基于第二代小波变换的滚动轴承故障诊断方法与实验研究含程序
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Signal Processing 69 (1998) 177182Remarks on the unsubsampled wavelet transform andthe lifting schemeAlexander Stoffel*Fachhochschule Ko (ln, Fachbereich Nachrichtentechnik, Betzdorfer Strae 2, D-50679 Ko (ln, GermanyReceived 10 November 1997; received in revised form 5 May 1998AbstractThe lifting scheme, which is known to be a very useful tool for the wavelet transform, is adapted to the calculation ofthe unsubsampled wavelet coefficients. It is shown that as in the subsampled case the two-band transform for eachFIR filter pair without common zeros can be performed by a finite number of lifting steps. Inverting each step, one getsa perfect reconstruction equivalent to the arithmetic mean of the result of two different reconstruction filters. Because ofthis easy inversion, the lifting scheme offers a great flexibility in treating boundary conditions in the case of finitely manydata. ( 1998 Elsevier Science B.V. All rights reserved.ZusammenfassungDas Lifting-Schema, das als ein sehr nu tzliches Werkzeug fu r die Wavelet-Transformation bekannt ist, wird an dieBerechnung der Wavelet-Koeffizienten ohne “Subsampling” angepat. Es wird gezeigt, da wie im Fall mit Subsamp-ling fu r jedes FIR-Filterpaar ohne gemeinsame Nullstellen die Zwei-Band-Transformation mit einer endlichen Zahlvon Lifting-Schritten durchgefu hrt werden kann. Indem man jeden Schritt invertiert, erha lt man eine perfekte Rekon-struktion, die a quivalent ist zum arithmetischen Mittel des Ergebnisses zweier unterschiedlicher Rekonstruktionsfilter.Aufgrund dieser einfachen Invertierung bietet das Lifting-Schema eine groe Flexibilita t bei der Behandlung vonRandbedingungen im Fall endlich vieler Daten. ( 1998 Elsevier Science B.V. All rights reserved.Re sume Le sche ma du soule vement (lifting scheme), qui est connu comme un outil tre s utile pour la transformationdondelettes, est adapte au calcul des coefficients dondelettes sans sous-e chantillonnage. Il est de montre que commedans le cas du sous-e chantillonnage pour chaque paire de filtres FIR sans ze ros communs, la transformation en deuxsous-bandes peut e tre effectue e par un nombre fini de pas de soule vement (lifting). En invertissant chaque pas desoule vement, on obtient une reconstruction parfaite qui est e quivalente a la moyenne arithme tique de deux filtres dereconstruction diffe rents. Gra ce a cette inversion facile, le sche ma du soule vement offre une grande flexibilite pour letraitement de conditions aux limites au cas dun nombre fini de donne es. ( 1998 Elsevier Science B.V. All rightsreserved.Keywords: Wavelet; Filter bank; Lifting scheme; Finite length signals*Tel.: #4922182752431; fax: #4922182752836; e-mail: stoffelnebsy.nt.fh-koeln.de.0165-1684/98/$ see front matter ( 1998 Elsevier Science B.V. All rights reserved.PII: S0 1 6 5 -1 68 4 (9 8 ) 00 0 9 9- 11. IntroductionOnly wavelets constructed in the framework ofa multiresolutionanalysis (MRA) withscaling func-tion u will be considered here. We fix the normaliz-ation condition +hk1, such that the two-scalerelation readsu(x)2+k|Zhku(2x!k),(1)and the wavelet fulfillst(x)2+k|Zgku(2x!k).(2)Let f be a signal. We shall study here the redundantcoefficientsri(k)2iPf(x)t(2ix!2ik)dx,(3)si(k): 2iPf(x)u(2ix!2ik)dx,(4)which fulfill the following recursion formulae:si(k)+l|Zhlsi1(k#2i1l),(5)ri(k)+l|Zglsi1(k#2i1l).Their advantage is a simple behaviour, if the signalis translated (usually referred as translation invari-ance).We shall denote the z-transform of a sequence bythe corresponding capital letter as for exampleSi(z): +k|Zsi(k)zk.(6)The above recursion formulae then readSi(z)H(z2i1)Si1(z),(7)Ri(z)G(z2i1)Ri1(z).If we have recursion filters H I (z) and G I (z) satisfyingthe following condition of perfect reconstruction:H I (z)H(z1)#G I (z)G(z1)1,(8)Fig. 1. Perfect reconstructionfilter bank (withoutsubsampling).the coefficients may be successively reconstructedbySi1(z)H I (z2i1)Si(z)#G I (z2i1)Ri(z).