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Digital Signal Processing 20 (2010) 276288Contents lists available at ScienceDirectDigital Signal Processing/locate/dspMechanical equipment fault diagnosis based on redundant secondgeneration wavelet packet transformRui Zhou, Wen Bao, Ning Li, Xin Huang, Daren YuHarbin Institute of Technology, 458#, No. 92, West Da-Zhi Street, Harbin, Heilongjiang, PR Chinaa r t i c l ei n f oa b s t r a c tArticle history:Available online 3 May 2009Keywords:Second generation wavelet packet transformLifting schemeFeature extractionFault diagnosisWavelet transform has been widely used for the vibration signal based mechanicalequipment fault diagnosis. However, the decomposition results of the discrete wavelettransform do not possess time invariant property, which may result in the loss of usefulinformation and decrease the classification accuracy of fault diagnosis. To overcome thisdeficiency, a novel fault diagnosis method based on the redundant second generationwavelet packet transform is proposed. Firstly, the redundant second generation waveletpacket transform is constructed on the basis of second generation wavelet transform andredundant lifting scheme. Secondly, the vibration signals are decomposed by redundantsecond generation wavelet packet transform and then the faulty features are extracted fromthe resultant wavelet packet coefficients. Finally, the extracted fault features are given asinput to classifiers for identification. The proposed method is applied for the fault diagnosisof gearbox and gasoline engine valve trains. Test results indicate that a better classificationperformance can be obtained by using the proposed fault diagnosis method in comparisonwith using second generation wavelet packet transform based method.2009 Elsevier Inc. All rights reserved.1. IntroductionGrowing demand for high quality production requires that deviation of machine conditions from its normal setting shouldbe identified and fixed promptly to reduce costly machine downtime and maintain high productivity. As a result, researchon effective mechanical equipments health monitoring and diagnosis has been enhanced in recently years 1,2. Since thevibration signal collected from these equipments during operation contains valuable information about the machine orpart condition, vibration analysis has been adopted widely as a means for machine failure identification 3. The vibrationsignal is often a mixture signal which simultaneously contains stationary, non-stationary and noisy components. Therefore,the information for maintenance decisions is not readily available from these vibration data unless the appropriate signalprocessing techniques are chosen 4.The wavelet transform (WT), as a state-of-the-art tool for signal processing, has focused much attention on both theoreticanalysis and engineering applications in many fields. WT can be used for multi-scale analysis of a signal through dilation andtranslation, so it can extract signal features from both time domain and frequency domain effectively. Consequently, WT hasbeen successfully applied for the condition monitoring and fault diagnosis of electromechanical equipment 5. A drawbackof WT is that the frequency resolution is rather poor in the high-frequency subband which the faulty characteristics alwaysexist in. The wavelet packet transform (WPT), a generalization of wavelet bases, is alternative bases formed by taking linearcombinations of usual wavelet functions 6,7. WPT divides the frequency space into various parts and allows a better*Corresponding authors. Fax: +86 451 86413241.E-mail addresses: hit_ (R. Zhou), yudaren (D.R. Yu).1051-2004/$ see front matter2009 Elsevier Inc. All rights reserved.doi:10.1016/j.dsp.2009.04.005R. Zhou et al. / Digital Signal Processing 20 (2010) 276288277timefrequency localization of signals. In recent years, WPT has been used as a popular method in the field of conditionmonitoring and fault diagnosis 812. In application of WT and WPT, it is crucial for selecting a proper wavelet functionfor a special problem and the engineering experiences show that the wavelet function should be selected according to thefault feature to be detected 13.The second generation wavelet transform (SGWT) is a new wavelet construction method using lifting scheme in the timedomain. It abandons the Fourier transform as design tool for wavelets, and wavelets are no longer defined as translatesand dilates of one