ARTICLE IN PRESS

Mechanical Systemsand

Signal ProcessingMechanical Systems and Signal Processing 20 (2006) 1992–2006

0888-3270/$ -

doi:10.1016/j.

�CorresponE-mail add

www.elsevier.com/locate/jnlabr/ymssp

Gearbox fault diagnosis using adaptive redundantLifting Scheme

Jiang Hongkaia,�, He Zhengjiab, Duan Chendonga, Chen Pengc

aSchool of Mechanical Engineering, Xi’an Jiaotong University, 710049, Xi’an, ChinabState Key Lab for Manufacturing Systems Engineering, Xi’an Jiaotong University, 710049, Xi’an, China

cDepartment of Environmental Science and Engineering, Faculty of Bioresources, Mie University, 1515 Kamihama-cho,

Tsu, Mie 514-8507, Japan

Received 30 January 2005; received in revised form 9 June 2005; accepted 18 June 2005

Available online 10 August 2005

Abstract

Vibration signals acquired from a gearbox usually are complex, and it is difficult to detect the symptomsof an inherent fault in a gearbox. In this paper, an adaptive redundant lifting scheme for the fault diagnosisof gearboxes is developed. It adopts data-based optimisation algorithm to lock on to the dominantstructure of the signal, and well reveal the transient components of the vibration signal in time domain.Both lifting scheme and adaptive redundant lifting scheme are applied to analyse the experimental signalfrom a gearbox with wear fault and the practical vibration signal from a large air compressor. The resultsconfirm that adaptive redundant lifting scheme is quite effective in extracting impulse and modulationfeature components from the complex background.r 2005 Elsevier Ltd. All rights reserved.

Keywords: Adaptive redundant lifting scheme; Optimisation algorithm; Vibration signal; Fault diagnosis

1. Introduction

Gearboxes are key parts in a wide range of mechanical systems. It is very important to detectincipient fault symptoms from gearboxes. Usually, vibration signals are acquired from

see front matter r 2005 Elsevier Ltd. All rights reserved.

ymssp.2005.06.001

ding author

ress: chenxf@mail.xjtu.edu.cn (J. Hongkai).

ARTICLE IN PRESS

J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–2006 1993

accelerometers mounted on the outer surface of a bearing housing. The signals consist ofvibrations from the meshing gears, shafts, bearings, and other components. The usefulinformation is corrupted and it is difficult to diagnose a gearbox from such vibration signals.Wavelet theory is a powerful tool for non-stationary signal analysis, and it has been successfullyused in gearbox diagnosis. Boulahbal et al. [1] used the amplitude and phase maps of continuouswavelet transform together to extract fault features of a gearbox and obtained a more positiveassessment of a tooth condition. Lin et al. [2,3] utilised the similarity between Morlet wavelet andan impulse shape, and detected tooth crack symptoms of a gearbox immersed in the noise.Loutridis et al. [4] used continuous wavelet transform and Hodlder exponents to classify gearfaults and got better performance.In all the wavelet techniques mentioned above, researchers usually selected an appropriate

wavelet function from a library of previously designed wavelet functions to match a specific faultsymptom. Different types of mechanical faults have different waveform characteristics, even onewavelet function is selected, and it is not always the best wavelet function to detect a specific faultsymptom in a gearbox. New wavelet method is needed to overcome the drawback.Lifting scheme is a spatial domain construction of biorthogonal wavelets developed by

Sweldens [5–7]. It abandons the Fourier transform as design tool for wavelets, wavelets are nolonger defined as translates and dilates of one fixed function. Compared with classical wavelettransform, Lifting scheme possesses several advantages, e.g. possibility of adaptive design, in-place calculations, irregular samples and integers-to-integers wavelet transforms. Lifting schemeprovides a great deal of flexibility, it can be designed according to the properties of the givensignal, and it ensures that the resulting transform is invertible.In this paper, we develop a new wavelet method called adaptive redundant lifting scheme.

It is based on lifting scheme and designed to capture the transient components of the gear-box vibration by adaptive decomposition. In Section 2, the theory of lifting scheme is reviewedbriefly. The data-based optimisation algorithm is described in Section 3. The method of adap-tive redundant lifting scheme is presented in Section 4. In Section 5, adaptive redundantlifting scheme is applied to analyse the vibration signals of an experimental gearbox and a largecompressor gearbox. Comparison with lifting scheme is also shown. Conclusions are givenin Section 6.

