Translation sensitivity is a significant defect of classical discrete wavelet decomposition, which often results in false features in the decomposition results. Maximal overlap decomposition strategy can avoid this defect, but the computational efficiency is significantly reduced. Dual tree wavelet transform (DTWT), proposed by Kingsbury, achieves a good trade off between accuracy and efficiency and the merit of translation invariance (TI) is realized. The wavelet of DTWT is a complex-valued function, as follows. (1)ψC(t)=ψRe(t)+j⋅ψIm(t)where j is the imaginary number, defined as j=−1 and the two wavelet generators construct a Hilbert transform pair, which is given as: (2)ψIm(t)=Hilbert{ψRe(t)}

Although DTWT can alleviate the distortion of TV to the extracted features, it cannot solve the problem of transition band feature extraction in dyadic wavelet subspace. To address this problem, centralized multiresolution (CMR) is proposed by Chen [33]. The essential idea of CMR is the construction of an implicit wavelet packet (IWP). Let {x(n)|n=1,…,N} be a digitized signal of length N and dyadic wavelet packets (DWPs) at j-stage decomposition be {dwpj,1,…,dwpj,2j} with (3)CF{dwpj,1}<CF{dwpj,2}<⋯<CF{dwpj,2j}

IWPs can be generated using (4)iwpj−1,k(n)=dwpj,2k(n)+dwpj,2k+1(n)

A (j+1)-stage DTWT can generate 2j−1 IWPs. The spectral support and center frequencies of WP and IWP are demonstrated in Table 1. The CF of the IWP is just located at the edge of the spectral passband of the DWP, which can make up for the requirement of transition band feature extraction.

As shown in Eq. (4), the wavelet generators of IWP are constructed based on those of WPs. Therefore, the property of TI can be preserved. The distribution of IWPs as the scale of analysis deepens is shown in Fig. 1. There are IWP sets in which the IWPs share an identical CF and their spectral resolutions are constantly refined. For example, CF{iwp0,1}=CF{iwpj,2j−1} and Support{iwpj+1,2j}=Support{iwpj,2j−1}/2 hold or j∈Z+ (see Fig. 2).

Compared with the classical basis expansion method, the sparse representation (SR) allows the addition of other optimization constraints, which can better suppress the monitoring noise. For a discrete signal {x(n)}, the ℓ1−norm norm, ℓ2−norm, and ∞−norm are expressed as below: (5)||x||1=∑i=0N|x(n)| (6)||x||22=∑i=0N|x(n)|2 (7)||x||∞=max1≤n≤N,n∈Z|x(n)|

Let w be the representation coefficient vector to be solved, a typical optimization problem of SR is formulated as (8)arg minx||w||1s.t.y=Awwhere the matrix AN×K (N≪K) is a redundant dictionary with predetermined atoms and y is the observation signal. As stated above, the existence of noise is inevitable in the condition monitoring of EVs, a more feasible problem P1ε can be formulated as

where λ is the Lagrangian parameter. This kind of problem is called the basis pursuit problem in the literature.

In fast Fourier transform (FFT), an orthonormal basis is used for decomposing the input signal. The spectral interval for adjacent sinusoidal atoms is Δf=fs/N. The basis for FFT can be expressed as (10)A=[ϕ1ϕ2…ϕN]N×N

The column vector ϕi=exp⁡(j⋅2π(i−1)n/N), in which 1≤n≤N, is a complex-valued sinusoidal atom. The Picket fence effect will occur for signals that are sampled at non-integer periods. In order to overcome this disadvantage, an over-complete Fourier dictionary (OFD), shown in Eq. (11), is proposed to represent the signal. (11)ARFFT=[exp⁡(j2πNmn)]N×Mwhere m=R−1k(0≤k≤RN−1) and R is a positive integer. Equivalently AR is an OFD with redundancy R. In this paper, to solve the P1ε problem, the strategy of split augmented Lagrangian shrinkage algorithm (SALSA) can be employed. (12)w^=arg minw12||y−ARFFTw||22+||λ⊙w||1where λ is a vector, which contains Lagrangian parameters, with (λ⊙w)i=λi⋅wi. To solve this problem, a strategy of variable splitting via introducing new variables, can be utilized and is given as: (13)wopt=arg minw,u12||y−ARFFTw||22+||λ⊙u||1s.t.u−w=0

On the basis of augmented Lagrangian theory, the above problem has an equivalent matrix form, given as: (14)arg minz1,,z212||y−ARFFTz1||22+||λ⊙z2||1s.t.Cz−b=0where C=[I−I], b=0, and z=[z1z2].

Let H be the complex conjugate of a matrix and the thresholding function Soft_Thres(x,TV) be defined as in Eq. (15), the SFD algorithm can be summarized as in Table 2. Although iterations are employed in the algorithm, practices have proved that the whole algorithm can be completed in tens to hundreds of microseconds for signals with less than 10000 samples. (15){y=max([|x|−TV,0])y=xy/(y+TV)

A sinusoidal signal y(t)=Accos⁡(2πfct+π/6), in which Ac=1 and fc=(500+0.21)Hz, is synthesized as the dynamic signal without measurement noise. The sampling rate and the sampling number are set as 1000 Hz and 1000. The signal y(t) in the time domain and spectral domain are shown in Fig. 3. Because this signal is not a positive periodic sampled, there is a significant picket fence effect in the FFT spectrum.