fixed function. Compared with classical WT, SGWT possesses several advantages, including possibility ofadaptive and nonlinear design, in-place calculations, irregular samples and integral transform 1416. Recently, the appli-cations of SGWT and second generation wavelet packet transform (SGWPT) to condition monitoring and fault diagnosis ofelectromechanical equipment deserve more attentions 1720.Unfortunately, SGWT does not have the property of time invariant. Using SGWT, the decomposition results of a delayedsignal are not the time-shifted version of those of the input signal, which may lead to the loss of useful faulty informationfor feature extraction and fault diagnosis. The redundant lifting scheme possesses time invariant property and overcomesthe disadvantage of lifting scheme by getting rid of the split step and zero padding of prediction operator and updateoperator. The approximation and detail signals at all levels are the same length as the input signal in the redundant liftingscheme 2123.In this paper, on the basis of SGWT and redundant lifting scheme, the redundant second generation wavelet packettransform (RSGWPT) is constructed, and then fault diagnoses of mechanical equipments are performed by using the pro-posed RSGWPT. The rest of the paper is organized as follows. In Section 2, the fundamental theory of SGWT and SGWPTis reviewed briefly. In Section 3, the construction method of RSGWPT is introduced. The fault diagnosis method based onRSGWPT is described in Section 4. In Section 5, the proposed fault diagnosis method is applied to diagnose different statesof a gearbox and the valve trains on a gasoline engine. The comparison results with SGWPT based fault diagnosis methodare also shown. Finally the conclusions have been drawn in Section 6.2. Review of second generation wavelet transform2.1. Second generation wavelet transformSecond generation wavelet transform, proposed by Wim Sweldens, is a new wavelet construction method using liftingscheme. It can be seen as an alternate implementation of classical discrete wavelet transform. The main feature of the secondgeneration wavelet transform is that it provides an entirely spatial domain interpretation of the transform, as opposed tothe traditional frequency domain based constructions 15. The decomposition stage of SGWT consists of three steps: split,prediction and update.In the split step, an approximate signal alat level l is split into even samples and odd samples.al+1=al(2i),dl+1=al(2i+1)(1)In the prediction step, a prediction operator P is designed and applied on al+1to predict dl+1. The resultant predictionerror dl+1is regarded as the detail coefficients of al.dl+1(i) =dl+1(i)M/2?r=M/2+1pral+1(i+r)(2)where prare coefficients of P and M is the length of pr.In the update step, a designed update operator U is applied on dl+1. Adding the result to the even samples, the resultantal+1is regarded as the approximate coefficients of al.al+1(i) =al+1(i)+N/2?j=N/2+1ujdl+1(i+j1)(3)where ujare coefficients of U and N is the length of uj.Iteration of the above three steps on the output a, and then the detail and approximation coefficients at different levelsare generated.The reconstruction stage of SGWT is a reverse procedure of the decomposition stage, which includes inverse update step,inverse prediction step and merging step.al+1(i) =al+1(i)N/2?j=N/2+1ujdl+1(i+j1)dl+1(i) =dl+1(i)+M/2?r=M/2+1pral+1(i+r)al(2i) =al+1,al(2i+1) =dl+1(4)278R. Zhou et al. / Digital Signal Processing 20 (2010) 276288Fig. 1. Structure of second generation wavelet transform, both analysis side and synthesis side.The operators P and U are built by means of interpolating subdivision method (ISM) 16. Choosing different P and Uis equivalent to choosing different biorthogonal wavelet filters 24. Fig. 1 depicts the structure of SGWT. The computationalcosts of the forward and inverse transform are exactly the same.2.2. Second generation wavelet packet transformThe timefrequency resolution of SGWT varies with the decomposition levels. It gives good time and poor frequencyresolution at high frequency subband, and good frequency and poor time resolution at low frequency subband. In order toobtain a higher resolution in the high frequency subband, SGWPT has been constructed and hence the detail coefficientsat each level are further decomposed to obtain their approximation and detail components 25,26. The decomposition andreconstruction stages of SGWPT are described as