Wavelet-Based Method for Fog Signal Denoising

Transcription

1 Jurnal f Autmatin and Cntrl Enginring, Vl., N. 2, Jun 23 Wavlt-Basd Mthd fr Fg Signal Dnising Yu Zu Cllg f Autmatin, Harbin Enginring Univrsity, Harbin, China andrw22.zu@gmail.cm Jinchu Ca Dpartmnt f Educatin, Univrsity f Nvada, Rn, USA cainchu@gmail.cm Abstract Fibr-Optic Gyrscp (FOG) has bn widly usd t masur th angl rat f vhicl in rcnt yars. Bing as unprdictabl and unmasurd rrr, randm drift gnratd frm FOG instability crat sriusly bad influnc n prcisin f FOG utput, as wll as Inrtial Navigatin Systm (INS). Althugh wavlt-basd tchniqu has mad cnsidrabl prgrss in FOG signal dnising, almst all th achivmnts ar basd n ff-lin analysis that givs littl cntributin t practical applicatin. This papr prsnts a rvisd plan n accunt f prvius rsarch n ral tim dnising, xplaining th cmputatinal cmplxity rductin frm thry. Thrugh simulatin and static FOG xprimnt using tim-frquncy analysis and Allan Varianc as th prfrmanc valuatin standard, th dnising ffctivnss cmpard t using traditinal mthd has bn prfd imprving t a larg xtnt. Indx Trms FOG signal, Scnd gnratin wavlt transfrm (SGWT), Dnising, Ral tim, Sliding windw I. INTRODUCTION Fibr-Optic Gyrscp (FOG) applid in Inrtial Navigatin Systm (INS) cntains randm drift gnratd by uncrtain disturbanc. Hwvr, it s hard t us prpr mdl t rprsnt and cmpnsat it. Cnsquntly, FOG signal dnis may wll b th ky fr imprving navigatin prcisin. Whn applid t FOG signal ral-tim dnis, th supririty f wavlt cmpard with slf-adaptiv filtr has bn discussd basd n ff-lin analysis in []-[2]. By using nfrcd d-nising tchniqu, th spd f rspns will b nhancd. Aftr that, thrugh utilizing sliding windw, s fr cncrt algrithm [3]-[4], wavlt transfrm can b implmntd in a fixd xtnt as th windw mving in rdr t cnstruct th ral-tim d-nising algrithm. Furthrmr, aftr prpsing a nw principl fr thrshld dtrminatin, nvl tactics cncrning abut wavlt cfficint prcss has bn substitutd in []-[6]. In INS basd n FOG as n f chif Inrtial Masurmnt Unit (IMU), raw signal dnising shuld b cnsidrd bth n ral-tim applicatin and satisfactry ffctivnss. Althugh nfrcd dnising prpsd in Manuscript rcivd Sptmbr, 22; rvisd Dcmbr 22, 22 [] wakns th cmplxity f thrshld principl, it has bad influnc n dynamic fatur, lading t rmving plnty f usful signal as wll whn trying t clan up th nis. Th mthd prpsd in [3] cmbins sliding windw with traditinal wavlt transfrm. Hwvr, th cmplicatd apprach rfrring t windw lngth dtrminatin rstricts it applid t ral-tim dnising. Mrvr, thrshld-basd d-nising mthd was rvisd in []. Th cmplxity still kps a high lvl s that th cmputatinal spd will b prmtd t a limitd xtnt. W prps t us SGWT t simplify th dcmpsitin and rcnstructin prcss, trmndusly nhancing cmputatinal spd frm thry. Spcifically, cmbind with SGWT, svral tchniqus statd abv has bn usd, which mainly including sliding windw, nfrcd dnising aimd at wavlt cfficint in first lvl and hard thrshld principl. This nw ral-tim dnising plan has bn prvd having mr ntabl dnising ffctivnss than traditinal wavlt basd mthd prpsd in [] frm simulatin. By daling with practical FOG signal, w indicat th viability f this plan by taking statistic xprimnt, shwing that nvl plan ptimiz attitud infrmatin f