Adaptive Amplitude Demodulation (AAD) as an order tracking method

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1 Adptive Amplitude Demodultion (AAD) s n order trcking metod Piet Vn Vlierberge, Krl Jnssens, Hermn Vn der Auwerer LMS Interntionl, Interleuvenln 68, B-31 Leuven, Belgium, pieter.vnvlierberge@lmsintl.com ABSTRACT Wit te incresing complexity of modern mcinery, mny rotting system problems require nlysis wit ig order resolution. Tody s order trcking tecniques (e.g. frequency domin metod, order domin metod, etc.) only provide decent order vlues wen te order content does not cnge during te observtion intervl. So to get te best out of tese lgoritms, te idel mesurement sequence would consist of truly constnt RPM during ec observtion period, lso known s semi-sttionry mesurement. Suc n pproc s severl disdvntges: (i) te totl mesurement time is fr iger tn in run-up mesurement; (ii) spurious effects occurring only during te trnsition from one RPM to te next go unnoticed, giving rise to n uncceptble number of semi-sttionry levels; (iii) t ig RPMs, even te sligtest oscilltions in RPM result in lrge pse sifts. Tis pper presents new Adptive Amplitude Demodultion (AAD) lgoritm wic llows seprtion of order components in run-up mesurements were te forementioned metods fil. First, n overview is given of some of te existing order trcking tecniques. Ten, te principles of te new AAD lgoritm re outlined. Finlly, te results of bencmrk study re presented in wic te new lgoritm is compred to existing order trcking tecniques for (i) slow nd fst run-up mesurements nd for (ii) orders wit slowly nd rpidly vrying mplitude nd pse profile. 1 THEORETICAL MODEL OF ORDER ANALYSIS A structure under nlysis contins number of xles A wit te number of te xle. All excittion in te structure is supposed to be cused by te rottion of te xles, in suc wy tt if te rottionl speed of n xle is kept constnt, te excittion cused by tt xle is periodic. However, te period does not necessrily correspond to single rottion of te xle, e.g. four-stroke combustion engine s period of two revolutions, ssuming tt te cylinders re not perfectly symmetric. Some xles re linked by gers, suc tt teir rottionl speeds differ only by constnt rtionl fctor. In tis cse, te period of te excittions divided by te period of one xle of te group cn be ny rtionl number. 1.1 Constnt ngulr velocity First, suppose te ngulr velocity to be constnt. Tis llows us to model te excittion x (t) of n xle group represented by A, t constnt ngulr velocity ω, s te liner combintion of number of oscilltors, one for ec rmonic. But te rmonics re not simply n integer multiple of bse frequency, becuse of ger rtios nd symmetries. So we will just model te excittion s loose combintion of oscilltors, tereby disregrding te rtios between te frequencies: x (t) X ( ω )e jρ α (t) (1) were X ρ X ρ *, α (t) ω t

2 Here: X is complex number tt represents te mplitude nd pse of te t rmonic. It is not constnt, s it depends on te structure s regime, i.e. torque, rottionl speed, temperture etc. Most of tese regime prmeters re kept constnt during test, but te rottionl speed is typiclly swept over rnge. So for given test regime, X cn be considered s function of te rottionl speed ω. ρ is te t order number, i.e. te rtio between te period of te A s rottion nd te period of te rmonic oscilltor. So te iger te order number, te iger te frequency of te ccompnying oscilltor. Te excittion gives rise to response in ll points P p of te structure. Assuming liner structure, te responses re given by trnsfer function. But in opertionl conditions, te excittion points cnnot be loclized. Moreover, te rmonic content of every excittion point my differ, e.g. bering defects will cuse oter rmonic content tn combustion. In order nlysis, no ttempt is mde to trce bck responses to excittion points. Rter, order nlysis focuses on finding te reltionsip between rmonic oscilltors nd responses. At constnt rottionl speed, trnsient effects in te responses will ve dwindled nd te response will rec stedy stte. Terefore, te response cn be modeled simply s follows: y p (t) Hp X ( ω )e jρα(t) (2) Here: y p (t) is te response in point p. H p H p (ρ ω ) is te trnsfer function between te t rmonic oscilltor nd te response point. Order nlysis is performed in opertionl conditions. So neiter H p (ω) nor X is known. Rter, order nlysis will ttempt to split up y p (t) into its constituents: y p (t) Yp ( ω )e jρα (t) (3) were Y H p p X Te order numbers ρ re often known in dvnce. So obtining te vlues of te function Y p is esy t constnt rottionl speed ω : you need to observe te response y p (t) over period tt is n integer multiple of every oscilltor s period, nd n FFT on tese dt will yield te vlues Y p (ω ) on te spectrl lines corresponding to te order number ρ. 1.2 Non-constnt ngulr velocity Tings complicte wen te rottionl speed is no longer constnt: y p (t) Yp ( ω (t))e jρα (t) (4) were α (t) Y p H ω p (t)dt X

