Performance Analysis of FFT Filter to Measure Displacement Signal in Road Roughness Profiler

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1 International Journal of Computer and Electrical Engineering, Vol. 5, No. 4, August 3 Performance Analysis of FFT Filter to Measure Displacement Signal in Road Roughness Profiler Thai Minh Do and Thong Chi Le Abstract In the application of pavement roughness test, accelerometer sensors are used to measure vertical acceleration of a vehicle body. The displacement of that object is then determined by using double integrating acceleration data from the accelerometer sensor. High pass filters are used to remove DC components to avoid integrated errors. Fast Fourier Transform (FFT) Filter is known to provide a higher accuracy compared to Infinite Impulse Response (IIR) Filter and Finite Impulse Response (FIR) Filter. In this research, we analyze the performance of FFT Filter with experiment in MATLAB. The results show that FFT filters give results more accurate than do FIR and IIR not only for single frequency signal but also for multi-frequency signal. The standard error and peak error of FFT Filter are less than those of FIR and IIR Filter. Moreover the error of FFT Filter can be reduced when increasing the cut frequency of FFT filter but it is remained less than the frequency of acceleration signal. Index Terms Road profiler, FFT, accelerometer, double integration. I. INTRODUCTION In a road profiler it is important to increase the accuracy and the reliability of the road profile signal. Because of earth gravity effect, DC component signal always exist at the same -g constant. Furthermore, the two initial conditions (velocity and position) must be known to avoid integration errors. However, the only way to get these initial conditions is thought direct measurement which is often impractical []. The small DC bias in the acceleration signal will result in drift associated with accelerometer. This paper will describe an accurate algorithm which is developed to measure displacement from acceleration signal without initial conditions. In the simple way, HPFs is used to remove DC component signal after each integrator. A. Road profiler system High-speed profilers(usually referred to as inertial profilers demonstrated in Fig. ) combine reference elevation, height relative to the reference and longitudinal distance to produce the true road surface profile []. A third device used to measure road roughness is known as a road roughness profiling device which measures the longitudinal profile of the road. This particular profilometer uses an accelerometer to create an inertial reference that defines the height of the accelerometer located on the vehicle. A height sensor which is most commonly a laser sensor is used to determine the height to the pavement surface from the Manuscript received December 9, ; revised March 9, 3. The authors are with Faculty of Electrical & Electronics Engineering, Ho Chi Minh City University of Technology, VNU-HCM, Ho Chi Minh City, Viet Nam ( thai.dominh@yahoo.com, chithong@hcmut.edu.vn). vehicle. Distance encoder is used to pick up distance road. The processing is done on a computer located inside the vehicle. The computations are performed in real time as the vehicle is moving. It can operate at speeds between and 7 miles per hour [3]. Fig.. A van equipped with an inertial profilometer [4] In Fig., The road profile is reconstructed from laser and accelerometer readings according to the following equation: zt () is the acceleration. w( t) z( t) dtdt h( t) () ht () is the height measured by the laser sensor. Fig.. The relationship among quantities of interest in a road profile [5] The w (t) road profile signal is then simulated in a quarter car model at speed 8 km/h. The parameters of this car must be set to represent the Golden Car. The IRI is calculated in (). LV /.. IRI Z sz u dt L () Z.. s : springmass velocity, Z u : unspring mass velocity L: length profile, V: speed (8 km/h, 5 mph) DOI:.7763/IJCEE.3.V

International Journal of Computer and Electrical Engineering, Vol. 5, No. 4, August 3 B. Double Integration Process A block diagram of the double integration process is shown in Fig. 3. at () af () t t vt () vf () t t xt () a() d v() d High-Pass Filter Acceleration High-Pass Filter Fig.

3Included with two stages of integration are three stages of high-pass filtering. The other way measures displacement that is used optical or laser [6]. II. DIGITAL FILTERING FOR DOUBLE INTEGRATION A.

h(t) is impulse response vector h h, h h M [,..., ] (4) In Fig. 4, this is a demonstrated frequency response of FIR Filter. f Fig. 5.

2 International Journal of Computer and Electrical Engineering, Vol. 5, No. 4, August 3 B. Double Integration Process A block diagram of the double integration process is shown in Fig. 3. at () af () t t vt () vf () t t xt () a() d v() d High-Pass Filter Acceleration High-Pass Filter Fig. 3. Block diagram of double integration process High-Pass Filter x () t Displacement This diagram shows displacement calculating process from acceleration which is received from accelerometer. Fig. 3Included with two stages of integration are three stages of high-pass filtering. The other way measures displacement that is used optical or laser [6]. II. DIGITAL FILTERING FOR DOUBLE INTEGRATION A. FIR Filter The Equation (3) describes a FIR Filter described in [7]: yn hx nhx n... hm xn M (3) where y is the output, x is the input and M is the order of the filter. h(t) is impulse response vector h h, h h M [,..., ] (4) In Fig. 4, this is a demonstrated frequency response of FIR Filter. f Fig. 5. Frequency response of an IIR filter [] Because the order of IIR Filter is lower than FIR Filter s order, it can reduce the number of calculations. In other hand, this filter has a nonlinear phase response (Fig. 5) thus it will cause distortion of output signal. C. FFT Filter FFR filtering technique uses the FFT to remove low frequency content near DC. This method is suggested firstly by Ribeiro [9] and improved secondly by Slifka [].Both The low frequency coefficients located at the beginning and the ones at the end of the FFT sequence must be changed to equal the conjugate together, because the FFT sequence must be conjugate symmetric for the signal of interest to remain real. The modification algorithm can be described below []: X fftx; Xf X ; Xf Xf k; (6) for i : k Xf i i Xf k; Xf N i conjxf i; xf RealIFFT Xf ; where N is the size of the FFT, k is the index number of the FFT coefficient representing the cutoff frequency, and the i s are filtering coefficients specified by the user. This Filter can be demonstrated in Fig. 6. Fig. 4. Frequency response of FIR filter [] This type of Filter is commonly used inside the double integration process in road roughness profiler, and it is recommended by Ribeiro [8]. The advantage of this Filter is linear phase response and real time calculating. But, its disadvantage is the order can be very high to achieve requirement. B. IIR Filter IIR Filter, an alternative approach, uses a recursive difference equation to represent the filter. M L i j (5) i j yn a yni b xn j where y is the output, x is the input and M is order of IIR filter. Fig. 6. Illustrate FFT filters The DC component of magnitude spectrum X(f) in Fig. 6.b must be set equal zero. After that, The inversion of FFT is performed on that signal to receive the output signal similar with original signal x(t) without DC components. III. INTEGRATION METHODS A. Analog Integration At the first times, analog integration is used to calculate displacement signal. Basic principle is processing double 357

3 International Journal of Computer and Electrical Engineering, Vol. 5, No. 4, August 3 integrator with RC or opamp circuit demonstrated in Fig. 7, but all of them have a number of errors. Ribiero did a study using analog circuitry to perform the double integration and found that the errors were unacceptable []. Vi R C Fig. 7. Double integrator amplifier used to obtain displacement from acceleration data [] B. Digital Integration Currently, the development of computer and chip technologies helps to calculate integration accurately with using digital integrator. Analog signals is sampled and digitalized by high speed ADCs. In this research, Trapezoidal method is used to calculate integration signal in Fig. 3. This method can be described in (7). yn yn xn xn, n f s Trapezoidal method is chosen to calculate in integrators because real time road surface analyzer will be performed in future work. These integrated methods are supported by MATLAB. IV. ANALYSIS OF ERRORS To estimate quantify and accuracy of filters, it is important to compare the ideal displacement signal and the results. The signals created in MATLAB with different frequencies and multi harmonicsare imported into the integrated processing. Standard error and peak error are used to estimate the calculated displacement data. A. Standard Error This parameter is used to indicate the accuracy of double integration processing. It is given by (8) described in []. n ^ X i i i % e n X N is the number of data points. is the double integrated position data R X i ^ X i is the ideal data. R3 From above equation, it is easy to show that Standard error is smaller, that method is better. When measuring standard error, the two signals, Ideal and calculated signal, must be matched very well. However, the calculated signal commonly meets delay causing filter and integrated process. When this happens, the calculated signal is desired to synchronize with the reference signal. C C R4 U 6 Vo (8) (7) So, a little standard error value can be accepted about less than % suggested by Slifka []. B. Peak Error This is another method to estimate number of errors in the calculated displacement signal, and it is also valuable. Therefore, it is important to determine the peaks and valleys in the position waveform. For the peak error, the peak points higher than a certain threshold will be measured and calculated. Usually, a threshold of 5% of the maximum peak that is suggested by Slifka[]is used in this paper. There are two parameters in the peak errors that are maximum and average peak error. The maximum peak error is calculated from two peaks having the most distant together, and the average of peak error is measured between all peaks. V. EXPERIMENTAL RESULTS We assumed that the original acceleration signal is A( t) a( t) d (9) Mathematically, the calculation of displacements x(t) from a measured acceleration a(t) is simple described in []: xt xt v( t ) t dt dt ad () t t t x(t ) is the initial displacement, v(t ) is the initial velocity, d is the acceleration drift, x(t) is the calculated displacement From (), we can see that the output signal can be unbounded over time because of ramp (v ) and parabola (d ) components. In this paper, the signals, single and multifrequency signal which are created in MATLAB, are imported into Fig. 3. After that, the output displacement signal is compared to the ideal position signal which is calculated from theory. A. Experiment with a Single Frequency Signal Experiment ideas: We have an original acceleration signal: t Asin t a After the st integrating, the velocity (assuming no initial conditions) is A t cos t v After the nd integrating, the displacement (assuming no initial conditions) is A t sin t x () () (3) 358

3, it is clear to see that the maximum peak error of IIR Filter is very large because of nonlinear phase response of this filter and the best result in term of the maximum peak error is from FFT

hz Fig.. Average peak error of filters when using single frequency signal. Fig.3. Maximum peak error of filters when using single frequency signal Fig.

8, the displacement when using FIR Filter is distortional at the beginning due to no initial conditions. In Fig.

Therefore we can use FFT Filter in calculating the displacement by using double integration without initial conditions. In Fig., Fig. and Fig.

4 International Journal of Computer and Electrical Engineering, Vol. 5, No. 4, August 3 To reject delay of the output signal, filtfiltfunction which is supported in MATLAB is used to replace filter function in double integrated process., Fig. ). In Fig. 3, it is clear to see that the maximum peak error of IIR Filter is very large because of nonlinear phase response of this filter and the best result in term of the maximum peak error is from FFT Filter. Fig. 8. Double integration using FIR Filters, with M=5, fc=5.hz Fig.. Standard error of filters when using single frequency signal Fig. 9. Double integration using IIR Filters, with M=4, fc=5.hz Fig.. Average peak error of filters when using single frequency signal. Fig.3. Maximum peak error of filters when using single frequency signal Fig.. Double integration using FFT Filters, with k=8, α=.775 Fig. 8, Fig. 9 and Fig. show the results when using different filters. In Fig. 8, the displacement when using FIR Filter is distortional at the beginning due to no initial conditions. In Fig. 9, the IIR Filter gives a big error in the displacement because of the nonlinear phase response of this filter. The best calculated displacement signal is the result of FFT Filter (Fig. ). Therefore we can use FFT Filter in calculating the displacement by using double integration without initial conditions. In Fig., Fig. and Fig. 3, the standard error, the average peak error, and the maximum peak error of different filters are presented. The standard error and the average peak error of FIR Filter is almost the same as those of FFT Filter especially at the high frequency, but the errors of IIR Filter is much larger than those of FFT Filter (Fig. B. Experiment with a Multi-Harmonic Signal In this case the acceleration signal with multi-frequency is applied because the real signal in a road profiler is random data. The acceleration signal is given in (4). a( t) d A sin( f t) A sin( f dt) min A sin( f t) 3 max mid (4) We did experiments in three cases. The first case uses FIR Filter with M=5, f c =5.Hz, f min =Hz, f max =4Hz. The second case uses IIR Filter with M=4, f c =5.Hz, f min =Hz, f max =4Hz. The third case uses FFT Filter with k=5 and α=.775. The displacement signal calculated from the multi-frequency acceleration signal when using FFT Filter is the most accurate result (Fig. 4). 359

International Journal of Computer and Electrical Engineering, Vol. 5, No. 4, August 3 A. Experiment with Changing Cut Frequency of Filters Fig. 8.

When cut frequency of FFT Filters is bigger, the error seems smaller. However, this cut frequency must be set much less than the smallest frequency of acceleration signal. Fig. 5.

CONCLUSIONS In this paper we analyzed the performance of FFT Filter which is applied to calculate the displacement from the signal measured from the accelerometer.

The standard error, the average peak error, and the maximum peak error of FFT Filter are less than those of FIR Filter and IIR Filter.

Average peak error of filters when using multi-frequency signal Fig. 7. Maximum peak error of filters when using multi-frequency signal From the results presented in Fig. 5, Fig. 6 and Fig.

