De-noising Method Based on Kernel Adaptive Filtering for Telemetry Vibration Signal of the Vehicle Test Kejun ZENG

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1 6th Internatonal Conference on Mechatroncs, Materals, Botechnology and Envronment (ICMMBE 6) De-nosng Method Based on Kernel Adaptve Flterng for elemetry Vbraton Sgnal of the Vehcle est Kejun ZEG PLA 955 Unt 94, Laonng Dalan, 63, Chna Keywords: elemetry; Kernel Adaptve Flterng; Vbraton Sgnal; De-nosng Abstract. A kernel adaptve flterng de-nosng method for telemetry sgnal was proposed. he telemetry vbraton sgnal s often nterfered wth Gaussan nose and mpulse nose at the same tme, the tradton de-nosng method cannot obtan good de-nosng performance. In the vehcle test, the vbraton sgnal has typcal non-statonary and non-lnear characterstcs, for whch lnear flterng method s falure. Kernel adaptve flterng transforms the nput data to hgh dmenson feature space and desgns a lnear method to solve the nonlnear problem. he proposed method uses kernel adaptve flterng to suppress the nose n the vbraton sgnal of the vehcle test under norm cost functon, whch can keep effectve under non-statonary and non-lnear condtons. he smulaton and test data processng results show the good performance of the proposed method whch can apply n the telemetry vbraton sgnal processng. Introducton elemetry sgnal s the man bass for the vehcle nteror ballstcs analyss and detecton, n whch the vbraton sgnal s the most representatve that can drectly reflect the response of the vehcle structure state under dfferent ncentve condtons, and t processng results can be used to determne the health status of the vehcle structure []. he telemetry sgnal s often nterfered wth nose durng the transmsson and recepton process by transmttng and recevng equpment and the transmsson of the external envronment nfluence. he nose n the sgnal decrease the average varance estmaton precson n tme doman analyss, furthermore, serous nose nterference can lead to the dffculty of frequency doman analyss and tme doman analyss. herefore, de-nosng s an mportant part n the preprocessng of telemetry vbraton sgnal. he analyss of lots of the expermental data show that the nose n the telemetry sgnal s often not a smple Gaussan nose, for the hgh bt error generated durng the transmsson process also lead to mpulse nose. herefore, the nose n the telemetry sgnal usually shows the mxed nose form of Gaussan nose and mpulse nose []. radton de-nosng method ncludes frequency doman flterng, medan flterng, wavelet de-nosng, sngular value decomposton (SVD) and emprcal mode decomposton (EMD) based on local wave analyss, n whch the wavelet threshold de-nosng method s most wdely used n engneerng. In the de-nosng methods, frequency doman flterng request sgnal meets the statonary dstrbuton, medan flterng s sutable for mpulse nose, the wavelet threshold de-nosng and SVD needs to set artfcal parameters whch s dffcult to obtan deal performance, EMD de-nosng takes the hgh ntrnsc mode functon (IMF) as the nose whch lack of theory bass. Meanwhle, all knds of the de-nosng methods meet the dffculty that to fnd the best combnaton ponts of nose suppresson and sgnal loss of detal nformaton. For the mxed nose form de-nosng, the sngle de-nosng method has poor performance, and n vehcle test data detecton proves that the telemetry sgnals often appear non-statonary characterstcs [3]. Kernel adaptve flterng provdes a new method for sgnal de-nosng, whch can map the nput sgnal to the hgh dmenson feature space to obtan good flterng performance under nonlnear condton. By usng the kernel adaptve flterng method, a new method of telemetry vbraton sgnal de-nosng was proposed, whch can solve the de-nosng problem for mxed nose form of Gaussan nose and mpulse nose and has good performance for non-statonary telemetry vbraton de-nosng. he computer smulaton and expermental data processng results show the 6. he authors - Publshed by Atlants Press 468