3 Here: Te model fils to cpture te trnsient beviour of te system s trnsfer functions. Tis is fundmentl limittion of te model, but is less importnt if dmping rtios re ig enoug. Te time signl is no longer periodic over te observtion intervl becuse e complex oscilltor: te frequency is not constnt. j ρ α (t) is no longer regulr Y p (ω ) is not constnt ny more over te observtion intervl, resulting in n dditionl source of periodicity. Tis is most prominent in coustic mesurements were te pse of te trnsfer function vries rpidly over te frequency xis. 2 EXISTING TECHNIQUES Let us look t te current metods for order extrction, nd teir ree of success in fitting te teoreticl model. 2.1 Frequency domin metod Te frequency domin or FFT-bsed metod is te most commonly used order trcking metod. Tis metod works s follows: y p (t) is smpled wit fixed smpling frequency, over n observtion intervl T. Te smpling frequency f s is t lest twice te frequency of ny of te involved oscilltors: : f s > ωρ π (5) Te observtion intervl T determines te frequency resolution. Te frequency resolution must be smller tn te smllest difference between order oscilltor frequencies:, 1 2 : ω < T ρ 1 2π ρ 2 (6) Periodic signl over te observtion intervl If ω is eld constnt, tere will be n observtion intervl over wic te function is periodic if ll rtios between ll ρ re rtionl numbers. In tis cse, tere exists T suc tt: : n n : T ωρ 2π, were n is positive integer (7) Any integer multiple of T will do s well, so we cn pick multiple tt stisfies te order resolution in eqution 6. A FFT of te discretized function yp (ti) will yield Y p (f i ) wit f i i * f s. From te definition of FFT, it cn be understood tt Y p (ω ) Y p (f ) wit: f ωρ 2π (8) From teoreticl point of view, te estimtion is exct. But in prctice, smpling frequencies nd observtion time intervls cnnot be picked rbitrrily. And if none of te smpling frequencies nd/or observtion intervls vilble to you fit te bove requirements, your signl will not be periodic.

