THE HILBERT-HUANG TRANSFORM AND THE ... - CiteSeerX

Albert Y. Ayenu-Prah, Stephen A. Mensah, and Nii O. Attoh-Okine

THE HILBERT-HUANG TRANSFORM AND THE FOURIER TRANSFORM IN THE ANALYSIS OF PAVEMENT PROFILES Albert Y. Ayenu-Prah* Graduate Student Department of Civil and Environmental Engineering University of Delaware Newark, DE 19716 USA Telephone: (302) 831-4391 Fax: (302) 831-3640 Email: [email protected] Stephen A. Mensah Graduate Student Department of Civil and Environmental Engineering University of Delaware Newark, DE 19716 USA Telephone: (302) 831-4391 Fax: (302) 831-3640 Email: [email protected] Nii O. Attoh-Okine, Ph.D., P.E. Associate Professor Department of Civil and Environmental Engineering University of Delaware Newark DE 19716 USA Telephone: (302) 831-4532 Fax: (302) 831-3640 Email: [email protected] Submitted: July 2005 Word Count: 3156 *Corresponding author.


ABSTRACT The present paper employs the Hilbert-Huang transform (HHT) and the Fourier transform to analyze the road surface profile of two flexible pavements in varying conditions. The central idea of HHT is the empirical mode decomposition (EMD), which is then decomposed into a set of intrinsic mode functions (IMFs). The Hilbert transforms can then be applied to the IMFs. The strength of HHT is the ability to process non-stationary and non-linear data. Unlike the Fourier transform, which transforms information from the time domain into the frequency domain, HHT does not move from the time domain into the frequency domain – Information is maintained in the time domain. The paper tries to identify which of the two transforms is better able to do both quantitative and qualitative identification of the profile type from field data. In performing the analyses the nature and behavior of road profiles as indicated by the literature are taken into account, that road profiles are non-stationary and are inherently non-Gaussian.


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INTRODUCTION Pavement surface roughness determines ride quality and it is a good indicator of the quality of construction. Roughness Index is used for the assessment of the acceptability of newly constructed pavement as well as in pavement management decision process for maintenance scheduling. Relatively less rough roads are desirable since the pavement performs better. It also enhances vehicle performance. Road roughness would be the surface profile undulations as opposed to the surface roughness that determines skid resistance. Two sets of digitized flexible pavement profile data are analyzed with the Fourier transform and with the Hilbert-Huang transform (HHT). The data is in the form of spatial acceleration, which is the double derivative of elevation. According to the literature, road profiles are fundamentally non-Gaussian and nonstationary (Bruscella et al. 1999). Compared to the HHT the Fourier Transform has its limitations. Fourier analysis requires that the signal must be strictly periodic or stationary otherwise the resulting Fourier spectrum will make little physical sense (Peng 2005). An attempt is made to determine which of the two analysis techniques produces better road profile descriptions. THE FOURIER TRANSFORM The Fourier series is made up of sines and cosines; the Fourier transform is a generalization of the Fourier series, and is made up of exponentials and complex numbers. The Fourier analysis has wide applications in mathematics and engineering, used in modeling diverse physical phenomena. Examples of some physical phenomena include heat transfer, wave propagation, circuit analysis, electronic circuit analysis, and vibrations. Interesting to note is the Fourier kernel, exp[2πiwt], which is a solution to an nth-order linear differential equation which, in turn, is used to model various physical phenomena; it is one reason why Fourier analysis has such wide applications (Weaver, 1989). On the interval [-π, π], any arbitrary function f(t) which is periodic and single-valued could be represented by the trigonometric series ∞

∞

n =1

n =1

f (t ) = a 0 + ∑ a n cos nt + ∑ bn sin nt

(1)

For a periodic interval, T, ∞

∞

n =1

n =1

f (t ) = a0 + ∑ a n cos nωt + ∑ bn sin nωt where

(2)