Applying the SFD algorithm on the synthesized signal by setting R=10 and NITR=100, the associated spectrum is shown in Fig. 4. Only three spectral lines (250.1, 250.2 and 250.3 Hz) with a frequency close to the actual 250.21 Hz have large amplitudes. The amplitudes of other frequencies in the range (245, 255) are smaller than 10−3. The reason for this phenomenon is that SFD is described as a P1ε problem, which makes most of the linear representation coefficients non-zero.

In contrast to the FFT spectrum where the energy of the harmonic components leaks in the entire frequency domain, the energy of the signal in the SFD spectrum is compressed in a narrow band with a bandwidth of only 0.2 Hz. An approximation signal y~(t) can be reconstructed using three spectral lines. The energy ratio of y~(t) to y(t) is 99.89%, and the related quantization error is ||y(t)−y^(t)||∞=0.0275.

In order to demonstrate the performance of SFD, the spectrum is compared with the FFT spectrum and the FFT spectrum with Hanning window. As shown in Fig. 5, the bandwidth of the main lobe of the SFD spectrum is the smallest, and the decay rate is the fastest near the main lobe. On the other hand, it is found that the amplitude of the side lobe in the SFD spectrum is only about 1/1000 of that in the FFT spectrum. This shows that the SFD spectrum, based on the redundant Fourier dictionary, has a good sparse representation ability for the harmonic components.

To analyze the impact of the redundancy of ARFFT on sparse representation results, different redundancy values were tested. The SFD spectra with different values of redundancy are shown in Fig. 6. In the case of R=2, there are some side-lobes with large energy in the SFD spectrum. With the increase of dictionary redundancy, the attenuation rate of side-lobe is accelerated, while the energy occupied by the main-lobe becomes more prominent. On the other hand, the width of the main-lobe also decreases with the increase of redundancy, and the number of spectral lines representing harmonic components alone does not decrease. For example, when R=50, the number of main-lobe lines is 5.

In order to test the ability of SFD spectra to characterize noisy harmonic components, white noise is added to the simulation signal in the formula. The time domain waveform of a noisy signal yn(t) with SNR=10dB is shown in Fig. 7. Because of the noise, the characteristic information of the harmonic wave cannot be well identified.

The SFD spectrum, generated by the proposed method, is shown in Fig. 8. Due to the existence of white noise, there are some prominent energy concentration regions in the SFD spectrum (Fig. 8a). However, the spectral component of the harmonic component is still dominant, and there are only three spectral lines in the main lobe (Fig. 8b). Comparing the spectral lines of the main-lobes in Figs. 4 and 8b, they are almost the same. An approximation signal y~n(t) can be reconstructed using the spectral lines in the main-lobe. The related quantization error between y~n(t) and y(t) is ||y(t)−y^n(t)||∞=0.0542. The above analysis shows that the presence of noise does not affect the effectiveness of the SFD method. That is, the harmonic components can still be sparsely represented.

In this subsection, the performance of SFD on PICs is validated. In the time domain, a typical PIC can be modeled as

where NI is the number of impulses in the signal and T is the interval between adjacent impulses. The impulse in the PIC can be expressed as (17)imp(t)=e−βtsin⁡(2πfrest)where β>0 is the decaying rate and fres is the ringing frequency of the impulse. Let β=60, fres=160+π, T=0.9, a synthesized PIC and its noisy version (SNR=20dB) are shown in Fig. 9. The SFD spectra of the two simulated signals are shown in Fig. 10. The bandwidth of the main-lobes of each harmonic component is still very narrow, and the side lobes are rapidly attenuated. It can be concluded from the above results that SFD still has a good sparse representation ability for cycle-stationary processes such as PIC.

To verify the effectiveness of the proposed approach, a case study using actual signals from engineering experiments, is investigated. The tested mechanical part is a roller element bearing with slight peeling on the outer race. Specifications of the test bearing are shown in Table 3. This test bearing was removed from a certain type of electric drive vehicle. It provides mechanical support for the drive shaft of the AC motor and works under heavy load. In this test, the bearing was placed in a hydraulically driven loading device. Schematic diagrams of the test set-up are shown in Fig. 11.

The time domain waveform and the FFT spectrum of a record of vibration signals are shown in Fig. 12. It can be seen from the figure that there is a lot of noise, which complicates the identification of fault features. The proposed method is applied to the acceleration signal. In the fault diagnosis algorithm, fc=57.8 Hz, TPIC=0.5. The evaluated impulsiveness values of the decomposed wavelet subspaces are shown in Fig. 13. An optimal wavelet subspace is selected. It is an implicit wavelet packet. The central frequency and the theoretical passband are 400 Hz and [200,600] Hz. The kurtosis value of this IWP is 4.633. The associated time domain waveform and its envelope spectrum are shown in Fig. 14. In the time domain waveform, it can be found that the frequency of the periodic impact is very close to the ball pass frequency of outer-race (BPFO, Fig. 14a), and the energy proportion of the PIC component in the envelope spectrum is 0.68 (see Fig. 14b).

If the indicator of PER is not calculated in the algorithm, the processing results are shown in Fig. 15. The central frequency and the theoretical passband of the selected wavelet subspace are 5750 Hz and [5750,5800] Hz. The kurtosis value of the extracted feature is 9.00. Although the subspace extracted by the comparison method is significantly larger than that of the method proposed in this paper, it is not a periodic impact feature to characterize the failure of mechanical parts. In the envelope spectrum, even if the PSD algorithm proposed in this paper is used, there is no energy concentration region characterizing the harmonic components. The PER indicator of this wavelet subspace is calculated as 0.17. This value is significantly less than 0.5, so the wavelet subspace is identified as a non-periodic impact component and filtered out in the method proposed in this paper.