2 THE FIRST AND SECOND GENERATION OF THE VK-FILTER

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1 ůma R. he passband Width of the Vold-Kalman Order racking Filter. Sborník vědeckých prací VŠB-U Ostrava řada stroní r. LI 5. č. paper No. 485 pp ISSN ISBN X. Jiří ŮMA * HE PASSBAND WIDH OF HE VOLD-KALMAN ORDER RAKING FILER ŠÍŘKA PROPUSNÉHO PÁSMA VOLD-KALMANOVA ŘÁDOVÉHO FILRU Abstract Even thogh the basic principle of the Vold-Kalman (VK) order tracking filter was pblished many times the isse dealing with the problem of the filter passband setting was omitted as yet. It is known that the filtration effect of the VK-filter is achieved by solving of the linear system eqations. he system matrix is sparse and the eqation system soltion can be based on the holesky factoriation of the system matrix reslting in the soltion of the forward redction and backward sbstittion. Redction and sbstittion can be considered as a filtering process. his is a main idea how to evalate the dependence of the filter bandwidth on a parameter determining the VK-filter property. Abstrakt Ačkoliv ákladní princip Vold-Kalmanova filtr byl iž několikrát pblikován problém stanovení šířky propstného pásma by opomíen. Je námo že filtrační efekt vyplývá řešení sostavy lineárních rovnic. Matice sostavy rovnic e řídká a řešení příslšné sostavy může být aloženo na holeskyho faktoriaci s dopředno redkcí a pětno sbstitci. Obě tyto operace představí filtrační proces. oto e ákladní idea pro výpočet šířky propstného pásma Vold- Kalmanova filtr.. INRODUION Some particlar class of signals consists of harmonic components that are all (or the most dominant of them) related in freqency to the fndamental freqency e.g. engine rotational speed. hese components are designated as sper- or sb- harmonics (the so-called orders) of the fndamental freqency in RPM which is measred. he paper deals with the Vold-Kalman (VK) order-tracking filter of two generations []. After describing the theoretical principle main attention is focsed on the control of the absolte and relative VK-filter bandwidth. Withot loss of the generality the analysis is dealing with the tracking of st one order. Under condition that the orders are not close or crossing the mltiple orders can be tracked individally in a step-by-step way. HE FIRS AND SEOND GENERAION OF HE VK-FILER Similarly as for the Kalman filter which is based on the process and measrement eqations the VK-filter is based on the strctral and data eqations that play the similar role in the filtration effect. Both these eqations are excited by the nknown fnctions on its right side. It is assmed that for the Kalman filter these fnctions are stochastic with known covariance while for the VK-filter a * Prof. Ing. Sc. Department of ontrol Systems and Instrmentation Faclty of Mechanical Engineering VŠB echnical University of Ostrava 7. listopad 5 Ostrava tel. (+4) iri.tma@vsb.c

2 ser sets only the relationship between them. he strctral eqation for the first and second generation of the VK-filter takes the form x n cos ω t x n + + x n + ε n x n x n x n + ε n () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x( n) x( n) x( n + ) + x( n + ) ε( n) where: n index ( n) x( n) x( n + ) + x( n + ) x( n + ) ε( n) x x ( n) amplitde and phase modlated harmonic signal as the filter otpt of the first generation VK filter or the envelope of the filter otpt for the second generation VK filter t sampling interval of the inpt data samples ω instantaneos anglar freqency of the otpt modlated harmonic signal ε ( n) error term enabling a slightly change of the filter otpt amplitde and freqency over the time samples involved in the eqation (). he strctral eqation of the second generation VK filter are shown in three possible forms differing in the order of the difference which is eqal to the error term ε ( n). he order of the differences pls nity gives the nmber of the filter poles p. he system of the strctral eqations () containing all the samples n... N takes the following form which can be rewritten in the matrix form A x ε () Instead of observing the sampled sinsoidal signal or its envelope x ( n) only samples ( n) recorded. he signal ( n) y are y is combined from both the signals satisfying the strctral eqation () as well as random noise and other sinsoidal components differing in the freqency with the sinsoidal signal x ( n) for the first generation filter or the envelope for the second generation filter. he random η. noise and other sinsoidal are combined into the signal ( n) y n x n + η n ( ) ( ) ( ) y n ( n) x( n) exp ( Θ( n) ) + η( n) Θ( n) ω ( i) where Θ ( n) is a signal phase as a reslt of the anglar freqency integration. Formally it can be written as N eqations arranged into the matrix form y x η y x η (4) i t { exp( Θ( ) )exp( Θ( ) )... ( Θ( N ))} diag exp GLOBAL SOLUION OF BOH HE GENERAION VK-FILERS he system of the data eqations (4) and the strctral eqations () is an nderdetermined system for the nknown waveform x ( n). he additional condition for the eqation soltion is that the variances of the non-homogeneity terms ε ( n) and the other sinsoidal components and backgrond random noise η ( n) have to be minimal while maintaining the given relationship between them. he global soltion can be fond sing the standard least sqare techniqe. he sm of the sqares of all ()