INS. Th papr is rganizd as fllws. In sctin 2, w rviw basic thry abut SGWT including sm ssntial cncpt. In sctin 3, w discuss th nw plan fr ral-tim dnis frm thr facts which ar sliding windw, cmputatinal cmplxity and simplificatin f thrshld principl. Simulatd signal xprimnt has bn dscribd in sctin 4. In sctin, FOG static xprimnt is takn. And w us tim-frquncy analysis as wll as Alln Varianc t udg th prfrmanc f nw plan. Finally, summry f cnclusins ar prsntd in sctin 6. II. SECOND GENERATION WAVELET TRANSFORM A. Lifting Schm Lifting, a spac-dmain cnstructin f birthgnal wavlts dvlpd by Swldns [7] cnsists f th itratin f th fllwing thr basic pratins: Split: Divid th riginal data xn [ ] int tw disint substs, fr xampl, vn sampl x[ n] and dd sampl x[] n. 23 Enginring and Tchnlgy Publishing di:.272/ac

2 Jurnal f Autmatin and Cntrl Enginring, Vl., N. 2, Jun 23 x [ n] x[2 n], x [ n] x[2n ] Prdictin: Dsign th prdictin pratr P t dal with x[ n ]. Gnrat wavlt cfficint dnas [ ] th rrr in prdicting x[] n frm x[ n ]. () d[ n] x [ n] P( x [ n]) (2) Updat: Dsign th uplad pratr U t dal with dn. [ ] Thn add t x[ n] t gt scaling cfficint rprsnting cars apprximatin. c[ n] x [ n] U( d[ n]) (3) Th prcss f rcnstructin f lifting schm is invrtd stps f dcmpsitin. Stps ar carrid ut by rarranging f pratins statd abv: x [ ] [ ] ( [ ]) n c n U d n (4) x [ n] d[ n] P( x [ n]) () x[2 n] x [ n], x[2n ] x [ n] (6) B. Intrplatin Subdivisin Intrplating subdivisin is a kind f frcast mthd usd in SGWT. Using prdictin pratr cfficint P as th xampl, its basic stratgy is t utiliz th vn sampls nighbring th dd sampls t stimat th frmr during th prcss f dcmpsitin r rcnstructin. Hr w tak th [4, 4] wavlt rprsnting fur prdictin and fur uplad cfficints applying t th nxt xprimnt. Tw kinds f cfficints will b pr-cmputd by th mthd prpsd in [2]. Th filtr cfficints ar -.62,.62,.62, -.62 (prdictin cfficint) and -.32,.282,.282, -.32 (uplad cfficint). As this tchniqu ds nt rly n th Furir transfrm [8], SGWT-basd transfrm nt nly avids signal transmitting btwn tim and frqunt dmain, but nly cntains th simplst fur fundamntal pratins f arithmtic. In additin t guarant that th systm will b rspns fr input signal within a shrt tim, rfraining frm liminatin f signal assmbld in lw and mdium frquncy is imprtant as wll. Cnsquntly, th nxt sctin cncntrats n rvisd plan dvlpd frm statmnt abv. C. Rvisd Plan f Ral-tim Dnising ) Sliding Windw Undr nrmal cnditins, thrshld dnising is a kind f ff-lin signal prcss, which mans taking ntir raw signal as a st f data fr d-nising manipulatin. Obviusly, this pratin cannt satisfy th practical rquisitin that th utput signal is ndd t prcssd n-lin. Cnsidring th rstrictin f wavlt transfrm in ral-tim signal prcss, sliding data windw has bn usd. Th data in th first pr-spcifid xtnt f windw d nt crss th d-nising systm. Whil th numbr f data accumulats t th lngth f windw, th st f data bgins t intrduc int d-nising prcss. Aftr that, as th data is rcivd frm FOG, th windw kping a fixd lngth slids t gt th latst n th data fr wavlt dnising. Aftr vry tims f rcnstructin, th n th data is takn as th dnisd utput transmittd t th nxt calculatin. 