(9)For i1, this is visualised in Fig. 1.It is well known that, given any FIR (FiniteImpulse Response) filters H(z) and G(z) with nocommon zeros, Euclids algorithm always furnishesFIR filters H I (z) and G I (z) fulfilling Eq. (8).The reconstruction filters are not unique, wehave the following proposition.Proposition 1. Suppose that a particular solutionH I0-$(z) and G I0-$(z) of Eq. (8) for a fixed pair of analy-sis filtersH(z) and G(z) is given. hen, using any FIRfilter (z), we always get new reconstruction filtersbyH I/%8(z)H I0-$(z)#(z)G(z1),(10)G I/%8(z)G I0-$(z)!(z)H(z1),(11)and any FIR reconstruction filters may be construc-ted in this way, starting from a given pair H I0-$(z) andG I0-$(z).That H I/%8(z) and G I/%8(z) fulfill Eq. (8) is elemen-tary, conversely the Laurent polynomial (z) maybe constructedanalogouslyto the proof of Proposi-tion 2.2. Lifting schemeThe lifting scheme is a very powerful tool for thewavelet transform with subsampling, see 1,57. Itis shown in this section, how it may be adapted tothe unsubsampled case. As the analog of 1,(Theorem 3) we get the following proposition.178A. Stoffel / Signal Processing 69 (1998) 177182Proposition2(lifting).GivenanysolutionH0-$(z), G0-$(z), H I0-$(z), G I0-$(z) of Eq. (8), then forany new solution fulfilling H I/%8(z)H I0-$(z) andG/%8(z)G0-$(z), there is an FIR filter (z) withH/%8(z)H0-$(z)!(z1)G0-$(z),(12)G I/%8(z)G I0-$(z)#(z)H I0-$(z).(13)Conversely, any FIR filter (z) defines a new solu-tion by the above equations.Proof. Let(z): H/%8(z)!H0-$(z),(14)R(z): G I/%8(z)!G I0-$(z).Subtracting the condition Eq. (8) for the new filtersand for the old ones furnishesH I0-$(z)(z1)#R(z)G0-$(z1)0.(15)Because of Eq. (8), the greatest common divisor ofH I0-$(z) and G0-$(z1) is a Laurent polynomial ofdegree zero (i.e. of the form c)zk, k3Z). ThereforeH I0-$(z) divides R(z), and we may construct a Laur-ent polynomial with the desired properties by thedivision algorithm(z): R(z)H I0-$(z).(16)The converse statement may be verified by elemen-tary calculations.hThe following proposition, which corresponds to1 (Theorem 4), may be proved analogously.Proposition 3 (dual lifting). Given any “old” solutionofEq. (8),forany“new”solutionfulfillingH/%8(z)H0-$(z) and G I/%8(z)G I0-$(z), there is anFIR filter (z) withH I/%8(z)H I0-$(z)#(z)G I0-$(z),(17)G/%8(z)G0-$(z)!(z1)H0-$(z).(18)Conversely, any FIR filter (z) defines a new solu-tion by the above equations.We shall now describe the modifications of thefactoring algorithm in 1, which allow to expressthe transform by an arbitrary FIR filter pairH(z), G(z) without common zeros (except possiblyat zero or infinity) as a product of lifting and duallifting steps. Thereby one starts from a particularsimple filter pair which in the unsubsampled case is connected with the “lazy” wavelet. Here, the“lazy” transform we propose corresponds to thetrivial filtersHL(z)GL(z)H IL(z)G IL(z)1/J2.(19)Those factors can be put together in a final factor 12after the addition step as is shown in the scheme ofFig. 2.In our description we shall suppose that thedegree of the Laurent polynomial H(z) is greaterthan or equal to the degree of G(z) (if this is not thecase, one has simply to reverse the role of the twofilters). Euclids algorithm has to be performed,starting with A0(z): H(z1) and B0(z): G(z1).Then Qi1(z) is defined by the division algorithm asthe quotient polynomial of Ai(z) by Bi(z), andBi1(z) as the remainder, such thatAi(z)Bi(z)Qi1(z)#Bi1(z),(20)and such that the degree of the remainder Bi1(z) issmaller than that of Bi(z). Furthermore, we defineAi1(z): Bi(z),andthisiscontinueduntilBi(z)0. Let ni be the first index i when thishappens (except for taking capitals, we kept thenotation of 1). WithV(z): 1i/nC011!Qi(z)D,(21)we haveCAn(z)0DV(z)CA0(z)B0(z)DV(z)CH(z1)G(z1)D.