4 2.1.2 Aperiodic signl over te observtion intervl If ω is no longer constnt, tere will probbly not be n intervl over wic te signl is periodic, nd lekge will occur: spectrl lines will be smered out over te frequency xis, nd te crispness of te single spectrl line per order will fde. Tis results in te following dverse effects on order estimtion: Picking out single spectrl line will result in n energy loss: energy tt leked into djcent spectrl lines is not ccounted for in te fitted model, so te order mplitude will be n underestimtion. To counter tis, te energy could be summed over frequency bnd round te order s frequency line, suc tt te leked energy is soked bck up. But tis could result in n overestimtion, since energy from djcent order components my ve leked into tis order, resulting in energy being counted twice. Tis penomenon is known s order cross-tlk. If you sum te energy over frequency bnd, wt bout te pse? No mtter wt tecnique is used, te pse will lwys be errnt. Te effects of lekge cnnot be voided, but tey cn be lessened in two wys: By pplying windowing tecniques. Altoug tis does decent job in reducing lekge, it lso produces some mplitude indeterminism. Window correction fctors cn be used, but tese re exct only for sine, wic our signl certinly is not since it is periodic over te observtion intervl. By enlrging te observtion intervl. Tis increses te frequency resolution, tereby diminising te effect of cross-tlk on first sigt, becuse tere re more frequency lines now between djcent orders. But it does little to improve te pse estimte, becuse te lekge ppens over te sme number of lines. Moreover, if te mesurement t nd is run-up, incresing te observtion intervl lso mens bigger difference in frequency of ll oscilltors over te observtion intervl, so even bigger periodicity. 2.2 Order domin metod Te order domin or resmpling-bsed metod is more dedicted pproc. Tis pproc overcomes mny of te limittions of te frequency domin metod [1-2]. Te order domin is to te ngle domin wt te frequency domin is to te time domin. So function FFT(f(α)) is n order domin function if f(α) is n ngle domin function. Obtining n ngle domin function from time domin function cn be done in severl wys, but is outside te scope of tis pper. Suffice it to sy tt te outcome of tis is discrete function f(α i ) wit α i i * α. Te metod is very similr to te frequency domin metod. It uses reformulted version of te order estimtion definition: y' p ( α) Yp ( ω ( α))e jρ α (9) 1 were ω ( α) t α Remrk tt te prime in y p indictes tt tis is function of te ngle, rter tn function of time Periodic signl over te observtion intervl Now, weter or not ω is eld constnt, tere will be n ngulr observtion intervl over wic te functions jρ α e re ll periodic, if ll rtios between ll ρ re rtionl numbers. In tis cse, tere exists n ngulr observtion intervl γ suc tt:

5 : n n : γ ρ 2π, were n is positive integer (1) Suppose tt ω is eld constnt. A FFT of te discretized function y' p ( α i) will yield Y p(n i ). Suppose α i i * α wit γ α n nd n is positive integer. From te definition of FFT, it cn be understood tt Y p (ω ) Y p (n ) wit: n ρ γ 2 π (11) Using tis lgoritm, bot γ nd α re independent of te rottionl speed ω, so te sme computtions ppen t different vlues of ω. Tis lgoritm provides decent results if ω is kept constnt, since te oscilltors re gurnteed to be periodic Aperiodic signl over te observtion intervl It is tempting to ssume tt tis lgoritm will work eqully well for vrying ω since ll order oscilltors remin periodic during te observtion intervl γ. Tis is owever not te cse. A closer look t eqution 9 revels wy: jρ te term α e is definitely periodic in γ, but wt bout Y ( ω ( α))? p Tis term represents te order mplitude nd pse. Since te system s trnsfer functions vry wit frequency, te order mplitude nd pse re not constnt over te observtion intervl in run-up mesurement. Orders consequently lek into djcent orders nd cn not longer be seprted. Suc order cross-tlk typiclly occurs in cse te order components ve rpidly vrying mplitude nd pse over te RPM xis or for exmple in fst run-ups wen te frequency cnges significntly during ec observtion period. To void cross-tlk, one cn decrese te order bndwidt by incresing te number of periods per observtion intervl. For exmple, for full-speed veicle ccelertions, we typiclly need t lest 8 periods to seprte te lf order components. If te orders would ve flt mplitude nd pse profile, only 2 periods per observtion intervl would be sufficient. By incresing te number of periods, order cross-tlk is indeed reduced, but we obtin less order trcking points over te RPM rnge. Te obtined order estimtion profiles re consequently fr too smoot, s will be illustrted furter in tis pper. 3 ADAPTIVE AMPLITUDE DEMODULATION Recently, new Adptive Amplitude Demodultion (AAD) metod ws developed tt llows seprtion of orders in run-up mesurements were te forementioned tecniques fil. Tis new order trcking lgoritm suffers less from order cross-tlk nd cn better ndle fst run-ups nd signls wit rpidly vrying order content. Te proposed lgoritm is bsed on following observtion: eqution 9 cn be red s composite mplitude jρ modultion, in wic Yp ( ω( α)) re te signls being modulted nd α e re te crrier wves. Tis nlogy clrifies were te previous lgoritms go wrong: imgine n AM receiver tt uses n FFT on fixed time intervl round t, ten sums up ll te energy in te frequency bnd of interest, nd produces tt s n output for time t.