2 effectveness of the proposed method. Basc Prncples of Kernel Adaptve Flterng Kernel method provdes a robust nonlnear parameter modelng method [4]. So far, the kernel method has been successfully appled to support vector machne (SVM), kernel prncpal component analyss (KPCA) and Fsher dscrmnant analyss. he basc dea of kernel methods s that mappng the nput data to a hgh dmenson feature space by a reproducng kernel, such that the nner product n the feature space can be more effcent computed by kernel evaluaton. In the feature space, the non-lnear problem can be solved by lnear flterng method f the algorthm can be expressed as nner product form, whch can avod the complex computaton process. he mappng of the nput sgnal and hgh dmenson feature space s shown n Fg.. Where ϕ s the feature mappng and ϕ( u) s the feature vector n the feature space, thus the nner product can be expressed by kernel as follow ϕ( u) ϕ ( u) = k( uu, ) () Fg. onlnear mappng from nput space to feature space he nner product space and reproducng kernel Hlbert space (RKHS) space s essentally the same, and the most commonly used method s the development of kernel flters n RKHS, whch can mplement the lnear adaptve flterng by the lnear structure of RKHS. Compared wth the neural network, the kernel adaptve flter has unversal approxmaton ablty and convex optmzaton feature, and has no local mnmum. Kernel adaptve flterng s a memory ntensve operaton and the computatonal complexty wth dmenson exponental growth. However, under the lmted nput data, kernel adaptve flterng method can not only complete data processng off-lne, also can solve the problem of learnng onlne. Functon estmator Adaptve flter algorthm Fg. Basc prncple of LMS adaptve flter In the tradtonal adaptve flterng algorthm, LMS algorthm s robust, smple structure and the most commonly used, f the problem to be solved s lnear problems, LMS usually can get the 469

3 desred convergence performance. Fg. shows the typcal LMS flterng block dagram [5]. Let the nput-output sample data s { u (), d() },{ u (), d() },,{ u ( ), d( ) }, there has contnuous nput and output mappng relatons between contnuous samples f : U R. As long as the flterng algorthm s ft to the mappng relatonshp, the LMS adaptve flter can complete the fttng, dentfcaton and classfcaton etc. LMS algorthm s usually to mnmze the emprcal rsk objectve functon to acheve the optmal functon of the parameters of the estmator, whch s to obtan the optmal flter weghts, the emprcal rsk cost functon s J( w) = d () () wu () = Accordng to the stochastc gradent descent algorthm, the LMS can update the weght coeffcent usng the nstantaneous gradent as follow J ( w) = u() d () wu() (3) w = hus the nstantaneous gradent at teratve tmes s J( w ) = u () d ()- w ( -) u () (4) Accordng to the steepest descent algorthm, the LMS algorthm can be expressed as follow w () = w ( ) + µ u () d () w ( ) u () (5) Where µ s the study step sze whch controls the convergence rate and the convergence precson of LMS adaptve flterng. If the nstantaneous error s defned as follow e () = d () w( ) u () (6) hen the update formula of the LMS algorthm can be wrtten as [6] w() = w( ) +µ e () u () (7) LMS algorthm assumes that the flter s a lnear fnte mpulse response flter, f the mappng relatonshp between the nput and output samples s hghly non-lnear, and then the performance of LMS algorthm wll be drastcally decreased. Accordng to the prncple of kernel adaptve flterng, f the samples can be mapped to hgh dmenson feature space, the teratve algorthm can mplement as lnear problem. Let the mappng relatonshp s u() ϕ( u ()), for the sake of smplcty, let ϕ () = ϕ( u()), thus n the hgh dmensonal space, a new nput and output sample sequence s obtaned as{ ϕ (), d ()}. he LMS algorthm can be used n the hgh dmensonal feature space as follow e () = d () ω( ) ϕ () (8) ω() = ω( ) +µ e () ϕ () (9) However, the dmenson of ϕ n the feature space s too hgh and mplct expresson. herefore, an alternatve method s needed to complete the calculaton, and the teratve process of the teratve Eq.9 can be obtaned the follows concluson. ω() = ω( ) + µ e () j() = [ ω( ) + µ e ( ) j( ) ] + µ e ( ) j( ) = ω( ) + µ [ e ( ) j( ) + e ( ) j( ) ] = () = ω() + µ e( j) j( j) = µ e( j) j( j) 47

4 In Eq., t s assumed that ω () =, whch can meet most of the applcaton condtons. If the system get a new nput, the output of the system can be expressed as nner product form as follow. ω( ) ϕ( u ) = µ e( j) ϕ( j) ϕ( u ) () = µ e( j) j( j) j( u ) In ths way, the output of the flter can be evaluated very effcently by usng the kernel technque n the nput space. ω( ) ϕ( u ) = µ e( jk ) ( ϕ( j) ϕ ( u )) () In kernel adaptve flterng, t s no longer nvolved to update the weghts, the flterng operator s completed by all prevous estmate error and kernel evaluaton functon. Kernel adaptve flterng algorthm only requres the nner product operaton. he algorthm can save a lot of tme, but regardless of the LMS algorthm n the nput space and the feature space s essentally the same. Let f s the nonlnear mappng of the nput and output, the followng sequental learnng rules can be formed for the kernel adaptve flterng algorthm. - f- = µ e( jk ) ( u( j),.) = f-( u( )) = µ e( jk ) ( u( j), u( )) (3) e () = d ()- f- ( u()) f = f- + µ ek () ( u(),.) Eq.3 called kernel LMS (KLMS) algorthm, whch s the LMS algorthm n the RHKS. KLMS assgns a new kernel unt to the new tranng data, and the center of the new kernel unt s the nput data u (), the kernel coeffcent s µ e (). In the tranng process, the coeffcent and the center are stored n the memory. Fg.3 shows the KLMS algorthm structure. Where a() s the coeffcent vector at the teratve tmes and a j () s the j th component of a (). he system output for the nput ucan be gven by f( u) = µ e( jk ) ( u( j), u ) (4) Fg.3 he network topology of KLMS at teratve tmes 47