2. Lifting scheme principle

Lifting scheme is an entirely space domain wavelet construction. A typical lifting schemeprocedure consists of three basic steps: split, predict and update [5–7].

Split: Let x(n) be an original data set. In this step, x(n) is divided into two disjoint even subsetxe(n) and odd subset x0(n), where xeðnÞ ¼ xð2nÞ and x0ðnÞ ¼ xð2nþ 1Þ.

Predict: If the original signal has a local correlation structure, then the even and odd subsets arehighly correlated. We predict the odd coefficients x0(n) from the neighbouring even coefficientsxe(n), and the prediction differences d(n) are defined as detail signal

dðnÞ ¼ x0ðnÞ � PðxeðnÞÞ, (1)

Where P ¼ ½pð1Þ; . . . ; pðNÞ�T is the prediction operator.

ARTICLE IN PRESS

J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–20061994

Update: This step forms a coarse approximation to the original signal. We combine the evencoefficients and the linear combination of the prediction differences, and then the approximationsignal c(n) are obtained

cðnÞ ¼ xeðnÞ þUðdðnÞÞ, (2)

Where U ¼ ½uð1Þ; . . . ; uð ~NÞ� is the update operator.Iteration of the three steps on the output c(n), and then the detail signal and approximation

signal at different levels are generated. Scaling function f(x) and wavelet function c(x) of liftingscheme can be derived from P and U by iteration algorithm. The program flow chart is illustratedin Fig. 1.The scaling function and wavelet function with N ¼ 6 and ~N ¼ 6 are shown in Fig. 2. The

scaling function and wavelet function are symmetrical and compactly supported. The shape of thewavelet function is very similar to an impulse, and it is desirable to detect the transientcomponents in the vibration signal. We can choose P and U according to the properties of thegiven signal.Since the design of lifting scheme without reference to Fourier techniques, each lifting step is

always invertible, and the inverse transform of lifting scheme is the inversion of the lifting steps.

Fig. 1. The program flow diagram of scaling and wavelet functions.

Fig. 2. Scaling function and wavelet function with N ¼ 6 and ~N ¼ 6.

ARTICLE IN PRESS

J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–2006 1995

Assuming the same P and U are chosen for the forward and inverse transform, the liftingconstruction guarantees perfect reconstruction for any P and U [8].

3. Initial prediction operator and update operator design

Lifting scheme increases the flexibility to extract the features of a given data set. We canoptimise the prediction operator and update operator to well reveal the feature components of thedata set. In this section, we adopt Claypoole’s optimisation algorithm [8,9] to design the initialprediction operator P and update operator U, and make them adapt to the dominating structuresof the data set at the corresponding level.The design of initial prediction operator and update operator based on the data characteristics

is accomplished by the following procedures.A N-point initial prediction operator P is designed to suppress polynomial components upto

order MoN, and the remaining N–M degrees of freedom are used to match the given signal. Theinitial prediction operator is as below

P ¼ ½ pð1Þ; . . . ; pðNÞ�T . (3)

Construct a M�N matrix V, its element is

½V �i;j ¼ ½2j �N � 1�i�1, (4)

where i ¼ 1; 2; . . . ;M; j ¼ 1; 2; . . . ;N.We require that

VP ¼ ½1; 0; 0; . . . 0�T , (5)

The vector of prediction differences can be expressed as follows:

e ¼ X 0 � X eP. (6)

The goal is to obtain the prediction coefficients that minimise the sum of squared predictiondifferences, namely

minP

X 0 � X ePk k2. (7)

We solve Eqs. (5) and (7), then the initial prediction operator P that locks on to the dominantstructure of a signal at the corresponding level is obtained.A ~N-point initial update operator U is designed by the following procedures, where ~NpN.Construct a 2N–1 -dimensional vector q, its element is defined as

i ¼ 1; . . .N : qð2i � 1Þ ¼ pðiÞ;

i ¼ 1; . . .N � 1 : qð2iÞ ¼ 0; qðNÞ ¼ 1.