2π T t +T 2 a n = ∫ f (t ) cos nωtdt T t

ω=

bn =

2 T

t +T

∫ f (t ) sin nωtdt t

T = period of f(t). For a signal or function, f(t), the Fourier transform is defined as

F (ω ) =

∞

∫ f (t )e

−iωt

dt ,

(3)

−∞

and the inverse Fourier transform to recover the original signal or function is defined as

f (t ) =

1 2π

∞

∫ F (ω )e

−∞

iωt

dω

(4)

These two Fourier transform equations are called the Fourier transform pair. Evidently seen in the two equations is the complex number, i, after transformation of the original function. The original function is in the time domain, t, which is transformed into the frequency domain, ω, after Fourier transformation. The resulting transformed signal gives an indication of the frequency components that contribute to the original signal when a


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spectral plot of amplitude versus frequency is constructed. Amplitude is the absolute value of the complex number in the Fourier (frequency) domain, and is usually the vertical axis. f(t) can be continuous or discrete. The preceding definition for the Fourier transform intrinsically assumes f(t) to be continuous. We can speak of a discrete f(t) over a certain finite interval, then we talk about the discrete Fourier transform, DFT, of f(t), defined as

Fn =

1 T

T 2

∫ f (t )e

−inωt

dt ,

(5)

T − 2

and the inverse discrete Fourier transform, as

f (t ) =

∞

∑F e n

inωt

,

(6)

n = −∞

where n is an index. Therefore, DFT is a digital tool, used to analyze the frequency components of discrete signals while Fourier transform is an analog tool, used to analyze the frequency components of continuous signals. A fast Fourier transform, FFT, is an algorithm that efficiently implements the DFT on a computer. Usually, power spectral density (PSD) plots are preferred to Fourier transform plots; PSD is equivalent to the square of the amplitude of the Fourier transform, which has a continuous dependence on absolute frequency (or in this case, wavenumber (cycles/m), since pavement data uses wavenumber instead of frequency in cycles/sec). If P denotes PSD, then

P=kA

2

(7)

where A = amplitude of Fourier transform k = appropriate scaling constant. A property of the PSD is given as follows,

∫

∞

0

P( f )df = σ 2 (8)

where σ2 = variance of original signal. In other words, the total variance of the signal is recovered upon integrating plotted spectral values over the frequency range. THE HILBERT-HUANG TRANSFORM The HHT consists of two parts: the empirical mode decomposition (EMD) and the Hilbert Spectral Analysis (HSA). The EMD generally separates nonlinear, non-stationary data into locally non-overlapping time scale components. According to Peng et al (2005) the signal decomposition process will break down the signal into a set of complete and almost orthogonal components called the IMF, which is almost mono-component. An IMF is a function that satisfies two of the following conditions (Peng et al. 2005): The number of extrema and the number of zero crossings must either equal or differ at most by one in whole data sets; the mean value of the envelop defined by the local maxima and the envelop defined by the local minima is zero at every point. The Hilbert transform of the IMFs will yield a full energy-frequency-time distribution of the signal known as the Hilbert-Huang spectrum. The HHT has the following advantages that make it desirable for signal analysis: • The most computationally intensive step is the EMD operation, which doesn’t involve convolution and other time-consuming operations and makes HHT ideal for signals of large size. • The Hilbert-Huang spectrum does not involve the concept of the frequency resolution but the instantaneous frequency. On the other hand, there are some drawbacks in the application of HHT. The EMD will generate undesired low amplitude IMFs at the low frequency region and raise some undesired frequency components. The first IMF


