Accurate Instantaneous Frequency Estimation with Iterated Hilbert Transform and Its Application

Transcription

1 Proceedngs of the 7th WSEAS Interntonl Conference on SIGAL PROCESSIG, ROBOTICS nd AUTOMATIO (ISPRA '8) Unversty of Cmbrdge, UK, Februry -, 8 Accurte Instntneous Frequency Estmton wth Iterted Hlbert Trnsform nd Its Applcton SHURE QI, YI QI, YOGFAG MAO Mechncl Engneerng Deprtment Chongqng Unversty Chongqng, 43 CHIA Abstrct: - Iterted Hlbert trnsform (IHT) s new method for multcomponent demodulton. The prncple of IHT s ntroduced, nd some of ts propertes re reserched. Although the mpltudes of the demodulted components obtned by IHT re ccurte, there re lmttons n the drect estmton of nstntneous frequences v the extrcted phses, so smoothed nstntneous frequency estmton (SIFE) method bsed on dfference opertor nd zero-phse dgtl low-pss flterng s proposed. The smulton results show tht the proposed pproch hs hgher performnce thn the dptve segmentton lgorthm nd Hlbert-Hung trnsform. Fnlly, IHT nd SIFE re ppled to fult dgnoss of rollng berng, nd the results show tht the multcomponent fult vbrton sgnl cn be demodulted correctly nd the we feture of fult sgnls cn be extrcted effcently wth IHT nd SIFE. Key-Words: - Multcomponent demodulton, Hlbert trnsform, Ampltude envelopes, Flterng, Instntneous frequency estmton, Fult dgnoss. Introducton When there s fult on the rotry component n the mechncl system, perodcl mpulse force wll occur, whch mes the vbrton sgnls present the feture of modulton. The mpltude envelopes nd phse sgnls of those modulted sgnls contn rch fult nformton. Addtonlly, most mchnery fult vbrton sgnls re multcomponent AM-FM sgnls, therefore, multcomponent demodulton s n effectve wy to extrct the fult chrcterstcs nd dgnose the fult type. Currently, the typcl multcomponent demodulton methods nclude: () Multbnd energy seprton lgorthm (MESA) [; () Perodc lgebrc seprton energy demodulton (PASED) [; () Hlbert-Hung trnsform (HHT) [3,4; (v) Iterted Hlbert trnsform (IHT) [5, whch s new method for multcomponent nlyss proposed by Gnfelc etc., hence clled Gnfelc trnsform n ths pper. Compred wth the former three methods, Gnfelc trnsform s of hgher demodulton ccurcy, nd of lower computtonl complexty. By the reserch of Gnfelc trnsform, the uthors fnd tht lthough exct mpltude envelope of ech component cn be obtned wth Gnfelc trnsform, however, to drectly compute the nstntneous frequences (IFs) from the extrcted phse sgnls preserves some lmttons. And the lmttons re theoretclly nlyzed tng generc two-component AM-FM sgnl s n exmple. Then smoothed nstntneous frequency estmton (SIFE) method s proposed, whch computes the dervtve of the phse sgnl obtned by Gnfelc trnsform, nd then crres out zero-phse dgtl low-pss flterng. Smulton shows tht ths method cn exctly estmte the IF of correspondng component. Fnlly, Gnfelc trnsform nd SIFE s ppled to fult dgnoss for rollng berng, whose results ndcte tht the method s ble to cheve multcomponent demodulton ccurtely nd extrct the we fult chrcterstcs n the vbrton sgnl. Gnfelc Trnsform. Hlbert Trnsform For n rbtrry sgnl x (, ts Hlbert trnsform s defned s [6 ( τ ) [ ( ) d τ π x H x t () t τ And the nlytcl sgnl z ( of x ( cn be defned s z( H[ A( e () In ths cse, the envelope A ( s gven by [ A z( x H (3) ISS: ISB:

2 Proceedngs of the 7th WSEAS Interntonl Conference on SIGAL PROCESSIG, ROBOTICS nd AUTOMATIO (ISPRA '8) Unversty of Cmbrdge, UK, Februry -, 8 nd the phse functon ( ( t ) rg[ z( (4) The nstntneous frequency cn be further computed by d( f ω (5) π π dt. Multcomponent AM-FM Decomposton Gven n rbtrry sgnl x (, nd tng dvntge of () t cn be represented s x Re[ z( cos[ α (6) where Re[ represents the rel prt of complex number, mpltude s z(, nd phse s α ( t ) rg[ z(. In order to obtn the multcomponent model, we could tertvely operte Hlbert trnsform to the mpltude envelope. Assume the mpltude envelope fter tertons s (. It s esy to get tht ( wll exhbt osclltng behvor nd be lwys non-negtve f the sgnl contns mxture of snusods. Therefore, ( cn be decomposed nto the trend ( nd lterntng component ~ ( t ) by mens of certn flterng lgorthm, tht s ~ (7) In order to ssure tht the result of the next Hlbert trnsform s resonble, ~ ( t ) should be zero-men osclltng sgnl, nd t cn be obtned by pplyng hgh-pss flterng to (. Then the trend ( cn be esly computed by vrtue of (7). Formlly, strtng wth s the frst step of ths tertve lgorthm, from (6) nd (7) we obtn cos[ (8) [ ( ) ~ t cos[ where α. Denote the nlytcl sgnl z ( of the lterntng component ~ ( t ) s z ~ ( ) [ ~ ( t H (9) And denote z ( () α rg[ z ( () Thus, ~ ( t ) cn be rewrtten s ~ cos[ α ( ) () t When, there s ~ ( cos[ α( (3) Substtute the bove equton nto (8), yelds cos[ ( {cos[ α( cos[ } (4) And wth Werner trgonometry formul, we obtn ( cos[ cos[ (5) where α( (6) α( (7) After the frst terton hs been done, contnue to del wth ( n the bove wy to fnsh the second terton. In the sme wy, fter tertons nd ( s fltered, we cn obtn [5 ~ x cos[ cos[ (8) where ( cn be tertvely clculted s l l α ( (9) l l α () ( Where l,, L, nd,, L, Assume tht ~ r cos[ () nd defne the norm s F( ω ) F( ω) dt () π - Then, let be the flter energy loss, defne s A ( ω) H( ω) (3) A ( ω) ~ where A ( ω ) nd A (ω) re respectvely the Fourer trnsform of ~ ( t ) nd (, H (ω) s the trnsfer functon of the flter whch s ppled for flterng (. And t cn be proved tht r when nd < / [5. Therefore, (8) cn be rewrtten s cos[ A cos[ Φ (4) where Φ (5) ISS: ISB:

3 Proceedngs of the 7th WSEAS Interntonl Conference on SIGAL PROCESSIG, ROBOTICS nd AUTOMATIO (ISPRA '8) Unversty of Cmbrdge, UK, Februry -, 8 A < (6) Equton (4) s n symptotclly exct decomposton of the sgnl n terms of mpltude nd phse envelopes. It s worth notng tht ( s the mpltude envelope of one component n the orgnl sgnl. Plese refer to [5 for the dscusson of convergence of (4)..3 Further Reserch The hgh-pss flter used n the tertve procedure cn be desgned wth FDATool n MATLAB, nd < / must be stsfed. For certn nd of sgnl such s mechncl vbrton sgnls, speech sgnls, erthque sgnls, rdr sgnls, experments cn be done to te the sutble prmeters for the flter nd n the terton, ust dopt the sme flter. Addtonlly, zero-phse dgtl flterng s the only flterng wy n the bove procedure of Gnfelc trnsform [7. Snce < /, the decomposton s to contnuous extrct the component wth gret energy,.e. the components re rnged ccordng to the energy from gret to smll. The number of components to be extrcted s determned by the number of tertons. Equton (4) represents components re extrcted, nd the envelope of th component s (, whle the nstntneous frequency s gven by one or more phse sgnls mong {,,, L }, whch wll be shown n secton 3 wth exmples. 