Assume that K ¼ 4N � 3, construct a K � ~N matrix H, its element is 0, except

Hðð2i � 1Þ : ð2i þ 2N � 3Þ; iÞ ¼ q, (8)

where i ¼ 1; . . . ; ~N.

ARTICLE IN PRESS

J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–20061996

And then, a ~N � K matrix W is constructed, in which element is

½W �m;n ¼ ½n�N � ~N þ 1�m�1, (9)

where m ¼ 1; 2; . . . ; ~N, n ¼ 1; 2; . . . ;K .We calculate U from the following expression:

WHU ¼ ½1; 0; 0; . . . 0�T . (10)

Since matrix H contains the initial prediction operator P, the initial update operator U alsoadapts to the dominant signal structure at the corresponding level.

4. Adaptive redundant lifting scheme design algorithm

Now, we explain how to design adaptive redundant lifting scheme. The design includes twoparts, the design of redundant prediction operator and update operator, and the construction ofadaptive redundant lifting scheme.

4.1. Redundant prediction operator and update operator design

The design of redundant prediction operator P[l] and redundant update operator U[l] (l is thedecomposition level) is the key step in adaptive redundant lifting scheme. P[l] and U[l] are obtainedby padding the initial prediction operator P and update operator U with zeros at thecorresponding level l. Their design algorithm is as follows.Assuming the initial prediction operator P and update operator U at level l are designed with

the algorithm in Section 3, where P ¼ fpmg, m ¼ 1; 2; . . . ;N; U ¼ fung, n ¼ 1; 2; . . . ; ~N, N and ~Nare the coefficient number of P and U, respectively. Then the redundant prediction operator P[l]

and redundant update operator U[l] at level l are designed by padding the prediction coefficient pm

and update coefficient un with zeros. The redundant prediction coefficient p½l�j is expressed as below

p½l�j ¼

pm; j ¼ 2l �m

0; ja2l �m

(; j ¼ 1; . . . 2lN. (11)

The redundant update coefficient u½l�i is expressed as below

u½l�i ¼

un; i ¼ 2l � n

0; ia2l � n; i ¼ 1; . . . 2l ~N:

((12)

The expressions of redundant prediction operator P[l] and update operator U[l] at level l are asbelow P½l� ¼ fp

½l�j g, j ¼ 1; . . . ; 2lN; U ½l� ¼ fu

½l�i g, i ¼ 1; . . . ; 2l ~N.

4.2. Adaptive redundant lifting scheme construction

We convert lifting scheme into adaptive redundant lifting scheme by getting rid of the splittingstep. In adaptive redundant lifting scheme, the signal is predicted and updated with redundantprediction operator and update operator directly, and the length of approximation signal anddetail signal for all levels is the same as the original signal. Adaptive redundant lifting scheme

ARTICLE IN PRESS

J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–2006 1997

possesses time invariant property, which well keeps the information of approximation signal anddetail signal, and is particularly important for fault feature extraction.An approximation signal cl at level l decomposed with adaptive redundant lifting scheme is

presented by following equations:

dlþ1 ¼ cl � P½l�cl ,

clþ1 ¼ cl �U ½l�dlþ1, ð13Þ

where cl+1 and dl+1 are approximation signal and detail signal at level l+1.The decomposition procedures of adaptive redundant lifting scheme are shown in Fig. 3.Adaptive redundant lifting scheme is easily invertible, and the reconstruction procedure is

directly achieved from the inverse transform of adaptive redundant lifting scheme decomposition.

5. Applications

To diagnose the gearbox faults effectively, the most important thing is to isolate the transientcomponents from original complex vibration signals. We present several application examples todemonstrate the performance of the adaptive redundant lifting scheme method.

5.1. Experiment signal analysis

In this section, adaptive redundant lifting scheme is used to analyse the experimental signals ofa rotating machine. Fig. 4 shows the experimental test rig.The specification of the gears is shown in Table 1.

− P [l] U [l]

c1+l

d1+l

cl

Fig. 3. Adaptive redundant lifting scheme decomposition.

Fig. 4. Rotating machine for test.