(22)Fig. 2. “Lazy” transform in the unsubsampled case.A. Stoffel / Signal Processing 69 (1998) 177182179By hypothesis, the greatest common divisor An(z)has degree zero, and the division may be performedsuch that this is a constant,i.e. An(z)c. We denotethe matrix elements of V(z) by ij(z). Because ofEq. (22), we get a reconstruction filter pair byH I0(z): 1c11(z),G I0(z): 1c12(z).(23)Using Eq. (22) furnishes for the z-transform of thecoefficientsCS1(z)R1(z)DV1(z)Cc)S0(z)0D,(24)where V1(z) denotes the inverse matrix of V(z).The matrix V1(z) has to be factored as in 1, buthere the last factor has to be changed. In the case ofeven n, we introduce We(z) as an abbreviation forthe first n!2 factors of V1(z):We(z): n1i/1C1Q2i1(z)01DC10Q2i(z)1D(25)(and We(z): I, the 22 identity matrix, if n2).We get the following factorization in lifting anddual lifting steps:CS1(z)R1(z)DWe(z)C1Qn1(z)01DC10Qn(z)!11DCcS0(z)cS0(z)D.(26)In the case of odd n, one has to proceed analog-ously.The above considerations prove as the analog of1, Theorem 7 the following proposition.Proposition 4. Given any FIR filter pair H(z) andG(z) without common zeros (except possibly at zeroor infinity), the calculation of the coefficients S1(z)and R1(z) by Eq. (7) can always be done composinga finite number of lifting and dual lifting steps of theform (26).This means that the left-hand side of Fig. 1 may in the case of an even number of factors obtainedby Euclids algorithm be replaced by the schemeshown in Fig. 3 (the modification for the odd caseis obvious). AsC1Q(z)01D1C1!Q(z)01D,(27)the reconstruction using the lifting scheme becomestrivial. We simply have to reverse the sign of all theLaurent polynomials, and to multiply the final re-sult by c1 (to balancethe constant) and 12(from thelazy transform), in order to get the reconstructedsignal. This is shown in Fig. 4.Therefore, the factorization of the analysis filtersinto lifting steps immediately furnishes reconstruc-tion filters. One may put the lifting and dual liftingsteps of Fig. 4 together, and express them by recon-struction filters H I (z) and G I (z), as it corresponds tothe right-hand side of Fig. 1. Let us have a closerlook, which reconstruction filters one gets by thisprocedure.If we take the inverses of the matrices of theright-hand side of Eq. (26), only the first factordiffers from the product, which defines the matrixV(z) in Eq. (21). A little calculation furnishes foreven nCS0(z)S0(z)D1cC1011DV(z)CS1(z)R1(z)D,(28)Fig. 3. Calculating the unsubsampled wavelet coefficients usingthe lifting scheme (in the case of an even number of factorsobtained by Euclids algorithm).Fig. 4. Reconstruction with the lifting scheme (in the case of aneven number of factors obtained by Euclids algorithm).180A. Stoffel / Signal Processing 69 (1998) 177182and for odd nCS0(z)S0(z)D1cC1110DV(z)CS1(z)R1(z)D.(29)In both cases we get a vector whose two compo-nents agree. Comparison of the two different ex-pressionsfor the componentson the righthand sidefor even and odd n shows, that they are just inter-changed. It turns out that one component is cal-culated by the reconstruction filters furnished byEuclids algorithm (23). The other is calculated bythe following new reconstruction filters:H I1(z)H I0(z)#1c21(z),(30)G I1(z)G I0(z)#1c22(z).(31)Our choice of the lazy transform in Fig. 2 meansthat we take the arithmetic mean of the data recon-structed by the two different filters (with index0 and 1). This might have some stabilizing effect, ifa manipulation of the coefficients takes place (forinstance in the case of denoising). The resultingequivalent filter bank is shown in Fig. 5. However,to save calculation time, one may simply take 1/ctimes one of the terms before the last summation inFig. 