HHT Based Analysis of Non Stationary Signals and Metal Structures

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1 Iteratioal Joural o Recet ad Iovatio Treds i Computig ad Commuicatio ISSN: Volume: Issue: HHT Based Aalysis of No Statioary Sigals ad Metal Structures Yashwath N Assistat Professor Departmet of Electroics ad Commuicatio Rajeev Istitute of Techology Hassa, Idia yashwathriths@gmail.com Harish M PhD Scholar Departmet of Electrical Egieerig Uiversity of Wyomig, Wyomig, USA harishmuralidhar@gmail.com Sujatha B R Associate Professor Departmet of Electroics ad Commuicatio Malad College of Egieerig Hassa, Idia brs@mcehassa.ac.i Abstract Hilbert-Huag Trasform (HHT) is a iovative data-processig techique for aalyzig o statioary ad oliear sigals. The aalysis of these sigals is to trasform the time-domai data to frequecy versus time data istead of the amplitude versus frequecy.this paper ivestigates techiques to apply HHT for locatig the istataeous frequecy i a sigal. This sigal processig techique helps i idetifyig several frequecy compoets which are the idicators of the problems preset i the system uder test. This ca be applied for aalyzig aircrafts body structure, biomedical sigals ad seismic sigals. Moitorig of civil structures such as bridges ad buildigs is critical for logterm operatioal cost ad safety of agig structures. Applyig HHT to these sigals obtaied from the various sesors placed i the viciity of evet or etity, it is possible to idetify the problems. Keywords Hilbert-Huag Trasform, Itrisic Mode Fuctios, Hilbert Trasform, Empirical Mode Decompositio ***** I. INTRODUCTION Hilbert-Huag Trasform (HHT) [1] is a ostatioary sigal processig method preseted by Professor Norde E. Huag from the Uited States i 1998, ad improved i The mai iovatios of this method are itrisic mode fuctio (IMF) ad empirical mode decompositio (EMD). Through EMD, the sigal is decomposed ito several IMFs (geerally for a limited umber), to each of which Hilbert Trasform is applied to get meaigful istataeous frequecy; the frequecy gives the exact expressio of time-frequecy spectrum i ostatioary sigals. Sigal processig techiques employig HHT fids may applicatios. Oe such applicatio is detectio of huma activity behid barriers such as walls ad debris whe lookig for earthquake survivors. The preferred sesors are radars sice they have the ability to peetrate deep through dielectric barriers. These sesors are used to recogize sigs of life by recogizig micro-doppler sigatures of huma activity, such as arm swigig, breathig etc., Such movemets iduce differet types of Doppler spectra depedig o the maer i which the limbs ad other body parts move, which ca be aalyzed by several well-kow time-frequecy approaches, icludig the empirical mode decompositio ad Hilbert-Huag Trasform. HHT applied to ay oliear sigal or ostatioary sigal obtaied from the sesors will yield a good time-frequecy plot so as to detect the huma activity. As majority of real-world sigals are ostatioary, Fourier aalysis provides usatisfyig results sice the frequecy cotet chages with time. Determiatio of the frequecy cotet of such sigals dictates to perform a aalysis across a spa of time (basis fuctio), ad the move to aother time positio. The major drawback of most trasforms is that the basisfuctios are fixed, ad do ot ecessarily match the varyig ature of sigals. The et effect of these operatios is to trasform the time-domai data to frequecy versus time data istead of amplitude versus frequecy variatio that the FFT provides. The Hilbert trasform is a well-kow method for computig the istataeous frequecy of ay sigal, uder the assumptio that oly oe frequecy will be preset at ay time. Such method caot directly be applied to a complex sigal, cotaiig several frequecies at ay give time. EMD techique proposes to decompose a multi-modal sigal ito a sum of moo-cotributio fuctios called Itrisic Mode Fuctios (IMFs) [3]. The EMD is a iterative method that picks out the highest frequecy compoets that remais i the sigal at each iteratio. The frequecy aalysis based o Hilbert-Huag Trasform has basically three steps of processig. To begi with the sigal is decomposedito a umber of IMFs usig EMD. I practice, it ca be demostrated that this decompositio process is complete, adaptive ad local.the secod step applies the Hilbert trasform to each IMF to compute the istataeous frequecy at each time. The third ad the fial step computes the eergy cotet of the cosidered frequecy bad at ay give time. II. HILBERT HUANG TRANSFORM The empirical mode decompositio is the core of HHT which whe combied with the Hilbert Trasform [1] completes the HHT process. This has bee developed from the simple assumptio that every sigal cosists of differet simple itrisic idepedet modes of oscillatios. Each liear or o-liear mode will have the same umber of IJRITCC October 14, 337

