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Research ArticleApplications of Fractional Lower Order SynchrosqueezingTransform Time Frequency Technology to MachineFault Diagnosis

Haibin Wang 1 and Junbo Long 2

1College of Information Science and Engineering Technology, Jiujiang University, Jiujiang, China2College of Electronics and Engineering, Jiujiang University, Jiujiang, China

Correspondence should be addressed to Junbo Long; [email protected]

Received 13 May 2020; Revised 28 June 2020; Accepted 13 July 2020; Published 3 August 2020

Academic Editor: Rosa M. Benito

Copyright © 2020 Haibin Wang and Junbo Long. +is is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work isproperly cited.

Synchrosqueezing transform (SST) is a high resolution time frequency representation technology for nonstationary signalanalysis. +e short time Fourier transform-based synchrosqueezing transform (FSST) and the S transform-based syn-chrosqueezing transform (SSST) time frequency methods are effective tools for bearing fault signal analysis. +e fault signalsbelong to a non-Gaussian and nonstationary alpha (α) stable distribution with 1< α< 2 and even the noises being also α stabledistribution. +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 STFTand 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 inverseFLOSSST. +e simulation results show that both FLOFSST and FLOSSST perform better than the conventional FSSST and SSSTunder α stable distribution noise in instantaneous frequency estimation and signal reconstruction. Finally, FLOFSST andFLOSSSTare applied to analyze the time frequency distribution of the outer race fault signal. Our results show that FLOFSSTandFLOSSST extract the fault features well under symmetric stable (SαS) distribution noise.

1. Introduction

Synchrosqueezing transform is a new time frequencyanalysis technology for the nonstationary signals. Its prin-ciple is to calculate time frequency distribution of the signal,then squeeze the frequency of the signal in time frequencydomain, and rearrange its time frequency energy, so as toimprove time frequency resolution greatly. Syn-chrosqueezing transform mainly includes continuouswavelet transform-based synchrosqueezing transform [1],short time Fourier transform-based synchrosqueezingtransform [2], and S transform-based synchrosqueezingtransform [3]. Synchrosqueezing transform methods havebeen widely applied in seismic signal analysis [4], biomedicalsignal processing [5, 6], radar imaging [7], mechanical faultdiagnosis, and other fields [8–11].

Daubechies et al. firstly gave synchrosqueezing trans-form concept based on the continuous wavelet transformand proposed a continuous wavelet transform-based syn-chrosqueezing transform (WSST) time frequency repre-sentation method and its inverse method. +e methodsqueezes the time frequency energy of continuous wavelettransform in a certain frequency range to nearby instanta-neous frequency of the signal, and the time frequencyresolution was improved effectively [12], whereafter anadaptive wavelet transform-based synchrosqueezing trans-form based onWSSTwas brought up by Li et al. who applieda time-varying parameter to control the widths of the timefrequency localization window according to the character-istics of signals [13]. +e demodulated WSST and FSSTmethods have been proposed for instantaneous frequencyestimation in [14, 15]. To improve the ability of processing

HindawiMathematical Problems in EngineeringVolume 2020, Article ID 3983242, 19 pageshttps://doi.org/10.1155/2020/3983242

mailto:[email protected]://orcid.org/0000-0002-5531-597Xhttps://orcid.org/0000-0002-8395-4533https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/3983242

the nonstationary signals with fast varying instantaneousfrequency, a new demodulated high order synchrosqueezingtransform method was presented in [11, 16], which caneffectively show the time frequency distribution of the faultsignal. Fourer et al. proposed a FSSTmethod employing thesynchrosqueezing transform and the Levenberg Marquardtreassignment in [17]; the idea of themethod is very similar tothe WSST method, which is reversible and adjustable. Yuet al. subsequently presented a synchroextracting short timeFourier transform, which is a postprocessing procedure ofSTFT [18]. Recently, they proposed an improved localmaximum synchrosqueezing transform, which can discovermore detailed features of the fault signals [19]. To compressand rearrange the S transform time frequency distribution ofthe signal, Huang et al. proposed a new S transform-basedsynchrosqueezing transform time frequency methodemploying synchrosqueezing transform and S transform,which can greatly improve time frequency resolution of Stransform [20, 21]. Subsequently, they modified the calcu-lation formula of the instantaneous frequency of the SSSTtime frequency method by using the second derivative oftime frequency spectrum to time and frequency and pro-posed a new second-order S transform-based syn-chrosqueezing transform method, which can obtain hightime frequency resolution for the nonstationary signalswhose instantaneous frequency varies nonlinearly with thetime [22]. Recently, an adaptive short time Fourier trans-form-based synchrosqueezing transform method has beenproposed with a time-varying parameter, and the corre-sponding 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 obvioustrail, which is a nonstationary and non-Gaussian distribu-tion and belongs to α stable distribution (0< α< 2); even thenoises are also α stable distribution [25–28]. +e perfor-mance of the above-mentioned methods degenerates underα stable distribution environment, which even fail. Some ofthe ways they apply are the fractional low order time fre-quency representation methods to analyze the signals, suchas fractional low order short time Fourier transform(FLOSTFT) [29, 30], fractional low order S transform(FLOST) [31, 32], and fractional low order Wigner-Villedistribution [30]. However, the time frequency resolution ofthe methods is not very ideal and depends jointly on thegeometry of the signal and the window function; even falsespectral energies would be observed on the time frequencydistribution at the locations where no spectral energiesshould exist. Hence, we propose the improved fractional loworder short time Fourier transform-based synchrosqueezingtransform (FLOFSST) and fractional low order S transform-based synchrosqueezing transform (FLOSSST) methodsinspired by the FSSTand SSSTmethods in this paper, and thederivation procedures of the inverse FLOFSST and inverseFLOSSST are introduced. +e simulation results show thatthe performances of the FLOFSST and FLOSSST time fre-quency representation methods are superior to the existingones under α stable distribution noise environment; theyhave higher time frequency resolution than the existing

FLOSTFTand FLOSTmethods and can better be suitable forthe impulse noise environment than the FSST and SSTmethods. +e IFLOFSST and IFLOSSST methods havesmaller reconstruction MSEs than the IFSST and ISSTmethods under different α(α< 2) and GSNR. Finally, weapply the FLOFSST and FLOSSST time frequency repre-sentation methods to analyze the bearing out race faultsignal. +e simulation results show that the FLOFSST andFLOSSSTmethods can work in Gaussian noise and α stabledistribution noise environment and extract the features ofthe outer race fault signal, which have some robustness; theirperformances are better than the existing FSST and SSSTmethods.