ARTICLE IN PRESS

Table 1

Specification of gears

Module 2

Width of the tooth 20 (mm)

Pressure angle (a) 201

Number of teeth (normal) 55

Number of teeth (test) 75

Backlash of the normal gear 0.5 (mm)

Load torque 1.5 (Nm)

Fig. 5. Vibration signal of the test gear.

J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–20061998

For a gear transmission, the meshing frequency fz is calculated by the formula

f z ¼ nz=60, (14)

where, n is the rotating speed of the test gear, z is the number of the test teeth. In this experiment,n ¼ 120 rpm, z ¼ 75. It follows from formula (14) that the meshing frequency of the test gear is150Hz.The vibration signal was acquired by an accelerometer mounted on the outer case of the

gearbox. The sampling frequency was 25.6 kHz. The signal was low-pass filtered at 6 kHz. Fig. 5shows the vibration signal of the test gear, in which the test gear was in wear state, and the datanumber is 1024. The vibration signal is complex, and transient component is hidden among manyirrelevant components.The FFT spectrum of the vibration signal is illustrated in Fig. 6. The frequency components is

abundant, the main frequency components are 567, 690 and 3410Hz. We do not find the featurefrequency components, namely, the meshing frequency 150Hz and its harmonics.The vibration signal is analysed by using lifting scheme. The prediction operator and update

operator are [�0.0625, 0.5625, 0.5625, �0.0625] and [�0.0313, 0.2813, 0.2813, �0.0313],respectively. Fig. 7 presents the approximation signals (a1–a2) and detail signals (d1–d2). Noobvious feature components appear.We analyse the same vibration signal by adopting adaptive redundant lifting scheme. The

prediction operator with 4 prediction coefficients and 3 vanishing moments is chosen, and thenumber of update coefficients is 4. Using the optimisation algorithm in Section 3, the initial

ARTICLE IN PRESS

Fig. 6. Spectrum of the vibration signal.

Fig. 7. Lifting scheme result for the vibration signal.

J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–2006 1999

prediction operator and update operator which adapted to the signal characteristics at thecorresponding level were calculated. The initial prediction operator and update operator atthe first level are [0.1650, �0.1200, 1.2450, �0.2900] and [�0.0881, 0.4519, 0.1106, 0.0256], andthe initial prediction operator and update operator at the second level are [�0.0701, 0.5854,0.5396, �0.0549] and [�0.0293, 0.2755, 0.2870, �0.0332]. Fig. 8 shows the decomposition resultusing adaptive redundant lifting scheme. The detail signal at level 2 distinctly discloses periodicimpulses. The period is just about 0.0067 s, and the frequency is 150Hz, which is equal to themeshing frequency of the test gear.

ARTICLE IN PRESS

Fig. 8. Adaptive redundant lifting scheme result for the vibration signal.

Fig. 9. Envelope spectrum of the detail signal at level 2.

J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–20062000

The envelope spectrum of the detail signal at level 2 is illustrated in Fig. 9. It mainly consists ofthe meshing frequency 150Hz and the second harmonic 300Hz, which coincide with thecharacteristic frequency of the gear wear fault.

5.2. Practical signal analysis

A large air compressor in a refinery consists of motor, gearbox, and compressor. Its structuresketch is shown in Fig. 10.The rotating speed of the motor is 2985 rpm, the rotating frequency of the high gear is 213Hz

and the transmitting ratio of the gearbox is 4.28125. Accelerometers were mounted to acquire

ARTICLE IN PRESS

Fig. 10. The structure sketch of air compressor.

Fig. 11. The vibration signal acquired from bearing bridge 5].

J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–2006 2001

vibration signals at the corresponding measurement points on bearing cases 3#, 4#, 5# and 6#.The sampling frequency was 15 kHz.After overhaul the air compressor was running again. It was found that the vibration of

gearbox was intense with ear-piercing noise. The vibration signal acquired from bearing case 5] isshown in Fig. 11, the data number is 512. The transient components are hidden in the signal. Wecan hardly find any useful diagnosis information from the vibration signal.The FFT spectrum of vibration signal acquired from bearing case 5] is shown in Fig. 12. It is

