Applications of Fractional Lower Order Synchrosqueezing Transform Time Frequency Technology to Machine Fault Diagnosis

Synchrosqueezing transform (SST) is a high resolution time frequency representation technology for nonstationary signal analysis. +e short time Fourier transform-based synchrosqueezing transform (FSST) and the S transform-based synchrosqueezing transform (SSST) time frequency methods are effective tools for bearing fault signal analysis. +e fault signals belong to a non-Gaussian and nonstationary alpha (α) stable distribution with 1< α< 2 and even the noises being also α stable distribution. +e conventional FSST and SSSTmethods degenerate and even fail under α stable distribution noisy environment. Motivated by the fact that fractional low order STFT and fractional low order S-transform work better than the traditional STFT and S-transformmethods under α stable distribution noise environment, we propose in this paper the fractional lower order FSST (FLOFSST) and the fractional lower order SSST (FLOSSST). In addition, we derive the corresponding inverse FLOSSTand inverse FLOSSST. +e simulation results show that both FLOFSST and FLOSSST perform better than the conventional FSSST and SSST under α stable distribution noise in instantaneous frequency estimation and signal reconstruction. Finally, FLOFSST and FLOSSSTare applied to analyze the time frequency distribution of the outer race fault signal. Our results show that FLOFSSTand FLOSSST extract the fault features well under symmetric stable (SαS) distribution noise.


Introduction
Synchrosqueezing transform is a new time frequency analysis technology for the nonstationary signals. Its principle is to calculate time frequency distribution of the signal, then squeeze the frequency of the signal in time frequency domain, and rearrange its time frequency energy, so as to improve time frequency resolution greatly. Synchrosqueezing transform mainly includes continuous wavelet transform-based synchrosqueezing transform [1], short time Fourier transform-based synchrosqueezing transform [2], and S transform-based synchrosqueezing transform [3]. Synchrosqueezing transform methods have been widely applied in seismic signal analysis [4], biomedical signal processing [5,6], radar imaging [7], mechanical fault diagnosis, and other fields [8][9][10][11].
Daubechies et al. firstly gave synchrosqueezing transform concept based on the continuous wavelet transform and proposed a continuous wavelet transform-based synchrosqueezing transform (WSST) time frequency representation method and its inverse method. e method squeezes the time frequency energy of continuous wavelet transform in a certain frequency range to nearby instantaneous frequency of the signal, and the time frequency resolution was improved effectively [12], whereafter an adaptive wavelet transform-based synchrosqueezing transform based on WSST was brought up by Li et al. who applied a time-varying parameter to control the widths of the time frequency localization window according to the characteristics of signals [13]. e demodulated WSST and FSST methods have been proposed for instantaneous frequency estimation in [14,15]. To improve the ability of processing the nonstationary signals with fast varying instantaneous frequency, a new demodulated high order synchrosqueezing transform method was presented in [11,16], which can effectively show the time frequency distribution of the fault signal. Fourer et al. proposed a FSST method employing the synchrosqueezing transform and the Levenberg Marquardt reassignment in [17]; the idea of the method is very similar to the WSST method, which is reversible and adjustable. Yu et al. subsequently presented a synchroextracting short time Fourier transform, which is a postprocessing procedure of STFT [18]. Recently, they proposed an improved local maximum synchrosqueezing transform, which can discover more detailed features of the fault signals [19]. To compress and rearrange the S transform time frequency distribution of the signal, Huang et al. proposed a new S transform-based synchrosqueezing transform time frequency method employing synchrosqueezing transform and S transform, which can greatly improve time frequency resolution of S transform [20,21]. Subsequently, they modified the calculation formula of the instantaneous frequency of the SSST time frequency method by using the second derivative of time frequency spectrum to time and frequency and proposed a new second-order S transform-based synchrosqueezing transform method, which can obtain high time frequency resolution for the nonstationary signals whose instantaneous frequency varies nonlinearly with the time [22]. Recently, an adaptive short time Fourier transform-based synchrosqueezing transform method has been proposed with a time-varying parameter, and the corresponding 2nd-order adaptive FSST was also present in [23,24].
Recently, it is verified that probability density function (PDF) of the mechanical bearing fault signals has an obvious trail, which is a nonstationary and non-Gaussian distribution and belongs to α stable distribution (0 < α < 2); even the noises are also α stable distribution [25][26][27][28].
e performance of the above-mentioned methods degenerates under α stable distribution environment, which even fail. Some of the ways they apply are the fractional low order time frequency representation methods to analyze the signals, such as fractional low order short time Fourier transform (FLOSTFT) [29,30], fractional low order S transform (FLOST) [31,32], and fractional low order Wigner-Ville distribution [30]. However, the time frequency resolution of the methods is not very ideal and depends jointly on the geometry of the signal and the window function; even false spectral energies would be observed on the time frequency distribution at the locations where no spectral energies should exist. Hence, we propose the improved fractional low order short time Fourier transform-based synchrosqueezing transform (FLOFSST) and fractional low order S transformbased synchrosqueezing transform (FLOSSST) methods inspired by the FSSTand SSSTmethods in this paper, and the derivation procedures of the inverse FLOFSST and inverse FLOSSST are introduced. e simulation results show that the performances of the FLOFSST and FLOSSST time frequency representation methods are superior to the existing ones under α stable distribution noise environment; they have higher time frequency resolution than the existing FLOSTFT and FLOST methods and can better be suitable for the impulse noise environment than the FSST and SST methods. e IFLOFSST and IFLOSSST methods have smaller reconstruction MSEs than the IFSST and ISST methods under different α(α < 2) and GSNR. Finally, we apply the FLOFSST and FLOSSST time frequency representation methods to analyze the bearing out race fault signal. e simulation results show that the FLOFSST and FLOSSST methods can work in Gaussian noise and α stable distribution noise environment and extract the features of the outer race fault signal, which have some robustness; their performances are better than the existing FSST and SSST methods.
In this paper, the improved FLOFSST and FLOSSST time frequency representation technologies based on fractional lower order statistics and synchrosqueezing transform are presented for the bearing fault diagnosis under Gaussian and α stable distribution environment. e paper is structured in the following manner. α stable distribution and the bearing fault signals are introduced in Section 2. e improved fractional lower order synchrosqueezing transform methods and their inverse transforms are demonstrated in Section 3, and simulation comparisons with the existing time frequency representation methods based on second-order statistics are performed to demonstrate superiority of the improved methods. Applications of the improved methods for the outer race fault signals diagnosis are demonstrated in Section 4. Finally, conclusions and future research are given in Section 5.