Magitude Magitude Iteratioal Joural o Recet ad Iovatio Treds i Computig ad Commuicatio ISSN: 31-8169 Volume: Issue: 1 337 34 extrema ad zero crossigs ad oly oe extremum betwee oticed that two

2 Magitude Magitude Iteratioal Joural o Recet ad Iovatio Treds i Computig ad Commuicatio ISSN: Volume: Issue: extrema ad zero crossigs ad oly oe extremum betwee oticed that two frequecy compoets 4Hz ad 15 Hz successivezero-crossigs. I this way, each sigal could be exists but its presece is ot idicated i the time domai. decomposed ito a umber of itrisic mode fuctios Now applyig STFT with a Gaussia widow for (IMFs). I cotrast to the Fourier spectral aalysis i which the time-frequecy localizatio, its spectrogram plot is a series of sie ad cosie fuctios havig varyig show i this figure for differet widow legths of 3, 64, amplitudes are used to represet each costituet frequecy 18 ad 56. It is observed that the resolutio i the compoets i the sigal, the HHT techique is based o the frequecy domai icreases as the width of the widow istataeous frequecy calculatio that results from the icreases ad the time resolutio decreases. Hilbert trasform of the sigal.the Hilbert trasform H[x Equatio () ca be rewritte i a polar coordiate system as (t)] for ay sigal x (t) is defied as H[x (t)] = y (t) = 1 π P ( x u + ) du (1) t u Z (t) = a(t) e iθ (t) (3) where P idicates the pricipal value of the sigular itegral.the Hilbert Trasform ca also be iterpreted as a atural π/phase shifter, which cosists of passig x(t) through a system that leaves the magitude uchaged, but chages thephase of all frequecy compoets by π/. With this defiitio, y(t) forms thecomplex cojugate of x(t) ad the aalytical sigalz(t) is defied as z (t) = x(t) + i y(t) () As a example,the time ad frequecy spectrum of asigal, is as show i Figure 1. From the frequecy spectrum, it is Wherea(t) = x(t) + y(t) is the amplitudead θ(t) = arc ta ( y(t) ) is the phase (4) x(t) Rewritig i the polar co-ordiate form x(t) = R(z(t)) = R(a(t)e j w(t)dt ) (5) The Hilbert ad the istataeous frequecy is calculated from the above equatios The sigal decompositio is based o the followig assumptios which eed to be satisfied by the EMD process [14] usig the algorithm show i Figure. 15 Short time fourier trasform with widow legth 3 15 Short time fourier trasform with widow legth 64 1 Iput sigal 14 FFT plot Short time fourier trasform with widow legth Short time fourier trasform with widow legth Figure1. ad domai represetatio ad Spectrogram for differet widow legths Figure. Flow chart of the EMD process IJRITCC October 14, 338