distinct that there exist three frequencies of 1480, 2960 and 4231Hz, and 213Hz sidebands aroundthem. While these three frequencies cannot be found in gearbox or compressor based onmechanical rotating condition. The 213Hz is just equal to the rotating frequency of high gear andthe compressor. It indicates that there existed modulation fault corresponding to the 213Hzrotating frequency. What property does the modulation fault possess? It is difficult to give ananswer according to the FFT spectrum shown in Fig. 12.The vibration signal acquired from bearing case 5] is analysed via lifting scheme. The

prediction operator and update operator are [0.0117, �0.0977, 0.5859, 0.5859, �0.0977, 0.0117]and [0.0059, �0.0488, 0.2930, 0.2930, �0.0488, 0.0059], respectively. The lifting schemeapproximations (a1�a2) and details (d1�d2) are shown in Fig. 13, no obvious amplitude

ARTICLE IN PRESS

Fig. 12. The FFT spectrum of vibration signal acquired from bearing bridge 5].

Fig. 13. Lifting scheme result for the vibration signal.

J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–20062002

modulation signals and periodic impulses are found. As a result, it is difficult to draw anyconclusive result from Fig. 13.We use adaptive redundant lifting scheme to analyse the same vibration signal. The prediction

operator with 6 prediction coefficients and five vanishing moments is adopted, and the number ofupdate coefficients is 6. Using the algorithm in Section 3, we calculated the initial predictionoperator and update operator that adapted to the signal characteristics at the corresponding level.The initial prediction operator and update operator at the first level are [0.0059, �0.0684, 0.5274,0.6445, �0.1269, 0.0176] and [0.0073, �0.0561, 0.3076, 0.2783, �0.0415, 0.0044], respectively; andthe initial prediction operator and update operator at the second level are [0.0449, �0.2636,0.9179, 0.2540, 0.0683, �0.0215] and [�0.0024, �0.0073, 0.2100, 0.37600, �0.0903, 0.0142],

ARTICLE IN PRESS

J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–2006 2003

respectively. The approximation signal at level 2 is shown in Fig. 14. From Fig. 14, the amplitudemodulation in the gearbox vibration is clearly revealed, and there are seven amplitude modulationsignals. The seven amplitude modulation signals spaced regularly at the time intervals correspondto the seven rotating period of the high gear.Fig. 15 illustrates the detail signal at level 2. From Fig. 15, distinct evenly spaced impulses can

be observed from the detail signal, and the frequency of the impulses is identified as 213Hz, whichexactly is in accordance with the rotating frequency of high gear.Adopting adaptive redundant lifting scheme, we are able to identify the amplitude modulation

signals and the time locations of the leading impulses in the gearbox vibration and determine thatthe fault occurred in the gearbox.In order to further find the gearbox fault, we use Hilbert envelope analysis to demodulate the

approximation signal at level 2. The envelope of the approximation signal, as shown in Fig. 16,highlights periodicity in the amplitude modulation signals.The envelope spectrum of the approximation signal at level 2 is shown in Fig. 17. It is

dominated by the impulse frequency 213Hz, and its second and fifth harmonic. From Fig. 17, wecan conclude that the impulse frequency 213Hz, which corresponds to the rotating frequency ofthe high gear, is the modulation frequency of the gearbox fault.What resulted in the occurrence of an impact in every rotating period of high gear? We study

the structure of the gearbox in Fig. 10. Helical gears were adopted to transmit force in the aircompressor. The thrust plate was mounted on the high gear, which transmits axial force to the low

Fig. 14. Adaptive redundant lifting scheme decomposition approximation signal at level 2.

Fig. 15. Adaptive redundant lifting scheme decomposition detail signal at level 2.

ARTICLE IN PRESS

Fig. 17. The envelope spectrum of the approximation signal at level 2.

Fig. 18. Gears assembled improperly.

Fig. 16. The envelope of the approximation signal at level 2.