α Stable Distribution and Bearing
Fault Signals 2.1. α Stable Distribution. Probability density function (PDF) of α stable distribution is expressed as where Gaussian process, α is its characteristic index, and its variance is infinite. When α � 2, which is Gaussian distribution, and when 0 < α < 2, it is low order stable distribution. μ is the location parameter and c is the dispersion coefficient, respectively. β is the symmetry parameter, when β � 0, which is called the symmetric α stable distribution (SαS). e waveforms of SαS stable distribution are shown in Figure 1 under α � 0.5, 1.0, 1.5, and 2.0, and their PDFs are demonstrated in Figure 2.   Mathematical Problems in Engineering experiments are conducted with a 2 HP reliance electric motor, and the acceleration data are measured at the proximal and distal of the motor bearings; the points include the drive end accelerometer (DE), fan end accelerometer (FE), and base accelerometer (BA). e normal signal is given in Figure 3(a), and the fault signals of inner race, ball, and outer race are shown in Figures 3(b)-3(d), respectively. We can know that the waveforms of the fault signals have a certain impulse.
In order to further verify the pulse characteristics of bearing failure signals, we use α stable distribution statistical model to estimate the parameters of inner race fault, ball fault, and outer race fault signals; the results are given in Table 1. As it can be seen, the characteristic index of the normal bearing signal is equal to 2, which is Gaussian distribution. However, the characteristic index of the bearing fault signals is greater than 1 but smaller than 2, and it belongs to non-Gaussian α stable distribution (α < 2).
PDFs  Table 1, and Figure 4 shows that PDFs of the fault signals are near symmetric. Hence, SαS distribution is a more concise and accurate statistical model for the bearing fault signals.