4, and gets s0(k) as well.The lifting scheme has a remarkable property forfilters satisfyingH(z)#G(z)1.(32)Examples of such filters are explicitly given by 4,(Eq. (5.75), their properties and advantages in fea-ture extraction are described in 2,3. For thoseFig. 5. Filter bank equivalent to the reconstruction by the lift-ing scheme in Fig. 4.Fig. 6. Analysis and reconstruction with the lifting scheme inthe case of filters satisfying Eq. (32).filters the factorization into lifting steps is trivial.We always get n2, c1, Q1(z)!1 andQ2(z)1!H(z1). The correspondinglifting stepsfor analysis and reconstruction are shown inFig. 6. This scheme furnishes the reconstructionfilters H I0(z)G I0(z)1, H I1(z)H(z1) and G I1(z)H(z1)#1.A similar investigation as in 1 shows, that inour case there is no reduction of calculation costusing lifting. The main advantage of lifting in theunsubsampled case will be its great flexibility toadopt boundary conditions in the case of finitelymany data, which is discussed in the next section.Until now in this section, all considerations wererestricted to the calculation of S1(k) and R1(k), i.e.to the first step in scale. Because of Eq. (7), liftingcan also be applied to the general case. For thesteps i1, one simply has to replace all the factorsQk(z) by Qk(z2i1).3. How to treat finite length signalsAs the wavelet transform is linear, we shall de-scribe it in this subsection by matrices. Instead ofthe sequences si(k) and ri(k), we consider vectorssiand ri3RN and identifys0(x0,x1,2,xN1) withthe N original data points. We may put the vectorss1and r1together to a single vector in R2N. Con-volution operators described by matrices, as theyappear on the right hand side of (25), whichcorrespond to lifting or dual lifting steps, will herebecome2N2Nmatrices. Theyhavea blockstruc-ture of the following form (I denotes the NNidentity matrix):CI0QIDrespectivelyCIQ0ID.A. Stoffel / Signal Processing 69 (1998) 177182181In the infinite case, the operator Q corresponds toa convolution with a sequence q(k) whose z-trans-form is Q(z), i.e. it can be considered as an infinitematrix with matrix elementsQikq(i!k).(33)AsCIQ0ID1CI!Q0ID,(34)we have total freedom, in choosing a finite matrix here again denoted by Q to replace the originalconvolutionoperator.And at the sametime, perfectreconstruction is maintained. Thus in the finitecase, Q will be an NN matrix with the followingstructure:QQLLQLI0QILQIIQIR0QRIQRR.(35)The matrix elements of all the inner submatricessatisfy Eq. (33), as they should agree with the infi-nite case. We have only to change the boundarymatricesQLLand QRR. Indeed,we may choose themarbitrarily.One of the simplest choices would be, tolet all the matrix elements satisfy Eq. (33), whichcorresponds to zero padding with subsequent pro-jection to RN (note that this does not correspond tozero padding for the whole wavelet transform, ascan be easily seen by calculating examples).For an even number of steps in Euclids algo-rithm, the wavelet transform using lifting now hasthe formCs1r1DcCIQ10IDCI0Q2ID2CI0Qn!IIDCs0s0D(36)(with an obvious modification in the case of odd n).Practically, this means that using lifting, we maychoose any boundary condition we want evendifferent ones for different lifting steps as far as wetake exactly the same boundary condition in thecorresponding inverse lifting step for reconstruc-tion.The above formula concerns the calculation forthe scale step i1. For general i1, the size of theboundary matrices QLLand QRRwill be increasedby a factor 2i1. We have to fill in 2i1!1 zerosbetween the matrix elements, and each row has tobe repeated 2i1 times, each time shifted by oneposition to the right. To obtain perfect reconstruc-tion, one has only to replace the whole block sub-matrix Q by !Q (respectively Q!I by !Q#I).The lifting scheme makes it therefore easy toimplement more
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