5 De-nosng Based on KLMS Kernel adaptve flterng s appled to non-lnear mappng, whch has a unque advantage n the processng of telemetry envronment parameters. elemetry envronmental parameters de-nosng can take the collected data as the tme sequence, whch s the sequental learnng sample, to realze parametrc fttng to complete de-nosng by usng kernel adaptve flterng. Let telemetry vbraton sgnal s xt () = xt () + vt () (5) Where xt () s the observed sgnal, xt () s the useful sgnal and vt () s the nose whch commonly s the mxed nose of Gaussan nose and mpulse nose. Let the cost functon s [ ] J( w) = yt () xt () (6) Where yt () s the output of the flter. Based on the cost functon Eq.6, the vbraton sgnal can be de-nosng by KLMS, where the kernel functon s selected as Gaussan kernel as follow k( uu, ) = exp α u u (7) he smulaton frequency hoppng sgnal s used to analyze the performance of the de-nosng method, whch has typcal non-statonary characterstc. he smulaton telemetry vbraton sgnal has the practcal applcaton background, that s, when the vehcle structure s abnormal, the frequency of the acquston sgnal s changed. he sample frequency of the smulaton sgnal s f s = 48 Hz, the center frequency of two sne sgnal are f = 5 Hz and f = 5 Hz respectvely. Let the SR=3dB. he waveforms before de-nosng by KLMS are shown n Fg.4 and Fg.5 respectvely. Fg.5 shows KLMS has good de-nosng performance for the smulaton sgnal. 4 Ampltude A me t/s Fg.4 Smulaton frequencey hoppng sgnal wth nose Ampltude A me t/s Fg.5 Smulaton frequencey hoppng sgnal after de-nosng o further llustrate the practcalty of the algorthm, hgh frequency vbraton sgnal collected by a vehcle test s used for processng, whch the sample frequency s 5kHz and waveform and frequency spectrum s shown n Fg.6. Fg.7 shows the de-nosng result of KLMS. he results show that KLMS can suppress the nose n the vbraton sgnal effectvely. 47

6 Ampltude A/g - Y(f) me t/s -4 Frequency f/hz Fg.6 Hgh frequency vbraton sgnal and ts frequency spectrum 4 Ampltude A/g - Y(f) me t/s -4 Frequency f/hz Fg.7 Hgh frequency vbraton sgnal after de-nosng Concluson hs work proposed a de-nosng method for telemetry sgnal based on kernel adaptve flterng. For the telemetry envronmental parameters n the vehcle test s often nterference wth mxed nose, the proposed method can obtan good de-nosng performance under non-statonary and non-lnear condton, whch has practcal applcaton sgnfcance n the telemetry vbraton sgnal de-nosng processng. he effectveness of the proposed method s proved by the smulaton sgnal and the measured sgnal processng results. References [] LIU X., LIAG H., XUA Z.W. elemetry vbraton sgnal anomaly detecton method based on fuzzy entropy of wavelet modulus maxma [J]. Journal of Vbraton and Shock, 6, 35(9): [] LIU X., XUA Z. W., LIAG H. elemetry vbraton sgnal abnormalty detecton method based on mult-scale spectral kurtoss map [J]. Journal of Projectles, Rockets, Mssles and Gudance, 5, 35(5): [3] Anton J., Randall R. B. he spectral kurtoss: a useful tool for characterzng non-statonary sgnals [J]. Mechancal Systems and Sgnal Processng, 6, (): [4] ZHAO J., LIAO X. F., WAG S. Y. Kernel least mean square wth sngle feedback [J]. IEEE Sgnal Processng Letter, 5, (7): [5] LIU J. C., ZHAO H. Z., QUA H. D., et al. Iteraton based varable step sze LMS algorthm and ts performance analyss [J]. Journal of Electroncs & Informaton echnology, 37(7): [6] Kwong R. H., Johnston E. W. A varable step sze LMS algorthm [J]. IEEE ransactons on Sgnal Processng, 99, 4(7):