In this paper, the improved FLOFSSTand FLOSSST timefrequency representation technologies based on fractionallower order statistics and synchrosqueezing transform arepresented for the bearing fault diagnosis under Gaussian andα stable distribution environment.+e paper is structured inthe following manner. α stable distribution and the bearingfault signals are introduced in Section 2. +e improvedfractional lower order synchrosqueezing transform methodsand their inverse transforms are demonstrated in Section 3,and simulation comparisons with the existing time fre-quency representation methods based on second-orderstatistics are performed to demonstrate superiority of theimproved methods. Applications of the improved methodsfor the outer race fault signals diagnosis are demonstrated inSection 4. Finally, conclusions and future research are givenin Section 5.

2. α Stable Distribution and BearingFault Signals

2.1. α Stable Distribution. Probability density function(PDF) of α stable distribution is expressed as

φ(t) � exp jμt − c|t|α[1 + jβ sign(t)ω(τ, α)] , (1)

where sign(t) �1, t> 0,0, t � 0,− 1, t< 0,

⎧⎪⎨

⎪⎩, ω(τ, α) �

tan(απ/2), α≠ 1,(2/π)log|τ|, α � 1, . α stable distribution is a generalized

Gaussian process, α is its characteristic index, and its var-iance is infinite. When α � 2, which is Gaussian distribution,and when 0< α< 2, it is low order stable distribution. μ is thelocation parameter and c is the dispersion coefficient, re-spectively. β is the symmetry parameter, when β � 0, whichis called the symmetric α stable distribution (SαS). +ewaveforms of SαS stable distribution are shown in Figure 1under α � 0.5, 1.0, 1.5, and 2.0, and their PDFs are dem-onstrated in Figure 2.

2.2. Bearing Fault Signals. +e real bearing fault signals dataare obtained from the Case Western Reserve University(CWRU) bearing data center [33]. +e experimentalequipment adopts 6205-2RS JEM SKF type bearing, theouter race diameter is 20.472 inches, and the inner race andthe ball diameter are 0.9843 inches and 0.3126 inches,

2 Mathematical Problems in Engineering

respectively. +e bearing outer race thickness is 0.5906inches, motor load is 0HP, and motor speed is 1797 rpm.+e bearing faults of inner race, ball, and outer race are set,

and the fault diameters are all 0.021 inches.+e fault data arecollected at 12,000 samples per second, and the outer raceposition relative to load zone centered at 6:00. +e

0 100 200 300 400 500 600 700 800 900 1000–1.5

–1–0.5

00.5

11.5

22.5

33.5

×105

N

Am

plitu

de

(a)

0 100 200 300 400 500 600 700 800 900 1000–500

0

500

1000

N

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(b)

0 100 200 300 400 500 600 700 800 900 1000–80

–60

–40

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plitu

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(c)

0 100 200 300 400 500 600 700 800 900 1000–5–4–3–2–1

012345

N

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(d)

Figure 1: +e waveform of SαS stable distribution when (a) α � 0.5, (b) α� 1.0, (c) α� 1.5, and (d) α� 2.0.

–5 –4 –3 –2 –1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

Symmetric α-stable densities, β = 0, γ = 1, and δ = 0

α = 0.5

α = 1.0

α = 1.5

α = 2.0

(a)

Symmetric α-stable densities, β = 0, γ = 1, and δ = 0

α = 0.5

α = 1.0

α = 1.5

α = 2.0

1.5 2 2.5 3 3.5 4 4.5 50

0.020.040.060.08

0.10.120.140.160.18

0.2

(b)

Figure 2: PDFs of SαS stable distribution under different α.

Mathematical Problems in Engineering 3

experiments are conducted with a 2HP reliance electricmotor, and the acceleration data are measured at theproximal and distal of the motor bearings; the points includethe drive end accelerometer (DE), fan end accelerometer(FE), and base accelerometer (BA). +e normal signal isgiven 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 acertain impulse.

In order to further verify the pulse characteristics ofbearing failure signals, we use α stable distribution statisticalmodel to estimate the parameters of inner race fault, ballfault, and outer race fault signals; the results are given inTable 1. As it can be seen, the characteristic index of thenormal bearing signal is equal to 2, which is Gaussiandistribution. However, the characteristic index of thebearing fault signals is greater than 1 but smaller than 2, andit belongs to non-Gaussian α stable distribution (α< 2).

PDFs of the inner race fault, ball fault, and outer racefault signals are shown in Figures 4(a)–4(f), respectively.Figures 4(b), 4(d), and 4(f ) show that the PDFs of the normalhave no tail, but the PDFs of the fault signals have heavy tail,

and the PDFs of the fault signals in DE are especially serious.+e parameters β of the fault signals are approximately equalto zero in Table 1, and Figure 4 shows that PDFs of the faultsignals are near symmetric. Hence, SαS distribution is amore concise and accurate statistical model for the bearingfault signals.

3. Fractional Lower Order SynchrosqueezingTransform Methods

3.1. Fractional Lower Order STFT Transform-BasedSynchrosqueezing Transform Method

3.1.1. Principle. Short time Fourier transform (STFT) of thefault machinery vibration signal contaminated by SαS dis-tribution noise or Gaussian noise y(t) can be written as

STFTy(τ, f) � +∞

− ∞y(t)h(t − τ)e− j2πftdt, (2)

and its fractional low order short time Fourier transform(FLOSTFT) is given by [30]

FLOSTFTx(τ, f) � Fτ⟶fx

〈p〉(t)h(t − τ) �

+∞

− ∞x

〈p〉(t)h(t − τ)e− j2πftdt, (3)

where f is the frequency parameter; t and τ are the timeparameter. τ denotes the displacement parameter on thetime axis. h(t − τ) is the Gaussian window function relatedto the time. 〈p〉 denotes p order moment of the signal y(t)(0≤p − 1< α/2). When y(t) is a real signal, y〈p〉(t) �

|y(t)|p− 1 · sign[y(t)], sign[y(t)] �1 y(t)> 00 y(t) � 0− 1 y(t)< 0

⎧⎪⎨

⎪⎩, and

when y(t) is a complex signal, y〈p〉(t) � |y(t)|p− 1 · y′(t). αis the characteristic exponent of SαS distribution, and y′ isthe complex conjugate operation.