Fractional Lower
and its fractional low order short time Fourier transform (FLOSTFT) is given by [30] FLOSTFT where f is the frequency parameter; t and τ are the time parameter. τ denotes the displacement parameter on the time axis. h(t − τ) is the Gaussian window function related to the time. 〈p〉 denotes p order moment of the signal y(t) , and when y(t) is a complex signal, y 〈p〉 (t) � |y(t)| p− 1 · y ′ (t). α is the characteristic exponent of SαS distribution, and y ′ is the complex conjugate operation. Letting according to Plancherel's theorem and Fourier transform, we have where Y(λ) � ∞ − ∞ y 〈p〉 (t)e − j2πλt dt is fractional lower order Fourier transform (FLOFT) of y 〈p〉 (t) and λ denotes fre- (4) is converted from the time domain to the frequency domain.
Letting t − τ � η, ψ(λ) can be written as Substituting (5) into (4), we have Letting y(t) � A cos(2πf 0 t) and ω 0 � 2πf 0 , then y 〈p〉 (t) � |A cos(2πf 0 t)| p− 1 · sign[A cos(2πf 0 t)], and its FLOFT can be expressed as    Mathematical Problems in Engineering Substituting (7) into (6), then Fourier transform of ψ(t)and ψ(t) makes λ cluster around ω 0 and FLOSTFT y (τ, ω) will be concentrated e instantaneous frequency (IF) formula of FLOSTFT y (τ, f) can be written as After synchrosqueezing the frequency in (10), the discrete values FLOSTFT y (τ, f ℓ ) can be obtained. Letting the frequency points in FLOSTFT time frequency spectrum, f l (l � 1, 2, . . . , K) and Δf l � f l − f l− 1 . By centering f ℓ and ; then, fractional lower order STFT-based synchrosqueezing transform can be defined as e FLOFSST can "squeeze" a frequency interval to a frequency point in the time frequency domain; therefore, the process can greatly improve the time frequency resolution.
A multicomponent signal can be expressed as where k � 1, 2, . . . , N. en, the FLOSTFT of kth component can be expressed as FLOSTFT is just as linear as STFT; then e instantaneous frequency (IF) calculation method of kth component y k (t) may be written as Mathematical Problems in Engineering e corresponding instantaneous frequency calculation method of y(t) may be obtained by By substituting (15) and (17) into (12), we can obtain the fractional low order STFT-based synchrosqueezing transform of the multicomponent signal.
According to the definition of inverse STFT-based synchrosqueezing transform in [18,19], inverse fractional lower order STFT transform-based synchrosqueezing transform (IFLOFSST) of knd signal can be written by and the signal y(t) can be gotten employing y(t) � k k�1 y k (t).

Application Review.
In this section, we design the following experiments to compare the proposed FLOFSST method with the existing STFT, FLOSTFT, and FSST methods. e simulation signal y(n) contaminated by the noise is defined as x 1 (n) � cos(18πn), x 2 (n) � [1 + 0.3 cos(2n)] · cos 10πn + 1.2πn 1.8 + 0.6π sin(n) · e − 0.05n , where v(n) is SαS distribution noise or Gaussian noise. When the noise v(n) is SαS distribution noise, generalized signal to noise ratio (GSNR) can be used instead of SNR, which is expressed as where c is the dispersion coefficient of SαS distribution noise. According to the given GSNR, the amplitude of the signal x(n) can be written as Let SNR � − 5 dB, GSNR � 22 dB, and α � 0.8. e waveforms of x(n) and y(n) in time domain are shown in Figure 5. We apply the fractional lower order STFT transform-based synchrosqueezing transform method, the existing the STFT transform-based synchrosqueezing transform method, the fractional lower order STFT method, and traditional STFT method to estimate time frequency distribution of the signal x(n) under Gaussian distribution noise and SαS stable distribution noise; the simulation results are shown in Figures 6 and 7.
In order to compare the effectiveness of the IFSST and IFLOFSST methods, where K is the number of Monte-Carlo experiment, x(n) is the reconstructed signal employing the IFSST method or the IFLOFSST method. Letting GSNR � 20, the signal x(n) is reconstructed employing the IFSST and IFLOFSST methods under different α; their MSEs are shown in Figure 8(a). We apply the IFSST and IFLOFSST methods to reconstruct the signal x(n) when α � 1; GSNR changes from 14 dB to 24 dB; the simulations are shown in Figure 8  Gaussian noise environment (SNR � − 5 dB), but the synchrosqueezing methods have better performance. e STFT method in Figure 7(a) and FSST time frequency method in Figure 7(c) fail under SαS noise environment (GSNR � 22 dB and α � 0.8); the FLOSTFT method in Figure 7(b) can estimate out the time frequency representation of the signal x(n), but its effect is not very ideal. e improved FLOFSST method in Figure 7(d) can better get the time frequency representation of the signal x(n) under SαS noise environment, which has good toughness. e STFT and FSST are unsuitable for SαS noise environment, and the FLOSTFT method can work under SαS noise environment, but has poor time frequency resolution and is controlled by the window function. e FSST method has better time frequency resolution, but cannot work in SαS noise environment. e improved FLOFSST method can work under SαS noise environment and has high time frequency resolution. As a result, the FSST time frequency method is only suitable to analyze the signals under Gaussian noise environment, but the improved FLOFSST can work under Gaussian and SαS noise environment, which is robust. Figure 8(a) is MSE comparison under GSNR � 20 dB and different α; the experimental results show that MSEs of the IFSST method change from 2 dB to 230 dB when α changes from 0.2 to 2, but MSEs of the IFLOFSST method are 2 dB. Hence, the IFLOFSST method has obvious advantage in reconstructing the signal under different α; particularly, the advantage of the IFLOFSST method is more obvious when α < 1. Figure 8(b) is MSE comparison under α � 1 and different GSNR; we can know that reconstruction MSEs of the IFSST method have a large variation, which changes from 14 dB to 78 dB; however, MSEs of the IFLOFSST method are stable in 2 dB. Hence, the improved IFLOFSST method has better stability in reconstructing the original signal.