3 the nknown non-homogeneity terms for the first and second-generation algorithm can be expressed as a scalar prodct ε ε xa A x (5) where a row vector ε is a transpose of the colmn vector ε. he sm of the sqares of the signal η n in both the VK-filter generations can be written in the form ( ) η ( η y x )( y x) ( H H H η η y x )( y x ) where the pper index H designates the complex congate qantities. As the matrix is complex both the vectors x η are complex as well. he weighted sm of the particlar sms (5) and (6) gives the loss fnction (6) J r ε ε + η η (7) where r is a weighting factor [ 4]. he choice of a large vale for the weighting factor r leads to the highly selective filtration in the freqency domain that takes a long time to converge in amplitde. In contrast fast convergence with low freqency resoltion is achieved by choosing r small. he first derivative of the loss fnction (7) with respect to the vector x gives a condition for the minimm of this fnction which is called a normal eqation. J J A A x + ( x y ) H r r A A x + x x ( x y) x (8) H ( r A A + E) y x ( r A A + E) y (9) he matrix eqations (8) are of the same form bt for an exception. he vector y is mltiplied H by which shifts the freqency of the tracked components toward to ero. he pass band filter becomes the low pass filter. he nknown waveform in the case of the first generation VK-filter and the nknown envelope in the case of the second-generation VK-filter reslt from the eqations (9). he prodct of the matrixes A and A gives a symmetric positive semidefinite matrix. he matrix B r A A + E becomes the symmetric positive definite matrix by adding the nity matrix E. he matrix B consists of the limit nmber of the non-ero diagonals. herefore it is easy to invert it. he nmber of the non-ero diagonals of the matrix B for the VK-filter of the second-generation is eqal to p + where p is the nmber of the filter poles. his nmber of the non-ero diagonals of the matrix B for the VK-filter of the first-generation can be designated as well by p + where p. Employing the holesky factoriation of the matrix B into the matrix prodct B L U where L is a lower-trianglar matrix and U L is an pper-trianglar matrix is the easiest way how to solve the eqation system (8). he only condition for the holesky factorisation is that all the main minor determinants are eqal to a positive vale what can be easily proved. he main advantage of the holesky factorisation algorithm is that it saves the nmber of the non-ero diagonals in the trianglar matrices at the vale p +. he soltion of the system (8) is broken down into two linear eqation systems the forward redction and backward sbstittion. Forward redction (first system) Backward sbstittion (second system)ter y ( y ) x N N N x x () ( ) N N N N N N N

4 For p +... N : y... ( p p ) For N ( p + )... : ( + x px + p ) x () In the forward redction the linear eqation system L y ( U y ) for an nknown vector is solved while in the backward sbstittion the nknown vector x of the eqation system U x is evalated. he vale of weighting factor r has to be limited not to lose the effect of adding nity to main matrix diagonal on the positive definiteness by ronding the diagonal elements de to the limit bit nmber for saving qantities in a compter memory. aking into consideration the maximal vale of the diagonal components of the matrix r A A and 4 decimal places for doble-precision compterarithmetic the limit vale r MAX for the weighting factor is shown in the table. his table gives also the lower limit for the relative VK-filter bandwidth which is introdced in the next chapter. ab. Limit vale of weighting coefficient and bandwidth Pole nmber: p p p p 4 ( r A A ) i i r 6r r 7r r 7x 6 4x 6 x 6.x 6 MAX f % > 5x -6 %.5 %.5 % % 4 BANDWIDH OF HE SEOND GENERAION VK FILER he main idea how to evalate the dependence of the filter bandwidth on a parameter determining the VK-filter property reslts from eqation (). aking into accont the reverse order of the samples x ( N )... x( ) in the backward sbstittion the filtration process is based on the same transfer fnction as for the forward redction. Altogether the forward redction and backward sbstittion reslts in ero-phase digital filtering analogos to the filtfilt fnction in Matlab. he rolloff of the second generation filter is eqal to 4 p db per decade. he dependence of the filter bandwidth on the weighting factor is evalated for the second generation VK filter namely for the one-pole filter version. he matrix prodct A A is eqal to A A () he vales of the elements of the matrix B r A A + E are given by b b r + b b b r + () he holesky factoriation of the matrix B into the matrix prodct B L U where L is a lower-trianglar matrix and U L is an pper-trianglar matrix reslts in the formlas b for... N : b b. (4) he linear eqation systems for the forward redction and backward sbstittion are given by ( ) y( ) for... N ( N ) ( N ) N N x for N... : ( ) y( ) ( ) 4 ( ) ( x + ). (5) : ( ) ( ) ( ) x +. (6)