2) Wavlt Cmputatinal Cmplxity In a cmmn xprssin, wavlt cmputatinal cmplxity is usd t rprsntd hw cmplicatd th wavlt cmputatin is. It is masurd by th numbr f multiplicatin and additin in a pair f cfficint ( cl, d l). Frm [9], cmpard t nrmal algrithm, wavlt transfrm using lift schm lwr th lvl f cmplxity sharply undr th sam data quantity. Tab. shws th cmparisn btwn standard algrithm and lifting schm. As it shwn, spd incras is mr than 6 prcntag avrag. TABLE I. COMPUTATIONAL COMPLEXITY OF STANDARD VS. LIFTING ALGORITHM Wavlt Standard Lifting Spdup Haar 3 3 % Db % Db % [2,2] 6 67% [4,4] % 3) Thrshld Principal Simplificatin T simplify th thrshld cmputatin, w us th cmmn thrshld: T 2ln(2 ) (7) whrt is thrshld in lvl, is standard varianc f nis in lvl. can b calculatd frm th fllwing quatin: mdian( d ) (8).674 Th quantizatin mthd usually cntains hard thrshld, sft thrshld and smisft thrshld which taks int accunt bth hard and sft thrshld s advantags. Exprimnt indicats that traditinal wavlt using sft thrshld fr FOG signal dnising can gt mr bnfits than using hard thrshld. Hwvr, if using SGWT, d-nising ffct is almst th sam. Hnc, cnsidring th cncis functin, w accpt hard thrshld as th principl fr thrshld prcss. Signal aftr dcmpsitin frm prvius lvl will b dividd int tw parts: th dtails and apprachs which rprsntd by wavlt cfficint and scaling cfficint rspctivly. Sinc th lvl dcmpsitin is takn n accunt f scaling part f + lvl. S as lng as raw signal is intrducd int d-nising systm, nis xtractin and liminatin will b takn frm highfrquncy t lw-frquncy. Whn applid t INS installd n ship, FOG signal cncntrats n lw r 87

3 Jurnal f Autmatin and Cntrl Enginring, Vl., N. 2, Jun 23 mdium frquncy, spcially th z-axis FOG. It is diffrnt frm using in plan r sm thr fast vhicl. Thrfr, bcaus f this kind f slw changd signal, nfrcd dnising, which mans that st all f wavlt cfficints t zr aftr th first dcmpsitin, is rasnabl. Hwvr, th principl backs t nrmal way that is hard-thrshld aftr scnd, third, furth dcmpsitin. D. Simulatin and Analysis Gnrat a squar wav with 2s prid, unit amplitud and sampls using MATLAB. And randm nis has bn suprimpsd n it. First, w utiliz db4 as th basis functin in th mthd prpsd in [] fr simulatd ral-tim dnising. Aftr that, w tak advantag f [4,4] wavlt basd rvisd plan fr th sam simulatd xprimnt. Essntial paramtrs ar st as fllwing: Dcmpsitin lvls: 3; sliding windw lngth: 24; Thrshld principl: Univrsal mthd + hard-thrshld. W intrduc Varianc v and Signal t Nis Rati (SNR) R SN as th main standards fr stimating dnsing prfrmanc. Th cmparisn f nis rductin mthds is shwn in Tab. 2. Th dgr that dnisd signal divrgs frm Expctatin (E) aftr using rvisd plan w prpsd is smallr than using traditinal plan. And SNR is imprvd by apprximat 2dB cmparing t traditinal mthd and 6.6dB cmparing t raw signal. Fig. and Fig.2 shw th simulatd rsult in tim-dmain and frquncy-dmain rspctivly. Varianc 4 ( ) TABLE II. COMPARISON OF VARIANCE AND SNR Raw signal Traditinal mthd Rvisd mthd Imprvd prcntag % SNR(db) % Nrmalizd amplititud raw squar signal "db4" ral-tim dnis rvisd mthd ral-tim dnis Frquncy/Hz Figur 2. Frquncy-dmain diagram f simulatd signal dnising E. Prfrmanc Evaluatin fr FOG Signal Dnising ) Static Exprimnt Raw signal is rcivd thrugh srial cmmunicatin with Hz sampl frquncy frm a static FOG. W