Fast Wavelet Transform Algorithm

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    ICSP'O4 ProceedingsFAST WAVELET TRANSFORM ALGORITHM AND T APPLTCATION

    IN HEALTH MONITORING SYSTEMWANG Bo, TAKATSUBO Junji, AKIMUNE Yoshio, TSUDA Hiroshi

    (National Institute of Advanced Industrial S cience and Technology,Tsukuba Central 2, Umezono 1-1-1, Tsukuba 305-8568, Japan.)

    Email: bo-aaiw@"t 40 D

    Abst rac t : Th e wavelet transform, as an h q " t tool of signdprocessing,has heen applied to many fields. In this paper, a fastwavelet transform (FWT) algorithm and a source locationalgorithm based OIL the FWT are presented. Based on thesealgoritluns, a health monitoring system for aniso@opicmaterials t ruches was developed. The algorithms can improve thepcis ion and s p a 1 of s o m e locat ion . The heal th monitorings y s t e m can locate The unpact damage and identify the hnpactload historywtha good precision in real-time.Key words: Fe;t usvelet tramform; Impact damageidentification,shuchlrd health monitaring.

    I . IntroductionIn recent years, the structural health monitoring

    (SHM) ha s become an important research topic. SHMcan detect the damage at early stage, inmase thes a f e t y of structures, decrease the costs of periodicalinspections and enable us to evalua te different inannersof repair. The most important SHM technique is thedamage identification. Carbon fiber reinforced plastics(CFRP), s the representative of advanced comp osites,are widely used in the aerospace industry bmause oftheir specific strength and stflness. However, CFRPstructures are susceptible to impact during service,which causes h e m a l damages (e.g. matrix crack,delamination, and fiber breakage). Therefore theimpact damage identification is veqy important fo rCFRP structure!;. The impact damage identificationparameters are mostly the location and size of damage,and the location is the parameter that should be firstidentified. Tlie impact damage can be located by using

    244-7803-84O6-7/04/$20.000 2004 IEEE.

    the arrival time differences of So - m o d e Lamb wavesignals, and the arrival time usually is determined bythreshold crossing technique [ ' I . Though the thresholdtechnique spends a shorter computation time, thelocation precision is easily afected by the noise inresponse signal and the dispersion of wave. In order toovercome this problem, we presented a fast wavelettransform (FWT) algorithm and a source locationalgorithm, and have applied them to our developedhealth mo nitoring system.

    In this paper, authors will present the FWTalgorithm and source location algorithm, and alsointroduce the developed health monitoring system.

    2. Fast Wavelet Transform AlgorithmThe continuous wavelet transform (WT) of an

    arbitrary signal d t ) s defined as [21) s ( t ) d t (1)1 * t - bw ( a , b )= xJ-m Y (- (1

    al

    where NUJ) nd v l t ) are the wavelet transformcoefficient and mother wavelet, respectively. w * ( I isthe complex conjugate of IU ( t ) . a is a positivenumber called scale, and b the time-shift variabie(L? and b are also known as the dilation and th etranslation parameters, respectively). The small LZcorresponds to a high frequency and vice versa, whileb simply shifts th e wavelet with respect to timewithout altering the frequency content. Let the SCf)Y (f and W ( a ,f 1 denote the Fourier transformsof s ( t ) , y ( t ) a n d w (a , t ) , respect ively, and theFou rier transform pair are defined as

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    S(f)e'2"f'df, (3)Using the Fourier transform formulas 13], the Eq.(l) canbe rewritten as follows when a is fixed.

    J -w

    = rm ( Q . ).eJZxpdf ,J --OD (4)where

    Usually, the Gabor function is used as the motherwavelet y / ( t) because it provides the best resolution intime as well as in frequency domain. The Gaborfunction is espre ssed as 1

    and its Fourier transform is

    where y is a positive constant given asy =~ ( 2 / I n 2 ) ~ ' ~5 .3364 , which was chosen to

    nearly satisfy the so-called adniissibility condition. tlief , is the center frequency which is usually se%asf, = 1 / ( & A t , ) , where A t , is the samplinginterval of signal s( t ) The Gabor function (6) may beconsidered as a complex sinusoidal function with aGaussian window centered at f = O and its Fouriertransfonn (7) centered at f =f,. Substituting (7)into (9, e obtain

    Since WT is a time-frequency analysis essentially, itcan be rewritten as foIlowing equations using thefrequency f,(=f,/a) ha t is relative to the scale a ,

    where the H(f,,f)can be considered as the systemtransfer function of wavelet transform system (WTS),and th e wavelet transform coefficient w ( f a , t ) as theoutput response of the WTS corresponding to an inputsignal s t ) .This relationship is shown in Fig. 1.

    Fig. 1. Wavelet transform systemIn numerical computation, H ( f , , f ) and w( f , , f )

    are 2D m x n matrices, where nz and n are thediscrete numbers of fn and f espectively.'In orderto obtain the wavelet transform coefficient w ( f n , t )of a signal s ( t ) at fast speed, the followingalgorithm was adopte d:1) Calculating the H(f,,flrom Eq.(9a) ;2) Calculating the Fourier transform s(f) f the

    signal S ( ) using the FFT algorithm;3) Calculating the W ( f , ,f) rom Eq. 9b);4) Calculating the wavelet transform coefficient

    w(f, , t ) using the IFFT (Inverse FFT) algorithm.We name this algorithm the fast wavelet transfonn

    (FWT). The algorithm flow chart is shown in Fig.2.The primary advantage of FWT is extreme saving incomputation time than the direct wavelet transform asshown in equation (1). So, it can be used to th e realtime processing system, such as SHM ystem.