5 he steady-state vales of the non-ero elements of the matrix U can be evalated as lim lim + (7) he digital filter transfer fnction in Z-transform reslting from () is defined as Z Y ( ) ( ) + It can be written for the steady-state vales of the elements of the matrix U that are given by (4) (8) b b () After rearrangement it is obtained b b Sbstitting the complex qantity by the term ( Ω) + () exp where ω t freqency response fnction based on (8) is given by Ω Ω Z ( ) ( e ) LP Ω Ω Y ( e ) + e G e Ω for ( π +π) Ω the For the reason explained at the beginning of this chapter (two-stage filtration process) the db ct-off freqency of the one-pole filter reslts from the following condition Ω ( e ) () G LP. () + e Ω he relative bandwidth corresponding to the absolte ct-off freqency f H of the low pass filter is given by the formla f f H f. he sbstittion of the exponentional term reslts in e Ω S ( π f ) cos( π f ) sin( π f ) he low pass filter freqency response fnction () is given by + e Ω b + b cos + exp (4) ( cos( π f ) sin( π f )) + + cos( π f ) ( π f ) r + r cos( π f ) + r ( cos( π f )) he last formla and eqation () reslts in a formla describing the dependence of the r on f (5) r ( cos( π f )) (6) 5 GENERAL FORMULAS FOR EVALUAION HE FILER BANDWIDH he bandwidth as the exact fnction of the weighting factor r for evalation f is given for the first generation VK-filter by the following formla f arccos cos ( Ω t) r arccos cos ( Ω t) + r (7) π he approximation of the mentioned exact fnction can be performed by sing sbstittion Ω ω t + Ω followed by simplification the formla by sing sbstittion cos( Ω ) and sin Ω. he approximation formla for evalation the weighting factor r is given by ( ) Ω 5

6 r (8) π f ( cos ( ω t) ) For the second generation VK-filter the weighting factor r as the exact fnction of the bandwidth and the approximation of this fnction sing the estimating formla ( π f ) π f are smmaried in the table. ( ) cos ( ) ab. Weighting factor as a fnction of the bandwidth for the nd generation of the VK filter Nmber of poles Soltion of the eqation Approximation r ( cos( π f )) r.4864 f (9) r 6 8 cos ( π f ) + cos( π f ) r f () r cos ( π f ) + cos( π f ) cos( π f ).7569 r f () In the case when the freqency of the tracked order is not eqal to a constant vale and it is not possible to evalate the appropriate weighting factor the loss fnction (7) is transferred to the form J ε R R ε + η η where R is a sqare diagonal matrix with diagonal elements which are eqal to the weighting factors determined for the instantaneos order freqency and the filter bandwidth e.g. ri i r( f f ). he VK-filtration can work with both the absolte bandwidth f in H and the relative bandwidth f f in percentage to remain a constant vale. ONLUSIONS he main reslts of the paper are formlas describing the dependence of the weighting factor on the VK-filter bandwidth. hese formlas play a key role in the software for the VK-filter and allow the setting of the filter bandwidth in either absolte vale in H or relative vale in percentage. REFERENES [] VOLD H. & LEURIDAN J. Order racking at Extreme Slew Rates Using Kalman racking Filters. SAE Paper Nmber 988. [] FELDBAUER h. & HOLDRIH R. Realisation of a Vold-Kalman racking Filter A Least Sqare Problem Proceedings of the OS G-6 onference on Digital Adio Effects (DAFX- Verona Italy December 7-9. [] UMA J. Vold-Kalman filtration in MALAB (in ech). In Proceedings of Eleventh MALAB onference. Praha : Hmsoft Praha 5.. s [4] UMA J. Dedopplerisation in Vehicle External Noise Measrements. In Proceedings of Eleventh International ongress on Sond and Vibration. St. Petersbrg : IIAV P his research has been done at the VSB-echnical University of Ostrava as a part of the proect No. /4/5 and has been spported by the ech Grant Agency. 6