Algorithms for the Vold-Kalman multiorder tracking filter

Transcription

1 Algoths fo the Vold-Kalan ultode tacng flte Jří ůa VSB echncal Unvest of Ostava Facult of Mechancal Engneeng 7 lstopadu Ostava zech Republc Jtua@vsbcz Abstact he pape deals wth the Vold-Kalan (VK ultode tacng flte he basc pncple of the Vold- Kalan (VK ode tacng flte fo onl one snusodal coponent of a sgnal was publshed an tes whle an algoth of tacng oe than one haonc coponent was not publshed et It s assued that the fequenc of the taced coponent s nown fo each saple of the sgnal he ultode tacng fltaton allows decouplng cossng odes e two haonc sgnals whch ae dffeng n fequenc ecept a cetan te oent when both sgnals can have the sae fequenc he analtcal calculaton ethod can tac onl one o two odes whle the G ethod s a tool fo oe than the two odes Kewods Vold-Kalan flte; tacng flte; cossng odes; G algoth; decouplng odes I INRODUION Soe patcula class of sgnals conssts of haonc coponents that ae all (o the ost donant of the elated n fequenc to the fundaental fequenc eg engne otatonal speed hese coponents ae desgnated as supe- o sub- haoncs (the so-called odes of the fundaental fequenc n RM whch s easued and ntepolated fo eve saple of a sgnal est engnees focus the nteest on the apltude and phase of these odes as a functon of te o oe fequentl as a functon of RM hs analss technque s called as the ode tacng fltaton he new algoth fo ths wa of fltaton was nvented b Vold and Leudan n 993 [] Late ths new fltaton technque was ncopoated nto softwae whch s nown unde nae I-DEAS (MS Sstes opoaton and LabShop ULSE (Büel & Kjæ opan All the papes wtten b Vold and Leudan gave onl basc nfoaton on the pncples of ths flte Much oe opened pape fo eades was publshed b Feldbaue and oldch n [] he ntoduced the dea how to solve the global sstes of the lnea equatons and how to solve a poble wth cossng odes he Vold-Kalan flte s a bandpass flte he fst gude how to set the bandwdth of ths flte whch s based on an analtcal calculaton was publshed b the autho of ths pape [5] he Vold-Kalan flte was developed n two geneatons he output of the fst geneaton s the flteed sgnal whle the output of the second one s the envelope of the flteed sgnal of the desed fequenc hs pape focuses onl on envelopes taced sgnal coponents Even though the basc pncple of the Vold-Kalan (VK ode tacng flte was publshed an tes the ssue dealng wth an algoth of ultode tacng of oe than one haonc coponent was not publshed et hee ae two ethods fo soluton of the ultode tacng fltaton he fst ethod s based on the analtcal appoach and the second one uses the G algoth that cobnes the dect and teatve soluton he analtcal appoach allows decouplng onl two cossng odes e two haonc sgnals whch ae dffeng n fequenc ecept a cetan te oent when both coponents can have the sae fequenc In contast to the analtcal appoach the G ethod s a tool fo oe than the two odes II ROSING ORDER SOLUION If the VK-flte s sultaneousl focused at the haonc coponents of a sgnal then the data equaton can be wtten n the geneal fo ( n ( n ( jθ ( n + η ( n ep ( whee ( n s a saple of the nput sgnal whch s easued at the n-th saplng step ( n s an envelope of the -th coponent Θ ( n s a phase of the -th coponent and η ( n s an eo te Fo fltaton t s assued that the nstantaneous fequenc of the taced coponents s nown hs fequenc s usuall a ultple of the otatonal speed of a ota achne he aveage otatonal speed due to the easueent ethod s nown onl fo the peod of otaton of a shaft and theefoe ts nstantaneous values ae ntepolated usng a splne functon hs technque does not deal wth ths pape he phase of the taced sgnal n ( s gven b Θ n ( n ( ω t ( In addton to the data equaton the VK flte also descbe stuctual equatons whose coposton depends on the nube of poles p of the VK flte he VK flte wth one o two poles s descbed b the followng stuctual equatons