Letting ψ(t) � h(t − τ)ej2πft, its complex conjugatefunction is ψ′(t) � h(t − τ)e− j2πft; according to Plancherel’stheorem and Fourier transform, we have

FLOSTFTy(τ, f) � ∞

− ∞y

〈p〉(t)ψ(t)dt �

12π

∞

− ∞Y(λ)ψ(λ)dλ,

(4)

where Y(λ) � ∞− ∞ y

〈p〉(t)e− j2πλtdt is fractional lower orderFourier transform (FLOFT) of y〈p〉(t) and λ denotes fre-quency variant. ψ(λ) � ∞

− ∞ ψ(t)e− j2πλtdt is Fourier trans-

form of ψ(t), and ψ(t) is complex conjugate of ψ(λ). +e

right side of (4) is converted from the time domain to thefrequency domain.

Letting t − τ � η, ψ(λ) can be written as

ψ(λ) � ∞

− ∞h(t − τ)ej2πfte− j2πλtdt

� ∞

− ∞h(η)ej2πf(τ+η)e− j2πλ(τ+η)dη

� e− j2πτ(λ− f)

∞

− ∞h(η)ej2πη(f− λ)dη

� ψ(f − λ)e− j2πτ(λ− f)

� ψ(f − λ)ej2πτ(λ− f).

(5)

Substituting (5) into (4), we have

FLOSTFTy(τ, f) �12π

∞

− ∞Y(λ)ψ(f − λ)ej2πτ(λ− f)dλ.

(6)

Letting y(t) � A cos(2πf0t) and ω0 � 2πf0, theny〈p〉(t) � |A cos(2πf0t)|

p− 1 · sign[A cos(2πf0t)], and itsFLOFT can be expressed as

Y(λ) � ∞

− ∞y

〈p〉(t)e

− j2πftdt � |A|p− 1πp− 1sign(A) δ λ + ω0( + δ λ − ω0( . (7)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–0.4–0.2

00.20.4

Normal (DE)

Time (s)

Am

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00.20.4

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Time (s)

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(a)

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0.5Inner race (BA)

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Inner race (DE)

Time (s)

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Time (s)

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(b)

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0.2Ball (BA)

Time (s)

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Ball (DE)

Time (s)

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0

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Time (s)

Am

plitu

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(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–1

0

1Outer race (BA)

Time (s)

Am

plitu

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–10

010

Outer race (DE)

Time (s)

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–2

0

2Outer race (FE)

Time (s)

Am

plitu

de

(d)

Figure 3: +e waveform of the bearing fault signals. (a) +e normal signals in DE and FE; (b) the inner race fault signals in BA, DE, and FE;(c) the ball fault signals in BA, DE, and FE; (d) the outer race fault signals in BA, DE, and FE.

Table 1: +e parameters of the bearing fault signals based on α stable distribution.

Parameters α β c μ

Normal DE 2.000 − 0.2863 0.0532 0.0121FE 2.000 1.000 0.0583 0.0236

Inner race faultBA 1.7682 0.0872 0.0590 0.0062DE 1.4195 0.0155 0.2407 0.0175FE 1.8350 0.0322 0.1495 0.0291

Ball faultBA 1.9790 0.0592 0.0293 0.0055DE 1.8697 0.1215 0.0772 0.0193FE 1.998 − 0.0371 0.0674 0.0321

Outer race faultBA 1.6077 − 0.1731 0.0530 0.0012DE 1.1096 0.0433 0.1341 0.0367FE 1.5435 − 0.0169 0.0968 0.0296


PDF

Normal PDFBA PDF

DE PDFFE PDF

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Inner race PDF

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(a)

Normal PDFBA PDF

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Inner race PDF

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PDF

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DE PDFFE PDF

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PDF

–1 –0.5 0 0.5 1x

(e)

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DE PDFFE PDF

1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2Outer race PDF

x

PDF

(f )

Figure 4: PDFs of the normal and bearing fault signals. (a), (b) PDFs of the normal and inner race fault signals in DE, FE, and BA; (c), (d)PDFs of the normal and ball fault signals in DE, FE, and BA; (e), (f ) PDFs of the normal and outer race fault signals in DE, FE, and BA.


Substituting (7) into (6), then

FLOSTFTy(τ,ω) �|A|p− 1πp− 2sign(A)

2∞

− ∞δ λ + ω0( + δ λ − ω0( ψ(ω − λ)e

jτ(λ− ω)dλ �|A|p− 1πp− 2sign(A)

2e

− j ω− ω0( )τ ψ ω − ω0( .

(8)

Fourier transform of ψ(t)and ψ(t) makes λ clusteraround ω0 and FLOSTFTy(τ,ω) will be concentratedaround τ � ω0/ω. Substituting f � ω/2π into (8), then

FLOSTFTy(τ, f) �|A|p− 1πp− 2sign(A)

2e

− j2πτ f− f0( ) ψ 2πf − 2πf0( .

(9)

+e instantaneous frequency (IF) formula ofFLOSTFTy(τ, f) can be written as

Fy(τ, f) � f +1

j2πFLOSTFTy(τ, f)z FLOSTFTy(τ, f)

zτ, (10)

z FLOSTFTy(τ, f) zτ

� − j|A|p− 1πp− 1sign(A) f − f0( e

− j2πτ f− f0( ) ψ 2πf − 2πf0( . (11)

After synchrosqueezing the frequency in (10), the dis-crete values FLOSTFTy(τ, fℓ) can be obtained. Letting thefrequency points in FLOSTFT time frequency spectrum,fl(l � 1, 2, . . . , K) and Δfl � fl − fl− 1. By centering fℓ and

letting Δf � fℓ − fℓ− 1, the synchrosqueezing calculation isextended to the successive bins[fℓ − (1/2)Δf, fℓ + (1/2)Δf]; then, fractional lower orderSTFT-based synchrosqueezing transform can be defined as

FLOFSSTy τ, fℓ( � (Δf)− 1

fl: F τ,fl( )− fℓ| |≤Δf/2

FLOSTFTy τ, fl( flΔfl. (12)

+e FLOFSST can “squeeze” a frequency interval to afrequency point in the time frequency domain; therefore, theprocess can greatly improve the time frequency resolution.