below.In the decomposition stage, Xl,kis split into even samples Xl,keand odd samples Xl,ko,Xl,ke=Xl,k(2i),Xl,ko=Xl,k(2i+1)(5)where Xl,krepresents the coefficients of the kth node at level l.Then calculate each subband coefficients at level l+1.Xl+1,2=Xl,1oP(Xl,1e)Xl+1,1=Xl,1e+U(Xl+1,2).Xl+1,2l+1=Xl,2loP(Xl,2le)Xl+1,2l+11=Xl,2le+U(Xl+1,2l+1)(6)In the reconstruction stage, the subband coefficients to be reconstructed are reserved, and then other subband coefficientsare set to be zeroes. Finally, the reconstructed results are obtained by the following formula.Xl,2le=Xl+1,2l+11U(Xl+1,2l+1)Xl,2lo=Xl+1,2l+1+P(Xl,2le)Xl,2l(2i) =Xl,2leXl,2l(2i+1) =Xl,2lo.Xl,1e=Xl+1,1U(Xl+1,2)Xl,1o=Xl+1,2+P(Xl,1e)Xl,1(2i) =Xl,1eXl,1(2i+1) =Xl,1o(7)Overall, the decomposition and reconstruction stages of SGWPT are shown in Figs. 2 and 3.R. Zhou et al. / Digital Signal Processing 20 (2010) 276288279Fig. 2. The decomposition stage of SGWPT.Fig. 3. The reconstruction stage of SGWPT.3. Redundant second generation wavelet packet transform3.1. Redundant lifting schemeIn the redundant lifting scheme, the split step is discarded. Assuming Pland Ulrepresent the prediction and updateoperators of the redundant lifting scheme at level l, the coefficients of Pland Ulare obtained by padding Prand Ujofinitial operator P and U with zeroes 21.pli=p00,0,.,0? ? ?2l1,p01,0,.,0? ? ?2l1,p02,.,p0M20,.,0? ? ?2l1,p0M1(8)ulj=u00,0,.,0? ? ?2l1,u01,0,.,0? ? ?2l1,u02,.,u0N20,.,0? ? ?2l1,u0N1(9)The decomposition results of an approximation signal alat level l via redundant lifting scheme are expressed by followingequations.280R. Zhou et al. / Digital Signal Processing 20 (2010) 276288Fig. 4. The forward and inverse transform of redundant lifting scheme.?dl+1=alPl+1alal+1=al+Ul+1dl+1(10)where al+1and dl+1are approximation coefficients and detail coefficients of alat level l+1.The reconstruction procedure of redundant lifting scheme is directly achieved from its forward transform, which isexpressed as below.al=12?al+1Ul+1dl+1+dl+1+Pl+1?al+1Ul+1dl+1?(11)The forward and inverse transform of redundant lifting scheme is shown in Fig. 4.3.2. Construction of RSGWPTWith the redundant lifting scheme and SGWPT, the RSGWPT is easily to be constructed. The prediction step and updatestep of RSGWPT at level l are performed by using Pland Ul, which are expressed as follows.Xl+1,2=Xl,1Pl+1(Xl,1)Xl+1,1=Xl,1+Ul+1(Xl+1,2).Xl+1,2l+1=Xl,2lPl+1(Xl,2l)Xl+1,2l+11=Xl,2l+Ul+1(Xl+1,2l+1)(12)The reconstruction stage of RSGWPT can be obtained from its decomposition stage and expressed by following equations.Xl,2l=12?Xl+1,2l+11Ul+1(Xl+1,2l+1)+Xl+1,2l+1+Pl+1?Xl+1,2l+11Ul+1(Xl+1,2l+1)?.Xl,1=12?Xl+1,1Ul+1(Xl+1,2)+Xl+1,2+Pl+1?Xl+1,1Ul+1(Xl+1,2)?(13)The forward and inverse transform of RSGWPT are shown in Figs. 5 and 6. Owing to without the split operation in thedecomposition stage of RSGWPT, the approximation and detail coefficients at all levels have the same length as that of theinput signal. Consequently, the decomposition results of RSGWPT possess time invariant property and keep the informationof the raw signal perfectly.4. The proposed fault diagnosis methodFig. 7 shows the flow diagram of the proposed fault diagnosis method. The vibration signals are acquired from themonitoring mechanical equipments. After A/D conversion, the sampled vibration data are decomposed by RSGWPT.In the feature extraction stage, nine statistical characteristics (i.e. peak value, mean, standard deviation, root mean square,shape factor, skewness, kurtosis, crest factor and pulse index) are calculated from each of the resultant subband waveletpacket coefficients.R. Zhou et al. / Digital Signal Processing 20 (2010) 276288281Fig. 5. The decomposition stage of RSGWPT.Fig. 6. The reconstruction stage of RSGWPT.For classification, three classical machine learning algorithms including C4.5 decision tree (C4.5), radial basis functionneural network (RBFNN) and support vector machine (SVM) are employed. In the following, the principles and some basicmathematical expressions for such algorithms are briefly reviewed.The C4.5 decision tree classifier is a type of induction algorithm 27. An example of a decision tree is shown in Fig. 8.A test node represents the selected predictor, which is used to divide the samples into subsets. Each branch descendingfrom that node corresponds to one of the possible values for this predictor. Finally, the leaf node provides the classificationof samples in the subsets. Generally