2 pole : poles : ( n ( n + ε ( n ( n ( n + + ( n + ε ( n (3 he at B the vecto of the unnown vaables and the vecto b have the followng stuctue whee ( n ε s an eo te All sapled vaables can be aanged nto vectos of the sze N fo vectos η o N + p fo the vecto ε ( [ η [ η ( ( N ] [ ( ( N ] ( η( N ] ε [ ε ( ε ( N + p ] he at fo of the stuctual equatons s as follows A ε he unnown envelopes ae nzng the object functon J ε ε A + η A η + (4 (5 (6 B B B he sze of the at B s vectos and b s N b (9 B N N and the sze of the he band atces B wth the eal entes ae setc and postve defnte Fo the cople dagonal at t s vald that ( j j j theefoe B s a etan at sste of (8 he stuctue of non-zeo entes of the at wth the fou blocs 4 s shown n Fg Indvdual blocs ae squae atces wth the nube of lnes eachng tens of thousands he bloc atces on the an dagonal ae taced at wth seveal nonzeo dagonals (the lt of a few dozen he bloc atces outsde of the an dagonal ae the dagonal atces whee s a paaete detenng the bandwdth of the VK flte Fo nstuctons on how to set the value of ths paaete s ncluded n the publcaton [5] he sste at A fo the -pole and -pole fltes dffes n the nube of coluns pole : A poles : A (7 o detene the object functon nu the fst devatve wth espect to the unnown vectos has to be evaluated Afte puttng the devatve to zeo the unnown envelopes ae obtaned as a soluton of the followng sste of equatons J B + whee ae used the followng substtutons B A A + E and E As A A s a setc at then B s also setc and postve defnte due to the addton of E to the an dagonal [8] he atces B a not be dentcal due to the value of whch depends on the ode fequenc he dect soluton of the sste of (8 s as follows A b but an teatve ethod s oe sutable due to the lage sze of atces and vectos (8 Fgue Space sste at Fo the at whch s coposed fo oe than two blocs t s alost possble to fnd an eplct foula fo solvng the sste of equatons It s uch bette to solve ths sste of equatons usng an teatve algoth hs pocedue fo the Vold-Kalan flte was also poposed eale b Feldbaue h & oldch R [] Geneall thee s a lot of the othe Iteatve ethods fo solvng the sste of lnea equatons A onogaph descbng the teatve soluton of sstes of lnea equatons [6] o the Matlab help ecoend the conjugate gadent ethod wth pecondtone at whch s naed as the econdtoned onjugate Gadent (G algoth hs algoth s patculal sutable fo solvng the lnea equatons wth the spase postvel defnte setc (SD at he pocedue pesented n ths pape copaed wth Feldbaue & oldch dffes n the choce of the pecondtone at and the othe detals III ANALYIAL SOLUION FOR Fo the sste of (8 s splfed to B B b B (

3 whee and ae dagonal atces whch ae defned b { ep( jθ( ep( jθ( N } ep( jθ ( ep( jθ ( N dag dag{ } ( he dect soluton of the sste ( n the fo of B b s as follows [8] ( B B ( B E ( B B ( B E ( he analtcal soluton s napplcable to oe than two coponents (odes IV IERAIVE MEODS he G ethod s a cobnaton of the conjugate gadent ethod (G and the technque based on usng the pecondtone at he G ethod was fst descbed b authos estenes MR & E Stefel n 95 [7] he outlne of the pncple of the G ethod can bng n shot Saad's onogaph wthout poof just follows Fst howeve wll be defned b the dot poduct of two cople vectos u and v N ( u v u v (3 whee the ba ove the second eleent n the dot poduct epesents the cople conjugate If ( u v then the vectos u and v ae called othogonal Futheoe the vectos u and v ae called as A -conjugate f t s vald ( Au v he last epesson eans othogonalt of vectos Au and v Fo a descpton of the calculaton pocedue we assue that the poble s to solve the sste of equatons A b n whch the spase sste at A s a setc ( A A postvel defnte at A > fo all non-zeo vectos of the N sze he sste of lnea equatons n the vecto epesson can be ntepeted geoetcall so that the vecto b at the ght sdes of the equaton ust agee wth the vecto A that esults fo the tansfoaton of the vecto hs tansfoaton of the vecto b the at A n the N- densonal space eans otaton and scalng (that s enlagng o shnng he dect soluton of the sste of lnea equatons s based on the evese otaton and shnng o enlagng of the vecto b b the nvese of the at A Due to the denson of the at A whch s too lage ths pocedue of calculaton s not techncall possble It s theefoe necessa to popose a soluton appoach that s based onl on successve appoatons of the soluton e the teatve calculaton whch contan onl a vecto-at ultplcaton wthout the nveson About 5 eas ago estenes M R & E Stefel suggested a ethod fo solvng a sste of lnea equatons wth the use of the successve seach fo the soluton fo an ntal guess called the conjugate gadent (G ethod he consecutve dectons of the seach n the N-densonal space ae utuall A -conjugated Solvng the entoned lnea sste of equatons usng the G ethod s equvalent to seachng fo nu of the followng object functon f (4 ( A b he gadent of ths object functon s as follows ( f f ( A b (5 he esdual whch s defned b b A s a negatve gadent vecto at the value of If b A then the gadent s equal to zeo f ( the nu of the object functon s eached A new estate of the vecto usng an teatve pocess n the -th step can be wtten as + + α p (6 whee α s a scala paaete whch odfes the length of the vecto p whch detenes the decton n whch ths ethod tes to fnd a new appoaton of the soluton of the sste of equatons If the esdual vecto n the -th teaton step of seachng fo soluton of A b s aed b b A (7 then the new sze of the esdual vecto n the net teaton step wth espect to (7 s gven b the foula ( + α p α + b A + b A (8 If the consecutve esdual vectos and α ae to be othogonal e the dot poduct s equal to zeo α then t can be deved that ( ( ( α (9 he net seach decton of the efned appoaton of soluton can be epessed as a lnea cobnaton of the pevous decton of fndng solutons to sstes of equatons and the new value of the esdual vecto p+ + + p (

