THE HILBERT-HUANG TRANSFORM AND THE FOURIER TRANSFORM IN THE ANALYSIS OF PAVEMENT PROFILES

Transcription

1 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie THE HILBERT-HUANG TRANSFORM AND THE FOURIER TRANSFORM IN THE ANALYSIS OF PAVEMENT PROFILES Albert Y. Ayeu-Prah* Graduate Studet Departmet of Civil ad Evirometal Egieerig Uiversity of Delaware Newark, DE 976 USA Telephoe: (30) Fax: (30) Stephe A. Mesah Graduate Studet Departmet of Civil ad Evirometal Egieerig Uiversity of Delaware Newark, DE 976 USA Telephoe: (30) Fax: (30) Nii O. Attoh-Okie, Ph.D., P.E. Associate Professor Departmet of Civil ad Evirometal Egieerig Uiversity of Delaware Newark DE 976 USA Telephoe: (30) Fax: (30) Submitted: July 005 Word Cout: 356 *Correspodig author.

2 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie ABSTRACT The preset paper employs the Hilbert-Huag trasform (HHT) ad the Fourier trasform to aalyze the road surface profile of two flexible pavemets i varyig coditios. The cetral idea of HHT is the empirical mode decompositio (EMD), which is the decomposed ito a set of itrisic mode fuctios (IMFs). The Hilbert trasforms ca the be applied to the IMFs. The stregth of HHT is the ability to process o-statioary ad o-liear data. Ulike the Fourier trasform, which trasforms iformatio from the time domai ito the frequecy domai, HHT does ot move from the time domai ito the frequecy domai Iformatio is maitaied i the time domai. The paper tries to idetify which of the two trasforms is better able to do both quatitative ad qualitative idetificatio of the profile type from field data. I performig the aalyses the ature ad behavior of road profiles as idicated by the literature are take ito accout, that road profiles are o-statioary ad are iheretly o-gaussia.

3 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie INTRODUCTION Pavemet surface roughess determies ride quality ad it is a good idicator of the quality of costructio. Roughess Idex is used for the assessmet of the acceptability of ewly costructed pavemet as well as i pavemet maagemet decisio process for maiteace schedulig. Relatively less rough roads are desirable sice the pavemet performs better. It also ehaces vehicle performace. Road roughess would be the surface profile udulatios as opposed to the surface roughess that determies skid resistace. Two sets of digitized flexible pavemet profile data are aalyzed with the Fourier trasform ad with the Hilbert-Huag trasform (HHT). The data is i the form of spatial acceleratio, which is the double derivative of elevatio. Accordig to the literature, road profiles are fudametally o-gaussia ad ostatioary (Bruscella et al. 999). Compared to the HHT the Fourier Trasform has its limitatios. Fourier aalysis requires that the sigal must be strictly periodic or statioary otherwise the resultig Fourier spectrum will make little physical sese (Peg 005). A attempt is made to determie which of the two aalysis techiques produces better road profile descriptios. THE FOURIER TRANSFORM The Fourier series is made up of sies ad cosies; the Fourier trasform is a geeralizatio of the Fourier series, ad is made up of expoetials ad complex umbers. The Fourier aalysis has wide applicatios i mathematics ad egieerig, used i modelig diverse physical pheomea. Examples of some physical pheomea iclude heat trasfer, wave propagatio, circuit aalysis, electroic circuit aalysis, ad vibratios. Iterestig to ote is the Fourier kerel, exp[πiwt], which is a solutio to a th-order liear differetial equatio which, i tur, is used to model various physical pheomea; it is oe reaso why Fourier aalysis has such wide applicatios (Weaver, 989). O the iterval [-π, π], ay arbitrary fuctio f(t) which is periodic ad sigle-valued could be represeted by the trigoometric series f ( t) = a0 + a cos t + b si t () = = For a periodic iterval, T, f ( t) = a0 + a cos ω t + b si ωt () where = = ω = π T t+ a = T t T t+ T f ( t)cos ωtdt b = f ( t)si ωtdt T t T = period of f(t). For a sigal or fuctio, f(t), the Fourier trasform is defied as iωt = f ( t e dt F( ω ) ), (3) ad the iverse Fourier trasform to recover the origial sigal or fuctio is defied as i = ω ω t f ( t) F( ) e dω (4) π These two Fourier trasform equatios are called the Fourier trasform pair. Evidetly see i the two equatios is the complex umber, i, after trasformatio of the origial fuctio. The origial fuctio is i the time domai, t, which is trasformed ito the frequecy domai, ω, after Fourier trasformatio. The resultig trasformed sigal gives a idicatio of the frequecy compoets that cotribute to the origial sigal whe a