speaking, a decision tree is constructed from a set of samples by using the divide-and-conquer strategy; that is, a best predictor is selected in each test node to split samples into the smaller subsets. The qualityof predictors is evaluated by using the information gain ratio 28. When the dataset is divided into several subsets by thepredictor, the information gain ratio is used to measure the reduction of the uncertainty of samples associated with thisprocess. This means that the samples with the same classification are sorted into the same subset as much as possible. For acontinuous predictor P, the threshold t in P?t should be found to maximize the information gain ratio 29. The samplesare sorted on their values of predictor P to give ordered distinct values v1,v2,.,vN. Every pair of adjacent values suggestsa potential threshold t= (vi+vi+1)/2. When the best predictor and its threshold are determined, the predictor is used totest these samples at the root node of the tree. A descendant of the root node is then created according to the thresholdof this predictor, and the training samples are sorted into an appropriate descendant node. C4.5 employs the top-down andrecursive splitting technique to produce the subtree. When samples in the subset have the same classification or all possibletests have the same class distribution, the leaf node is generated. The classification of the samples in the leaf node is thesame as the most frequent classification in this leaf node.282R. Zhou et al. / Digital Signal Processing 20 (2010) 276288Fig. 7. Flow chart of fault diagnostic procedure.Fig. 8. Example of a decision free.The RBFNN has a feed forward architecture with an input layer, a hidden layer and an output layer as shown in Fig. 9.The radial basis functions are embedded into a two-layer feed forward neural network. Such a network is characterized by aset of inputs and a set of outputs. In between the inputs and outputs there is a layer of processing units called hidden units.Each of them implements a radial basis function 30. The input layer of this network has NIunits for a NIdimensionalinput vector. The input units are fully connected to the NHhidden layer units, which are in turn fully connected to the NCoutput layer units, where NCis the number of output classes. The activation functions of the hidden layer were chosen tobe Gaussians, and are characterized by their mean vectorsi, and covariance matrices Ci=2iI, i=1,2,.,NH. Then theactivation function of the ith hidden unit for an input vector xjis given bygi(xj) =exp?xji?222i?(14)Theiand2iare calculated by using suitable clustering algorithm. The hidden layer units are fully connected to theNCoutput layer through weights wik. The output units are linear, and the response of the kth output for an input xjisgiven byyk(xj) =NH?i=0wikgi(xj),k=1,2,.,NC(15)Training the RBF involves two stages. First, the basis functions must be established using an algorithm to cluster data inthe training set. Typical ways to do this include Kohohen self-organizing maps, K-Means clustering, decision trees, geneticalgorithms or orthogonal least squares and Max-Min algorithms 31,32. Next, it is necessary to fix the weights linkingthe hidden and the output layers. If neurons in the output layer contain linear activation functions, these weights can beR. Zhou et al. / Digital Signal Processing 20 (2010) 276288283Fig. 9. Architecture of a radial basis function neural network.Fig. 10. (A) A separating plane with small margin; (B) A separating plane with larger margin.calculated directly using matrix inversion and matrix multiplication. Because of the direct calculation of weights in an RBF,it is usually much quicker to train than an equivalent multilayer perceptron training algorithm.The SVM is a statistic machine learning technique that has been widely applied in the pattern recognition area 3336.Let(xi,yi),i=1,.,Nbe a training sample set S and each sample xibelongs to a class by yi 1,1. The goal of SVMis to find a hyperplane which divides S, such that all the points with the same label are on the same side of the hyperplanewhile maximizing the distance between the two classes A, B and the hyperplane. An example of the optimal hyperplaneof two data sets is presented in Fig. 10. As shown in Fig. 10, rings and diamonds stand for these two classes of samplepoints respectively; H is a separating plane. H1and H2are the planes that are parallel to H and respectively pass throughthe