4 whee s a scala paaete As t was stated befoe the consecutve dectons of the seach s to be A -conjugated theefoe s othogonal to p A consequence of the above equaton s as follows ( A p ( p p ( p ( p ples that he othogonalt condton ( + the scala paaete s gven b the foula ( + ( p ( Snce we can epess accodng to (8 + α ( then the paaete can be calculated usng the foula ( + ( + ( ( + ( + (3 (4 α he fnal veson of the G algoth accodng to Saad's onogaph [6] s shown n ab I he fst decton of the seach s gven b the negatve gadent p as t s shown n Fg Fgue Iteaton pocedue fo solvng the sste of equatons ABLE I G ALGORIM opute: b A p Fo untl convegence Do: 3 α ( ( p α p α p p 6 ( ( 8 EndDo V G IERAIVE MEOD Intoducton of the pecondtone at M whch sutabl appoates the at A the geneal sste of equatons A b can be used to an eas dect soluton of the altenatve sste M u assung eas nveson of the pecondtone at M As alead entoned the G ethod solves a sste of equatons wth a at sste whch s setc and postve defnte hs popet s also epected fo the at M he teatve ethod s sutable to solve a sste of equatons A M u b (5 he pecondtone sste M u enable ou to fnd a bette statng guess whch s based on the eplaceent of the at A b the at M he fst vaant of poveent of the G ethod to the G ethod esults n speedng up calculaton that s based on the possblt of eas nveson of the at M A new decton of fndng bette appoaton of solutons usng the foula ( s a lnea cobnaton of the esdual vecto z whch s the soluton pecondtone sste e z M and an ognal decton of seachng fo the pevous appoaton of the soluton he algoth of calculaton s shown n ab II ABLE II G ALGORIM VERSION opute: b A z M p Fo untl convegence Do: 3 α ( z ( p αp α z + M + + z+ 8 p+ z+ + p 7 ( ( z 9 EndDo z he second vaant of the G ethod s based on the holes decoposton of the pecondtone at M LL (6 Fo ths vaant t s necessa to defne the followng aula vectos p L p L z u L L A L AL (7 It s possble to deduce that

