performance analysis of fft filter to measure …ijcee.org/papers/731-et0006.pdf · performance...

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 1-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 [1]. 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. 1) combine reference

elevation, height relative to the reference and longitudinal

distance to produce the true road surface profile [2]. 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 19, 2012; revised March 9, 2013.

The authors are with Faculty of Electrical & Electronics Engineering,

Ho Chi Minh City University of Technology, VNU-HCM, Ho Chi Minh

City, Viet Nam (e-mail: [email protected],

[email protected]).

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 10 and

70 miles per hour [3].

Fig. 1. A van equipped with an inertial profilometer [4]

In Fig. 2, The road profile is reconstructed from laser and

accelerometer readings according to the following equation:

( ) ( ) ( )w t z t dtdt h t

(1)

where:

( )z t is the acceleration.

( )h t is the height measured by the laser sensor.

Fig. 2. 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 80 km/h. The parameters of this car

must be set to represent the Golden Car. The IRI is

calculated in (2).

/ . .

0

1| |

L V

s uIRI Z Z dtL

(2)

where: .

sZ : springmass velocity, .

uZ : unspring mass velocity

L: length profile, V: speed (80 km/h, 50 mph)

Performance Analysis of FFT Filter to Measure

Displacement Signal in Road Roughness Profiler

Thai Minh Do and Thong Chi Le

356

International Journal of Computer and Electrical Engineering, Vol. 5, No. 4, August 2013

DOI: 10.7763/IJCEE.2013.V5.731

B. Double Integration Process

A block diagram of the double integration process is

shown in Fig. 3.

( )a t ( )fa t ( )v t ( )fv t ( )x t

Acceleration Displacement

( )fx tHigh-Pass

Filter0

( )t

a d High-Pass

Filter0

( )t

v d High-Pass

Filter

Fig. 3. Block diagram of double integration process

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]:

0 1 1 ... My n h x n h x n h x n 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

0, 1[ ,..., ]Mh h h h (4)

In Fig. 4, this is a demonstrated frequency response of

FIR Filter.

Fig. 4. Frequency response of FIR filter [1]

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.

1 0

M L

i j

i j

y n a y n i b x n j

(5)

where y is the output, x is the input and M is order of IIR

filter.

Fig. 5. Frequency response of an IIR filter [1]

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 [1].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 [1]:

0

;

;

0 ;

1: 1

; ;

;

i

X fft x

Xf X

Xf Xf k

for i k

Xf i Xf k Xf N i conj Xf i

xf Real IFFT Xf

(6)

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. 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


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 [10].

Fig. 7. Double integrator amplifier used to obtain displacement from

acceleration data [10]

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).

11 1 , 0

2 s

y n y n x n x n nf

(7)

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 [1].

21 ^

0% 1001

n

i i

i

X X

en

(8)

where:

N is the number of data points.

Xi is the double integrated position data ^

iXis the ideal data.

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.

So, a little standard error value can be accepted about less

than 10 % suggested by Slifka [1].

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 50% of the maximum peak that is suggested by

Slifka[1]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

0( ) ( )A t a t d

(9)

Mathematically, the calculation of displacements x(t)

from a measured acceleration a(t) is simple described in

[11]:

0 0

2

0 0 0

1( )

2

t

t t

x t x t v t t d t dt a d

(10)

where:

x(t0) is the initial displacement, v(t0) is the initial

velocity, d0is the acceleration drift, x(t) is the calculated

displacement

From (10), we can see that the output signal can be

unbounded over time because of ramp (v0) and parabola (d0)

components. In this paper, the signals, single and multi-

frequency 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:

tAta 1sin (11)

After the 1st integrating, the velocity (assuming no initial

conditions) is

tA

tv 1

1

cos

(12)

After the 2nd integrating, the displacement (assuming no

initial conditions) is

tA

tx 12

1

sin

(13)

R3

R2R1

R4

-

+

U1

3

26

C2

C1

C3

VoVi

358


To reject delay of the output signal, filtfiltfunction which

is supported in MATLAB is used to replace filter function

in double integrated process.