J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–20062004

gear when the helical gears mesh. If the gears are assembled improperly, or there existsmisalignment between the axles of high gear and the compressor, the plane of the thrust plate doesnot parallel the plane of low gear, as shown in Fig. 18. Then, rub-impact between the thrust plateand the low gear occurs once in every rotating period of the high gear, and it has excited the threedistinct frequencies of 1480Hz, 2960Hz and 4231Hz in Fig. 12, which are natural frequencies ofthe gearbox. It is obvious that rub-impact causes the intense vibration with ear-piercing noise inthe gearbox [10].After stoppage, the parallel condition of the gears and the alignment of axles of high gear and

compressor were inspected and assembled. The machine set was adjusted well. When it was

ARTICLE IN PRESS

J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–2006 2005

running again, the vibration became normal and there was no ear-piercing noise. A successfulmechanical fault diagnosis using adaptive redundant lifting scheme has been achieved.

6. Conclusions

In this paper, we have proposed an adaptive redundant lifting scheme for gearbox faultdiagnosis. Firstly, the initial prediction operator and update operator are designed by adoptingdata-based optimisation algorithm, they are padded with zeros, and the redundant predictionoperator and update operator are obtained, which adapt to the characteristics of the signal at thecorresponding level. Then, the adaptive redundant lifting scheme algorithm is constructed byusing the redundant prediction operator and update operator, in which the splitting step isomitted, and the signal at each level is predicted and updated directly, the data number for alllevels remains the same.The proposed method is tested with the analysis of the experimental signal of a gearbox and the

practical signal of a large air compressor. The results show that it performs better than liftingscheme that used the fixed prediction operator and update operator. Adaptive redundant liftingscheme clearly highlights the periodic impulses caused by wear fault in the gearbox vibration, andthe amplitude modulation signal and impulses caused by rub-impact in the compressor byadaptive decomposition. It is obvious that adaptive redundant lifting scheme is an effectivewavelet method to reveal the transient components hidden in the vibration signals.

Acknowledgements

This work was supported by the key project of National Natural Science Foundation of China(No. 50335030), Doctor Program Foundation of University of China (No. 20040698026),National Basic Research Program of China (2005CB724106) and Natural Science Foundation ofXi’an Jiaotong university.

Appendix A. Nomenclature

x original data setxe even subsetxo odd subsetc approximation signald detail signalP prediction operatorU update operatorf scaling functionC wavelet functionP[l] redundant prediction operator at level l

U[l] redundant update operator at level l

ARTICLE IN PRESS

J. Hongkai et al. / Mechanical Systems and Signal Processing 20 (2006) 1992–20062006

References

[1] D. Boulahbal, M. Frid Golnaraghi, F. Ismail, Amplitude and phase wavelet maps for the detection of cracks in

geared systems, Mechanical Systems and Signal Processing 13 (3) (1999) 423–426.

[2] J. Lin, L. Qu, Feature extraction based on Morlet wavelet and its application for mechanical fault diagnosis,

Journal of Sound and Vibration 234 (1) (2000) 135–148.

[3] J. Lin, M.J. Zuo, Gearbox fault diagnosis using adaptive wavelet filter, Mechanical System and Signal Processing

17 (6) (2003) 1259–1269.

[4] S. Loutridis, A. Trochidis, Classification of gear faults using Hoelder exponents, Mechanical System and Signal

Processing 18 (5) (2004) 1009–1030.

[5] W. Sweldens, The lifting scheme: a custom-design construction of biorthogonal wavelets, Applied Computer

Harmonic Analysis 3 (2) (1996) 186–200.

[6] W. Sweldens, The lifting scheme: A construction of second generation wavelets, SIAM Journal of Mathematical

Analysis 29 (2) (1997) 511–546.

[7] I. Daubechies, W. Sweldens, Factoring wavelet transform into lifting steps, Journal of Fourier Analysis

Applications 4 (3) (1998) 245–267.

[8] R. Claypoole, G. Davis, W. Sweldens, Nonlinear wavelet transforms for image coding via lifting, IEEE

Transactions on Image Processing 12 (12) (2003) 1449–1459.

[9] R.L. Claypoole Jr., Adaptive Wavelet Transforms via lifting, Thesis: Doctor of Philosophy, Electrical and

Computer Engineering, Rice University, Houston, Texas, October 1999.

[10] Z.J. He, Y.Y. Zi, Q.F. Meng, et al., Fault Diagnosis Principle of Non-stationary Signal and Applications to

Mechanical Equipment, Higher Education Press, Beijing, 2001, pp. 133–146.