3 Accurte Instntneous Frequency Estmton 3. Reserch on Lmttons of Drect Computton of IF The nstntneous frequency of ech component re computed ccordng to the ssocted phse sgnls n (4). A self-dptve segmentton lgorthm to compute the nstntneous frequency s used n [5. It s method for clcultng the dervtve of the unwrpped phse by lner regresson whch cn effectvely reduce the nfluence of nose upon IF estmton, but s mnly sutble for phse sgnl preservng pecewse lnerty. For some sgnls whose frequency modulton prts re usully snusodl sgnls, e.g. mechncl vbrton sgnls, f usng self-dptve segmentton lgorthm to estmte IF, the ccurcy wll declne. Besdes, n the model shown n (4), ech phse sgnl s the result of combnton of severl components phse, hence to drectly compute the IF from the dervtve of the phse sgnl hs certn lmttons. We wll show you the lmttons by nlyzng two-component AM-FM sgnl s phse chrcterstcs fter the Hlbert trnsform. A generc two-component AM-FM sgnl cn be denoted s x x (7) ( cos[ ( cos[ where for t, there s >, so the frst step of Gnfelc trnsform s to demodulte the component x (. Assume tht Bedrosn theorem s stsfed by both x ( t ) nd x [8, then the Hlbert trnsform of x ( s gven by H[ H[ x H[ x (8) ( sn[ ( sn[ And the correspondng nlytcl sgnl s z H[ (9) Accordng to the chrterstcs of Gnfelc trnsform, the phse sgnl used to estmte the nstntneous frequency of x ( t ) s H[ ( t ) rg( z( ) rctn C (3) where C s constnt correspondng to the sgn of H [ nd x (. Compute the dertves of both the two ends of (3) nd we get H& & [ H[ x& (3) z( From (7) nd (8), we cn obtn H& [ H[ x& & ( & (3) [ & ( ( & sn[ ( [ & & cos[ z( (33) cos[ ( Then ssume & & ( ε (34) And substtute (3), (33), (34) nto (3), yelds & ( t ) & ( E( (35) where E( [ ( sn[ (36) ε [ cos[ ISS: ISB:

4 Proceedngs of the 7th WSEAS Interntonl Conference on SIGAL PROCESSIG, ROBOTICS nd AUTOMATIO (ISPRA '8) Unversty of Cmbrdge, UK, Februry -, 8 From (35) nd (36) we cn see tht t wll rouse error to estmte the IF of x ( t ) by dfferenttng ( nd the error E ( preserves hgh frequency chrcterstcs. The lterntng component ~ ( t ) cn lso be seen s multcomponent sgnl, nd the phse obtned by Hlbert trnsform lso hve the smlr errors. Furthermore, from (9) nd (), t cn be seen tht the demodulton error by the former step wll spred nto the next decomposton. In the sme wy, these lmttons wll be extended to generc multcomponent sgnls,.e. the IFs computed wth the phze sgnl derved n (4) hs certn hgh frequency error. 4 Smulton nd Comprson Consder the two-component AM-FM sgnl x x x [.5cos(π sn[ πt.5 cos(π.3sn(π8 (38) where t [,, x ( t ) s the AM-FM sgnl, nd x ( t ) s the snusodl sgnl. The tme domn wveform of x ( s shown n Fg.. 3. Smoothed Instntneous Frequency Estmton In order to reduce the error of nstntneous frequency estmton by drectly computng the dervtve of the phse, sutble low-pss flter cn be used to flter the dervtve of the phse sgnl. Although the self-dptve segmentton lgorthm cn effectvely dmnsh the nfluence of nose, t nvolves lot of computton, hence not sutble for engneerng pplcton. Therefore, the dfference opertor s ppled to compute the dervtve of the phse sgnl. Assume the dscrete sgnl correspondng to the phse sgnl Φ ( n (4) to be Φ, then the dervtve of Φ ( cn be pproxmted by bcwrd dfference,.e. ω ( n) Φ ( n) Φ ( n ) (37) where ω s the dscrete nstntneous ngulr frequency correspondng to Φ. If ω s negtve, ts bsolute vlue ω s ten. Then crry out zero-phse dgtl low-pss flterng to ω to elmnte the hgh frequency error. And the reltvely exct dscrete nstntneous ngulr frequency ω s obtned. The method s nmed smoothed nstntneous frequency estmton(sife), nd ts