Electric vibrators find wide applications in reliability testing, waveform generation, and vibration simulation, making their noise characteristics a topic of significant interest. While Variational Mode Decomposition (VMD) and Empirical Wavelet Transform (EWT) offer valuable support for studying signal components, they also present certain limitations. This article integrates the strengths of both methods and proposes an enhanced approach that integrates VMD into the frequency band division principle of EWT. Initially, the method decomposes the signal using VMD, determining the mode count based on residuals, and subsequently employs EWT decomposition based on this information. This addresses mode aliasing issues in the original method while capitalizing on VMD’s adaptability. Feasibility was confirmed through simulation signals and ultimately applied to noise signals from vibrators. Experimental results demonstrate that the improved method not only resolves EWT frequency band division challenges but also effectively decomposes signal components compared to the VMD method.
The electric vibrator serves as a crucial tool for mechanical environment testing, enabling the generation of controllable waveforms. As the noise signal encapsulates substantial information concerning the operational characteristics of the vibrator, often exhibiting a chaotic nature [
In recent years, numerous non-stationary signal analysis methods have been introduced. For instance, Potter introduced the short-time Fourier transform in 1947, which elucidates the evolutionary characteristics of signal spectra [
To optimize EMD, two typical methods have emerged: Variational Mode Decomposition (VMD) and Empirical Wavelet Transform (EWT). VMD, proposed by Dragomiretskiy in 2014, formulates signal decomposition as a variational problem, offering a more solid mathematical foundation [
Therefore, this article addresses the similarities and respective shortcomings of the VMD and EWT algorithms by proposing an enhanced approach that integrates VMD into the frequency band division principle of EWT. The validity of this method is confirmed through simulations and experimental data obtained from the noise signals of an electric vibrator. Consequently, leveraging VMD’s adaptability to address EWT’s mode number selection problem presents a promising solution.
The method proposed in this article effectively addresses issues such as the lack of adaptive mode selection and mode aliasing during feature extraction from non-stationary signals, as exemplified by electric vibrator noise. This improvement enhances both the efficiency and efficacy of signal decomposition, rendering it of significant application value and engineering importance.
VMD is a non-recursive decomposition method capable of breaking down a signal into several finite bandwidth components. Assuming the noise signal f, its constrained variational model is depicted in Equation (1).
{ min { u k } , { ω k } { ∑ k ‖ ∂ t [ ( δ ( t ) + j π t ) ∗ u k ( t ) ] e − j ω k t ‖ 2 2 } s . t . ∑ k u k = f (1)
where, δ ( t ) represents the impulse function, u k stands for the k-th component obtained through decomposition, and ω k denotes the center frequency corresponding to u k . The optimal solution of Equation (1) can be obtained using the augmented Lagrange function, as follows:
L ( { u k } , { ω k } , { λ } ) = α ∑ k ‖ ∂ t [ ( δ ( t ) + j π t ) ∗ u k ( t ) ] e − j ω k t ‖ 2 2 + ‖ f ( t ) − ∑ k u k ( t ) ‖ 2 2 + 〈 λ ( t ) , f ( t ) − ∑ k u k ( t ) 〉 (2)
where, α represents the quadratic penalty factor, λ denotes the Lagrange parameter. Employing the alternating direction multiplier method to derive the optimal solution of Equation (2), the decomposition component u k and its corresponding center frequency ω k are obtained as
u ^ k n + 1 = f ( ω ^ ) − ∑ i ≠ k u i ( ω ^ ) + λ ( ω ^ ) / 2 1 + 2 α ( ω − ω k ) 2 (3)
ω k n + 1 = ∫ 0 ∞ ω | u k ( ω ^ ) | 2 d ω ∫ 0 ∞ | u k ( ω ^ ) | 2 d ω (4)
where, u ^ k n + 1 is the Wiener filter of f ( ω ^ ) − ∑ i ≠ k u i ( ω ^ ) , and ω k n + 1 is the center frequency.
EWT automatically segments the frequency spectrum of signal f, subsequently constructing an orthogonal filter bank to decompose f into several components. The detail coefficients of EWT decomposition are calculated as
W ( n , t ) = F − 1 ( f ^ ⋅ ψ ^ ¯ n ) (5)
where, n denotes the n-th frequency band after spectrum segmentation, f ^ represents the Fourier transform of f; F − 1 denotes the inverse Fourier transform; ψ ^ ¯ n stands for the conjugate of the Fourier transform of wavelet ψ n .
The approximate coefficients for EWT decomposition is as follows:
W ( 0 , t ) = F − 1 ( f ^ ⋅ φ ^ ¯ 1 ) (6)
where, φ ^ ¯ 1 is the conjugate of the Fourier transform of scaling function φ 1 .