Fractional Lower Order S Transform-Based
Synchrosqueezing Transform 3.2.1. Principle. e fault machinery vibration signal contaminated by the noise may be given by where x(t) is fault vibration signal and v(t) is SαS distribution noise or Gaussian noise. Its S transform can be written as (23) and its fractional lower order S transform was defined as [28] FLOST where f is the frequency parameter and t is the time parameter. τ denotes the displacement parameter on the time axis. h(τ − t, f) is the Gaussian window function related to the frequency. Equation (24) can be written as )e − (t 2 /2) e j2πft , and its complex conjugate function is ψ( e right side of (27) is converted from the time domain to the frequency domain based on Plancherel's theorem and Fourier transform; then we obtain
rough synchrosqueezing the frequency with (32), the discrete values FLOST y (τ, f ℓ ) can be gotten. Letting the frequency points in FLOST time frequency spectrum, f l (l � 1, 2, . . . , K) and Δf l � f l − f l− 1 . By centering f ℓ and letting Δf � f ℓ − f ℓ− 1 , extend the synchrosqueezing process to the successive bins [f ℓ − (1/2)Δf, f ℓ + (1/2)Δf]; then, fractional lower order S transform-based synchrosqueezing transform can be written as For the calculation of IF and FLOSSST of a multicomponent signal, y(t), we have where k � 1, 2, . . . , N. en, the FLOST of kth component can be expressed as FLOST is just as linear as ST; then e IF calculation method of kth component y k (t) may be written as (38) e corresponding IF calculation method of y(t) may be obtained by By substituting (37) and (39) into (34), fractional low order STFT-based synchrosqueezing transform time frequency representation of the multicomponent signal can be obtained.

Inverse Fractional Lower Order S Transform-Based Synchrosqueezing Transform.
Multiplying e j2πfτ f − 1 on both sides of (28) and taking the integral to f, then For the real signal y(t), In the piecewise constant approximation corresponding to the binning in f, we have FLOST of the signal y(t) in (26) can be written as where |FLOST(τ, f)| is modulo operation of |FLOST(τ, f)| and j(τ, f) is its phase position. Substituting (44) to (34), we obtain Multiplying e j2πfτ f − 2 l on both sides of (45) and letting Substituting (46) into (43), we can deduce the following expression: where y(t) is real signal. According to y 〈p〉 (t) � |y(t)| p− 1 · sign[y(t)], inverse fractional lower order S transform-based synchrosqueezing transform (IFLOSSST) of y(t) can be written as We can reconstruct the signal y(t) in FLOSSST time frequency domain employing (48).  (19) is used as the experiment signal. e proposed fractional lower order S transform-based synchrosqueezing transform method, the existing the S transform-based synchrosqueezing transform method, the fractional lower order S transform method, and traditional S transform method are used to display time frequency distribution of the signal x(n) under Gaussian distribution noise (SNR � − 5 dB) and SαS stable distribution noise (GSNR � 22 dBand α � 0.8); the simulation results are shown in Figures 9 and 10. Letting GSNR � 22 dB and α � 1.4, the ISSST and IFLOSSST methods are used to reconstruct the original signal; the results are shown in Figure 11. In order to further compare the effectiveness of the ISSST and IFLOSSST methods, letting GSNR � 20 dB, the signal x(n) is reconstructed employing the ISSST and IFLOSSST methods under different α; their MSEs are shown in Figure 12(a). We apply the ISSST and IFLOSSST methods to reconstruct the signal x(n) when α � 1; GSNR changes from 14 dB to 24 dB; the simulations are shown in Figure 12(b). Figure 9 Figure 10. e results show that the ST method in Figure 10(a) and SSST method in Figure 10(c) fail; the FLOST method in Figure 10(b) can estimate out the time frequency distribution of the signal x(n), but its effect is not very ideal. e improved FLOSSST method in Figure 10(d) can better get the time frequency representation of the signal x(n), which has higher time frequency resolution. e reconstructed signal x(n) employing the ISSST method is shown in Figure 11(b) under SαS noise environment (GSNR � 22 dBand α � 1.2); it can be seen that the signal x(n) is covered by SαS noise; the ISSST method fails. Figure 11(b) is the reconstructed signal x(n) based on the IFLOSSST method under the same conditions; it shows that the reconstructed signal x(n) is very similar to the original signal x(n). Figure 12(a) is reconstruction MSE comparison under GSNR � 20 dB when α changes from 0.2 to 2; the results show that the reconstruction MSEs of the IFSST method change from 1 dB to 290 dB, but the reconstruction MSEs of the IFLOFSST method have an obvious low level, which are stable at about 2 dB. Hence, the IFLOFSSTmethod has obvious advantage in reconstructing the signal under different α; particularly, the advantage of the IFLOFSST method is more obvious when α < 1. Figure 12(b) is the reconstruction MSE comparison under α � 1 when GSNR changes from 14 dB to 78 dB; it shows that the reconstruction MSEs of the IFSST method have a large variation, but the reconstruction MSEs of the IFLOFSST method change from − 2 dB to 8 dB. Hence, the improved IFLOFSST method has better stability in reconstructing the signal.