flowchrt s depcted n Fg.. Zero-phse Φ Dfference ω Sgned ω Dgtl ω Mgntude Opertor Low-pss Arthmetc Flterng Fg. Flowchrt of SIFE. The low-pss flter used n SIFE s desgned n the sme wy s the hgh-pss flter mentoned n the former secton. Usully, the low-pss flter nd the hgh-pss flter re complementry. Fg. Smulton Sgnl. After Gnfelc trnsform, we obtn the mpltude envelope (, ( t ) respectvely of x ( t ), x, whch s shown n Fg. 3 nd phse sgnl,,. Then usng SIFE, we cn compute the nstntneous frequences respectvely correspondng to, nd. Accordng to the spectrl chrcterstcs of x (, we cn udge tht s the flse phse sgnl. Therefore, the nstntneous frequences should be clculted by nd. The result s shown n Fg. 4. The nstntneous frequences of the two components extrcted by self-dptve segmentton lgorthm re shown n Fg. 5. From Fg. 4 nd Fg. 5, we cn see tht SIFE s pprently better thn self-dptve segmentton lgorthm. Although SIFE s of gret performnce, however, zero-phse dgtl flterng wll rouse fluctuton t the ends of the nstntneous frequency curve. Therefore, n occsons whch requre hgh precson, expnson t the two ends of tme seres should be done before zero-phse dgtl flterng. In order to better pprecte the vldty of Gnfelc trnsform nd SIFE, comprson wth the performnce of HHT s crred out. For the sme sgnl, the mpltude envelopes nd nstntneous frequences of the two components obtned by HHT re respectvely shown n Fg. 6 nd 7. As cn be seen from Fg. 3, 4, 6 nd 7, the curves obtned by Gnfelc trnsform nd SIFE s smoother nd of hgher precson. In ddton, snce there s fst lgorthm for Hlbert trnsform, the clculton speed of Gnfelc trnsform s fster. ISS: ISB:

5 Proceedngs of the 7th WSEAS Interntonl Conference on SIGAL PROCESSIG, ROBOTICS nd AUTOMATIO (ISPRA '8) Unversty of Cmbrdge, UK, Februry -, 8 compute one component s nstntneous frequency. Plese refer to [5 to see the exmple. Fg. 3 Ampltude envelopes of the two components n obtned by Gnfelc trnsform (sold the mpltude envelope estmton of x (, dshed the mpltude envelope estmton of x (). Fg. 4 Instntneous frequences of the two components n obtned by SIFE (sold he IF estmton of x (, dshed the IF estmton of x (). 5 Applcton to Fult Dgnoss When there s we dmge to the rollng berng, the sgnl contnng the fult chrcterstcs s usully drowned by the bcground sgnls relevnt to the rotry speed of the rotor nd other noses. In ths cse, t s hrd to extrct the fult chrcterstcs by envelope nlyss, nd multcomponent demodulton should be employed to do fult dgnoss. Fg. 8 shows the vbrton ccelerton sgnl of SKF63-type rollng berng wth out-rce fult. The smplng frequency s Hz, the rotry frequency (f r ) s 9.95 Hz, nd by clculton the out-rce fult chrcterstc frequency (f oc ) s 7.4 Hz. The spectrum of the sgnl s shown n Fg. 9, from whch we cn see tht there re severl frequency fmles (monocomponent sgnls), nd hence t s hrd to udge whether there s fult on the berng nd further nlyss s needed. Fg. 5 Instntneous frequences of the two components n obtned by self-dptve segmentton lgorthm (sold the IF estmton of x (, dshed the IF estmton of x (). Fg. 8 Vbrton sgnl of rollng berng wth out-rce fult. Fg. 6 Ampltude envelopes of the two components n obtned by HHT (sold the mpltude envelope estmton of x (, dshed the mpltude envelope estmton of x (). Fg. 7 Instntneous frequences of the two components n obtned by HHT (sold the IF estmton of x (, dshed the IF estmton of x (). In the bove exmple, only one phse sgnl s