4 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie spectral plot of amplitude versus frequecy is costructed. Amplitude is the absolute value of the complex umber i the Fourier (frequecy) domai, ad is usually the vertical axis. f(t) ca be cotiuous or discrete. The precedig defiitio for the Fourier trasform itrisically assumes f(t) to be cotiuous. We ca speak of a discrete f(t) over a certai fiite iterval, the we talk about the discrete Fourier trasform, DFT, of f(t), defied as F = T T T f ( t) e iωt ad the iverse discrete Fourier trasform, as = dt, (5) iωt f ( t) = F e, (6) where is a idex. Therefore, DFT is a digital tool, used to aalyze the frequecy compoets of discrete sigals while Fourier trasform is a aalog tool, used to aalyze the frequecy compoets of cotiuous sigals. A fast Fourier trasform, FFT, is a algorithm that efficietly implemets the DFT o a computer. Usually, power spectral desity (PSD) plots are preferred to Fourier trasform plots; PSD is equivalet to the square of the amplitude of the Fourier trasform, which has a cotiuous depedece o absolute frequecy (or i this case, waveumber (cycles/m), sice pavemet data uses waveumber istead of frequecy i cycles/sec). If P deotes PSD, the P = k A (7) where A = amplitude of Fourier trasform k = appropriate scalig costat. A property of the PSD is give as follows, 0 P ( f ) df = σ where σ = variace of origial sigal. I other words, the total variace of the sigal is recovered upo itegratig plotted spectral values over the frequecy rage. THE HILBERT-HUANG TRANSFORM The HHT cosists of two parts: the empirical mode decompositio (EMD) ad the Hilbert Spectral Aalysis (HSA). The EMD geerally separates oliear, o-statioary data ito locally o-overlappig time scale compoets. Accordig to Peg et al (005) the sigal decompositio process will break dow the sigal ito a set of complete ad almost orthogoal compoets called the IMF, which is almost moo-compoet. A IMF is a fuctio that satisfies two of the followig coditios (Peg et al. 005): The umber of extrema ad the umber of zero crossigs must either equal or differ at most by oe i whole data sets; the mea value of the evelop defied by the local maxima ad the evelop defied by the local miima is zero at every poit. The Hilbert trasform of the IMFs will yield a full eergy-frequecy-time distributio of the sigal kow as the Hilbert-Huag spectrum. The HHT has the followig advatages that make it desirable for sigal aalysis: The most computatioally itesive step is the EMD operatio, which does t ivolve covolutio ad other time-cosumig operatios ad makes HHT ideal for sigals of large size. The Hilbert-Huag spectrum does ot ivolve the cocept of the frequecy resolutio but the istataeous frequecy. O the other had, there are some drawbacks i the applicatio of HHT. The EMD will geerate udesired low amplitude IMFs at the low frequecy regio ad raise some udesired frequecy compoets. The first IMF (8)