sample points closest to H in these two classes. The distance between H1and H2is defined as margin. The optimalseparating plane that has the smallest generalization error is the one that not only correctly separates all sample pointsinto these two classes but also leaves the largest margin between H1and H2. SVM can be used in nonlinear classificationtasks with application of kernel functions. The basic idea is to transform input vectors into a high dimensional feature spaceusing a nonlinear transformation, and then to do a linear separation in feature space. To construct a nonlinear supportvector classifier, the inner product (xi,x)is replaced by a kernel function K(xi,x):f(x) =sgn?N?i=1iyiK(xi,x)+b?(16)284R. Zhou et al. / Digital Signal Processing 20 (2010) 276288Fig. 11. (A) Experimental setup; (B) accelerometer location; (C) broken teeth; and (D) slight-worn teeth.The SVM has two layers. During the learning process, the first layer selects the basis K(xi,x), i=1,2,.,N, from thegiven set of bases defined by the kernel; the second layer constructs a linear function in this space. This is completelyequivalent to constructing the optimal hyperplane in the corresponding feature space. The SVM algorithm can construct avariety of learning machines by use of different kernel functions such as linear, polynomial, Gaussian and Laplacian radialbasis function.All algorithms are implemented in WEKA (Waikato Environment for Knowledge Analysis) which is data mining softwarein java and can be obtained at http:/www.cs.waikato.ac.nz/ml/weka. Default setting of C4.5 whose eighth version is calledJ48 in WEKA is adopted. For RBFNN, K-Means clustering method is used and the RBFNN is set to five intermediate nodes.For SVM, the radial basis function kernel is selected and the complexity parameter C is set to 10. All algorithm are testedby ten fold cross validation which divided samples into ten folds and then 9 folds are used for training and the remaining1 fold for testing. The results of the 10 tests are given as a mean of all tests.5. ApplicationsTo demonstrate the performance of the proposed fault diagnosis method, two application examples for the diagnosis ofgearbox conditions and valve trains conditions on a gasoline engine are present in this section.5.1. Fault diagnosis for gearboxIn Ref. 11, the experimental setup consists of a four-speed motorcycle gearbox, an electrical motor with a constantnominal rotation speed of 1420 RPM, a load mechanism, multi-channel pulse analyzer system, a triaxial accelerometer,tachometer and four shock absorbers under the base of test-bed, as depicted in Fig. 11(A). All vibration signals were collectedfrom the experimental testing of gearbox using the accelerometer which was mounted on the outer surface of the bearingcase of input shaft of the gearbox as shown in Fig. 11(B). Three different fault conditions were tested that were slight-worn,medium-worn and broken-teeth of gear as shown in Fig. 11(C) and (D). Also a faultless condition was tested. The rotationalspeed of the system was measured by tachometer used as a measure to compensate for the fluctuations of rotational speedof the system due to uncertainties of the load mechanism which is a constant static load. The signals were sampled at16384 Hz lasting 8 s.The measured vibration signals is dimensionally synchronized by using piecewise cubic hermite interpolation methodfrom the revolution point of view which were not equal following the signals acquired by tachometer. After the synchro-nization of the vibration signals per each revolution, there were 704 samples points in each period of these four states.Finally, a dataset was obtained. The four states of the tested gearbox in one period are shown in Fig. 12. In Fig. 12, S-1,S-2, S-3 and S-4 represent faultless condition, slight-worn condition, medium-worn condition and broken-teeth conditionrespectively.In order to identify different working conditions of the monitoring gearbox, the proposed fault diagnosis method isperformed. Based on ISM, the initial operator P and U with M=4 and N=4 are calculated. The initial prediction operatoris0.0625,0.5625,0.5625,0.0625and the initial update operator is0.0313,0.2813,0.2813,0.0313. In this example,each sample was decomposed to level 3, then eight subband coefficients were obtained and 72 statistical characteristics foreach sample were extracted. The classification accuracies are reported in Table 1.R. Zhou et al. / Digital Signal