6 Rter, demodultion tecnique sould be used tt does not ssume periodicity of te signl being modulted, nd tt does not mingle wit te pse eiter. Tis cn be cieved by multiplying te signl per observtion intervl wit te crrier wve, nd using srp low pss filter wit little pse distortion on it: jρα ( sinc( ρ α) ( y' ( α) * e )( α ) Yp ( ω ( α )) p (12) were sin( α) sinc( α) α Te qulity of tis metod depends on by wic filter you replce te idel sinc filter. To obtin s little pse distortion s possible, FIR filter is pproprite. Te ctul design of te filter is trde-off between te: Amount of istory dt needed to run in te filters Amount of future dt needed to run out te filters Computtion effort required Accurcy of te filter Depending on te trde-off mde, tis FIR filter introduces some errors in te lgoritm. It is owever not te only source of errors. Remember tt te lgoritm ssumes dt to be expressed in te ngle domin. If te dt re only vilble t fixed smpling frequency, you will need to use n interpoltion filter to cross over to te ngle domin. Tis interpoltion filter needs to suppress lising nd s to prevent pse distortion s well. So te ctul error of te lgoritm is result of cscde of filters. Tis cn be circumvented by pplying te sme lgoritm, but now directly in te time domin: Y p ( ω ( α )) sinc( ρ ( α sinc( ρ ( α sinc( ρ ( α jρα ( sinc( ρ α) ( y' ( α) * e )( α)) α(t))) α(t))) jρα ( y' ( α) * e ) p dα jρα(t) ( y' ( α(t)) * e ) p jρ d α(t) α ( y (t) * e ) dt p p dt α ) dα dt dt (13) So, by trcking te ngle over time, for ny given ω during run-up, te corresponding ngle α cn be found. Subsequently, FIR filter cn be defined for α tt is n pproximtion of te idel filter: sinc( ρ ( α α(t)))e jρα(t) dα dt (14) Tis pproc sves you from te trnsition to te ngle domin nd te error incurred by doing so, nd yields FIR filter tt cn be computed once for ec desired ω, nd pplied on ll response points p. Te combined pproc of using n equivlent of dptive resmpling nd mplitude demodultion in single computtion run explins te nme of te new AAD order trcking metod.