5 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie 3 may cover a wide frequecy rage at the high-frequecy regio ad therefore caot satisfy the moocompoet defiitio well. The EMD operatio ofte caot separate some low-eergy compoets from the aalysis sigal therefore those compoets may ot appear i the frequecy-time plae. Ay fuctio ca be decomposed as follows:. Idetify all the local extrema, ad the coect all the local maxima by a cubic splie as the upper evelope.. Repeat the procedure for the miima to produce the lower evelope. The upper ad lower evelopes should cover all the data. If the mea is desigated as m ad the differece betwee the data ad m is the first compoet h, the x ( t) m = h (9) h is a IMF. The mea m is give by the sum of local extrema coected by the cubic splie: L + U m = (0) U is the local maxima ad L is the local miima. The IMF ca have both amplitude ad frequecy modulatios. I may cases there are overshoots ad udershoots after the first roud of processig, ad this is termed siftig. The siftig process serves two purposes (Huag et al. 00): It elimiates ridig sigals/profile, makig the sigal or profile more symmetric. The siftig process has to be repeated may times. I the secod siftig process, h is treated as the data ad as the first compoet. h is almost a IMF, except some error might be itroduced by the splie curve fittig process. To treat h as ew set of data, a ew mea is computed. The, h m = h () After repeatig the siftig process up to k times, h k becomes the IMF, that is h ( k ) m k = h k () Let h k = c, the first IMF from the data. c should cotai the fiest scale or the shortest period compoet of the data/profile. Now c ca be separated from the rest of the data by x ( t) c = r (3) Sice r is the residue, it cotais iformatio o loger period compoet; it is ow treated as the ew data ad subjected to the same siftig process. The procedure is repeated for all subsequet r j s ad the result is r c = r r c = r ;... (4) c is ow the secod IMF of the data. The siftig process ca be stopped by ay of the followig predetermiig criteria: Either whe the compoet c or the residue r becomes so small that it has a predetermied value of substatial cosequece, or Whe the residue r becomes a mootoic fuctio from which o IMF ca be extracted. Summig equatios (3) ad (4) yields the followig equatio: x( t) = j= c j + r c j is the jth IMF, = umber of sifted IMF. r ca be iterpreted as a tred i the sigal/profile. The c i have zero mea. Due to the iterative process oe of the sifted IMFs derived is i closed aalytical form (Schlurma, 00). The IMF ca be liear or oliear based o the characteristics of the data. The IMFs are almost orthogoal ad form a complete basis. Their sum equals the origial data (Salisbury ad Wimbush, 00). The EMD the picks out the highest-frequecy oscillatio that remais i the sigal. Fladri et al. (003) established how EMD could be used as a filter bak. (5)