A multicomponent signal can be expressed as

y(t) � k

k�1yk(t) �

N

k�1Ak(t)cos 2πfkt( , (13)

where k � 1, 2, . . . , N. +en, the FLOSTFT of kth compo-nent can be expressed as

FLOSTFTyk τ, fk( �12π

∞

− ∞Yk(λ)ψ fk − λ( e

j2πτ λ− fk( )dλ.

(14)

FLOSTFT is just as linear as STFT; then

FLOSTFTy(τ, f) � N

k�1FLOSTFTyk τ, fk( . (15)

+e instantaneous frequency (IF) calculation method ofkth component yk(t) may be written as

Fyk τ, fk( � fk +1

j2πFLOSTFTyk τ, fk( z FLOSTFTyk τ, fk(

zτ. (16)


+e corresponding instantaneous frequency calculationmethod of y(t) may be obtained by

Fy(τ, f) � N

k�1Fyk τ, fk( �

N

k�1δ f − fk( fk +

1j2πFLOSTFTyk τ, fk(

z FLOSTFTyk τ, fk( zτ

⎧⎨

⎩

⎫⎬

⎭. (17)

By substituting (15) and (17) into (12), we can obtain thefractional low order STFT-based synchrosqueezing trans-form of the multicomponent signal.

According to the definition of inverse STFT-basedsynchrosqueezing transform in [18, 19], inverse fractionallower order STFT transform-based synchrosqueezingtransform (IFLOFSST) of knd signal can be written by

yk(t) � y〈p〉

k (t)

1/p− 1

sign yk(t) ,

y〈p〉

k (t) � Re1

2πh(0)

f− Fykτ,fk( )

≤Δfl

FLOFSSTyk τ, fk( df⎡⎣ ⎤⎦,

(18)

and the signal y(t) can be gotten employingy(t) �

kk�1 yk(t).

3.1.2. ;e Steps of the FLOFSST Method

(i) Step 1: compute FLOSTFTyk(τ, fk) of each com-ponent yk(t) for the signal y(t) employing (14).

(ii) Step 2: compute instantaneous frequency Fyk(τ, fk)of each component yk(t) by substitutingFLOSTFTyk(τ, fk) to(16).

(iii) Step 3: solve Fy(τ, f) by (17).(iv) Step 4: solve the discrete values FLOSTFTy(τ, fℓ)

employing Fy(τ, f).(v) Step 5: compute FLOSTFTy(τ, fℓ) by substituting

FLOSTFTy(τ, fℓ) to (12).

3.1.3. Application Review. In this section, we design thefollowing experiments to compare the proposed FLOFSSTmethod with the existing STFT, FLOSTFT, and FSSTmethods. +e simulation signal y(n) contaminated by thenoise is defined as

y(n) � x(n) + v(n) � x1(n) + x2(n) + x3(n) + v(n),

x1(n) � cos(18πn),

x2(n) � [1 + 0.3 cos(2n)] · cos 10πn + 1.2πn1.8

+ 0.6π sin(n) · e− 0.05n,

x3(n) � [2 + 0.2 cos(n)] · cos[6πn + 1.2π cos(n)],

(19)

where v(n) is SαS distribution noise or Gaussian noise.When the noise v(n) is SαS distribution noise, generalizedsignal to noise ratio (GSNR) can be used instead of SNR,which is expressed as

GSNR � 10 log10E |x(n)|2

cα

⎧⎨

⎩

⎫⎬

⎭ � 10 log101

Ncα

N− 1

n�0|x(n)|

2,

(20)

where c is the dispersion coefficient of SαS distributionnoise. According to the given GSNR, the amplitude of thesignal x(n) can be written as

A �10(GSNR/10)

1/NN− 1n�0 |x(n)|2c

α

1/2

. (21)

Let SNR � − 5 dB, GSNR � 22 dB, and α � 0.8. +ewaveforms of x(n) and y(n) in time domain are shown inFigure 5. We apply the fractional lower order STFT trans-form-based synchrosqueezing transform method, theexisting the STFT transform-based synchrosqueezing

transform method, the fractional lower order STFTmethod,and traditional STFT method to estimate time frequencydistribution of the signal x(n) under Gaussian distributionnoise and SαS stable distribution noise; the simulation re-sults are shown in Figures 6 and 7.

In order to compare the effectiveness of the IFSST andIFLOFSST methods, letting MSE � (1/K) Kk�1 (1/N){

Nn�1 [x(n) − x(n)]

2}, where K is the number of Monte-Carlo experiment, x(n) is the reconstructed signalemploying the IFSST method or the IFLOFSST method.Letting GSNR � 20, the signal x(n) is reconstructedemploying the IFSST and IFLOFSST methods under dif-ferent α; their MSEs are shown in Figure 8(a). We apply theIFSSTand IFLOFSSTmethods to reconstruct the signal x(n)when α � 1; GSNR changes from 14 dB to 24 dB; the sim-ulations are shown in Figure 8(b).

3.1.4. Remarks. +e STFT, FLOSTFT, FSST, and FLOFSSTtime frequency methods in Figure 6 all can estimate out thetime frequency representation of the signal x(n) under


xA

mpl

itude

–5

0

5

0 1 2 3 4 5 6 7 8 9 10

Time (s)

(a)

x + Gaussian noise

Am

plitu

de

–10

0

10

0 1 2 3 4 5 6 7 8 9 10

Time (s)

(b)

–500

0

500x + alpha noise

0 1 2 3 4 5 6 7 8 9 10

Time (s)

Am

plitu

de

(c)

Figure 5: +e waveforms in time domain. (a) +e signal x(n); (b) the signal x(n) contaminated by Gaussian noise environment(SNR � − 5 dB); (c) the signal x(n) contaminated by SαS noise environment (GSNR � 22 dB and α � 0.8).

STFT method

Time (s)

Freq

uenc

y (H

z)

0 1 2 3 4 5 6 7 8 902468

101214161820

(a)

FLOSTFT method

Time (s)

Freq

uenc

y (H

z)

0 1 2 3 4 5 6 7 8 902468

101214161820

(b)

Time (N)

Freq

uenc

y (H

z)

FSST method

1 2 3 4 5 6 7 8 9 1002468

101214161820

(c)

Time (s)

Freq

uenc

y (H

z)

FLOFSST method

1 2 3 4 5 6 7 8 9 1002468

101214161820

(d)

Figure 6: Time frequency representations of the signal x(n) under Gaussian noise environment (SNR � − 5 dB and p � 1.8). (a) STFT timefrequency representation of the signal x(n); (b) FLOSTFT time frequency representation of the signal x(n); (c) FSST time frequencyrepresentation of the signal x(n); (d) FLOFSST time frequency representation of the signal x(n).


0

50

100

150

200

250

Err

ors p

ower

(dB)

IFSST method

IFLOFSST method

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

α

(a)

IFSST method

IFLOFSST method

14 15 16 17 18 19 20 21 22 23 24

GSNR

0

10

20

30

40

50

60

70

80

Erro

rs p

ower

(dB)

(b)

Figure 8: MSE comparisons of signal reconstruction of the IFSST and IFLOFSST algorithms under different α and GSNR (p � 1.2).(a) GSNR � 20 and MSE comparison under different α; (b) α � 1 and MSE comparison under different GSNR.

STFT method

Time (s)

Freq

uenc

y (H

z)

0 1 2 3 4 5 6 7 8 902468

101214161820

(a)

FLOSTFT method

Time (s)

Freq

uenc

y (H

z)

0 1 2 3 4 5 6 7 8 902468

101214161820

(b)

FSST method

Time (s)

Freq

uenc

y (H

z)

0 1 2 3 4 5 6 7 8 902468

101214161820

(c)

Time (s)

Freq

uenc

y (H

z)

FLOFSST method

1 2 3 4 5 6 7 8 9 1002468

101214161820

(d)

Figure 7: Time frequency representations of the signal x(n) under SαS noise environment (GSNR � 22 dB and α � 0.8, p � 1.2). (a) STFTtime frequency representation of the signal x(n); (b) FLOSTFT time frequency representation of the signal x(n); (c) FSST time frequencyrepresentation of the signal x(n); (d) FLOFSST time frequency representation of the signal x(n).


Gaussian noise environment (SNR � − 5 dB), but the syn-chrosqueezing methods have better performance. +e STFTmethod in Figure 7(a) and FSST time frequency method inFigure 7(c) fail under SαS noise environment(GSNR � 22 dB and α � 0.8); the FLOSTFT method inFigure 7(b) can estimate out the time frequency represen-tation of the signal x(n), but its effect is not very ideal. +eimproved FLOFSSTmethod in Figure 7(d) can better get thetime frequency representation of the signal x(n) under SαSnoise environment, which has good toughness.

+e STFT and FSST are unsuitable for SαS noise envi-ronment, and the FLOSTFT method can work under SαSnoise environment, but has poor time frequency resolutionand is controlled by the window function. +e FSSTmethodhas better time frequency resolution, but cannot work in SαSnoise environment. +e improved FLOFSST method canwork under SαS noise environment and has high timefrequency resolution. As a result, the FSST time frequencymethod is only suitable to analyze the signals underGaussian noise environment, but the improved FLOFSSTcan work under Gaussian and SαS noise environment, whichis robust.

Figure 8(a) is MSE comparison under GSNR � 20 dBand different α; the experimental results show that MSEs ofthe IFSST method change from 2 dB to 230 dB when αchanges from 0.2 to 2, but MSEs of the IFLOFSST methodare 2 dB. Hence, the IFLOFSST method has obvious ad-vantage in reconstructing the signal under different α;particularly, the advantage of the IFLOFSSTmethod is moreobvious when α< 1.

Figure 8(b) isMSE comparison under α � 1 and differentGSNR; we can know that reconstruction MSEs of the IFSSTmethod have a large variation, which changes from 14 dB to78 dB; however, MSEs of the IFLOFSSTmethod are stable in2 dB. Hence, the improved IFLOFSST method has betterstability in reconstructing the original signal.

3.2. Fractional Lower Order S Transform-BasedSynchrosqueezing Transform

3.2.1. Principle. +e fault machinery vibration signal con-taminated by the noise may be given by

y(t) � x(t) + v(t), (22)

where x(t) is fault vibration signal and v(t) is SαS distri-bution noise or Gaussian noise. Its S transform can bewritten as

ST(τ, f) � ∞

− ∞y(t)

|f|��2π

√ e− (τ− t)2f2( )/2( )e

− j2πftdt, (23)

and its fractional lower order S transformwas defined as [28]

FLOST(τ, f) � ∞

− ∞y

〈p〉(t)h(τ − t, f)e− j2πftdt, (24)

h(τ − t, f) �|f|��2π

√ e− (τ− t)2f2/2( ), (25)

where f is the frequency parameter and t is the time pa-rameter. τ denotes the displacement parameter on the timeaxis. h(τ − t, f) is the Gaussian window function related tothe frequency.

Equation (24) can be written as

FLOST(τ, f) �|f|��2π

√ ∞

− ∞y

〈p〉(t)e

− (t− τ)2f2/2( )e− j2πftdt.

(26)

Let ψ(t) � (1/��2π

√)e− (t

2/2)ej2πft, and its complex con-jugate function is ψ(t) � (1/

��2π

√)e− (t

2/2)e− j2πft. +en, (26)changes as

FLOST(τ, f) �|f|��2π

√ ∞

− ∞y

〈p〉(t)e

− (t− τ)2f2( )/2( )e− j2πftdt

�|f|��2π

√ ∞

− ∞y

〈p〉(t)e

− (t− τ)2f2( )/2( )e− j2πf(t− τ)

e− j2πfτdt

� |f| ∞

− ∞y

〈p〉(t)

e− f2(t− τ)2( )/2( )e− j2πf(t− τ)

��2π

√ e− j2πfτdt

� |f| ∞

− ∞y

〈p〉(t)ψ[f(t − τ)]e− j2πfτdt.

(27)

+e right side of (27) is converted from the time domainto the frequency domain based on Plancherel’s theorem andFourier transform; then we obtain

FLOST(τ, f) �12π

∞

− ∞Y(λ)ψ

λf

ejτ(λ− 2πf)dλ, (28)

where Y(λ) � ∞− ∞ y

〈p〉(t)e− j2πλtdt is fractional lower orderFourier transform (FLOFT) of y〈p〉(t) and λ denotes fre-quency variant. ψ(λ/f) is Fourier transform of ψ(λ/f), andψ(λ/f) is complex conjugate of ψ(λ/f).