7 4 BENCHMARK STUDY A bencmrk order trcking study ws orgnized to test nd evlute te new AAD lgoritm. Severl run-up sounds were syntesized using te order components of typicl 4-cylinder crs. A Virtul Cr Sound syntesis tool [3-4] ws used ereto. For systemtic nlysis, we generted sounds wit (i) different run-up speed (slow, medium nd very fst run-ups of respectively 25, 8 nd 2.5 s) nd wit (ii) order components crcterized by eiter slow or rpid mplitude nd pse vritions over te RPM xis. Trigger pulse signls wit 2 pulses per revolution were generted s well. Te dt were generted t 32 khz smpling frequency. For te nlysis, we considered te following 6 run-up exmples wit incresing order bndwidt nd difficulty to seprte te orders: Cse RPM rnge Run-up speed Order content RPM Slow: 25 s Orders.5-1, smoot order profiles RPM Slow: 25 s Orders.5-1, rpidly vrying order profiles RPM Medium: 8 s Orders.5-1, smoot order profiles RPM Medium: 8 s Orders.5-1, rpidly vrying order profiles RPM Very fst: 2.5 s Orders.5-1, smoot order profiles RPM Very fst: 2.5 s Orders.5-1, rpidly vrying order profiles Tble 1: Overview of te 6 engine run-up exmples. Te conventionl order domin metod nd te new AAD lgoritm were bot pplied to te 6 run-up sounds. In te AAD lgoritm, we used srp low pss filter wit order bndwidt of.5. In te order domin pproc, observtion intervls of minimum 4 (cses 1-2) or 8 periods (cses 3-6) were tken to minimize order cross-tlk. As n exmple, figures 1 nd 2 sow te order trcking results of te order domin metod nd AAD lgoritm for orders 2 (dominnt order) nd 7.5 (less significnt order) in slow, medium nd very fst run-up (cses 2,4,6). Te mplitude nd pse estimtion results of bot lgoritms (red nd blck line) re sown in comprison wit te s (blue line). Only prt of te RPM rnge is visulized to better igligt te differences between te order trcking results. For te dominnt order component 2, te AAD order trcking results re excellent, even in cse of very fst runup of only 2.5 s. Te AAD metod clerly performs muc better tn te order domin metod. Te order domin metod only works properly for slower run-ups. In cse of fst run-ups, it does not ccurtely trck te mplitude nd pse vritions nd te order estimtion results re fr too smoot. Te reson for tis is te use of longer observtion intervls required to void order cross-tlk. Similr results re obtined for te less significnt orders, like for exmple order 7.5 in figure 2. Te AAD metod performs extremely well, except for te lst run-up exmple sown in te lower grp. In tis extreme cse wit very fst run-up nd rpidly vrying orders, tere is inerent order cross-tlk nd it is impossible to seprte te orders. However, wen te order components ve smooter mplitude nd pse profile, te AAD lgoritm is very well cpble to ccurtely trck te order evolution in very fst run-ups. Tis is for exmple illustrted in figure 3 for order 7.5 in run-up cse 5. An overview of ll order trcking results is presented in tbles 2-4. Te tbles sow te mplitude nd pse errors of te order domin nd AAD metod for ll te order components.5-1 in te 6 run-up exmples. Te tbles confirm te fore discussed results. Te AAD metod is clerly superior to te order domin pproc, not only for te slow run-ups, but lso for te more cllenging exmples wit fster ccelertion nd rpidly cnging order content. It llows seprtion of order components in run-ups were te order domin metod fils. Only in te very extreme cses wit inerent order cross-tlk, it does not work properly, but order extrction is simply impossible in suc conditions.

8 9 Cse 2: order 2, run-up of 25 s Cse 1: order 2; run-up of 25 s; originl order profile 8 db 7 6 order domin metod new AAD lgoritm 5 2 Cse 1: order 2; run-up of 25 s; originl order profile 1-1 order domin metod new AAD lgoritm -2 9 Cse 4: order 2, run-up of 8 s Cse 3: order 2; run-up of 8 s; originl order profile 8 db 7 6 order domin metod new AAD lgoritm 5 2 Cse 3: order 2; run-up of 8 s; originl order profile 1-1 order domin metod new AAD lgoritm -2 9 Cse 6: order 2, run-up of 2.5 s Cse 5: order 2; run-up of 2.5 s; originl order profile 8 db 7 6 order domin metod new AAD lgoritm 5 2 Cse 5: order 2; run-up of 2.5 s; originl order profile 1-1 order domin metod new AAD lgoritm -2 Figure 1: Order 2 trcking results for run-up sounds 2 (slow: 25 s), 4 (medium: 8 s) nd 6 (very fst: 2.5 s) wit rpidly vrying order content: order domin metod (red), new AAD metod (blck), (blue).

9 6 Cse 2: order 7.5, run-up of 25 s Cse 1: order 7.5; run-up of 25 s; originl order profile 55 5 db order domin metod new AAD lgoritm 3 2 Cse 1: order 7.5; run-up of 25 s; originl order profile 1-1 order domin metod new AAD lgoritm -2 6 Cse 4: order 7.5, run-up of 8 s Cse 3: order 7.5; run-up of 8 s; originl order profile 55 5 db order domin metod new AAD lgoritm 3 2 Cse 3: order 7.5; run-up of 8 s; originl order profile 1-1 order domin metod new AAD lgoritm -2 6 Cse 6: order 7.5, run-up of 2.5 s Cse 5: order 7.5; run-up of 2.5 s; originl order profile 55 5 db order domin metod new AAD lgoritm 3 2 Cse 5: order 7.5; run-up of 2.5 s; originl order profile 1-1 order domin metod new AAD lgoritm -2 Figure 2: Order 7.5 trcking results for run-up sounds 2 (slow: 25 s), 4 (medium: 8 s) nd 6 (very fst: 2.5 s) wit rpidly vrying order content: order domin metod (red), new AAD metod (blck), (blue).