5 International Journal of Computer and Electrical Engineering, Vol. 5, No. 4, August 3 A. Experiment with Changing Cut Frequency of Filters Fig. 8. Standard errors of FFT Filter when altering filter cut frequency Fig. 4. Double integration with multi-harmornic signal At higher frequencies in Fig. 8, the error appears to increase with frequency. When cut frequency of FFT Filters is bigger, the error seems smaller. However, this cut frequency must be set much less than the smallest frequency of acceleration signal. Fig. 5. Standard error of filters when using multi- frequency signal VI. CONCLUSIONS In this paper we analyzed the performance of FFT Filter which is applied to calculate the displacement from the signal measured from the accelerometer. The experiments showed that FFT filters give results more accurate than do FIR and IIR not only for single frequency signal but also for multi-frequency signal. The standard error, the average peak error, and the maximum peak error of FFT Filter are less than those of FIR Filter and IIR Filter. Moreover the error of FFT Filter can be reduced when increasing the cut frequency of FFT filter, but it is remained much less than the frequency of acceleration signals. REFERENCES Fig. 6. Average peak error of filters when using multi-frequency signal Fig. 7. Maximum peak error of filters when using multi-frequency signal From the results presented in Fig. 5, Fig. 6 and Fig. 7, it is easy to see that the errors when using FFT Filter for the case of multi-harmonic signal are the least, and the position signals of filtering methods are more fail at higher frequencies. [] L. D. Slifka, "An accelerometer based approach to measuring displacement of a vehicle body," M.S. thesis, Dept. Elect. Comput. Eng., Michigan Univ., USA, 4. [] J. H. A. Qader, "High performance real-time embedded systems: design and implementation of road surface analyzer system," Ph.D. dissertation, Dept. Comput. Sci. and Eng.,Univ. Texas at Arlington, USA,. [3] A. DeMarco and C. Stedman, "Automated GPS Mapping of Road Roughness," Bachelor thesis, Worcester Polytechnic Inst., USA, 7. [4] M. W. Sayers and S. W. Karamihas, The Litte Book of Profiling, Michigan, USA: Michigan Univ., 996. [5] S. A. Dyer et al., "Refinement of Measurement Techniques of Road Profile and International Roughness Index to Support The KDOT Pavement Management System Annual Road-Condition Survey Research," Kansas State Univ., 5. [6] M. Harris and G. Piersol, Shock and Viration HandBook, 5th, Ed. New York: McGraw-Hill Book Company,. [7] S. J. Orfanidis, Introduction to Signal Processing, Prentice Hall, 9. [8] J. G. T. Ribeiro et al., "New improvements in the digital double integration filtering method to measure displacements using accelerometers," in Proc. 9th Int. Modal Analysis Conf., vol. 377, Orlando, Florida,, pp [9] J. G. T. Ribeiro et al., "Using the FFT- DDI method to measure displacements with piezoelectric, resistive and ICP accelerometers," in Conf. Exposition on Structural Dynamics, 3. [] J. G. T. Ribeiro et al., "Problems in analogue double integration to determine displacements from acceleration data," in Proc. 5th Int. Modal Analysis Conf., vol. 389, Orlando, Florida, 997, pp [] M. Arraigada el al., "Calculation of displacements of measured 36

International Journal of Computer and Electrical Engineering, Vol. 5, No. 4, August 3 accelerations analysis of two accelerometers and application in road," in STRC, 6.

He received Bachelor s Degree in Electronics and Telecommunications Engineering from University of Technical Education Ho Chi Minh City, Viet Nam in.

6 International Journal of Computer and Electrical Engineering, Vol. 5, No. 4, August 3 accelerations analysis of two accelerometers and application in road," in STRC, 6. Thai Minh Do was born in Dong Nai District, Viet Nam in 988. He received Bachelor s Degree in Electronics and Telecommunications Engineering from University of Technical Education Ho Chi Minh City, Viet Nam in. Now, he is studying for Master s Degree in Electronics Engineering from Ho Chi Minh City University of Technology. His thesis focuses on the development and evaluation of road roughness profiler system. Since, he has been an instructor in Ho Chi Minh City Vocational College. He is really interested in Signal Processing, Embedded System, and Road Profiler System. Mr. Do received certificated top student graduated Bachelor s Degree in, ODON VALLET Scholarship for Excellent Students in 9, NGUYEN THAI BINH Scholarship in 8. Thong Chi Le is faculty member of Ho Chi Minh City University of Technology, Vietnam for several years is a Ph.D. candidate in the Electrical Engineering Department at University of Arkansas. He is interested in digital electronics, microcontroller, and control system. His research includes using microcontroller to develop automatic control systems, and using HDL to design digital systems. He is also interested in embedded sensors using nanotechnology for industrial applications. He received his B.S. degree in Electronic Engineering from Ho Chi Minh City University of Technology, Vietnam in 993 and M.S. degree in Electronic Engineering from Ho Chi Minh City University of Science, Vietnam in 998. He got his PhD degree in Electrical Engineering from University of Arkansas, United States in 9. 36

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