Letting y(t) � A cos(2πf0 t) and ω0 � 2πf0, theny〈p〉(t) � |A cos(2πf0t)|

p− 1 · sign[A cos(2πf0t)], and itsFLOFT can be expressed as

Y(λ) � ∞

− ∞y

〈p〉(t)e

− j2πftdt � |A|p− 1πp− 1sign(A) δ λ + ω0( + δ λ − ω0( . (29)


By substituting (29) to (28), then

FLOSTy(τ,ω) �|A|p− 1πp− 2sign(A)

2∞

− ∞δ λ + ω0( + δ λ − ω0( ψ(λ/ω)e

jτ(λ− ω)dλ �|A|p− 1πp− 2sign(A)

2e

− j ω− ω0( )τ ψ 2πω0/ω( .

(30)

Fourier transform of ψ(t) and ψ(t) can assemble λ ataround ω0, and FLOSTy(τ,ω) will be concentrated aroundτ � ω0/ω. By substituting f � ω/2π to (30), then

FLOSTy(τ, f) �|A|p− 1|π|p− 2sign(A)

2e

− j2πτ f− f0( ) ψ2πf0

f .

(31)

+e instantaneous frequency of FLOSTy(τ, f) can bewritten as

Fy(τ, f) � f +1

j2πFLOSTy(τ, f)z FLOSTy(τ, f)

zτ, (32)

z FLOSTy(τ, f) zτ

� − j|A|p− 1

|π|p− 1sign(A) f − f0( e− j2πτ f− f0( ) ψ

2πf0f

. (33)

By substituting (31) and (33) to (32), we can obtain thesqueezed instantaneous frequency. +rough syn-chrosqueezing the frequency with (32), the discrete valuesFLOSTy(τ, fℓ) can be gotten. Letting the frequency pointsin FLOST time frequency spectrum, fl(l � 1, 2, . . . , K) and

Δfl � fl − fl− 1. By centering fℓ and letting Δf � fℓ − fℓ− 1,extend the synchrosqueezing process to the successive bins[fℓ − (1/2)Δf, fℓ + (1/2)Δf]; then, fractional lower order Stransform-based synchrosqueezing transform can be writtenas

FLOSSSTy τ, fℓ( � (Δf)− 1


FLOSTy τ, fl(

flΔfl. (34)

For the calculation of IF and FLOSSST of a multicom-ponent signal, y(t), we have

y(t) � k

k�1yk(t) �

N

k�1Ak(t)cos 2πfkt( , (35)

where k � 1, 2, . . . , N. +en, the FLOST of kth componentcan be expressed as

FLOSTyk τ, fk( � fk

∞

− ∞y

〈p〉

k (t)ψ fk(t − τ) e− j2πfkτdt.

(36)

FLOST is just as linear as ST; then

FLOSTy(τ, f) � N

k�1FLOSTyk τ, fk( . (37)

+e IF calculation method of kth component yk(t) maybe written as

Fyk τ, fk( � fk +1

j2πFLOSTyk τ, fk( z FLOSTyk τ, fk(

zτ.

(38)

+e corresponding IF calculation method of y(t) may beobtained by

Fy(τ, f) � N

k�1Fyk τ, fk( �

N

k�1δ f − fk( fk +

1j2πFLOSTyk τ, fk(

z FLOSTyk τ, fk( zτ

⎧⎨

⎩

⎫⎬

⎭. (39)


By substituting (37) and (39) into (34), fractional loworder STFT-based synchrosqueezing transform time fre-quency representation of the multicomponent signal can beobtained.

3.2.2. Inverse Fractional Lower Order S Transform-BasedSynchrosqueezing Transform. Multiplying ej2πfτf− 1 on bothsides of (28) and taking the integral to f, then

∞

0FLOST(τ, f)ej2πfτf− 1df �

12π

∞

− ∞∞

0Y(λ)ψ λf− 1( ejτλf− 1dλdf. (40)

Let ξ � λf− 1; then

∞

0FLOST(τ, f)ej2πfτf− 1df � −

12

∞

0ψ(ξ)ξ− 1dξ ·

1π

∞

0Y(λ)ejτλdλ . (41)

For the real signal y(t), letting − 1/2∞0 ψ(ξ)ξ− 1dξ � Γ,

we have

Re Γ− 1 ∞

0FLOST(τ, f)ej2πfτf− 1df � Re

1π

∞

0Y(λ)ejτλdλ � y〈p〉(τ). (42)

In the piecewise constant approximation correspondingto the binning in f, we have

y(τ) ≈ Re Γ− 1 l

FLOST τ, fl( ej2πflτf

− 1l Δfl⎤⎦

⎫⎬

⎭.⎡⎣

⎧⎨

⎩

(43)

FLOST of the signal y(t) in (26) can be written as

FLOST(τ, f) � |FLOST(τ, f)|ejj(τ,f), (44)

where |FLOST(τ, f)| is modulo operation of |FLOST(τ, f)|and j(τ, f) is its phase position. Substituting (44) to (34), weobtain

FLOSSSTy τ, fℓ( ejj(τ,f)

� (Δf)− 1 fl: F τ,fl( )− fℓ| |≤Δf/2

FLOSTy τ, fl( flΔfl. (45)

Multiplying ej2πfτf− 2l on both sides of (45) and lettingΦ � ej[2πfτ+j(τ,f)]f− 2l , then

FLOSSSTy τ, fℓ( ej[2πfτ+j(τ,f)]

f− 2l � (Δf)

− 1


FLOSTy τ, fl( ej2πfτ

f− 1l Δfl � FLOSSSTy τ, fℓ( Φ. (46)

Substituting (46) into (43), we can deduce the followingexpression:

y〈p〉

(τ) � Re ΦΓ− 1 ℓFLOSSSTy τ, fℓ( Δf⎤⎦,⎡⎣ (47)

where y(t) is real signal. According toy〈p〉(t) � |y(t)|p− 1 · sign[y(t)], inverse fractional lowerorder S transform-based synchrosqueezing transform(IFLOSSST) of y(t) can be written as


y(t) � y〈p〉

(t)

1/p− 1

sign y〈p〉

� Re ΦΓ− 1 ℓFLOSSSTy τ, fℓ( Δf⎤⎦

1/p− 1

sign Re ΦΓ− 1 ℓFLOSSSTy τ, fℓ( Δf⎤⎦

⎫⎬

⎭.⎡⎣

⎧⎨

⎩⎡⎢⎢⎢⎢⎣

(48)

We can reconstruct the signal y(t) in FLOSSST timefrequency domain employing (48).