10 db Cse 5: order 7.5, run-up of 2.5 s Cse 6: order 7.5; run-up of 2.5 s; LP filtered order profile order domin metod new AAD lgoritm Cse 6: order 7.5; run-up of 2.5 s; LP filtered order profile order domin metod new AAD lgoritm -1-2 Figure 3: Order 7.5 trcking results for run-up sound 5 (very fst: 2.5 s) wit slowly vrying order content: order domin metod (red), new AAD metod (blck), (blue). Cse 1: Cse 2: mplitude error (db) pse error () mplitude error (db) pse error () Tble 2: Amplitude nd pse errors of te order domin (OD) nd AAD metod for ll orders in run-up cses 1 (slow run-up of 25 s, smoot order profiles) nd 2 (slow run-up of 25 s, rpidly vrying order profiles). AAD errors in bold type wen t lest twice s smll s OD errors.

11 Cse 3: Cse 4: mplitude error (db) pse error () mplitude error (db) pse error () Tble 3: Amplitude nd pse errors of te order domin (OD) nd AAD metod for ll orders in run-up cses 3 (mid-speed run-up of 8 s, smoot order profiles) nd 4 (mid-speed run-up of 8 s, rpidly vrying order profiles). AAD errors in bold type wen t lest twice s smll s OD errors. Cse 5: Cse 6: mplitude error (db) pse error () mplitude error (db) pse error () Tble 4: Amplitude nd pse errors of te order domin (OD) nd AAD metod for ll orders in run-up cses 5 (very fst run-up of 2.5 s, smoot order profiles) nd 6 (very fst run-up of 2.5 s, rpidly vrying order profiles). AAD errors in bold type wen t lest twice s smll s OD errors.

12 5 CONCLUSIONS A new Adptive Amplitude Demodultion (AAD) lgoritm ws developed tt llows order extrction wit very ig ccurcy. Tis order trcking tecnique, wic combines dptive resmpling nd mplitude demodultion, performs significntly better tn tody s order trcking metods (e.g. frequency domin metod, order domin metod). Tese conventionl metods only provide decent order estimtion results wen te order content rdly cnges during te observtion intervl, for exmple in slow run-ups or in cse tt te order components ve reltively smoot mplitude nd pse profile. A bencmrk order trcking nlysis crried out on vrious engine run-up sounds reveled tt te AAD metod cn better ndle fster run-ups nd signls wit rpidly vrying order content. Only in te extreme cse of inerent order cross-tlk, it does not work properly, but order extrction is simply impossible under suc conditions. 6 ACKNOWLEDGMENTS Tis work ws crried out in te frme of te MEDEA+ project 2A24 SWANS Silicon pltforms for Wireless Advnced Networks of Sensors. Te finncil support of te Institute for te Promotion of Innovtion by Science nd Tecnology in Flnders (IWT) is grtefully cknowledged. REFERENCES [1] R. Potter nd M. Gribler, Computed Order Trcking Obsoletes Older Metods. Proceedings of te SAE N&V Conference, Trverse City, MI, US, SAE pper [2] P. Vn de Ponseele, H. Vn der Auwerer nd M. Mergey, Performnce Evlution of Advnced Signture Anlysis Tecniques. Proceedings of IMAC Conference 7, Ls Vegs, NV, US, p [3] K. Jnssens, M. Adms, P. Vn de Ponseele nd L. Vllejos, 22. Te Integrtion of Experimentl Models in Rel-time Virtul Cr Sound Engineering Environment. Proceedings of ISMA Conference, Leuven, Belgium. [4] P. Vn de Ponseele, M. Adms, K. Jnssens nd L. Vllejos, 22. Virtul Cr Sound Environment for Interctive, Rel-time Sound Qulity Evlution. Internoise Conference, Derborn, Micign, US.