The signal reconstruction equation for EWT is
f ( t ) = W ( 0 , t ) ∗ φ 1 + ∑ n = 1 N − 1 W ( n , t ) ∗ ψ n (7)
where, * represents the convolution operator. The expression for the decomposition component is as
f 0 ( t ) = W ( 0 , t ) ∗ φ 1 (8)
f k ( t ) = W ( k , t ) ∗ ψ k (9)
The recursive algorithms like EMD can introduce modal aliasing and endpoint effects, distorting the obtained physical information, while VMD offers improvements in these aspects. However, the preset parameters can significantly impact the center frequency and narrow bandwidth of each mode. Moreover, if both the penalty factor and sampling frequency are incorrectly set, detecting parameter errors becomes challenging due to VMD’s high adaptability, despite resulting in problematic decomposition outcomes. Similarly, when using EWT for decomposing noisy signals, reasonable division of signal spectrum boundaries becomes imperative. Only through such division can effective components be separated from noise interference. Essentially, the center frequency and bandwidth of the modes after VMD are influenced by parameter settings. Although EWT can adaptively select bandwidth, determining the number of pattern decompositions remains necessary.
Hence, leveraging VMD’s capability to determine the number of modal decompositions based on actual conditions can address the challenge of selecting the number of EWT modes. To tackle these issues, this article proposes an enhanced approach that integrates VMD into the frequency band division principle of EWT, referred to hereinafter as the Enhanced Approach.
Implementation Steps of the Enhanced Approach:
Step 1: Employ VMD to decompose the noise signals and vary the number of modal decompositions. Monitor the residuals, and upon stabilization with a sudden decrease, record the number of modal decompositions at stability as Ns. The EWT frequency band division number is then N = Ns − 1.
Step 2: Reconstruct the noise signal decomposed by VMD with a modal number of N, denoted as S.
Step 3: Set the number of frequency bands to N, and utilize EWT to decompose the reconstructed signal S.
In this section, we will utilize the enhanced approach to analyze harmonic superposition signals, comparing them with VMD to illustrate the superiority of it.
To construct the f s i g ( t ) , we can use the following expression:
f s i g ( t ) = cos ( 8 π t ) + 1 / 3 cos ( 32 π t ) + 1 / 9 cos ( 96 π t ) + n ( t ) (10)
The simulated signal comprises three cosine signals with frequencies of 4 Hz, 16 Hz, and 48 Hz, respectively, alongside white noise with an amplitude of 1/50 and a power of -20 dB. Sampling points are set to N = 2000, and the sampling frequency to fs = 1000 Hz. The waveform of the simulated signal is depicted in the following
Utilizing VMD for signal decomposition, following the principles outlined in section 2.3, we observe that with a modal number set to 4, the peak residual value is 0.0049. Increasing the modal number to 5 yields a peak residual value of 0.0048, while further raising it to 6 results in a sharp drop to 0.0037. Consequently, the mode decomposition number indicating stability is 5. For EWT, the determined mode number by VMD is 4, with the corresponding frequency band division number set to 4.
The decomposition outcomes of the analog signal using VMD and EWT are presented in
In general, predicting the number of modes N for a signal segment without prior information can be challenging. Hence, Gilles proposed a straightforward N-value estimation method: detecting M maximum points in the Fourier spectrum of the signal and assembling them into a set {Mi}. These Mi values are then arranged in descending order within the set, yielding M1 > M2 > M3 > ... > MM. Setting a threshold at MM + β (M1 − MM), where β represents the relative amplitude ratio, the number of extreme points surpassing this threshold plus 1 corresponds to the value of N.
Following this principle, the spectrum and the calculated threshold are depicted in
The structure of the EDM-3200 electric vibration system depicted in
The time-domain signal of the noise is depicted in
The octave frequency spectrum of the noise at a specific measuring point on the electric vibrator is illustrated in
Similarly,
The VMD algorithm was utilized to decompose the noise signal, resulting in the extraction of IMFs and their corresponding frequency spectra, as illustrated in
determined by VMD is 9, with the corresponding frequency band division number set to 9. Among these decomposed components, low-frequency signals exhibit the largest proportion and pertain to the fundamental frequency signal. However, the second largest proportion occurs near 250 Hz. The obtained result is not in line with the conclusions drawn from the preceding analysis.
Hence, we utilize the EWT with a frequency band division of 9 to decompose the reconstructed signal after VMD, resulting in the extraction of Multi-resolution Analyses (MRAs) and their corresponding frequency spectrum, as illustrated in
Moreover, to validate the significance of frequency band division for the EWT method, we processed the signal using EWT algorithms with N = 8 and N = 10, respectively.
This article tackles the challenges posed by VMD mode aliasing and EWT’s limited adaptability in partitioning frequency bands during signal decomposition. Building upon these two algorithms, an enhanced approach that integrates VMD into the frequency band division principle of EWT was proposed. This method initially decomposes the signal through VMD, determines the number of modes based on residuals, and subsequently reconstructs the signal after VMD decomposition using EWT decomposition. The results demonstrate that the proposed method effectively leverages the advantages of both VMD and EWT, enabling efficient decomposition of the noise signal generated by the electric vibrator. Furthermore, this method mitigates the drawback of the EWT method’s lack of a basis for frequency band division. In comparison to the VMD method, this approach accurately decomposes noise signals into corresponding frequency components.
The authors declare no conflicts of interest regarding the publication of this paper.
Xu, Z.Y. and Chen, Z.W. (2024) Variational Mode Decomposition-Informed Empirical Wavelet Transform for Electric Vibrator Noise Analysis. Journal of Applied Mathematics and Physics, 12, 2320-2332. https://doi.org/10.4236/jamp.2024.126138