3.2.3. ;e Steps of the FLOSST Time Frequency Method

(i) Step 1: compute FLOSTyk(τ, fk) of each componentyk(t) for the signal y(t) employing (31).

(ii) Step 2: solve FLOSTy(τ, f) by substitutingFLOSTyk(τ, fk) of each component yk(t) to (37).

(iii) Step 3: compute instantaneous frequency Fyk(τ, fk)of each component yk(t) by substitutingFLOSTyk(τ, fk) to (38).

(iv) Step 4: solve Fy(τ, f) of the signal yk(t) bysubstituting Fyk(τ, fk) to (39).

(v) Step 5: compute the discrete values FLOSTy(τ, f)employing Fy(τ, f).

(vi) Step 6: solve FLOSSST of the signal y(t) bysubstituting FLOSTy(τ, fℓ) to (34).

3.2.4. Application Review. In this section, x(n) in (19) isused as the experiment signal. +e proposed fractional lowerorder S transform-based synchrosqueezing transformmethod, the existing the S transform-based syn-chrosqueezing transform method, the fractional lower orderS transformmethod, and traditional S transformmethod areused to display time frequency distribution of the signal x(n)under Gaussian distribution noise (SNR � − 5 dB) and SαSstable distribution noise (GSNR � 22 dBand α � 0.8); thesimulation results are shown in Figures 9 and 10.

Letting GSNR � 22 dB and α � 1.4, the ISSST andIFLOSSST methods are used to reconstruct the originalsignal; the results are shown in Figure 11. In order to furthercompare the effectiveness of the ISSST and IFLOSSSTmethods, letting GSNR � 20 dB, the signal x(n) is recon-structed employing the ISSSTand IFLOSSSTmethods underdifferent α; their MSEs are shown in Figure 12(a). We applythe ISSST and IFLOSSST methods to reconstruct the signalx(n) when α � 1; GSNR changes from 14 dB to 24 dB; thesimulations are shown in Figure 12(b).

3.2.5. Remarks. Figure 9 is the time frequency representa-tions of the signal x(n) under Gaussian noise environment(SNR � − 5 dB) employing the ST, FLOST, SSST, andFLOSSST methods, respectively. We can see that all themethods can estimate out the time frequency distribution ofthe signal x(n), but the synchrosqueezing transformmethods have obvious advantages in time frequency reso-lution.+e time frequency representations of the signal x(n)employing the ST, FLOST, SSST, and FLOSSST methodsunder SαS noise environment (GSNR � 22 dB and α � 0.8)

are shown in Figure 10. +e results show that the STmethodin Figure 10(a) and SSST method in Figure 10(c) fail; theFLOST method in Figure 10(b) can estimate out the timefrequency distribution of the signal x(n), but its effect is notvery ideal. +e improved FLOSSSTmethod in Figure 10(d)can better get the time frequency representation of the signalx(n), which has higher time frequency resolution.

+e reconstructed signal x(n) employing the ISSSTmethod is shown in Figure 11(b) under SαS noise envi-ronment (GSNR � 22 dBand α � 1.2); it can be seen that thesignal x(n) is covered by SαS noise; the ISSSTmethod fails.Figure 11(b) is the reconstructed signal x(n) based on theIFLOSSSTmethod under the same conditions; it shows thatthe reconstructed signal x(n) is very similar to the originalsignal x(n). Figure 12(a) is reconstruction MSE comparisonunder GSNR � 20 dB when α changes from 0.2 to 2; theresults show that the reconstruction MSEs of the IFSSTmethod change from 1 dB to 290 dB, but the reconstructionMSEs of the IFLOFSST method have an obvious low level,which are stable at about 2 dB. Hence, the IFLOFSSTmethodhas obvious advantage in reconstructing the signal underdifferent α; particularly, the advantage of the IFLOFSSTmethod is more obvious when α< 1. Figure 12(b) is thereconstruction MSE comparison under α � 1 when GSNRchanges from 14 dB to 78 dB; it shows that the recon-struction MSEs of the IFSSTmethod have a large variation,but the reconstruction MSEs of the IFLOFSST methodchange from − 2 dB to 8 dB. Hence, the improved IFLOFSSTmethod has better stability in reconstructing the signal.

As a result, the SSST time frequency method and theISSST signal reconstruction method are only suitable toanalyze and reconstruct the signals under Gaussian noiseenvironment, but the improved FLOSSST and IFLOSSSTmethods can work in Gaussian and α stable distributionnoise environment, which are robust.

4. Application Simulations

In this simulation, the experiment signal adopts the bearingouter race fault signal (DE) in Section 2. 0.2 seconds’ data isselected as the test signal, which is collected at 12,000samples per second, and N � 2400. +e improved FLOFSSTand FLOSSSTmethods are applied to analyze time frequencydistribution of the outer race fault signal; the simulationresults are shown in Figure 13.

Figures 13(a) and 13(b) are the time frequency repre-sentations of the outer race fault signal employing theFLOFSSTand FLOSSSTmethods, respectively. It can be seenthat two methods have a good lateral resolution, the low-frequency shock pulse mainly includes 0Hz to 4000Hz, andthe dominant frequency of the vibration components isapproximately 600Hz, 2800Hz, and 3500Hz. All the


Time (s)

Freq

uenc

y (H

z)

ST method

0 1 2 3 4 5 6 7 8 902468

101214161820

(a)

Time (s)

Freq

uenc

y (H

z)

FLOST method

0 1 2 3 4 5 6 7 8 902468

101214161820

(b)

SSST method

Time (s)

Freq

uenc

y (H

z)

0 1 2 3 4 5 6 7 8 902468

101214161820

(c)

FLOSSST method

Time (s)

Freq

uenc

y (H

z)

0 1 2 3 4 5 6 7 8 902468

101214161820

(d)

Figure 9: Time frequency representations of the signal x(n) under Gaussian noise environment (SNR � − 5 dB and p � 1.8). (a) ST timefrequency representation of the signal x(n); (b) FLOST time frequency representation of the signal x(n); (c) SSST time frequency rep-resentation of the signal x(n); (d) FLOSSST time frequency representation of the signal x(n).