Processing 20 (2010) 276288285Fig. 12. Gearbox vibration signals.Table 1Classification accuracies (%) of gearbox experiment.Features extracted by using RSGWPTFeatures extracted by using SGWPTC4.5RBFNNSVMC4.5RBFNNSVM99.376210099.937699.213899.937699.8971For the purpose of comparison, using the same P and U, each sample was decomposed to level 3 via SGWPT andthe same 72 statistical characteristics were calculated from the resultant eight subband wavelet packet coefficients. Theclassification accuracies are obtained and also listed in Table 1.From Table 1, a slight improvement can be obtained in the classification accuracies of all three classifiers when thefeatures extracted by using RSGWPT are given as input.5.2. Fault diagnosis for valve trains on gasoline engineIn the second example, we used the proposed method to identifying different states of valve trains on a gasoline engine.The experimental setup and the data acquisition system are shown in Fig. 13. The dead line of the fourth cylinder generatedby the pulse sensor was set as the basis point for all the signals. A working cycle of vibration signals are recorded, whichinclude 720 degree of crank angle. The vibration acceleration sensor is mounted on the cylinder head and the samplingfrequency is 24 kHz.Several typical faults of the valve trains were simulated, such as the intake valve clearance being too large and theexhaust valve clearance being too large. Five states including four faulty states and one normal state of valve trains weretested. The description of the five states of valve trains is shown in Table 2. Before identifying different working conditions,the synchronization of the obtained vibration signals per each revolution was performed and the method was the same asthat used in the first example. In this case there were 2160 samples points in each period. The vibration data in one cycleof different working conditions are shown in Fig. 14.Based on the proposed fault diagnosis method and using the same decomposition level and initial operators (P andU)employed in the first example, the classification accuracies of valve trains on the test gasoline engine were calculatedand reported in Table 3. Also, the classification results obtained by using the features extracted by SGWPT are listed in thesame table. From Table 3, we can find a considerable improvement in the classification accuracies compared to the resultsobtained by using the SGWPT based fault diagnosis method with all of the three classifiers.286R. Zhou et al. / Digital Signal Processing 20 (2010) 276288Fig. 13. Sketch of the experimental setup.Table 2Five states of valve trains on gasoline engine.State codeFaulty descriptionS-1Normal stateS-2Intake valve clearance of the 4th cylinder is too large and exhaust valve clearance of the 4th cylinder is too largeS-3Intake valve clearance of the 3rd and 4th cylinder is too large and exhaust valve clearance of the 2nd cylinder is too largeS-4Intake valve clearance of the 4th is too large and exhaust valve clearance of the 1st cylinder is too largeS-5Intake valve clearance of the 3rd and 4th is too large and exhaust valve clearance of the 1st and 2nd cylinder is too largeFig. 14. Valve trains vibration signals.5.3. DiscussionFrom the above two experiments, we can find that the classification accuracies obtained by using the proposed methodare higher than those obtained by using SGWPT based fault diagnosis method. But the degree of improvement is different.In the gearbox experiment, as for C4.5, RBFNN and SVM, there are increases of 0.1624%, 0.0624% and 0.0405% respectively.But in the valve trains experiment, the increases are 7.7295%, 4.3478% and 6.0386% respectively. We can explain the problemfrom the point of view of the complexity of condition classification. From Fig. 12, it can be found that the difference betweenthe waveforms of the four conditions is large. From Table 1, using SGWPT based method has already achieved very highclassification accuracies and therefore the room for improvement is little. As to the second experiment, we can clearly findthat the difference between the waveforms of the five conditions is small from Fig. 14. Thus, the complexity of conditionR. Zhou et al. / Digital Signal Processing 20 (2010) 276288287Table 3Classification accuracies (%) of valve trains experiment.Features extracted by using RSGWPTFeatures extracted by using SGWPTC4.5RBFNNSVMC4.5RBFNNSVM86.715091.787494.927578.985587.439688.8889Table 