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may cover a wide frequency range at the high-frequency region and therefore cannot satisfy the monocomponent definition well. The EMD operation often cannot separate some low-energy components from the analysis signal therefore those components may not appear in the frequency-time plane. Any function can be decomposed as follows: 1. Identify all the local extrema, and then connect all the local maxima by a cubic spline as the upper envelope. 2. Repeat the procedure for the minima to produce the lower envelope. The upper and lower envelopes should cover all the data. If the mean is designated as m1 and the difference between the data and m1 is the first component h1 , then

x(t ) − m1 = h1 (9) h1 is an IMF. The mean m1 is given by the sum of local extrema connected by the cubic spline: L +U (10) m1 = 2 U is the local maxima and L is the local minima. The IMF can have both amplitude and frequency modulations. In many cases there are overshoots and undershoots after the first round of processing, and this is termed sifting. The sifting process serves two purposes (Huang et al. 2001): It eliminates riding signals/profile, making the signal or profile more symmetric. The sifting process has to be repeated many times. In the second sifting process, h1 is treated as the data and as the first component. h1 is almost an IMF, except some error might be introduced by the spline curve fitting process. To treat h1 as new set of data, a new mean is computed. Then,

h1 − m11 = h11

(11)

After repeating the sifting process up to k times, h1k becomes the IMF, that is

h1( k −1) − m1k = h1k

(12)

Let h1k = c1 , the first IMF from the data. c1 should contain the finest scale or the shortest period component of the data/profile. Now c1 can be separated from the rest of the data by

x(t ) − c1 = r1

(13)

Since r1 is the residue, it contains information on longer period component; it is now treated as the new data and subjected to the same sifting process. The procedure is repeated for all subsequent r j ’s and the result is

r1 − c 2 = r2 ;...rn −1 − c n = rn

(14)

c 2 is now the second IMF of the data. The sifting process can be stopped by any of the following predetermining criteria: • Either when the component c n or the residue rn becomes so small that it has a predetermined value of substantial consequence, or • When the residue rn becomes a monotonic function from which no IMF can be extracted. Summing equations (13) and (14) yields the following equation: n

x(t ) = ∑ c j + rn j =1

(15)

c j is the jth IMF, n = number of sifted IMF. rn can be interpreted as a trend in the signal/profile. The ci have zero mean. Due to the iterative process none of the sifted IMFs derived is in closed analytical form (Schlurmann, 2002). The IMF can be linear or nonlinear based on the characteristics of the data. The IMFs are almost orthogonal and form a complete basis. Their sum equals the original data (Salisbury and Wimbush, 2002). The EMD then picks out the highest-frequency oscillation that remains in the signal. Flandrin et al. (2003) established how EMD could be used as a filter bank.


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The IMF components need to retain enough physical sense of both amplitude and frequency modulations. This can be achieved by limiting the size of the standard deviation (SD), computed from the two consecutive sifting results as T

SD = ∑ [

(h1( k −1)(t ) − hik (t ) )

2

h12( k −1)(t )

t =0

(16)

A value of SD between 0.2-0.3 was used. In the next step, the Hilbert transform is applied to each of the IMFs, subsequently providing the Hilbert amplitude spectra and significant instantaneous frequency. The Hilbert transform of each IMF is represented by n

x(t ) = Re ∑ a j (t ) exp(i ∫ ω (t )dt

(17)

j =1

Equation (17) gives both the amplitude and frequency of each component as a function of time. The Fourier representation would be as follows: ∞

x(t ) = Re ∑ a j exp

iω j t

(18)

j =1

with both

a j and ω j as constants. Further information on the Hilbert-Huang transform and Hilbert Spectral