Time (s)

Freq

uenc

y (H

z)

ST method

0 1 2 3 4 5 6 7 8 902468

101214161820

(a)

Time (s)

Freq

uenc

y (H

z)

FLOST method

0 1 2 3 4 5 6 7 8 902468

101214161820

(b)

Figure 10: Continued.


methods have a good vertical resolution; the gap between theimpacts can be clearly seen, which regularly change. +etime interval in A, B, C, D, E, and F is about 30ms; thecorresponding characteristic frequency of the outer racefault signal is about 33.333Hz.

In order to further prove the advantages of the improvedFLOFSST and FLOSSST methods, SαS distribution noise(α � 1 and GSNR � 22 dB) is added in the α stable distri-bution outer race fault signal as the background noise ofactual working environment. +e improved methods andexisting methods are applied to demonstrate time frequencyrepresentation of the outer race fault signal; the simulationsare shown in Figure 14. +e results show that the FSST

method in Figure 14(a) and SSTmethod in Figure 14(b) fail.However, the FLOFSST method in Figure 14(c) andFLOSSST method in Figure 14(d) can give out time fre-quency distribution of the fault signal under substandardconditions, which have certain ability in the horizontal andvertical time frequency representation, and we can know thedominant frequency and the time interval in A, B, C, D, E,and F, but the overall resolution is not high and needs toimprove. Hence, the improved fractional lower order syn-chrosqueezing methods have better performance superioritythan the existing synchrosqueezing methods, which aremore suitable for fault analysis in complex environment andare robust.

SSST method

Time (s)

Freq

uenc

y (H

z)

0 1 2 3 4 5 6 7 8 902468

101214161820

(c)

FLOSSST method

Time (s)

Freq

uenc

y (H

z)

0 1 2 3 4 5 6 7 8 902468

101214161820

(d)

Figure 10: Time frequency representations of the signal x(n) under SαS noise environment (GSNR � 22 dB, α � 0.8, andp � 1.2). (a) STtime frequency representation of the signal x(n); (b) FLOST time frequency representation of the signal x(n); (c) SSST time frequencyrepresentation of the signal x(n); (d) FLOSSST time frequency representation of the signal x(n).

Original signal

Am

plitu

de

–5

0

5

0 1 2 3 4 5 6 7 8 9 10

Time (s)

(a)

Am

plitu

de

Reconstructed signal employing ISST method

–50

0

50

0 1 2 3 4 5 6 7 8 9 10

Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 10–5

0

5

Am

plitu

de

Reconstructed signal employing IFLOSST method

Time (s)

(c)

Figure 11: +e waveforms of signal reconstruction under SαS noise environment (GSNR � 20 dB and α � 1.2). (a) +e original signal x(n);(b) the reconstructed signal x(n) employing the ISSST method; (c) (b) the reconstructed signal x(n) employing the IFLOSSST method.


0

50

100

150

200

250

300

α

Err

ors p

ower

(dB)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

ISSST method

IFLOSSST method

(a)

14 15 16 17 18 19 20 21 22 23 24–10

0

10

20

30

40

50

60

70

GSNR

Erro

rs p

ower

(dB)

ISSST method

IFLOSSST method

(b)

Figure 12: MSE comparisons of signal reconstruction of the ISSST and IFLOSSST algorithms under different α and GSNR (p � 1.2). (a)GSNR � 20 and MSE comparison under different α; (b) α � 1 and MSE comparison under different GSNR.

FLOFSST method

Freq

uenc

y (H

z)

0

1000

2000

3000

4000

5000

6000

A B C D E F

Time (N)

500 1000 1500 2000

(a)

Time (N)

Freq

uenc

y (H

z)

FLOSSST method

500 1000 1500 20000

1000

2000

3000

4000

5000

6000

A B C D E F

(b)

Figure 13: +e time frequency representations of the outer race fault signal (p � 1.8). (a) +e FLOFSST time frequency method; (b) theFLOSSST time frequency method.

Freq

uenc

y (H

z)

Time (N)

FSST method

500 1000 1500 20000

1000

2000

3000

4000

5000

6000

(a)

Time (N)

Freq

uenc

y (H

z)

SSST method

500 1000 1500 20000

1000

2000

3000

4000

5000

6000

(b)

Figure 14: Continued.


5. Conclusions

α stable distribution is a more appropriate statisticalmodel for the bearing fault signals. STFT transform-basedsynchrosqueezing transform and S transform-basedsynchrosqueezing transform are two new time frequencyrepresentation methods for nonstationary signal pro-cessing; their time frequency resolution can be greatlyimproved by rearranging the time frequency energy of thesignals. In order to make the FSST and SST methodsapplicable to Gaussian and α stable distribution noiseenvironment, the improved FLOFSST and FLOSSST timefrequency representation methods are proposed byemploying fractional low order statistics. +e perfor-mances of the FLOFSST and FLOSSST methods are su-perior to the existing time frequency analysis methods;they have higher time frequency resolution than theexisting FLOSTFT and FLOST methods because of thesynchrosqueezing processing and can better suppress theimpulse noise than the FSST and SST methods. +eIFLOFSST and IFLOSSST methods have smaller recon-struction MSEs than the IFSST and ISSSTmethods underdifferent α(α< 2) and GSNR. We can apply the improvedmethods to analyze the α stable distribution bearing faultsignal; even α stable distribution noise environment, thefault characteristic frequency, the dominant frequency,and the other fault frequency features of the fault signalscan be clearly obtained. In the future, we can also furtherstudy time frequency filtering technology based on theproposed IFLOFSST and IFLOSSST methods, and themethods have a good application prospect in the field ofthe bearing fault analysis and detection.

Data Availability

+e data used to support the findings of this study areprovided in the Supplementary Materials.

Conflicts of Interest

+e authors declare that they have no conflicts of interest.

Acknowledgments

+is work was financially supported by the Natural ScienceFoundation of China (61962029), the Natural ScienceFoundation of Jiangxi Province, China (20192BAB207002),the Science and Technology Project of Provincial EducationDepartment of Jiangxi (GJJ170954), and the Science andTechnology Project of Jiujiang University, China(2014SKYB009).

Supplementary Materials

+is section contains the original experimental data of thispaper. (Supplementary Materials)

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