Constructing a wavelet-based envelope function for vibration signal analysis
Introduction
The Hilbert transform is usually applied to demodulate the vibration signal for extracting the signal envelope. The filter-based Hilbert transform posses the capability of real-time signal processing. But there is more serious distortion in the phase response. Furthermore, it is inconvenient in the filter design to derive the parameters corresponding to the cut-off frequencies of the passband. On the other hand, the FFT-based Hilbert transform is most commonly used by signal analyser and software. According to the FFT-based Hilbert transform, the traditional method for deriving the signal envelope has the advantage of high computing speed. Therefore, the FFT-based Hilbert transform is adaptable to signal analysis on line. But it is given in closed form, which has no time–frequency localisation feature [1]. In addition, the leakage error also exists in the FFT-based Hilbert transform [2].
The recent method for detecting vibration envelope is based on the wavelet transform. Because of the time–frequency localisation feature and the fast waveform convergence, the wavelet-based Hilbert transform is an effective method for vibration envelope detection [1], [4]. Selesnick [3] and Kingsbury [5] propose a dual-tree discrete wavelet to form an approximate Hilbert transform pair. They try to construct a Hilbert transform pair that calls for two wavelet transforms, where one wavelet is the Hilbert transform of the other [3]. Depending on the application, the discrete wavelet transform needs to decide the number of zero wavelet moments and the degree of approximation to the half sample delay. On the other hand, Morlet wavelet is often applied to constructing a continuous wavelet to perform the Hilbert transform. Liu and Qiu [1] construct an envelope function from Morlet wavelet, but it is only suitable for a single-frequency harmonic signal. Huang [2] proposes an algorithm for the linear combination of Morlet wavelets to apply in detecting vibration signal envelope of a mechanical system. However, it is inconvenient to apply in practice for the transformation function expressed in the combination function form. In addition, the normalised coefficient for the linear combination of analysis wavelets also needs to be determined, as the case may be.
In this paper, a wavelet-based envelope function is proposed to apply in vibration signal analysis. It is a complex function where the imaginary part is the Hilbert transform of the real part. By properly setting the function parameters for the wavelet-based envelope function, the satisfactory properties of fast waveform convergence, constant passband gain and less phase distortion can be obtained. Thus, it would be useful in the real-time signal processing.
Section snippets
Analysis wavelet for Hilbert transform
According to Morlet wavelet [1], the mother wavelet gi(t)∈L2(R) isThe Fourier transform of above wavelet can be expressed aswhich is a impulse frequency response with arbitrary centre frequency fi. When we let f=0, the frequency response is =0. It is implied that . Thus, the mother analysis wavelet gi(t) satisfies the admissibility condition. In the case of the wavelet transform, those elementary functions
Construction of envelope function
Considering the wavelet transform as a type of linear operation, the linear combination of the appropriate analysis wavelets gi,a,0(t) in Eq. (5) can be expressed as [2]with the constrain fL⩾1. For a signal f(t)∈L2(R), the wavelet transform is can be defined as [4]where Ga*(t) is a complex conjugate of Ga(t). It is easy to verify that the imaginary part of Ga*(t) is orthogonal to its own real part. So that, Ga*(t) can be used to form a
Experimental study
The resonance demodulation technique has been extensively used for the diagnosis of roller bearing defect. The purpose of this technique is to demodulate the vibration signal on a selected passband. Thus, the impulse response produced by defect could be returned to the original condition. In the following, the envelope function , as shown in Eq. (12), would be applied to demodulate the vibration signals of tapered roller bearings (SKF type 32208). There is an initial defect occurring on
Conclusion
This paper proposes a wavelet-based envelope function derived from Morlet wavelet. This envelope function is a complex function with the four parameters that are the low-cut-off frequency, the high-cut-off frequency, the dilation and the frequency difference. The passband of this envelope function can be arbitrarily determined by setting the low-cut-off frequency and the high-cut-off frequency. When designing the passband, the band slope is affected by the dilation and the flutter in passband
References (6)
- Liu ChongChun, Qiu Zhengding, A method based Morlet wavelet for extracting vibration signal envelope, Proceedings of...
- Huang Dishan, A wavelet-based algorithm for the Hilbert transform, Mechanical Systems and Signal Processing 10(2)...
Hilbert transform pairs of wavelet bases
IEEE Signal Processing Letters
(2001)
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2015, Mechanical Systems and Signal ProcessingCitation Excerpt :This is because the fault-related periodic impulse is a modulator to the high natural frequencies of the machine [5]. To demodulate the impact impulses, some wavelet-based demodulation methods have been studied [6–10] and confirmed to outperform the conventional filter-based or FFT-based Hilbert transform [7,9]. However, the extracted envelope by those methods is either at a single scale or among a scale band, which still confronts the contamination of in-band noise.