chs th x-axis (astrn axis) as th rsarch axis. And cut ff an xtnt f data cntaining sampls. Th lngth f sliding windw quals t th lngth f 24 sampls. Db4 and [4, 4] wavlt hav bn usd with dcmpsitin lvl 3. Entir xprimnt is actually cmpltd ff-lin with th hlp f MATLAB. Nnthlss, it has simulatd th prcss f ral-tim dnising and its rsults ar still valuabl and maningful. Fig.3 and Fig.4 plt th riginal static FOG signal and dnisd signal prsntd in tim and frquncy dmain rspctivly. In fact, th pint whs nrmalizd amplitud is dsn t lcat xactly in zr-frquncy, but appraching t zr. Th rasn is that FOG may wll b influncd by slw drift f its wn r glbal slfrtatin dtctin drivd frm installatin rrr. Amplitud raw squar signal "db4" ral-tim dnis rvisd mthd ral-tim dnis scnd/s Figur. Tim-dmain diagram f simulatd signal dnising Angl Rat dgr/h... riginal FOG signal "db4" ral-tim dnis rvisd mthd ral-tim dnis tim /s Figur 3. Tim-dmain diagram f FOG signal dnising 88

4 Jurnal f Autmatin and Cntrl Enginring, Vl., N. 2, Jun 23 Nrmalizd Amplitud riginal FOG signal "db4" ral-tim dnis rvisd mthd ral-tim dnis frquncy/hz Figur 4. Frquncy-dmain diagram f FOG signal dnising FOG is sttld n th singl-axis turn-tabl shwn in Fig.7. Th utput f FOG is gaind and strd. Thn tak it as input t th INS. At last, w gt attitud infrmatin abut currnt stat including rll, pinch and yaw. Data sts as fllws: h sampling with 98Hz frquncy, cutting ut a sctin f signal starting frm 3th hur with abut s lngth f signal dnisd by ur plan. Bcaus f n rfrnc psitin in singl-axis turn-tabl s that it may wll b affctd by fundatin supprt, w rgard,, 3 as th currnt attitud. Th rsult is plttd in Fig.6. Fig.6 shws th dcrasd amplitud f rrr aftr d-nising prcss using ur mthd. It s '' abut 6 dcrass frm riginal rll and pinch angl. Frm cmparisn in frquncy-dmain, th nis cmbind with signal xists n full band. Thrugh implmnting rvisd plan, nrmalizd amplitud f pints whs frquncy ar gratr than.hz dcrass t a crtain xtnt. And paks hav bn almst liminatd. Exprimnt indicats that th nis whs frquncy is gratr than.hz has bn rprssd. Hr w intrduc Allan Varianc in rdr t valuat d-nising ffct. Allan Varianc mthd is a tim dmain analysis tchniqu riginally dvlpd t study th frquncy stability f prcisin scillatrs []. It is usd t charactriz randm prcsss rspnsibl fr nis prsnt in data []. By analyzing Allan Varianc, diffrnt typs f nis can b rcgnizd in FOG signal. Tabl 3 lists sm kinds f randm rrr cfficints such as Bias stability, quantizatin nis and rat ramp. Figur 6. FOG signal prprcss bfr INS calculatin TABLE III. RESULT OF ALLEN VARIANCE ANALYSIS plan Varianc Bias stability ( / h ) Quantizatin nis ( rad ) Rat ramp 2 ( / h ) Raw signal Traditinal m thd Rvisd mth d ) Attitud Errr Exprimnt Th purps f d-nising prcss is fr th imprvmnt f attitud accuracy aftr INS calculatin. Fig. shws th psitin that d-nising prcss is in as a stp fr raw FOG signal prprcss. Original FOG signal D-nising prcss INS Acclrmtr signal Attitud utput Figur. FOG signal prprcss bfr INS calculatin Figur 7. FOG and singl-axis turn-tabl F. Summarizs A rvisd mthd fr FOG signal ral-tim dnising prpsd in this papr, basd n SGWT aviding tim t frquncy cnvrting, lads t lss cmputatinal cmplxity than using traditinal wavlt basd mthd. Sliding data windw, as wll as nfrcd dnis cmbind with hard-thrshld, ar tw ky tchniqus. Th frmr whs windw lngth has bn prfd having littl influnc n INS by rpatd simulatin, allws taking wavlt transfrm n-lin. Th latr simplifis thrshld principl in trms f charactristic f FOG signal, making furthr imprvmnt n cmputatin fficincy. Simulatin and static FOG xprimnt indicat that nis aggrgats n full band. And impuls disturbanc 89