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    function of propagation direction and cart be calculatedor measured beforehand. r is the radius of a sensor.

    and taiare the theoretical and measured arrivaltime from the impact source to the i-th sensor. Ifta i ( i =1,2,3,4) are known, the impact positioncoordinate ( 5 , ~ )an be found from the Eq.(lO) bythe global optimization algorithms 31 .

    Le t s ( t ) (Fig.4) indicate the impact response signaldetected by a sensor, then the arrivaI time t , fromsourc e to the sensor can be determined in the followingsteps:1) Calculating the wavelet transform. coefficient

    w ( k , ! t ) of the signal s(f),using.the FWT withvarying fa in a certain frequency rangedepending on the structure property (the contourplot of w (fa, ) s show n in the Fig.5);

    2) Selecting the special freque ncy f, which usuallycorrelates to the antisymmetrical Lamb wave4 - m o d e ) from w ( f , , l ) , and fmding thearrival time t , relative to the first peak ofwavelet coefficient w(I,t ) (shown in Fig.6).

    Sensor 1 . Sensor4

    . . . , . A , ( L 7 f t , -

    '

    I . Fig.2.Fast wavelet transform algorithm3. Source Location Algorithm Based on FWT

    Iii 2-dimension sourc e location, at least three sensorsare required. In order to improve the precision, foursensors are used. For isotropic materials, it is eEisy tolocate the' impact position..But in anisotropic ma terials,the source location becomes "ore difficult since thevelocity is a function of propagation direction in theplate. Let + ( X i : , Y i ) (i=1,2,3,4) and f ' (X ,Y) denotethe position coordinates of each sensor and impactsource respectively, then the following equations canbe derived from the relations shown in Fig. 3.

    Sensor 2 Sensor 3 0. 3 I I

    where v,(B,) is the group velocity of Lamb wavefrom the impact s;ource to the i-th sensor, which is a

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    I , !-0.3 I0 1000 2000 3000 4000Time (us)Fig.4. Impact response signal

    p 2010

    Fig.5. ontour plot of th e W T

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    0 1000 4 2000 3000 4000ta Time (us)

    Fig.6. WT at selected frequency4. Health Monitoring System

    The developed health monitoring system consists ofthe hardware and the software, whose C onfiguration iss h o w n in Fig: 7. T he hardware cons i s t s of a[0 /90] , C F P laminated plate ( 9 6 0 ~ 9 6 0 ~ 4 . 8m3 )with four surface-mounted piezoelectric sensors (FujiCeramics Inc. C-64, thickness: 200pm, diameter:j n i n i ) , a d ig i t a l o sc i l lo scope (T ek t ron ix Inc.TDS3014B), local network, a wireless LAN card and anotebook computer. The software, developed withLabVIEW and C language, includes th e waveformrecording, preprocessing, a source location algorithmbased on FWT, and an impact load identificationalgorithm based on inipact response inverse analysis(discussed in [ 5 ] ) . The piezoelectric sensors can detectthe impact response signals without any amplification.Th e source location algorithm can reduce the influenc eof noise in the impact response signal. When the CFRPplate is impacted, the oscilloscope promptly recordsthe impact response signals detected by mounted

    5r,Digital Oscilloscl

    Fig.7. Configuration of health monitoring system

    piezoelectric sensors, and the computer simultaneouslyreceives the signals via local network and wirelessLAN card from the oscilloscope. Then, the systemlocates the impact position and identifies the impactload history. All identifications can be performed inreal-time.

    5. ConclusionsIn order to improve the precision and speed of

    source location, a FWT algorithm and a source locationalgorithm were proposed. Based on these algorithms, ahealth monitoring system or anisotropic materialstructure was developed. Using this system, the impactposition and the impact toad history can be identified.The 1ocation.error is about 1.5%, and the load historyerror is about 4%. All identfications can b e completedwithin 0.2 second.

    AcknowledgementTlie research is supported by Japan Science and

    Technology Agency (JST).

    References1. Koo J. H., Kim B. N., Bnoki M. and Kishi T.,

    Aco ustic emission signal analysis in C/C composites,J. Acoustic Emission, Vol.15, p89-94, 1997.

    I

    2. Chui CK., n ntroduction to wavelets. San Diego,CA: Academic Press, 1992.

    3. William H. Press,et al., Numerical recipes in C, theart of scient$c compuring, Cambridge UniversityPress, 1992.

    4. Suzuki H. , et al., rcWmeiet transfirm o acousticemission signals, J. . Acoustic Emission, 14(2),pp69-84, 1996.

    5 . Wang B., Takatsubo J. and Akimune Y., Thedevelopment of remote impact damage identijkutionsystem. Proc. of th e 1 International Workshop onAdvanced Sensors, Structural Health Monitoring, andSmart Structures, apan, Nov.10-11,2003 (in press).

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