5 ( z ( L L ( L L ( ( ( p A L p L p ( L A L p p ( A p p (8 he pevous algoth can be ewtten n the followng sequence α ( ( A p p u + u + α p ( ( + α A p + + (9 p + p + + whch the fequences as a functon of te can concde wth each othe he envelopes n these te nstants ae calculated b usng the teatve pocess A Eaple It s assued that the sgnal s coposed of two haonc coponents of the sae apltude whch s equal to unt One of these two coponents has a constant fequenc of 5 z and the second one has a fequenc lneal nceasng fo to z he esultng sgnal and nstantaneous fequences as a functon of te ae shown n Fg 3 he G algoth s appled to the pecondtone sste of equatons A u L b (3 he algoth of calculaton s shown n ab III ABLE III G ALGORIM VERSION opute: b A L p Fo untl convegence Do: α A p p 3 ( ( α L A p + α p 6 ( ( p L p 8 EndDo L Fgue 3 e hsto of the sgnal and the nstantaneous fequenc VI EXAMLES he basc tc fo fast convegence conssts n the fact that ths pecondtone at was used fo the soluton of pactcal eaples B B M (3 B he advantage of ths choce of the pecondtone at M s that the esult of calculaton s the fst guess of the soluton n the fo of the sepaate envelopes fo each coponent assung that thee s no poble wth cossng odes heefoe we get M (3 b We assue that the fequences of the snusodal coponents dffe fo each othe but thee s a te nstant n Fgue 4 Envelopes and teatve pocess as a fucton of nde he esults of calculaton ae shown n Fg 4 he apltude of the two coponents as a esult of the calculaton dffes fo the tue apltude b to pecent he vaable RELRES s the elatve esdue NORM (b - B* / NORM (b

6 o calculate the envelopes we use thee G ethods he top daga n the ght colun n Fg 4 shows the esults of calculaton wth the G algoth ognatng fo MALAB whle the esult whch was obtaned wth the use of the second pocedue called as Mcg coesponds to ab II and the thd esult coesponds to the pocedue Mcg n ab III he calculaton esults ae alost dentcal he sepaaton of envelopes s efeed to as decouplng odes B Eaple he coposton of the sgnal n ths eaple s shown n the ultspectu n the Fg 5 hs sgnal contans a snusodal coponent wth fequenc of z and a lot of haoncs wth the fequences whch ae nceasng n te he fequences of the fou haonc coponents concde wth a constant fequenc Fgue 6 Envelopes of the sgnal coponents AKNOWLEDGMEN hs eseach has been suppoted b the zech Gant Agenc poject No //5 Actve vbaton dapng of oto wth the use of paaetc ectaton of jounal beangs and b Opeatonal ogae Educaton fo opettveness n the faewo of the poject Oppotunt fo oung eseaches eg no Z7/3/36 Fgue 5 Multspectu of a ulttonal sgnal he esults of the envelope calculaton ae shown n Fg 6 he envelope of the sgnal coponent of a constant fequenc s labeled b X whle the envelopes of the coponents of the fequences whch ae nceasng n te ae labeled b X X3 and X4 he envelope X contans a pea whch coesponds to the fouth haonc coponent whch s not a pat of the ultode odel fo calculaton VII ONLUSION he pape contans a descpton of the calculaton of the envelope coponents wth fequences whch a coss o be close he pape deonstates that the analtcal pocedue can onl calculate two envelopes whle the nuecal teatons can select the envelope of seveal snusodal coponents whose fequenc can be n the sae te nstant dentcal he G ethod s pefeed to use fo soluton of a lage sste of equatons REFERENES [] Vold J Leudan Ode acng at Etee Slew Rates Usng Kalan acng Fltes SAE ape Nube 9388 [] h Feldbaue R oldch Realsaton of a Vold-Kalan acng Flte A Least Squae oble oceedngs of the OS G-6 onfeence on Dgtal Audo Effects (DAFX- Veona Ital Decebe 7-9 [3] Jua Vold-Kalan fltaton n MALABu (n zech In oceedngs of Eleventh MALAB onfeence aha : uusoft aha 5 3 s [4] J ua Dedopplesaton n Vehcle Etenal Nose Measueents In oceedngs of Eleventh Intenatonal ongess on Sound and Vbaton St etesbug : IIAV [5] J ůa Settng the passband wdth n the Vold-Kalan ode tacng flte In: welfth Intenatonal ongess on Sound and Vbaton (ISV Lsabon Jul -4 5 ape 79 [6] Y Saad Iteatve Methods fo Spase Lnea Sstes Second Edton SIAM Januaz 3d [7] MR estenes and E Stefel "Methods of conjugate gadents fo solvng lnea sstes" J Reseach Nat Bu Standads 49 (95 pp [8] M Fedle Specální atce a jejch použtí v nuecé ateatce KI SNL aha 98