6 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie 4 The IMF compoets eed to retai eough physical sese of both amplitude ad frequecy modulatios. This ca be achieved by limitig the size of the stadard deviatio (SD), computed from the two cosecutive siftig results as SD = T ( h ( k )( t) hik ( t ) ) t= 0 h( k )( t) [ (6) A value of SD betwee was used. I the ext step, the Hilbert trasform is applied to each of the IMFs, subsequetly providig the Hilbert amplitude spectra ad sigificat istataeous frequecy. The Hilbert trasform of each IMF is represeted by j= x( t) = Re a ( t)exp( i ω ( t) dt (7) j Equatio (7) gives both the amplitude ad frequecy of each compoet as a fuctio of time. The Fourier represetatio would be as follows: x( t) = Re j= a j exp iω t j with both a j ad ω j as costats. Further iformatio o the Hilbert-Huag trasform ad Hilbert Spectral Aalysis ca be obtaied from available texts. The preset paper employs oly the EMD part of the HHT. The Fourier trasform represets the global rather tha ay local properties of the data/sigal. The major differece betwee the covetioal Hilbert trasform ad HHT is the defiitio of istataeous frequecy. The istataeous frequecy has more physical meaig oly through its defiitio of the IMF compoet, while the classical Hilbert trasform of the origial data might possess urealistic features (Huag et al. 998). This implies that the IMF represets a geeralized Fourier expasio. The variable amplitude ad istataeous frequecy eable the expasio to accommodate o-statioary data. Huag et al. (00) demostrated that the Fourier ad wavelets-based compoets ad spectra might ot have clear physical meaig like the HHT. For example, the wavelet-based iterpretatio of a pavemet profile is meaigful relative to selected mother wavelets (Attoh-Okie, 999). Furthermore, Hilbert aalysis (Log et al. 995) is based o almost o-causal sigular iformatio. Therefore at ay give time, data or sigal has oly oe amplitude ad frequecy, both of which ca be determied locally. This represets the best iformatio at that particular time. Physically, the defiitio of istataeous frequecy has true meaig for moo-compoet sigals, where there is oe frequecy, or at least a arrow rage of frequecies, varyig as a fuctio of time. Sice most data do ot show these ecessary characteristics, sometimes the Hilbert trasform makes little physical sese i practical applicatios. I the preset method, the EMD is used to decompose the sigal ito a series of moo compoet sigals. Furthermore, to extract sigificat iformatio from the time-frequecyamplitude joit distributio [ a ( t), ω ( t), t] could be developed. This joit distributio i 3D space ca be replaced by[ H ( ω, t) ], where x( t) = H ( ω, t). This fial represetatio is referred to as the Hilbert Spectrum. DATA AND RESULTS Power spectral desity (PSD) plots are developed to acquire the most domiat modes (or waveumbers) that make up the sigal. The sigal used i the preset paper is data acquired from a pavemet surface with a laser profilometer, which is digitized. The data has bee trasformed ito spatial acceleratio, which is the double derivative of road elevatio, ad it gives a idicatio of the vertical acceleratio of a vehicle travelig alog the pavemet. Accordig to Bruscella et al. (999), large road surface elevatio values do ot ecessarily cause large vehicle vibratios, ad that to idetify road profile trasiets a parameter that is sesitive to the spatial rate of chage i elevatio is required, ad ot the elevatio itself. Bruscella et al. (999) idicate that raod profile data are iheretly o-gaussia with a large umber of evets occurrig above ±3 stadard deviatios; they also metio that movig statistics show the o-statioary ature of road profile elevatio data. Furthermore, the properties of road surface elevatio data are ot affected whe elevatio data are trasformed ito spatial acceleratio. It was cocluded that spatial acceleratio is a better parameter for the characterizatio ad classificatio of road profiles tha elevatio because it is a more reliable idicator of road roughess. Trasiets (8)

7 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie 5 are easily idetified ad ca hece be aalyzed apart from the uderlyig road surface i the spatial acceleratio domai. Figures ad show PSD plots for profiles ad respectively. The plots show three peak values of PSD correspodig to three domiat modes (or waveumbers). Table gives domiat modes ad correspodig PSDs. Domiat waveumbers are about the same for both profiles. However, PSD values for profile are higher tha for profile for similar correspodig waveumbers. This would probably result i higher vertical acceleratios o profile tha o profile. I other words, the PSD plot presets a idicatio of the effect of the degree of profile. HHT results are preseted i the same domai as the origial sigal, ad hece would be easier to visualize. EMD recovers idividual moocompoet waveforms (IMF compoets) that make up the origial sigal. The last tred at the ed of each group of IMFs is the residue, which gives the basic tred of the road profile. Higher amplitude of residue idicates a profile of higher udulatios, ad, therefore, ituitively would result i higher vertical acceleratios. Figures 3 ad 4 give the IMF compoets of profiles ad respectively. The residue plot for profile has amplitude of about 0.6 mm/m, ad about 0. mm/m for profile. Therefore, profile would result i higher vertical acceleratios. CONCLUSION A simple compariso has bee made betwee Fourier spectral aalysis ad the Hilbert-Huag trasform aalysis (as far as the empirical mode decompositio) usig two data sets cotaiig road spatial acceleratio values from two flexible pavemet sectios. Results have show that the EMD gives results that are easier to visualize tha the Fourier aalysis regardig road profile descriptio. Power spectral desity (PSD) plots resulted i similar domiat modes for profiles ad. However, PSD values were higher for profile tha for profile, which would idicate that the effect o vertical acceleratio would be higher o profile tha o profile. EMD aalysis gives similar results but retais iformatio i the same domai as the origial sigal, resultig i easier basic profile visualizatio. REFERENCES Attoh-Okie, N. O. (999). Applicatio of wavelets i pavemet profile evaluatio ad assessmet. Proc. Estoia Academy of Sciece, Vol 5, pp Bruscella B. Aalysis of road surface profiles. Joural of Trasportatio Egieerig Vol. 5, No., Jauary/February 999. Fladri, P., Rillig, G. ad Gocalves, P. (003). Empirical mode decompositio as a filter bak. Paper to appear i IEEE Sigal Processig Letters. Huag, N. Cher, C. C., Huag, K., Salvio, L. W., Log, S ad Fa, K. L. (00). A ew spectral represetatio of earthquake data: Hilbert spectral aalysis of statio TCU 9, Chi-Chi, Taiwa, September. Bulleti of the Seismological Society of America, Vol 9, No. 5 pp Huag Norde E. A ew method for oliear ad ostatioary time series aalysis: Empirical mode decompositio ad Hilbert spectral aalysis. Proceedigs of SPIE Vol 4056 (000). Huag, N. E., She, Z., Log, S. R., Wu, Shih, M. C., Zheg, H-H., Q., Ye, N.-C., Tug C. C., ad Liu, H-H., (998). The empirical mode decompositio ad the Hilbert spectrum for oliear ad o-statioary time series aalysis. Proc. Royal Society. Lodo, Series A, Vol 454, Log, S. R., Huag, N. E., Tug, C. C., Wu, M. C., Li, R. Q., Mollo-Christese, E., Jua, Y. (995). The Hilbert spectrum for oliear ad o-statioary time series aalysis. Proc. Royal Society. Lodo, Series A, Vol 454, Nues J. C. ad Attoh-Okie N. O. Pavemet image aalysis usig the bidimesioal empirical mode decompositio.