3 Iteratioal Joural o Recet ad Iovatio Treds i Computig ad Commuicatio ISSN: Volume: Issue: III. EMD PROCEDURE Repeat these siftig procedure k times, util h 1k is a IMF, that is The EMD process is also kow as the shiftig process. I geeral, most of the data are ot aturally IMFs ad the Hilbert trasform caot provide the full descriptio of the frequecy cotet if the data ivolves more tha oe oscillatory mode at a give time. Hece, there is a eed to fid a way to decompose the data ito a set of idepedet IMF compoets. Huag itroduced a method to decompose a complicated data ito IMF compoets with meaigful istataeous frequecies. This ew method is ituitive, direct, a posterior ad adaptive. The decompositio is based o three assumptios: h 1k = h 1(k-1) m 1k (1) Whe the stop criterio is met, the IMF is defied as c 1 = h 1k (11) After the IMF c 1 is foud, defie the residue r 1 as the differece of this IMF ad the iputsigal r 1 = x (t) c 1 (1) a) The sigal has at least two extrema, oe maximum ad oe miimum. b) The characteristic time scale is defied by the time lapse betwee the extrema. c) If the data were totally devoid of extrema but cotaied oly iflectio poits, the it ca be differetiated oce or more times to reveal the extrema. To fid the IMFs of a sigal the siftig process cosists of several steps ad are described usig a arbitrary sigal deoted x(t). (1) Fid the positios ad amplitudes of all local maxima ad miima of the iput sigal[]. () Create the upper evelope by splie iterpolatio of the local maxima ad the lower evelope by splie iterpolatio of the local miima, deoted by e max (t) ad e mi (t). (3) Ata time istat t, calculate the mea of the upper evelope ad the lower evelope m 1. m 1 = (e max (t) + e mi (t)) / (6) (4) Subtract the evelope mea sigal from the iput sigal. h 1 (t) = x (t) m 1 (7) This is oe iteratio of the siftig process. The ext step is to check if the sigal h 1 (t) is a IMF or ot. I the origial work of Huag, the siftig process stops whe the differece betwee two cosecutive siftigs is smaller tha a selected threshold stadard deviatio (SD) [7], defied by SD = T [ (h 1 k 1 (t) h 1k (t) t= ] (8) h 1(k 1) (t) (5)If h 1 (t) is ot a IMF, iterate by repeatig the process from step (1) with the resultig sigal from step (4). Therefore i the secod siftig process, h 1 (t) is treated as the data resultig i, h 11 = h 1 m 11 (9) (6) The ext IMF is foud begiig from step(1), with the residue as the iput sigal. Steps (1) to (6) ca be repeated for all the subsequet residues r j ad the result is r 1 c = r,r -c 3 =r 3,, r -1 c = r (13) The EMD is completed whe the residue, ideally, does ot cotai ay extrema poits. This meas that it is either a costat or a mootoic fuctio. The sigal ca be expressed as the sum of IMFs ad the last residue x(t) = i=1 c i + r (14) The extracted IMFs are symmetric ad have a uique local frequecy ad differet IMFs do ot exhibit the same frequecy at the same time. IV. HHT ANALYSIS PROCEDURE HHT geeratio is doe based o equatios (1) through (4) ad (14), with equatio (5) modified as x(t) = R ( a i (t)e j w i(t)dt i=1 ) (15) i whicha i (t) = [c i t ] + H [c i (t)] adw i (t) = d dt (ta-1 H[c i t ] c i (t) ). The term r i (14) is ot icluded i (16) as it is a mootoicfuctio [5] ad does ot idicate the frequecy cotet of the sigal. Comparig (15) with the Fourier-based represetatio of a sigal x (t) give by x(t) = R ( a i e j Ω it i=1 ) (16) where both Ai ad Ωi are costat, it becomes evidet that the EMD process eables flexible represetatio of a dyamic sigal by revealig its time-depedet amplitude ad the characteristic frequecy compoets at various time istaces. The sigal is thus represeted by a time-frequecy IJRITCC October 14, 339