Analysis can be obtained from available texts. The present paper employs only the EMD part of the HHT. The Fourier transform represents the global rather than any local properties of the data/signal. The major difference between the conventional Hilbert transform and HHT is the definition of instantaneous frequency. The instantaneous frequency has more physical meaning only through its definition of the IMF component, while the classical Hilbert transform of the original data might possess unrealistic features (Huang et al. 1998). This implies that the IMF represents a generalized Fourier expansion. The variable amplitude and instantaneous frequency enable the expansion to accommodate non-stationary data. Huang et al. (2001) demonstrated that the Fourier and wavelets-based components and spectra might not have clear physical meaning like the HHT. For example, the wavelet-based interpretation of a pavement profile is meaningful relative to selected mother wavelets (Attoh-Okine, 1999). Furthermore, Hilbert analysis (Long et al. 1995) is based on almost non-causal singular information. Therefore at any given time, data or signal has only one amplitude and frequency, both of which can be determined locally. This represents the best information at that particular time. Physically, the definition of instantaneous frequency has true meaning for mono-component signals, where there is one frequency, or at least a narrow range of frequencies, varying as a function of time. Since most data do not show these necessary characteristics, sometimes the Hilbert transform makes little physical sense in practical applications. In the present method, the EMD is used to decompose the signal into a series of mono component signals. Furthermore, to extract significant information from the time-frequencyamplitude joint distribution a (t ), ω (t ), t could be developed. This joint distribution in 3D space can be

[

]

[

replaced by H (ω , t ) , where x(t )

]

= H (ω , t ) . This final representation is referred to as the Hilbert Spectrum.

DATA AND RESULTS Power spectral density (PSD) plots are developed to acquire the most dominant modes (or wavenumbers) that make up the signal. The signal used in the present paper is data acquired from a pavement surface with a laser profilometer, which is digitized. The data has been transformed into spatial acceleration, which is the double derivative of road elevation, and it gives an indication of the vertical acceleration of a vehicle traveling along the pavement. According to Bruscella et al. (1999), large road surface elevation values do not necessarily cause large vehicle vibrations, and that to identify road profile transients a parameter that is sensitive to the spatial rate of change in elevation is required, and not the elevation itself. Bruscella et al. (1999) indicate that raod profile data are inherently non-Gaussian with a large number of events occurring above ±3 standard deviations; they also mention that moving statistics show the non-stationary nature of road profile elevation data. Furthermore, the properties of road surface elevation data are not affected when elevation data are transformed into spatial acceleration. It was concluded that spatial acceleration is a better parameter for the characterization and classification of road profiles than elevation because it is a more reliable indicator of road roughness. Transients


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are easily identified and can hence be analyzed apart from the underlying road surface in the spatial acceleration domain. Figures 1 and 2 show PSD plots for profiles 1 and 2 respectively. The plots show three peak values of PSD corresponding to three dominant modes (or wavenumbers). Table 1 gives dominant modes and corresponding PSDs. Dominant wavenumbers are about the same for both profiles. However, PSD values for profile 1 are higher than for profile 2 for similar corresponding wavenumbers. This would probably result in higher vertical accelerations on profile 1 than on profile 2. In other words, the PSD plot presents an indication of the effect of the degree of profile. HHT results are presented in the same domain as the original signal, and hence would be easier to visualize. EMD recovers individual monocomponent waveforms (IMF components) that make up the original signal. The last trend at the end of each group of IMFs is the residue, which gives the basic trend of the road profile. Higher amplitude of residue indicates a profile of higher undulations, and, therefore, intuitively would result in higher vertical accelerations. Figures 3 and 4 give the IMF components of profiles 1 and 2 respectively. The residue plot for profile 1 has amplitude of about 0.6 mm/m2, and about 0.1 mm/m2 for profile 2. Therefore, profile 1 would result in higher vertical accelerations. CONCLUSION A simple comparison has been made between Fourier spectral analysis and the Hilbert-Huang transform analysis (as far as the empirical mode decomposition) using two data sets containing road spatial acceleration values from two flexible pavement sections. Results have shown that the EMD gives results that are easier to visualize than the Fourier analysis regarding road profile description. Power spectral density (PSD) plots resulted in similar dominant modes for profiles 1 and 2. However, PSD values were higher for profile 1 than for profile 2, which would indicate that the effect on vertical acceleration would be higher on profile 1 than on profile 2. EMD analysis gives similar results but retains information in the same domain as the original signal, resulting in easier basic profile visualization. REFERENCES Attoh-Okine, N. O. (1999). “Application of wavelets in pavement profile evaluation and assessment.” Proc. Estonian Academy of Science, Vol 5, pp 53-63. Bruscella B. “Analysis of road surface profiles”. Journal of Transportation Engineering Vol. 125, No. 1, January/February 1999. Flandrin, P., Rilling, G. and Goncalves, P. (2003). “Empirical mode decomposition as a filter bank”. Paper to appear in IEEE Signal Processing Letters. Huang, N. Chern, C. C., Huang, K., Salvino, L. W., Long, S and Fan, K. L. (2001). “A new spectral representation of earthquake data: Hilbert spectral analysis of station TCU 129, Chi-Chi, Taiwan, 21 September”. Bulletin of the Seismological Society of America, Vol 91, No. 5 pp 1310-1338. Huang Norden E. “A new method for nonlinear and nonstationary time series analysis: Empirical mode decomposition and Hilbert spectral analysis”. Proceedings of SPIE Vol 4056 (2000). Huang, N. E., Shen, Z., Long, S. R., Wu, Shih, M. C., Zheng, H-H., Q., Yen, N.-C., Tung C. C., and Liu, H-H., (1998). “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis”. Proc. Royal Society. London, Series A, Vol 454, 903-995. Long, S. R., Huang, N. E., Tung, C. C., Wu, M. C., Lin, R. Q., Mollo-Christensen, E., Juan, Y. (1995). “The Hilbert spectrum for nonlinear and non-stationary time series analysis”. Proc. Royal Society. London, Series A, Vol 454, 903-995. Nunes J. C. and Attoh-Okine N. O. “Pavement image analysis using the bidimensional empirical mode decomposition”.