8 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie 6 Peg Z. K., Tse Peter W. ad Chu F. L. A compariso study of improved Hilbert-Huag trasform ad wavelet trasform: Applicatio to fault diagosis for rollig bearig. Mechaical Systems ad Sigal Processig 9 (005) Salisbury, J. I. ad Wimbush, M. (00). Usig moder time series aalysis techiques to predict ENSO evets from the SOI time series. Noliear Processes i Geophysics, Vol 9, pp Schlurma, T. (00). Spectral aalysis of oliear water waves based o the Hilbert-Huag trasformatio. Trasactio of ASME Joural of Offshore Mechaics ad Artic Egieerig, Vol 4, pp -7. Weaver, H. J. Theory of discrete ad cotiuous fourier aalysis. Joh Wiley & Sos, Ic. 989.

9 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie 7 LIST OF TABLES TABLE Domiat Waveumbers ad PSD values. LIST OF FIGURES FIGURE (a) Origial sigal for profile, (b) PSD plot for profile. FIGURE (a) Origial sigal for profile, (b) PSD plot for profile. FIGURE 3 (a) IMF compoets for profile. FIGURE 3 (b) IMF compoets for profile (cotiuatio of Figure 3 (a)). FIGURE 4 (a) IMF compoets for profile. FIGURE 4 (b) IMF compoets for profile (cotiuatio of Figure 4 (a)).

10 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie 8 TABLE Domiat Waveumbers ad PSD Values. Domiat Waveumber (cycles/m) PSD (mm /m ) Profile , ,000,000 Profile ,000

11 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie 9 FIGURE (a) Origial sigal for profile, (b) PSD plot for profile.

12 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie 0 FIGURE (a) Origial sigal for profile, (b) PSD plot for profile.

13 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie FIGURE 3 (a) IMF compoets for profile.

14 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie FIGURE 3 (b) IMF compoets for profile (cotiuatio of Figure 3 (a)).

15 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie 3 FIGURE 4 (a) IMF compoets for profile.

16 Albert Y. Ayeu-Prah, Stephe A. Mesah, ad Nii O. Attoh-Okie 4 FIGURE 4 (b) IMF compoets for profile (cotiuatio of Figure 4 (a)).