4 Iteratioal Joural o Recet ad Iovatio Treds i Computig ad Commuicatio ISSN: Volume: Issue: distributio. The uderlyig HHT of the sigal is mathematically defied as, Case Study1: Gear fault aalysis i vehicles HHT (t,w)= i=1 HHT i t, w = i=1 a i t, w i (17) wherehht i (t, w) represets the time-frequecy distributio obtaied from the i th IMF of the sigal. V. EXPERIMENTAL VERIFICATION HHT aalysis is doe for two case studies. The first study examies the ratioale of HHT for aalyzig the gear fault aalysis i vehicles. The secod is metal plate data aalysis which studies the damage which occurs o the plate whe a bullet is shot from a air gu from a small distace. The basic cocept of HHT is first preseted where the empirical mode decompositio must be applied o the sigal usig a siftig process to obtai itrisic mode fuctios before the Hilbert spectral aalysis ca be meaigfully performed. The wavelet trasform was also compared with HHT i the previous work doe by us [14] which gave results similar to STFT. HHT is studied for two differet cases. To study the performace of HHT for the gear fault aalysis, the test sigal is geerated cosiderig the meshig frequecies at 3Hz ad 5Hz, with the Gaussia radom oise added. The presece of two impulses at.5 ad.1833 secods i the system are added to represet the fault as show i Figure 3. Thetwo frequecies that are preset for the etire duratio, with the impulse at the specified time are reflected i the HHT plot as show i Figure 4. Case Study: Metal plate data aalysis This case study illustrates the feasibility of the HHT as a sigalprocessig tool for locatig aaomaly, i the form of a crack, delamiatio, stiffess loss or boudary i metal plate, based o physically acquired propagatig wave sigals. Thisca be exteded to study the exteral body surface of aircrafts based o simple wave propagatiococepts usig flight times ad speed ad the correspodig frequecy chages. Figure 3. Iput sigal s(t) with impulse for gear defect aalysis Figure 4. HHT plot for the sigal s(t) with a impulse IJRITCC October 14, 34

5 Amplitude Amplitude Amplitude Amplitude Freq Iteratioal Joural o Recet ad Iovatio Treds i Computig ad Commuicatio ISSN: Volume: Issue: The data is obtaied from a metal plate fitted i the aalyzig the - plot. Figures 5.8 provides frot of a aircraft. For experimetal purpose, the plate is the spectrogram, EMD plots ad HHT plots for udamaged exposed to air gu shots at differet speeds ad the data ad damaged plate. It is see that as the extet of damage correspodig to the damage doe is collected. With respect icreases, the HHT plot shows the icreased umber of to the speed, the damage o the plate i percetage is frequecy compoets. By appropriate calibratio the extet measured.for three valuesof 5%, % ad 3%, the HHT of the damage is quatified. results are plotted to measure the extet of damage by Udamaged plate: 6 x 1-4 Udamaged plate iput 1 HHT result Figure5. Iput sigal, EMD plot ad HHT plot for the udamaged plate Damaged plate(5%,%,3%): 6 x 1-4 Plate with 5% damage 4 x 1-4 Plate with % damage 4 x 1-4 Plate with 3% damage Figure 6. Iput with 5%, % ad 3% damage Figure 7. EMD plot for 5%, % ad 3% damage IJRITCC October 14, 341

Freq Freq Freq Iteratioal Joural o Recet ad Iovatio Treds i Computig ad Commuicatio ISSN: 31-8169 Volume: Issue: 1 337 34 1 9 8 7 6 5 4 HHT result HHT result 1 HHT result 1 9 8 7 6 5 4 9 8 7 6 5 4 3

usig its multi-resolutio properties, cocateated sie sigals ca be decomposed to get the time-frequecy characteristics of sigals.

, I additio, usig EMD multi-scale filterig features, we ca effectively remove the oise, ad retai sufficiet characteristics of the sigal.