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Peng Z. K., Tse Peter W. and Chu F. L. “A comparison study of improved Hilbert-Huang transform and wavelet transform: Application to fault diagnosis for rolling bearing”. Mechanical Systems and Signal Processing 19 (2005) 974-988. Salisbury, J. I. and Wimbush, M. (2002). “Using modern time series analysis techniques to predict ENSO events from the SOI time series”. Nonlinear Processes in Geophysics, Vol 9, pp 341-345. Schlurmann, T. (2002). “Spectral analysis of nonlinear water waves based on the Hilbert-Huang transformation”. Transaction of ASME – Journal of Offshore Mechanics and Artic Engineering, Vol 124, pp 22-27. Weaver, H. J. “Theory of discrete and continuous fourier analysis”. John Wiley & Sons, Inc. 1989.


LIST OF TABLES TABLE 1 Dominant Wavenumbers and PSD values. LIST OF FIGURES FIGURE 1 (a) Original signal for profile 1, (b) PSD plot for profile 1. FIGURE 2 (a) Original signal for profile 2, (b) PSD plot for profile 2. FIGURE 3 (a) IMF components for profile 1. FIGURE 3 (b) IMF components for profile 1 (continuation of Figure 3 (a)). FIGURE 4 (a) IMF components for profile 2. FIGURE 4 (b) IMF components for profile 2 (continuation of Figure 4 (a)).

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TABLE 1 Dominant Wavenumbers and PSD Values. Dominant Wavenumber PSD (mm2/m2) (cycles/m) Profile 1 0.0033 50 0.0020 2,500 0.0010 5,000,000 Profile 2 0.0030 0.0017 0.0010

12 250 800,000

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FIGURE 1 (a) Original signal for profile 1, (b) PSD plot for profile 1.

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FIGURE 2 (a) Original signal for profile 2, (b) PSD plot for profile 2.

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FIGURE 3 (a) IMF components for profile 1.

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FIGURE 3 (b) IMF components for profile 1 (continuation of Figure 3 (a)).

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FIGURE 4 (a) IMF components for profile 2.

13


FIGURE 4 (b) IMF components for profile 2 (continuation of Figure 4 (a)).

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