Fig. 8. Double integration using FIR Filters, with M=1501, fc=5.0Hz

Fig. 9. Double integration using IIR Filters, with M=4, fc=5.0Hz

Fig. 10. Double integration using FFT Filters, with k=8, α=0.0775

Fig. 8, Fig. 9 and Fig. 10 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. 10). Therefore we can use FFT

Filter in calculating the displacement by using double

integration without initial conditions.

In Fig. 11, Fig. 12 and Fig. 13, 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.

11, Fig. 12). In Fig. 13, 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. 11. Standard error of filters when using single frequency signal

Fig. 12. Average peak error of filters when using single frequency signal.

Fig.13. Maximum peak error of filters when using single frequency signal

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 (14).

0 1 min 2

3 max

( ) sin(2 ) sin(2 )

sin(2 )

mida t d A f t A f dt

A f t

(14)

We did experiments in three cases. The first case uses

FIR Filter with M=1501, fc=5.0Hz, fmin=20Hz, fmax=40Hz.

The second case uses IIR Filter with M=4, fc=5.0Hz,

fmin=20Hz, fmax=40Hz. The third case uses FFT Filter with

k=5 and α=0.0775. The displacement signal calculated from

the multi-frequency acceleration signal when using FFT

Filter is the most accurate result (Fig. 14).

359


360


Fig. 14. Double integration with multi-harmornic signal

Fig. 15. Standard error of filters when using multi- frequency signal

Fig. 16. Average peak error of filters when using multi-frequency signal

Fig. 17. Maximum peak error of filters when using multi-frequency signal

From the results presented in Fig. 15, Fig. 16 and Fig. 17,

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.

Fig. 18. Standard errors of FFT Filter when altering filter cut frequency

At higher frequencies in Fig. 18, 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.

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

[1] L. D. Slifka, "An accelerometer based approach to measuring

displacement of a vehicle body," M.S. thesis, Dept. Elect. Comput.

Eng., Michigan Univ., USA, 2004.

[2] 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, 2010.

[3] A. DeMarco and C. Stedman, "Automated GPS Mapping of Road

Roughness," Bachelor thesis, Worcester Polytechnic Inst., USA,

2007.

[4] M. W. Sayers and S. W. Karamihas, The Litte Book of Profiling,

Michigan, USA: Michigan Univ., 1996.

[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., 2005.

[6] M. Harris and G. Piersol, Shock and Viration HandBook, 5th, Ed.

New York: McGraw-Hill Book Company, 2002.

[7] S. J. Orfanidis, Introduction to Signal Processing, Prentice Hall,

2009.

[8] J. G. T. Ribeiro et al., "New improvements in the digital double

integration filtering method to measure displacements using

accelerometers," in Proc. 19th Int. Modal Analysis Conf., vol. 3727,

Orlando, Florida, 2001, pp. 538-542.

[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, 2003.

[10] J. G. T. Ribeiro et al., "Problems in analogue double integration to

determine displacements from acceleration data," in Proc. 15th Int.

Modal Analysis Conf., vol. 3089, Orlando, Florida, 1997, pp. 930-

934.

[11] M. Arraigada el al., "Calculation of displacements of measured

A. Experiment with Changing Cut Frequency of Filters

361


accelerations analysis of two accelerometers and application in road ,"

in STRC, 2006.

Thai Minh Do was born in Dong Nai District, Viet

Nam in 1988. He received Bachelor’s Degree in

Electronics and Telecommunications Engineering

from University of Technical Education Ho Chi

Minh City, Viet Nam in 2011. 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 2011, 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 2011, ODON VALLET Scholarship for Excellent Students in 2009,

NGUYEN THAI BINH Scholarship in 2008.

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 1993 and M.S. degree in Electronic

Engineering from Ho Chi Minh City University of Science, Vietnam in

1998. He got his PhD degree in Electrical Engineering from University of

Arkansas, United States in 2009.

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