Remarks.
As a result, the SSST time frequency method and the ISSST signal reconstruction method are only suitable to analyze and reconstruct the signals under Gaussian noise environment, but the improved FLOSSST and IFLOSSST methods can work in Gaussian and α stable distribution noise environment, which are robust.

Application Simulations
In this simulation, the experiment signal adopts the bearing outer race fault signal (DE) in Section 2. 0.2 seconds' data is selected as the test signal, which is collected at 12,000 samples per second, and N � 2400. e improved FLOFSST and FLOSSST methods are applied to analyze time frequency distribution of the outer race fault signal; the simulation results are shown in Figure 13. methods have a good vertical resolution; the gap between the impacts can be clearly seen, which regularly change. e time interval in A, B, C, D, E, and F is about 30 ms; the corresponding characteristic frequency of the outer race fault signal is about 33.333 Hz.
In order to further prove the advantages of the improved FLOFSST and FLOSSST methods, SαS distribution noise (α � 1 and GSNR � 22 dB) is added in the α stable distribution outer race fault signal as the background noise of actual working environment. e improved methods and existing methods are applied to demonstrate time frequency representation of the outer race fault signal; the simulations are shown in Figure 14.
e results show that the FSST method in Figure 14(a) and SST method in Figure 14(b) fail. However, the FLOFSST method in Figure 14(c) and FLOSSST method in Figure 14(d) can give out time frequency distribution of the fault signal under substandard conditions, which have certain ability in the horizontal and vertical time frequency representation, and we can know the dominant frequency and the time interval in A, B, C, D, E, and F, but the overall resolution is not high and needs to improve. Hence, the improved fractional lower order synchrosqueezing methods have better performance superiority than the existing synchrosqueezing methods, which are more suitable for fault analysis in complex environment and are robust.

Conclusions
α stable distribution is a more appropriate statistical model for the bearing fault signals. STFT transform-based synchrosqueezing transform and S transform-based synchrosqueezing transform are two new time frequency representation methods for nonstationary signal processing; their time frequency resolution can be greatly improved by rearranging the time frequency energy of the signals. In order to make the FSST and SST methods applicable to Gaussian and α stable distribution noise environment, the improved FLOFSST and FLOSSST time frequency representation methods are proposed by employing fractional low order statistics. e performances of the FLOFSST and FLOSSST methods are superior to the existing time frequency analysis methods; they have higher time frequency resolution than the existing FLOSTFT and FLOST methods because of the synchrosqueezing processing and can better suppress the impulse noise than the FSST and SST methods. e IFLOFSST and IFLOSSST methods have smaller reconstruction MSEs than the IFSST and ISSST methods under different α(α < 2) and GSNR. We can apply the improved methods to analyze the α stable distribution bearing fault signal; even α stable distribution noise environment, the fault characteristic frequency, the dominant frequency, and the other fault frequency features of the fault signals can be clearly obtained. In the future, we can also further study time frequency filtering technology based on the proposed IFLOFSST and IFLOSSST methods, and the methods have a good application prospect in the field of the bearing fault analysis and detection.

Data Availability
e data used to support the findings of this study are provided in the Supplementary Materials.

Conflicts of Interest
e authors declare that they have no conflicts of interest.