6 Freq Freq Freq Iteratioal Joural o Recet ad Iovatio Treds i Computig ad Commuicatio ISSN: Volume: Issue: HHT result HHT result 1 HHT result Figure 8. HHT plot for 5%, % ad 3% damage VI. CONCLUSION HHT is a useful tool to get timefrequecyrepresetatio.usig its multi-resolutio properties, cocateated sie sigals ca be decomposed to get the time-frequecy characteristics of sigals.it ca also be applied for health moitorig of civil structures such as bridges, biomedical sigal such as EEG, etc., I additio, usig EMD multi-scale filterig features, we ca effectively remove the oise, ad retai sufficiet characteristics of the sigal. The future work will iclude sesor etworks for autoomous structural health moitorig that addresses the syergetic issues of itegratig a sesor etwork with avibratio-based SHM method. [11] Quek S, Wag Q, Zhag, L, OgK.H, Practical issues ithe detectio of damage i beams usigwavelets, Smart Materials ad Structures1,pp ,1. [1] Okafor, A., Dutta, A., Structural damage detectio ibeams by wavelettrasforms, Smart Materials adstructures9, pp ,. [13] Soh. H, Farrar, C.Hemez, F.Shuk, D. Stiemates, D,ad Nadler, B., AReview of Structural Health MoitorigLiterature: , Los AlamosNatioal LaboratoryReport, 3 [14] Yashwath N, B R Sujatha, represetatioofo-statioary real world sigals,ieee Iteratioal coferece o Impact of E Techology o US, PESIT, Bagalore, 1 11 Ja 14, Publishers TMHEducatio, pp [15] Li S, Yag J, ad Zhou L, Damage idetificatio of abechmark buildigfor structural health moitorig,smartmaterials ad Structures14, pp ,5. REFERENCES [1] RuqiagYa,Robert X. Gao, Hilbert Huag Trasform Based Vibratio Sigal Aalysis for Machie Health Moitorig IEEE Trasactios O Istrumetatio Ad Measuremet, Vol. 55, pp. 3-39, Dec 6. [] N. E. Huag, Z. She, S. R. Log, M. L.Wu, H. H. Shih, Q.Zheg, N. C.Ye, C. C. Tug, ad H. H. Liu, "The empiricalmodedecompositio ad Hilbert spectrum for oliear ado-statioary time series aalysis", Procedures of RoyalSociety of 998. [3] R. Ya ad R. Gao, Complexity as a measure for machiehealthevaluatio, IEEE Tras. Istrum. Meas.vol. 53, o.4, pp ,Aug. 4. [4] K. Mori, N. Kasashima, T. Yoshioka, ad Y. Ueo, Predictio of spallig o a ball bearig by applyig thediscrete wavelet trasform to vibratio sigals, Elsevier,Wear,vol. 195, pp , [5] R. Gao ad R. Ya, Nostatioary sigal processig forbearig health moitorig, It. J. Mauf. Res., vol. 1, o. 1, pp. 18 4, 6. [6] J. N. Yag, Y. Lei, S. Li, S. ad N. E. Huag, HilbertHuag based approach for structural damage detectio, J. Eg. Mechaics, vol. 13, o. 1, pp , 4. [7] Ruqiag Ya, Robert X. Gao Trasiet Sigal AalysisBased o Hilbert-Huag Trasform IMTC 5, Istrumetatio ad MeasuremetTechology Coferece Ottawa, Caada, pp May 5. [8] Moore, M., Phares, B., Graybeal B., RoladerD.Washer, G. Reliability of visual ispectioforhighway bridges Volume I: Fial Report FHWARD1-,1. [9] Jag, J-H, Yeo.I,Shi.S, Chag S-P Experimetalivestigatio of system idetificatio baseddamageassessmet o structures. Joural of StructuralEgieerig, 18(5), pp ,. [1] Su, Z. Chag, C.C. Structural damageassessmet based o wavelet packet trasform JouralofStructuralEgieerig, 18(1), pp ,. Authors Profile Yashwath N obtaied his B.E.degree from KIT, Tiptur i 1 ad M.Tech from SET, Jai Uiversity i 1. He is iterested i the field of Sigal Processig, VLSI Implemetatio ad Wireless Sesor Networks. Harish M obtaied his B.E.degree from MCE, Hassa i 8, M.S. from Uiv. of Wyomig i 1 ad is curretly pursuig PhD.His areas of iterest are Statistical Sigal Processig ad Wireless AdHoc Networks. Dr. B R Sujathaobtaied herb.e.degree from NIE, Mysore i 1983, M.E. from IISc, Bagalore i 199 ad Ph.D form VTU, Belgaum. Her areas of iterest are Computer Networks, Wireless AdHoc Networks ad Wireless Sesor Networks. IJRITCC October 14, 34

Comparison of Methods for Different Timefrequency Analysis of Vibration Signal

Comparison of Methods for Different Timefrequency Analysis of Vibration Signal 68 JOURNAL OF SOFTWARE, VOL. 7, NO., JANUARY 202 Compariso of Methods for Differet Timefrequecy Aalysis of Vibratio Sigal Lig Xiag Mechaical Egieerig Departmet, North Chia Electric Power Uiversity, Baodig