5 Jurnal f Autmatin and Cntrl Enginring, Vl., N. 2, Jun 23 can b rprssd ffctivly. In additinal, almst vry sampl will b prcssd s that th ability fr attitud rrr supprssin can b nhancd t a larg xtnt. ACKNOWLEDGMENT Th authrs wish t thank Mrs. Ca wh cllctd larg amunts f matrials assciatd with ur rsarch and mad cntributin t pattrn arrangmnt f th articl. REFERENCES [] R. M. Yuan, F. Sun, and H. Chn, Wavlt filtring mthd in signal prcssing f fibr ptic gyr, Jurnal f Harbin Institut f Tchnlgy, vl. 36, n. 9, pp , 24. [2] C. S. Qu, H. L. Xu, and Y. Tan, Signal dnising basd n scnd gnratin wavlt transfrm fr lasr gyr filtring, Infrard and Lasr Enginring, vl. 38, n. 2, pp , 29. [3] D. F. Jiang, M. A. Chn, Ral-tim wavlt d-nising algrithm, Chins Jurnal f Scintific Instrumnt, vl. 2, n. 6, pp , 24. [4] G. M. Sng, H. J. Wang, H. Liu t al., Analg circuit fault diagnsis using lifting wavlt transfrm and SVM, Jurnal f Elctrnic Masurmnt and Instrumnt, vl. 24, n., pp. 7-22, 2. [] P. Lv and J. Y. Liu, Rsarch n thrshld algrithm in raltim wavlt d-nising f FOG, Jurnal f Prctils, Rckts, Missils and Guidanc, vl. 29, n., pp. 8-22, 29. [6] J. H. Liu and K. Sh Indpndnt cmpnnt analysis algrithm using wavlt filtring, Jurnal f Elctrnic Masurmnt and Instrumnt, vl. 24, n., pp , 2. [7] W. Swldns, Th lifting schm: A custm-dsign cnstructin f birthgnal wavlts, J. Appl. Cmp. Harm. Anal. vl. 3, n. 2, pp. 86-2, 996. [8] W. Swldns, Th lifting schm: A cnstructin f scnd gnratin wavlt, Siam Jurnal n Mathmatical Analysis, vl. 29, n. 2, pp. -46, 997. [9] I. Daubchis and W. Swnldns, Factring wavlt transfrm int lifting stps, Jurnal f J Mathmatical Analysis Applicatins, vl. 4, n. 3, pp , 998. [] W. Allan, D and A. Barns, J, A Mdifid Allan Varianc with Incrasd Oscillatr Charactrizatin Ability, in Prcdings 3th Annu. Frq. Cntrl Symp, Baltimr, MD, USA, Nv. 98, pp [] L. Samrat and J.Nayak, Charactrzatin f fibr ptics gyr and nis cmpnsatin using discrt wavlt transfrm, in Prc. Scnd Intrnatinal Cnfrnc n Emrging Trnds in Enginring and Tchnlgy, 29, pp [2] R. L. Claypl, R. G. Baraniuk, and R. D Nwak, Adaptiv wavlt transfrms via lifting [EB/OL]. in Prc. IEEE Intrnatinal Cnfrnc n Acustics, Spch and Signal Prcssing, 998, pp Yu. Zu was brn at Biing, China in May 988. And rcivd his bachlr f nginring frm Harbin Enginring Univrsity, Harbin, China in 2. Nw his rsarch fild cntrs n intgratd navigatin systm, inrtial navigatin systm as wll as rlatd signal prcssing tchniqus. H is nw th assistant rsarchr at Marin Navigatin Rsarch Institut which is n f affiliatd instituts in Cllg f Autmatin, Harbin Enginring Univrsity. Main publishd paprs in rcnt n yar includ: GAO Wi, ZU Yu (crrspnding authr), tal, Rsarch n ral-tim d-nising f FOG basd n scnd gnratin wavlt transfrm[j]. Chins Jurnal f Scintific Instrumnt, 22(4), pp , Apr 22. 9