4Classification accuracies (%) under different decomposition levels.LevelGearbox experimentValve trains experimentC4.5RBFNNSVMC4.5RBFNNSVM199.122810010080.917983.091882.3671299.561410010084.299590.821389.1304399.561410010086.473493.719894.4783499.003510010087.956594.202997.3430599.342110010088.372094.444497.8261classification is increased. At this time, the RSGWPT based fault diagnosis method can achieve higher improvements becausethe time invariant property inhering in RSGWPT. It ensures that the decomposition results of RSGWPT can retain morefaulty information. Consequently, the statistical features extracted by RSGWPT have a greater ability to distinguish betweendifferent conditions compared with those extracted by SGWPT.It is to be noted that the synchronization of the vibration signals before classification is not necessary for the proposedfault diagnosis method. The purpose of preprocessing of the raw vibration signals is to make comparisons between RSGWPTand SGWPT based diagnosis methods. Actually, the RSGWPT of level l is well defined for any sample size, while the SGWPTof level l must restrict the sample size to an integer multiple of 2l. Consequently, the sample sizes of each revolution in theabove two experiments were synchronized to 704 and 2106 respectively.The selection of decomposition level l is a problem that should be noted in the wavelet-based fault diagnosis method. InTable 4, we list the classification accuracies obtained by using the proposed method under different decomposition levels.In this time, no preprocessing operation of the raw vibration signals is performed.In the gearbox experiment, the classification accuracies under different decomposition levels show no significant dif-ferences when using C4.5, while the classification accuracies are all 100% when using RBFNN and SVM. Thus there is noobvious relationship between decomposition level and classification accuracy can be found. In the valve trains experiment,as for all of the three classifiers, we can clearly find that the classification accuracies are increase with the decompositionlevel and at the same time the increase rates tends to be reduced. It appears to be that the increase of decomposition levelwill lead to higher classification performance. But we also know that the computation cost of RSGWPT and the numberof subband signals are increase with the decomposition level. On the other hand, more subband signals will lead to morestatistical characteristics, and thus increase the complexity of classification. So, when selecting the decomposition level fora specific fault diagnosis problem, there is an unavoidable tradeoff between classification accuracy and the complexity ofalgorithm.6. ConclusionIn this paper, we have proposed a novel fault diagnosis method for mechanical equipment based on redundant secondgeneration wavelet packet transform. In order to possess time invariant property, the redundant second generation waveletpacket is presented. Because the length of the coefficients at each level is equal to the length of raw signal after decompo-sition, the wavelet packet coefficients can retain more faulty information, and hence can be extracted more distinguishingstatistical features for classification.In order to investigate the effectiveness for practical applications, the proposed method is tested in identifying differentworking conditions of gearbox and valve trains on a gasoline engine. The test results show that the proposed method canachieve a higher classification accuracy rate than the result obtained by using second generation wavelet packet transformbased fault diagnosis method.The high performance of the proposed method is attributed primarily to RSGWPT s inherent time invariant property.Therefore the redundant second generation wavelet packet transform provides an effective tool to condition monitoring andfault diagnosis from the vibration signals for mechanical equipment.AcknowledgmentsThe authors would like to thank Dr. Javad Rafiee of Azad University for making the gearbox vibration data set availableand providing useful clarifications. This paper has benefited from the comments of two anonymous reviewers.288R. Zhou et al. / Digital Signal Processing 20 (2010) 276288References1 M. Cavacece, A. Introini, Analysis of damage of ball bearings of aeronautical transmissions by auto-power spectrum and cross-power spectrum, ASMEJ. Vibr. 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