used to compute ech component s nstntneous frequency. However, f some components frequency rnges overlp wth ech other, then more thn one phse sgnls should be ten nto ccount to Fg. 9 Ampltude spectrum of the sgnl shown n Fg. 8. Gnfelc trnsform nd SIFE re employed to demodulte the frst three components nd ther nstntneous frequences f ( t ), f nd ( ) f 3 t re respectvely shown n Fg. ()-(c), whle the correspondng envelopes re denoted wth ( t ), ( t ) nd ( ) 3 t. Then spectrum nlyss s crred out to ( t ), nd ( ) 3 t, nd the mpltude spectrums re shown n Fg. ()-(c) respectvely. From Fg. t s esy to note tht the center frequences of the 3 components re respectvely bout Hz, Hz nd 7 Hz. Combned wth Fg. 9, we cn see tht the method ntroduced n ths pper correctly extrcted the nformton of three frequency fmles n the orgnl sgnl. And ccordng to Fg. we lern tht the frst two components re AM-FM sgnls correspondng to the ISS: ISB:

6 Proceedngs of the 7th WSEAS Interntonl Conference on SIGAL PROCESSIG, ROBOTICS nd AUTOMATIO (ISPRA '8) Unversty of Cmbrdge, UK, Februry -, 8 rotry frequency, nd the thrd component s the AM-FM sgnl contnng the fult nformton. In Fg. (c), there s obvous spectrl lne t the out-rce fult chrcterstcs frequency f oc. () optmztons, hence hs gret pplcton potentl. The mn efforts of the pper re s follows: ) Introduces the prncple of Gnfelc trnsform, sttes how to desgn the flters, nd ponts out the decomposton s process of contnuously extrctng the component wth gret energy. ) Shows the lmttons of drectly computng the nstntneous frequency of ech component by theoretcl nlyss. Hence, ntroduces smoothed nstntneous frequency estmton method nd vldte ts effectveness v smulton. 3) Apples Gnfelc trnsform nd SIFE nto erly fult dgnoss for berng nd obtns good effect, thus provdng mechncl fult dgnoss wth new method. (b) (c) Fg.. Instntneous frequences of frst three components n the fult sgnl shown n Fg. 8: () IF f ( of frst component, (b) IF f ( of second component, (c) IF f 3 ( of thrd component. () (b) (c) Fg.. Ampltude spectrums of mpltude envelopes (, ( nd 3 (: () Ampltude spectrum of (, () Ampltude spectrum of (, (c) Ampltude spectrum of (. 6 Concluson Gnfelc trnsform (IHT) s new sgnl nlyss method for multcomponet sgnls. It hs hgher performnce nd doesn t need complex flter References: [ A.C. Bov, P. Mrgos, T.F. Quter, AM FM energy detecton nd seprton n nose usng multbnd energy opertors, IEEE Trnsctons on Sgnl Processng, Vol.4, o., 993, pp [ B. Snthnm, P. Mrgos, Multcomponent AM FM demodulton v perodcty-bsed lgebrc seprton nd energy-bsed demodulton, IEEE Trnsctons on Communctons, Vol.48, o.3,, pp [3.E. Hung, Z. Shen, S.R. Long, M.L.C. Wu, H.H. Shh, Q.. Zheng,.C. Yen, C.C. Tung, H.H. Lu, The emprcl mode decomposton nd the Hlbert spectrum for nonlner nd nonsttonry tme seres nlyss, Proc. R. Soc. Lond. A, Vol.454, 998, pp [4 S.R. Qn, Y.M. Zhong, A new envelope lgorthm of Hlbert Hung Trnsform, Mechncl System nd Sgnl Processng, Vol., o.8, 6, pp [5 F. Gnfelc, G. Bgett, P. Crpp, C. Turchett, Multcomponent AM-FM representtons: n symptotclly exct pproch, IEEE Trnsctons on Audo, Speech, nd Lnguge Processng, Vol., o.3, 7, pp [6 S. L. Hhn, Hlbert trnsforms n sgnl processng, Artech House, 996. [7 A. Antonou, Dgtl flters, nlyss, desgn, nd pplctons, nd ed, McGrwHll, 993. [8 B. Boshsh, Estmtng nd nterpretng the nstntneous frequency of sgnl Prt : Fundmentls, Proceedngs of the IEEE, Vol.8, o.4, 99, pp ISS: ISB: