this by a new exploitation of the analytic signal corresponding to each intrinsic mode function, resulting from onedimensional empirical mode decomposition of the fringe pattern. A theoretical analysis was carried out for this technique, followed by computer simulations and a real experimental fringe pattern for verification.

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On the quadrature approximation to the Hilbert transform of modulated signals

Cited by 220 publications

References 4 publications

Planetary Gearbox Fault diagnosis via Joint Amplitude and Frequency Demodulation Analysis Based on Variational Mode Decomposition

Planetary Gearbox Fault diagnosis via Joint Amplitude and Frequency Demodulation Analysis Based on Variational Mode Decomposition

When Interpolation-Induced Reflection Artifact Meets Time–Frequency Analysis

Single Frame Fringe Pattern Analysis for Phase Recovery with Analytic Signal

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